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Metamath Proof Explorer

Theorem psdmul

Description: Product rule for power series. An outline is available at https://github.com/icecream17/Stuff/blob/main/math/psdmul.pdf . (Contributed by SN, 25-Apr-2025)

Ref Expression
Hypotheses psdmul.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
psdmul.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
psdmul.p ⊢ + = ( +g ‘ 𝑆 )
psdmul.m ⊢ · = ( .r ‘ 𝑆 )
psdmul.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
psdmul.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
psdmul.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
psdmul.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
Assertion psdmul ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐹 · 𝐺 ) ) = ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) + ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) )

Proof

Step Hyp Ref Expression
1 psdmul.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psdmul.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
3 psdmul.p ⊢ + = ( +g ‘ 𝑆 )
4 psdmul.m ⊢ · = ( .r ‘ 𝑆 )
5 psdmul.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
6 psdmul.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
7 psdmul.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
8 psdmul.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
9 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
10 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
11 5 crngringd ⊢ ( 𝜑 → 𝑅 ∈ Ring )
12 11 ringcmnd ⊢ ( 𝜑 → 𝑅 ∈ CMnd )
13 12 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑅 ∈ CMnd )
14 simpr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
15 reldmpsr ⊢ Rel dom mPwSer
16 1 2 15 strov2rcl ⊢ ( 𝐹 ∈ 𝐵 → 𝐼 ∈ V )
17 7 16 syl ⊢ ( 𝜑 → 𝐼 ∈ V )
18 eqid ⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
19 18 psrbagsn ⊢ ( 𝐼 ∈ V → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
20 17 19 syl ⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
21 20 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
22 18 psrbagaddcl ⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
23 14 21 22 syl2anc ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
24 18 psrbaglefi ⊢ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∈ Fin )
25 23 24 syl ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∈ Fin )
26 eqid ⊢ ( .g ‘ 𝑅 ) = ( .g ‘ 𝑅 )
27 5 crnggrpd ⊢ ( 𝜑 → 𝑅 ∈ Grp )
28 27 grpmndd ⊢ ( 𝜑 → 𝑅 ∈ Mnd )
29 28 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝑅 ∈ Mnd )
30 18 psrbagf ⊢ ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑑 : 𝐼 ⟶ ℕ0 )
31 30 adantl ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 : 𝐼 ⟶ ℕ0 )
32 6 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑋 ∈ 𝐼 )
33 31 32 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 )
34 peano2nn0 ⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
35 33 34 syl ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
36 35 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
37 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
38 11 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝑅 ∈ Ring )
39 1 9 18 2 7 psrelbas ⊢ ( 𝜑 → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
40 39 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
41 elrabi ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
42 41 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
43 40 42 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) )
44 1 9 18 2 8 psrelbas ⊢ ( 𝜑 → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
45 44 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
46 eqid ⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) }
47 18 46 psrbagconcl ⊢ ( ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
48 23 47 sylan ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
49 elrabi ⊢ ( ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
50 48 49 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
51 45 50 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ∈ ( Base ‘ 𝑅 ) )
52 9 37 38 43 51 ringcld ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ ( Base ‘ 𝑅 ) )
53 9 26 29 36 52 mulgnn0cld ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
54 disjdifr ⊢ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∩ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) = ∅
55 54 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∩ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) = ∅ )
56 1nn0 ⊢ 1 ∈ ℕ0
57 0nn0 ⊢ 0 ∈ ℕ0
58 56 57 ifcli ⊢ if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℕ0
59 58 nn0ge0i ⊢ 0 ≤ if ( 𝑖 = 𝑋 , 1 , 0 )
60 31 ffvelcdmda ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
61 60 nn0red ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℝ )
62 58 nn0rei ⊢ if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℝ
63 62 a1i ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℝ )
64 61 63 addge01d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 0 ≤ if ( 𝑖 = 𝑋 , 1 , 0 ) ↔ ( 𝑑 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
65 59 64 mpbii ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
66 65 ralrimiva ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ∀ 𝑖 ∈ 𝐼 ( 𝑑 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
67 31 ffnd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 Fn 𝐼 )
68 56 57 ifcli ⊢ if ( 𝑦 = 𝑋 , 1 , 0 ) ∈ ℕ0
69 68 elexi ⊢ if ( 𝑦 = 𝑋 , 1 , 0 ) ∈ V
70 eqid ⊢ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) )
71 69 70 fnmpti ⊢ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼
72 71 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
73 17 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐼 ∈ V )
74 inidm ⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼
75 67 72 73 73 74 offn ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
76 eqidd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
77 eqeq1 ⊢ ( 𝑦 = 𝑖 → ( 𝑦 = 𝑋 ↔ 𝑖 = 𝑋 ) )
78 77 ifbid ⊢ ( 𝑦 = 𝑖 → if ( 𝑦 = 𝑋 , 1 , 0 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
79 58 elexi ⊢ if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ V
80 78 70 79 fvmpt ⊢ ( 𝑖 ∈ 𝐼 → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
81 80 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
82 67 72 73 73 74 76 81 ofval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
83 67 75 73 73 74 76 82 ofrfval ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑑 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
84 66 83 mpbird ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
85 84 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
86 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐼 ∈ V )
87 18 psrbagf ⊢ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑘 : 𝐼 ⟶ ℕ0 )
88 87 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑘 : 𝐼 ⟶ ℕ0 )
89 31 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 : 𝐼 ⟶ ℕ0 )
90 18 psrbagf ⊢ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
91 23 90 syl ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
92 91 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
93 nn0re ⊢ ( 𝑞 ∈ ℕ0 → 𝑞 ∈ ℝ )
94 nn0re ⊢ ( 𝑟 ∈ ℕ0 → 𝑟 ∈ ℝ )
95 nn0re ⊢ ( 𝑠 ∈ ℕ0 → 𝑠 ∈ ℝ )
96 letr ⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ ) → ( ( 𝑞 ≤ 𝑟 ∧ 𝑟 ≤ 𝑠 ) → 𝑞 ≤ 𝑠 ) )
97 93 94 95 96 syl3an ⊢ ( ( 𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( ( 𝑞 ≤ 𝑟 ∧ 𝑟 ≤ 𝑠 ) → 𝑞 ≤ 𝑠 ) )
98 97 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ) → ( ( 𝑞 ≤ 𝑟 ∧ 𝑟 ≤ 𝑠 ) → 𝑞 ≤ 𝑠 ) )
99 86 88 89 92 98 caoftrn ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑘 ∘r ≤ 𝑑 ∧ 𝑑 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) → 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
100 85 99 mpan2d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑘 ∘r ≤ 𝑑 → 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
101 100 ss2rabdv ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
102 undifr ⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↔ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
103 101 102 sylib ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
104 103 eqcomd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } = ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
105 9 10 13 25 53 55 104 gsummptfidmsplit ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
106 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
107 ovex ⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
108 107 rabex ⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V
109 108 rabex ⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∈ V
110 109 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∈ V )
111 ovex ⊢ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ V
112 eqid ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) )
113 111 112 fnmpti ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) }
114 113 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
115 fvexd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 0g ‘ 𝑅 ) ∈ V )
116 114 25 115 fndmfifsupp ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
117 9 106 26 110 52 116 13 35 gsummulg ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
118 difrab ⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑘 ∘r ≤ 𝑑 ) }
119 118 eleq2i ⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↔ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑘 ∘r ≤ 𝑑 ) } )
120 breq1 ⊢ ( 𝑘 = 𝑢 → ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
121 breq1 ⊢ ( 𝑘 = 𝑢 → ( 𝑘 ∘r ≤ 𝑑 ↔ 𝑢 ∘r ≤ 𝑑 ) )
122 121 notbid ⊢ ( 𝑘 = 𝑢 → ( ¬ 𝑘 ∘r ≤ 𝑑 ↔ ¬ 𝑢 ∘r ≤ 𝑑 ) )
123 120 122 anbi12d ⊢ ( 𝑘 = 𝑢 → ( ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑘 ∘r ≤ 𝑑 ) ↔ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑢 ∘r ≤ 𝑑 ) ) )
124 123 elrab ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑘 ∘r ≤ 𝑑 ) } ↔ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑢 ∘r ≤ 𝑑 ) ) )
125 18 psrbagf ⊢ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑢 : 𝐼 ⟶ ℕ0 )
126 125 ffnd ⊢ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑢 Fn 𝐼 )
127 126 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑢 Fn 𝐼 )
128 75 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
129 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐼 ∈ V )
130 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
131 67 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 Fn 𝐼 )
132 68 a1i ⊢ ( 𝑦 ∈ 𝐼 → if ( 𝑦 = 𝑋 , 1 , 0 ) ∈ ℕ0 )
133 70 132 fmpti ⊢ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0
134 133 a1i ⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 )
135 134 ffnd ⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
136 135 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
137 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
138 80 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
139 131 136 129 129 74 137 138 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
140 127 128 129 129 74 130 139 ofrfval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
141 127 131 129 129 74 130 137 ofrfval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∘r ≤ 𝑑 ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
142 141 notbid ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ¬ 𝑢 ∘r ≤ 𝑑 ↔ ¬ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
143 rexnal ⊢ ( ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ↔ ¬ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) )
144 142 143 bitr4di ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ¬ 𝑢 ∘r ≤ 𝑑 ↔ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
145 140 144 anbi12d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑢 ∘r ≤ 𝑑 ) ↔ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
146 33 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 )
147 125 adantl ⊢ ( ( 𝜑 ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑢 : 𝐼 ⟶ ℕ0 )
148 6 adantr ⊢ ( ( 𝜑 ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑋 ∈ 𝐼 )
149 147 148 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
150 149 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
151 150 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
152 nn0nlt0 ⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 → ¬ ( 𝑑 ‘ 𝑋 ) < 0 )
153 146 152 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ¬ ( 𝑑 ‘ 𝑋 ) < 0 )
154 31 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 : 𝐼 ⟶ ℕ0 )
155 154 ffvelcdmda ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
156 155 nn0cnd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℂ )
157 156 addridd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ‘ 𝑖 ) + 0 ) = ( 𝑑 ‘ 𝑖 ) )
158 157 breq2d ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + 0 ) ↔ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
159 158 biimpd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + 0 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
160 ifnefalse ⊢ ( 𝑖 ≠ 𝑋 → if ( 𝑖 = 𝑋 , 1 , 0 ) = 0 )
161 160 oveq2d ⊢ ( 𝑖 ≠ 𝑋 → ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( ( 𝑑 ‘ 𝑖 ) + 0 ) )
162 161 breq2d ⊢ ( 𝑖 ≠ 𝑋 → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ↔ ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + 0 ) ) )
163 162 imbi1d ⊢ ( 𝑖 ≠ 𝑋 → ( ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ↔ ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + 0 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
164 159 163 syl5ibrcom ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑖 ≠ 𝑋 → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
165 164 imp ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
166 165 impancom ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) → ( 𝑖 ≠ 𝑋 → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
167 166 necon1bd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) → ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → 𝑖 = 𝑋 ) )
168 167 ancrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) → ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
169 168 ex ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ) )
170 169 ralimdva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ) )
171 170 anim1d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) → ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
172 171 imp ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
173 rexim ⊢ ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ∃ 𝑖 ∈ 𝐼 ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) )
174 173 imp ⊢ ( ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) → ∃ 𝑖 ∈ 𝐼 ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
175 fveq2 ⊢ ( 𝑖 = 𝑋 → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑋 ) )
176 fveq2 ⊢ ( 𝑖 = 𝑋 → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑋 ) )
177 175 176 breq12d ⊢ ( 𝑖 = 𝑋 → ( ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ↔ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) ) )
178 177 notbid ⊢ ( 𝑖 = 𝑋 → ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ↔ ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) ) )
179 178 ceqsrexbv ⊢ ( ∃ 𝑖 ∈ 𝐼 ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ↔ ( 𝑋 ∈ 𝐼 ∧ ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) ) )
180 179 simprbi ⊢ ( ∃ 𝑖 ∈ 𝐼 ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) → ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) )
181 174 180 syl ⊢ ( ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) → ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) )
182 33 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 )
183 182 nn0red ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℝ )
184 150 nn0red ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℝ )
185 183 184 ltnled ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) ↔ ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) ) )
186 185 biimpar ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ¬ ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) ) → ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) )
187 181 186 sylan2 ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) → ( 𝑖 = 𝑋 ∧ ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) )
188 172 187 syldan ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) )
189 breq2 ⊢ ( ( 𝑢 ‘ 𝑋 ) = 0 → ( ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) ↔ ( 𝑑 ‘ 𝑋 ) < 0 ) )
190 188 189 syl5ibcom ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ( 𝑢 ‘ 𝑋 ) = 0 → ( 𝑑 ‘ 𝑋 ) < 0 ) )
191 153 190 mtod ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ¬ ( 𝑢 ‘ 𝑋 ) = 0 )
192 191 neqned ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑢 ‘ 𝑋 ) ≠ 0 )
193 elnnne0 ⊢ ( ( 𝑢 ‘ 𝑋 ) ∈ ℕ ↔ ( ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 ∧ ( 𝑢 ‘ 𝑋 ) ≠ 0 ) )
194 151 192 193 sylanbrc ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ )
195 elfzo0 ⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) ↔ ( ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 ∧ ( 𝑢 ‘ 𝑋 ) ∈ ℕ ∧ ( 𝑑 ‘ 𝑋 ) < ( 𝑢 ‘ 𝑋 ) ) )
196 146 194 188 195 syl3anbrc ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑑 ‘ 𝑋 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) )
197 fzostep1 ⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) ∨ ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) ) )
198 196 197 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) ∨ ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) ) )
199 151 nn0red ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℝ )
200 35 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
201 200 nn0red ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℝ )
202 6 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑋 ∈ 𝐼 )
203 iftrue ⊢ ( 𝑖 = 𝑋 → if ( 𝑖 = 𝑋 , 1 , 0 ) = 1 )
204 176 203 oveq12d ⊢ ( 𝑖 = 𝑋 → ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( ( 𝑑 ‘ 𝑋 ) + 1 ) )
205 175 204 breq12d ⊢ ( 𝑖 = 𝑋 → ( ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ↔ ( 𝑢 ‘ 𝑋 ) ≤ ( ( 𝑑 ‘ 𝑋 ) + 1 ) ) )
206 205 rspcv ⊢ ( 𝑋 ∈ 𝐼 → ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( 𝑢 ‘ 𝑋 ) ≤ ( ( 𝑑 ‘ 𝑋 ) + 1 ) ) )
207 202 206 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) → ( 𝑢 ‘ 𝑋 ) ≤ ( ( 𝑑 ‘ 𝑋 ) + 1 ) ) )
208 207 imp ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ‘ 𝑋 ) ≤ ( ( 𝑑 ‘ 𝑋 ) + 1 ) )
209 208 adantrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( 𝑢 ‘ 𝑋 ) ≤ ( ( 𝑑 ‘ 𝑋 ) + 1 ) )
210 199 201 209 lensymd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ¬ ( ( 𝑑 ‘ 𝑋 ) + 1 ) < ( 𝑢 ‘ 𝑋 ) )
211 210 intn3an3d ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ¬ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 ∧ ( 𝑢 ‘ 𝑋 ) ∈ ℕ ∧ ( ( 𝑑 ‘ 𝑋 ) + 1 ) < ( 𝑢 ‘ 𝑋 ) ) )
212 elfzo0 ⊢ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) ↔ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 ∧ ( 𝑢 ‘ 𝑋 ) ∈ ℕ ∧ ( ( 𝑑 ‘ 𝑋 ) + 1 ) < ( 𝑢 ‘ 𝑋 ) ) )
213 211 212 sylnibr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ¬ ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ( 0 ..^ ( 𝑢 ‘ 𝑋 ) ) )
214 198 213 orcnd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ∧ ∃ 𝑖 ∈ 𝐼 ¬ ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
215 145 214 sylbida ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑢 ∘r ≤ 𝑑 ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
216 215 anasss ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑢 ∘r ≤ 𝑑 ) ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
217 124 216 sylan2b ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ 𝑘 ∘r ≤ 𝑑 ) } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
218 119 217 sylan2b ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
219 218 oveq1d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
220 219 mpteq2dva ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
221 220 oveq2d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
222 18 psrbaglefi ⊢ ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∈ Fin )
223 222 adantl ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∈ Fin )
224 28 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑅 ∈ Mnd )
225 35 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
226 11 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑅 ∈ Ring )
227 elrabi ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
228 39 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
229 228 ffvelcdmda ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) )
230 227 229 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) )
231 44 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
232 31 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑑 : 𝐼 ⟶ ℕ0 )
233 232 ffvelcdmda ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
234 233 nn0cnd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℂ )
235 227 125 syl ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑢 : 𝐼 ⟶ ℕ0 )
236 235 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑢 : 𝐼 ⟶ ℕ0 )
237 236 ffvelcdmda ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℕ0 )
238 237 nn0cnd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℂ )
239 58 nn0cni ⊢ if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℂ
240 239 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℂ )
241 234 238 240 subadd23d ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑑 ‘ 𝑖 ) − ( 𝑢 ‘ 𝑖 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( ( 𝑑 ‘ 𝑖 ) + ( if ( 𝑖 = 𝑋 , 1 , 0 ) − ( 𝑢 ‘ 𝑖 ) ) ) )
242 234 240 238 addsubassd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) = ( ( 𝑑 ‘ 𝑖 ) + ( if ( 𝑖 = 𝑋 , 1 , 0 ) − ( 𝑢 ‘ 𝑖 ) ) ) )
243 241 242 eqtr4d ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑑 ‘ 𝑖 ) − ( 𝑢 ‘ 𝑖 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) )
244 243 mpteq2dva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) − ( 𝑢 ‘ 𝑖 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) ) )
245 eqid ⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 }
246 18 245 psrbagconcl ⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑢 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
247 elrabi ⊢ ( ( 𝑑 ∘f − 𝑢 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → ( 𝑑 ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
248 246 247 syl ⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
249 248 adantll ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
250 18 psrbagf ⊢ ( ( 𝑑 ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( 𝑑 ∘f − 𝑢 ) : 𝐼 ⟶ ℕ0 )
251 249 250 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑢 ) : 𝐼 ⟶ ℕ0 )
252 251 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑢 ) Fn 𝐼 )
253 71 a1i ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
254 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐼 ∈ V )
255 232 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑑 Fn 𝐼 )
256 236 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑢 Fn 𝐼 )
257 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
258 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
259 255 256 254 254 74 257 258 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) − ( 𝑢 ‘ 𝑖 ) ) )
260 80 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
261 252 253 254 254 74 259 260 offval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) − ( 𝑢 ‘ 𝑖 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
262 simplr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
263 20 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
264 262 263 22 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
265 264 90 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
266 265 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
267 255 253 254 254 74 257 260 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
268 266 256 254 254 74 267 258 offval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) ) )
269 244 261 268 3eqtr4d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) )
270 18 psrbagaddcl ⊢ ( ( ( 𝑑 ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
271 249 263 270 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
272 269 271 eqeltrrd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
273 231 272 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ∈ ( Base ‘ 𝑅 ) )
274 9 37 226 230 273 ringcld ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ ( Base ‘ 𝑅 ) )
275 9 26 224 225 274 mulgnn0cld ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
276 disjdifr ⊢ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∩ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) = ∅
277 276 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∩ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) = ∅ )
278 simpl ⊢ ( ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) → 𝑘 ∘r ≤ 𝑑 )
279 278 a1i ⊢ ( 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) → 𝑘 ∘r ≤ 𝑑 ) )
280 279 ss2rabi ⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 }
281 280 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
282 undifr ⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↔ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
283 281 282 sylib ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
284 283 eqcomd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } = ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∪ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) )
285 9 10 13 223 275 277 284 gsummptfidmsplit ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
286 eldifi ⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
287 6 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑋 ∈ 𝐼 )
288 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑋 ) = ( 𝑑 ‘ 𝑋 ) )
289 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑋 ) = ( 𝑢 ‘ 𝑋 ) )
290 255 256 254 254 74 288 289 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) = ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) )
291 287 290 mpdan ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) = ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) )
292 286 291 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) = ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) )
293 292 oveq2d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ) = ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) ) )
294 236 287 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
295 286 294 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
296 295 nn0cnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℂ )
297 33 nn0cnd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℂ )
298 297 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( 𝑑 ‘ 𝑋 ) ∈ ℂ )
299 296 298 pncan3d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) ) = ( 𝑑 ‘ 𝑋 ) )
300 293 299 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ) = ( 𝑑 ‘ 𝑋 ) )
301 300 oveq1d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ) + 1 ) = ( ( 𝑑 ‘ 𝑋 ) + 1 ) )
302 251 287 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ∈ ℕ0 )
303 286 302 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ∈ ℕ0 )
304 303 nn0cnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ∈ ℂ )
305 1cnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → 1 ∈ ℂ )
306 296 304 305 addassd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑢 ‘ 𝑋 ) + ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ) + 1 ) = ( ( 𝑢 ‘ 𝑋 ) + ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ) )
307 301 306 eqtr3d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( ( 𝑢 ‘ 𝑋 ) + ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ) )
308 307 oveq1d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( 𝑢 ‘ 𝑋 ) + ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
309 28 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → 𝑅 ∈ Mnd )
310 peano2nn0 ⊢ ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) ∈ ℕ0 → ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
311 302 310 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
312 286 311 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
313 286 274 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ ( Base ‘ 𝑅 ) )
314 9 26 10 mulgnn0dir ⊢ ( ( 𝑅 ∈ Mnd ∧ ( ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 ∧ ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℕ0 ∧ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ ( Base ‘ 𝑅 ) ) ) → ( ( ( 𝑢 ‘ 𝑋 ) + ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
315 309 295 312 313 314 syl13anc ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑢 ‘ 𝑋 ) + ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
316 308 315 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
317 316 mpteq2dva ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
318 317 oveq2d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
319 difssd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
320 223 319 ssfid ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∈ Fin )
321 9 26 224 294 274 mulgnn0cld ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
322 286 321 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
323 9 26 224 311 274 mulgnn0cld ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
324 286 323 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) → ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
325 eqid ⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
326 eqid ⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
327 9 10 13 320 322 324 325 326 gsummptfidmadd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ( +g ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
328 318 327 eqtrd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
329 6 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → 𝑋 ∈ 𝐼 )
330 67 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → 𝑑 Fn 𝐼 )
331 elrabi ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
332 331 126 syl ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } → 𝑢 Fn 𝐼 )
333 332 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → 𝑢 Fn 𝐼 )
334 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → 𝐼 ∈ V )
335 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑋 ) = ( 𝑑 ‘ 𝑋 ) )
336 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑋 ) = ( 𝑢 ‘ 𝑋 ) )
337 330 333 334 334 74 335 336 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) = ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) )
338 329 337 mpdan ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) = ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) )
339 fveq1 ⊢ ( 𝑘 = 𝑢 → ( 𝑘 ‘ 𝑋 ) = ( 𝑢 ‘ 𝑋 ) )
340 339 eqeq1d ⊢ ( 𝑘 = 𝑢 → ( ( 𝑘 ‘ 𝑋 ) = 0 ↔ ( 𝑢 ‘ 𝑋 ) = 0 ) )
341 121 340 anbi12d ⊢ ( 𝑘 = 𝑢 → ( ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ↔ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) )
342 341 elrab ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↔ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) )
343 342 simprbi ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } → ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) )
344 343 simprd ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } → ( 𝑢 ‘ 𝑋 ) = 0 )
345 344 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( 𝑢 ‘ 𝑋 ) = 0 )
346 345 oveq2d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( 𝑑 ‘ 𝑋 ) − ( 𝑢 ‘ 𝑋 ) ) = ( ( 𝑑 ‘ 𝑋 ) − 0 ) )
347 33 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 )
348 347 nn0cnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℂ )
349 348 subid1d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( 𝑑 ‘ 𝑋 ) − 0 ) = ( 𝑑 ‘ 𝑋 ) )
350 338 346 349 3eqtrrd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( 𝑑 ‘ 𝑋 ) = ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) )
351 350 oveq1d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) = ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) )
352 351 oveq1d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
353 352 mpteq2dva ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
354 353 oveq2d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
355 328 354 oveq12d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) = ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
356 27 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑅 ∈ Grp )
357 108 rabex ⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∈ V
358 357 difexi ⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∈ V
359 358 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∈ V )
360 322 fmpttd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) : ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⟶ ( Base ‘ 𝑅 ) )
361 ovex ⊢ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ V
362 361 325 fnmpti ⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } )
363 362 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) )
364 363 320 115 fndmfifsupp ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
365 9 106 13 359 360 364 gsumcl ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
366 324 fmpttd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) : ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⟶ ( Base ‘ 𝑅 ) )
367 ovex ⊢ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ V
368 367 326 fnmpti ⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } )
369 368 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) )
370 369 320 115 fndmfifsupp ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
371 9 106 13 359 366 370 gsumcl ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
372 108 rabex ⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ∈ V
373 372 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ∈ V )
374 280 sseli ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } → 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
375 374 323 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) → ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
376 375 fmpttd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) : { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ⟶ ( Base ‘ 𝑅 ) )
377 eqid ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
378 367 377 fnmpti ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) }
379 378 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } )
380 223 281 ssfid ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ∈ Fin )
381 379 380 115 fndmfifsupp ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
382 9 106 13 373 376 381 gsumcl ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
383 9 10 356 365 371 382 grpassd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) )
384 285 355 383 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) )
385 221 384 oveq12d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
386 105 117 385 3eqtr3d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
387 7 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐹 ∈ 𝐵 )
388 8 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐺 ∈ 𝐵 )
389 1 2 37 4 18 387 388 23 psrmulval ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐹 · 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
390 389 oveq2d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 · 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
391 109 difexi ⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∈ V
392 391 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∈ V )
393 eldifi ⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
394 41 125 syl ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → 𝑢 : 𝐼 ⟶ ℕ0 )
395 394 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝑢 : 𝐼 ⟶ ℕ0 )
396 6 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → 𝑋 ∈ 𝐼 )
397 395 396 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
398 9 26 29 397 52 mulgnn0cld ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
399 393 398 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
400 399 fmpttd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) : ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ⟶ ( Base ‘ 𝑅 ) )
401 eqid ⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
402 361 401 fnmpti ⊢ ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
403 402 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
404 difssd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
405 25 404 ssfid ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∈ Fin )
406 403 405 115 fndmfifsupp ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
407 9 106 13 392 400 406 gsumcl ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
408 9 10 356 371 382 grpcld ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
409 9 10 356 407 365 408 grpassd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
410 386 390 409 3eqtr4d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 · 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) )
411 410 mpteq2dva ⊢ ( 𝜑 → ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 · 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
412 1 2 4 11 7 8 psrmulcl ⊢ ( 𝜑 → ( 𝐹 · 𝐺 ) ∈ 𝐵 )
413 1 2 18 6 412 psdval ⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐹 · 𝐺 ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 · 𝐺 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) )
414 27 grpmgmd ⊢ ( 𝜑 → 𝑅 ∈ Mgm )
415 1 2 414 6 7 psdcl ⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∈ 𝐵 )
416 1 2 4 11 415 8 psrmulcl ⊢ ( 𝜑 → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) ∈ 𝐵 )
417 1 2 414 6 8 psdcl ⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ∈ 𝐵 )
418 1 2 4 11 7 417 psrmulcl ⊢ ( 𝜑 → ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ∈ 𝐵 )
419 1 2 10 3 416 418 psradd ⊢ ( 𝜑 → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) + ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) = ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) ∘f ( +g ‘ 𝑅 ) ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) )
420 1 9 18 2 416 psrelbas ⊢ ( 𝜑 → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
421 420 ffnd ⊢ ( 𝜑 → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
422 1 9 18 2 418 psrelbas ⊢ ( 𝜑 → ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
423 422 ffnd ⊢ ( 𝜑 → ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
424 108 a1i ⊢ ( 𝜑 → { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∈ V )
425 inidm ⊢ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∩ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
426 415 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∈ 𝐵 )
427 1 2 37 4 18 426 388 14 psrmulval ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) ‘ 𝑑 ) = ( 𝑅 Σg ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ) )
428 357 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∈ V )
429 11 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑅 ∈ Ring )
430 elrabi ⊢ ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
431 1 9 18 2 415 psrelbas ⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
432 431 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
433 432 ffvelcdmda ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ∈ ( Base ‘ 𝑅 ) )
434 430 433 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ∈ ( Base ‘ 𝑅 ) )
435 44 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
436 18 245 psrbagconcl ⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑏 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
437 436 adantll ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑏 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
438 elrabi ⊢ ( ( 𝑑 ∘f − 𝑏 ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → ( 𝑑 ∘f − 𝑏 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
439 437 438 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑑 ∘f − 𝑏 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
440 435 439 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ∈ ( Base ‘ 𝑅 ) )
441 9 37 429 434 440 ringcld ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ∈ ( Base ‘ 𝑅 ) )
442 441 fmpttd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) : { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ⟶ ( Base ‘ 𝑅 ) )
443 ovex ⊢ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ∈ V
444 eqid ⊢ ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) = ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) )
445 443 444 fnmpti ⊢ ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 }
446 445 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
447 446 223 115 fndmfifsupp ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
448 eqid ⊢ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
449 df-of ⊢ ∘f + = ( 𝑚 ∈ V , 𝑛 ∈ V ↦ ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) )
450 vex ⊢ 𝑢 ∈ V
451 450 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑢 ∈ V )
452 ssv ⊢ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ⊆ V
453 452 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ⊆ V )
454 ssv ⊢ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ⊆ V
455 454 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ⊆ V )
456 449 451 453 455 elimampo ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↔ ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ) )
457 456 biimpa ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) )
458 elrabi ⊢ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑚 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
459 18 psrbagf ⊢ ( 𝑚 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑚 : 𝐼 ⟶ ℕ0 )
460 459 ffund ⊢ ( 𝑚 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → Fun 𝑚 )
461 458 460 syl ⊢ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → Fun 𝑚 )
462 461 funfnd ⊢ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑚 Fn dom 𝑚 )
463 462 ad2antrl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → 𝑚 Fn dom 𝑚 )
464 velsn ⊢ ( 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ↔ 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) )
465 funmpt ⊢ Fun ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) )
466 funeq ⊢ ( 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) → ( Fun 𝑛 ↔ Fun ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
467 465 466 mpbiri ⊢ ( 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) → Fun 𝑛 )
468 467 funfnd ⊢ ( 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) → 𝑛 Fn dom 𝑛 )
469 464 468 sylbi ⊢ ( 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } → 𝑛 Fn dom 𝑛 )
470 469 ad2antll ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → 𝑛 Fn dom 𝑛 )
471 vex ⊢ 𝑚 ∈ V
472 471 dmex ⊢ dom 𝑚 ∈ V
473 472 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → dom 𝑚 ∈ V )
474 vex ⊢ 𝑛 ∈ V
475 474 dmex ⊢ dom 𝑛 ∈ V
476 475 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → dom 𝑛 ∈ V )
477 eqid ⊢ ( dom 𝑚 ∩ dom 𝑛 ) = ( dom 𝑚 ∩ dom 𝑛 )
478 eqidd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑜 ∈ dom 𝑚 ) → ( 𝑚 ‘ 𝑜 ) = ( 𝑚 ‘ 𝑜 ) )
479 eqidd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑜 ∈ dom 𝑛 ) → ( 𝑛 ‘ 𝑜 ) = ( 𝑛 ‘ 𝑜 ) )
480 463 470 473 476 477 478 479 offval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑚 ∘f + 𝑛 ) = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) )
481 480 eqeq2d ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ) )
482 elsni ⊢ ( 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } → 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) )
483 482 oveq2d ⊢ ( 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } → ( 𝑚 ∘f + 𝑛 ) = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
484 483 eqeq2d ⊢ ( 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
485 484 ad2antll ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
486 17 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐼 ∈ V )
487 458 459 syl ⊢ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑚 : 𝐼 ⟶ ℕ0 )
488 487 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑚 : 𝐼 ⟶ ℕ0 )
489 133 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 )
490 nn0cn ⊢ ( 𝑞 ∈ ℕ0 → 𝑞 ∈ ℂ )
491 nn0cn ⊢ ( 𝑟 ∈ ℕ0 → 𝑟 ∈ ℂ )
492 nn0cn ⊢ ( 𝑠 ∈ ℕ0 → 𝑠 ∈ ℂ )
493 addsubass ⊢ ( ( 𝑞 ∈ ℂ ∧ 𝑟 ∈ ℂ ∧ 𝑠 ∈ ℂ ) → ( ( 𝑞 + 𝑟 ) − 𝑠 ) = ( 𝑞 + ( 𝑟 − 𝑠 ) ) )
494 490 491 492 493 syl3an ⊢ ( ( 𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ) → ( ( 𝑞 + 𝑟 ) − 𝑠 ) = ( 𝑞 + ( 𝑟 − 𝑠 ) ) )
495 494 adantl ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ ( 𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ) → ( ( 𝑞 + 𝑟 ) − 𝑠 ) = ( 𝑞 + ( 𝑟 − 𝑠 ) ) )
496 486 488 489 489 495 caofass ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑚 ∘f + ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
497 simpr ⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝑖 ∈ 𝐼 )
498 58 a1i ⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℕ0 )
499 70 78 497 498 fvmptd3 ⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
500 135 135 17 17 74 499 499 offval ⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
501 500 oveq2d ⊢ ( 𝜑 → ( 𝑚 ∘f + ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑚 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) )
502 501 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑚 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) )
503 239 subidi ⊢ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) = 0
504 503 mpteq2i ⊢ ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ 0 )
505 fconstmpt ⊢ ( 𝐼 × { 0 } ) = ( 𝑖 ∈ 𝐼 ↦ 0 )
506 504 505 eqtr4i ⊢ ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( 𝐼 × { 0 } )
507 506 oveq2i ⊢ ( 𝑚 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑚 ∘f + ( 𝐼 × { 0 } ) )
508 0zd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 0 ∈ ℤ )
509 490 addridd ⊢ ( 𝑞 ∈ ℕ0 → ( 𝑞 + 0 ) = 𝑞 )
510 509 adantl ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑞 ∈ ℕ0 ) → ( 𝑞 + 0 ) = 𝑞 )
511 486 488 508 510 caofid0r ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝐼 × { 0 } ) ) = 𝑚 )
512 507 511 eqtrid ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = 𝑚 )
513 496 502 512 3eqtrd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = 𝑚 )
514 simpr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
515 513 514 eqeltrd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
516 oveq1 ⊢ ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
517 516 eleq1d ⊢ ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↔ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
518 515 517 syl5ibrcom ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
519 518 adantrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
520 485 519 sylbid ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
521 481 520 sylbird ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
522 521 rexlimdvva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
523 457 522 mpd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
524 simpr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
525 17 mptexd ⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ V )
526 elsng ⊢ ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ V → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ↔ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
527 525 526 syl ⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ↔ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
528 70 527 mpbiri ⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } )
529 528 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } )
530 449 mpofun ⊢ Fun ∘f +
531 530 a1i ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → Fun ∘f + )
532 xpss ⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ⊆ ( V × V )
533 472 inex1 ⊢ ( dom 𝑚 ∩ dom 𝑛 ) ∈ V
534 533 mptex ⊢ ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ∈ V
535 534 rgen2w ⊢ ∀ 𝑚 ∈ V ∀ 𝑛 ∈ V ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ∈ V
536 449 dmmpoga ⊢ ( ∀ 𝑚 ∈ V ∀ 𝑛 ∈ V ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ∈ V → dom ∘f + = ( V × V ) )
537 535 536 mp1i ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → dom ∘f + = ( V × V ) )
538 532 537 sseqtrrid ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ⊆ dom ∘f + )
539 524 529 531 538 elovimad ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) )
540 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝐼 ∈ V )
541 elrabi ⊢ ( 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑣 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
542 18 psrbagf ⊢ ( 𝑣 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑣 : 𝐼 ⟶ ℕ0 )
543 541 542 syl ⊢ ( 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → 𝑣 : 𝐼 ⟶ ℕ0 )
544 543 ad2antll ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑣 : 𝐼 ⟶ ℕ0 )
545 133 a1i ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 )
546 494 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ ( 𝑞 ∈ ℕ0 ∧ 𝑟 ∈ ℕ0 ∧ 𝑠 ∈ ℕ0 ) ) → ( ( 𝑞 + 𝑟 ) − 𝑠 ) = ( 𝑞 + ( 𝑟 − 𝑠 ) ) )
547 540 544 545 545 546 caofass ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑣 ∘f + ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
548 135 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
549 80 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
550 548 548 540 540 74 549 549 offval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
551 550 oveq2d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑣 ∘f + ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑣 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) )
552 506 oveq2i ⊢ ( 𝑣 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑣 ∘f + ( 𝐼 × { 0 } ) )
553 0zd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 0 ∈ ℤ )
554 nn0cn ⊢ ( 𝑝 ∈ ℕ0 → 𝑝 ∈ ℂ )
555 554 addridd ⊢ ( 𝑝 ∈ ℕ0 → ( 𝑝 + 0 ) = 𝑝 )
556 555 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑝 ∈ ℕ0 ) → ( 𝑝 + 0 ) = 𝑝 )
557 540 544 553 556 caofid0r ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑣 ∘f + ( 𝐼 × { 0 } ) ) = 𝑣 )
558 552 557 eqtrid ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑣 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = 𝑣 )
559 547 551 558 3eqtrrd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑣 = ( ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
560 oveq1 ⊢ ( 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
561 560 eqeq2d ⊢ ( 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑣 = ( ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
562 559 561 syl5ibrcom ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
563 20 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
564 18 psrbagaddcl ⊢ ( ( 𝑚 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
565 458 563 564 syl2an2 ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
566 18 psrbagf ⊢ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
567 565 566 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
568 567 adantrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 )
569 feq1 ⊢ ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 : 𝐼 ⟶ ℕ0 ↔ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) : 𝐼 ⟶ ℕ0 ) )
570 568 569 syl5ibrcom ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → 𝑢 : 𝐼 ⟶ ℕ0 ) )
571 485 570 sylbid ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) → 𝑢 : 𝐼 ⟶ ℕ0 ) )
572 481 571 sylbird ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → 𝑢 : 𝐼 ⟶ ℕ0 ) )
573 572 rexlimdvva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → 𝑢 : 𝐼 ⟶ ℕ0 ) )
574 457 573 mpd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑢 : 𝐼 ⟶ ℕ0 )
575 574 adantrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑢 : 𝐼 ⟶ ℕ0 )
576 575 ffvelcdmda ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℕ0 )
577 576 nn0cnd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℂ )
578 239 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℂ )
579 577 578 npcand ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( 𝑢 ‘ 𝑖 ) )
580 579 mpteq2dva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( 𝑢 ‘ 𝑖 ) ) )
581 575 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑢 Fn 𝐼 )
582 581 548 540 540 74 offn ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
583 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
584 581 548 540 540 74 583 549 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
585 582 548 540 540 74 584 549 offval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
586 575 feqmptd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑢 = ( 𝑖 ∈ 𝐼 ↦ ( 𝑢 ‘ 𝑖 ) ) )
587 580 585 586 3eqtr4rd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → 𝑢 = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
588 oveq1 ⊢ ( 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
589 588 eqeq2d ⊢ ( 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑢 = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
590 587 589 syl5ibrcom ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
591 562 590 impbid ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∧ 𝑣 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) → ( 𝑢 = ( 𝑣 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑣 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
592 448 523 539 591 f1o2d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) : ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) –1-1-onto→ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
593 9 106 13 428 442 447 592 gsumf1o ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ) = ( 𝑅 Σg ( ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ∘ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) )
594 555 adantl ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑝 ∈ ℕ0 ) → ( 𝑝 + 0 ) = 𝑝 )
595 486 488 508 594 caofid0r ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝐼 × { 0 } ) ) = 𝑚 )
596 507 595 eqtrid ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑖 ∈ 𝐼 ↦ ( if ( 𝑖 = 𝑋 , 1 , 0 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = 𝑚 )
597 496 502 596 3eqtrd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = 𝑚 )
598 597 514 eqeltrd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
599 598 517 syl5ibrcom ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
600 599 adantrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
601 485 600 sylbid ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
602 481 601 sylbird ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
603 602 rexlimdvva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) )
604 457 603 mpd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
605 eqidd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
606 eqidd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) = ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) )
607 fveq2 ⊢ ( 𝑏 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) = ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
608 oveq2 ⊢ ( 𝑏 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑑 ∘f − 𝑏 ) = ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
609 608 fveq2d ⊢ ( 𝑏 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) = ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )
610 607 609 oveq12d ⊢ ( 𝑏 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) = ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) )
611 604 605 606 610 fmptco ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ∘ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) )
612 6 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑋 ∈ 𝐼 )
613 7 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝐹 ∈ 𝐵 )
614 elrabi ⊢ ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
615 604 614 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
616 1 2 18 612 613 615 psdcoef ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( ( ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )
617 574 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑢 Fn 𝐼 )
618 133 a1i ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 )
619 618 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
620 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝐼 ∈ V )
621 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑋 ) = ( 𝑢 ‘ 𝑋 ) )
622 iftrue ⊢ ( 𝑦 = 𝑋 → if ( 𝑦 = 𝑋 , 1 , 0 ) = 1 )
623 1ex ⊢ 1 ∈ V
624 622 70 623 fvmpt ⊢ ( 𝑋 ∈ 𝐼 → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) = 1 )
625 624 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) = 1 )
626 617 619 620 620 74 621 625 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = ( ( 𝑢 ‘ 𝑋 ) − 1 ) )
627 612 626 mpdan ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = ( ( 𝑢 ‘ 𝑋 ) − 1 ) )
628 627 oveq1d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) + 1 ) = ( ( ( 𝑢 ‘ 𝑋 ) − 1 ) + 1 ) )
629 nn0sscn ⊢ ℕ0 ⊆ ℂ
630 629 a1i ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ℕ0 ⊆ ℂ )
631 574 630 fssd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑢 : 𝐼 ⟶ ℂ )
632 631 612 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℂ )
633 1cnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 1 ∈ ℂ )
634 632 633 npcand ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( 𝑢 ‘ 𝑋 ) − 1 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
635 628 634 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) + 1 ) = ( 𝑢 ‘ 𝑋 ) )
636 617 619 620 620 74 offn ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
637 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
638 80 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
639 617 619 620 620 74 637 638 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
640 574 ffvelcdmda ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℕ0 )
641 640 nn0cnd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℂ )
642 239 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℂ )
643 641 642 npcand ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( 𝑢 ‘ 𝑖 ) )
644 620 636 619 617 639 638 643 offveq ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = 𝑢 )
645 644 fveq2d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝐹 ‘ ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝐹 ‘ 𝑢 ) )
646 635 645 oveq12d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ 𝑢 ) ) )
647 616 646 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ 𝑢 ) ) )
648 30 ad2antlr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑑 : 𝐼 ⟶ ℕ0 )
649 648 ffvelcdmda ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
650 649 nn0cnd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℂ )
651 650 641 642 subsub3d ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ‘ 𝑖 ) − ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) )
652 651 mpteq2dva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑖 ) − ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) ) )
653 67 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑑 Fn 𝐼 )
654 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
655 653 636 620 620 74 654 639 offval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑖 ) − ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) ) )
656 653 619 620 620 74 offn ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
657 653 619 620 620 74 654 638 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
658 656 617 620 620 74 657 637 offval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) − ( 𝑢 ‘ 𝑖 ) ) ) )
659 652 655 658 3eqtr4d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) )
660 659 fveq2d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) )
661 647 660 oveq12d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ 𝑢 ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) )
662 11 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑅 ∈ Ring )
663 574 612 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
664 663 nn0zd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑢 ‘ 𝑋 ) ∈ ℤ )
665 39 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
666 simpllr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
667 20 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
668 simprl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
669 eqid ⊢ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } = { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) }
670 18 245 669 psrbagleadd1 ⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
671 666 667 668 670 syl3anc ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
672 eleq1 ⊢ ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↔ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) )
673 671 672 syl5ibrcom ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) )
674 485 673 sylbid ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) → 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) )
675 481 674 sylbird ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) ∧ ( 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) → ( 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) )
676 675 rexlimdvva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) → 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) )
677 457 676 mpd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
678 elrabi ⊢ ( 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
679 677 678 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
680 665 679 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) )
681 44 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝐺 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
682 23 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
683 18 669 psrbagconcl ⊢ ( ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑢 ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
684 682 677 683 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
685 elrabi ⊢ ( ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { 𝑙 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑙 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
686 684 685 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
687 681 686 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ∈ ( Base ‘ 𝑅 ) )
688 9 26 37 mulgass2 ⊢ ( ( 𝑅 ∈ Ring ∧ ( ( 𝑢 ‘ 𝑋 ) ∈ ℤ ∧ ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ∈ ( Base ‘ 𝑅 ) ) ) → ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ 𝑢 ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
689 662 664 680 687 688 syl13anc ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ 𝑢 ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
690 661 689 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
691 690 mpteq2dva ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) ) = ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
692 611 691 eqtrd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ∘ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
693 692 oveq2d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ∘ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
694 snex ⊢ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ∈ V
695 357 694 xpex ⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ∈ V
696 695 funimaex ⊢ ( Fun ∘f + → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∈ V )
697 530 696 mp1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ∈ V )
698 28 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → 𝑅 ∈ Mnd )
699 9 37 662 680 687 ringcld ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ∈ ( Base ‘ 𝑅 ) )
700 9 26 698 663 699 mulgnn0cld ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ) → ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ∈ ( Base ‘ 𝑅 ) )
701 eqid ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) = ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
702 361 701 fnmpti ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) }
703 702 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) Fn { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
704 703 25 115 fndmfifsupp ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
705 462 ad2antlr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → 𝑚 Fn dom 𝑚 )
706 469 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → 𝑛 Fn dom 𝑛 )
707 472 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → dom 𝑚 ∈ V )
708 475 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → dom 𝑛 ∈ V )
709 eqidd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ∧ 𝑜 ∈ dom 𝑚 ) → ( 𝑚 ‘ 𝑜 ) = ( 𝑚 ‘ 𝑜 ) )
710 eqidd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ∧ 𝑜 ∈ dom 𝑛 ) → ( 𝑛 ‘ 𝑜 ) = ( 𝑛 ‘ 𝑜 ) )
711 705 706 707 708 477 709 710 offval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → ( 𝑚 ∘f + 𝑛 ) = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) )
712 711 eqeq2d ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ) )
713 712 rexbidva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ) )
714 20 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
715 oveq2 ⊢ ( 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) → ( 𝑚 ∘f + 𝑛 ) = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
716 715 eqeq2d ⊢ ( 𝑛 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) → ( 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
717 716 rexsng ⊢ ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
718 714 717 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑚 ∘f + 𝑛 ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
719 713 718 bitr3d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ↔ 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
720 719 rexbidva ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∃ 𝑛 ∈ { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } 𝑢 = ( 𝑜 ∈ ( dom 𝑚 ∩ dom 𝑛 ) ↦ ( ( 𝑚 ‘ 𝑜 ) + ( 𝑛 ‘ 𝑜 ) ) ) ↔ ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
721 breq1 ⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
722 breq1 ⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑘 ∘r ≤ 𝑑 ↔ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ) )
723 fveq1 ⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑘 ‘ 𝑋 ) = ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) )
724 723 eqeq1d ⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( 𝑘 ‘ 𝑋 ) = 0 ↔ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) )
725 722 724 anbi12d ⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ↔ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ∧ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) ) )
726 725 notbid ⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ↔ ¬ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ∧ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) ) )
727 721 726 anbi12d ⊢ ( 𝑘 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) ↔ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ∧ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) ) ) )
728 458 714 564 syl2an2 ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
729 simplr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
730 simpr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
731 18 245 46 psrbagleadd1 ⊢ ( ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
732 729 714 730 731 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
733 721 elrab ⊢ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ↔ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
734 733 simprbi ⊢ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
735 732 734 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
736 6 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑋 ∈ 𝐼 )
737 487 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑚 : 𝐼 ⟶ ℕ0 )
738 737 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝑚 Fn 𝐼 )
739 135 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
740 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐼 ∈ V )
741 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( 𝑚 ‘ 𝑋 ) = ( 𝑚 ‘ 𝑋 ) )
742 624 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑋 ) = 1 )
743 738 739 740 740 74 741 742 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∧ 𝑋 ∈ 𝐼 ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = ( ( 𝑚 ‘ 𝑋 ) + 1 ) )
744 736 743 mpdan ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = ( ( 𝑚 ‘ 𝑋 ) + 1 ) )
745 737 736 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ‘ 𝑋 ) ∈ ℕ0 )
746 nn0p1nn ⊢ ( ( 𝑚 ‘ 𝑋 ) ∈ ℕ0 → ( ( 𝑚 ‘ 𝑋 ) + 1 ) ∈ ℕ )
747 745 746 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ‘ 𝑋 ) + 1 ) ∈ ℕ )
748 744 747 eqeltrd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) ∈ ℕ )
749 748 nnne0d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) ≠ 0 )
750 749 neneqd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ¬ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 )
751 750 intnand ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ¬ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ∧ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) )
752 735 751 jca ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ∧ ( ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑋 ) = 0 ) ) )
753 727 728 752 elrabd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } )
754 eleq1 ⊢ ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ↔ ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) )
755 753 754 syl5ibrcom ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) )
756 breq1 ⊢ ( 𝑘 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑘 ∘r ≤ 𝑑 ↔ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ) )
757 elrabi ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
758 757 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
759 133 a1i ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 )
760 757 125 syl ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → 𝑢 : 𝐼 ⟶ ℕ0 )
761 760 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 : 𝐼 ⟶ ℕ0 )
762 6 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑋 ∈ 𝐼 )
763 761 762 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ0 )
764 341 notbid ⊢ ( 𝑘 = 𝑢 → ( ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ↔ ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) )
765 120 764 anbi12d ⊢ ( 𝑘 = 𝑢 → ( ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) ↔ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) ) )
766 765 elrab ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ↔ ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) ) )
767 766 simprbi ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ) )
768 767 simpld ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
769 768 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
770 769 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
771 757 126 syl ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → 𝑢 Fn 𝐼 )
772 771 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 Fn 𝐼 )
773 772 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝑢 Fn 𝐼 )
774 23 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
775 90 ffnd ⊢ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
776 774 775 syl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
777 776 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
778 17 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝐼 ∈ V )
779 eqidd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
780 eqidd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) )
781 773 777 778 778 74 779 780 ofrfval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) ) )
782 770 781 mpbid ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) )
783 782 r19.21bi ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) )
784 783 adantr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) )
785 67 ad3antrrr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) → 𝑑 Fn 𝐼 )
786 71 a1i ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
787 17 ad4antr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) → 𝐼 ∈ V )
788 eqidd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
789 80 adantl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
790 785 786 787 787 74 788 789 ofval ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ≠ 𝑋 ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
791 790 an32s ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
792 160 adantl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) = 0 )
793 792 oveq2d ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( ( 𝑑 ‘ 𝑖 ) + 0 ) )
794 31 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝑑 : 𝐼 ⟶ ℕ0 )
795 794 ffvelcdmda ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
796 795 adantr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
797 796 nn0cnd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℂ )
798 797 addridd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( ( 𝑑 ‘ 𝑖 ) + 0 ) = ( 𝑑 ‘ 𝑖 ) )
799 791 793 798 3eqtrd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
800 784 799 breqtrd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 ≠ 𝑋 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) )
801 simpr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ( 𝑢 ‘ 𝑋 ) = 0 )
802 31 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑑 : 𝐼 ⟶ ℕ0 )
803 802 762 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 )
804 803 nn0ge0d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 0 ≤ ( 𝑑 ‘ 𝑋 ) )
805 804 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 0 ≤ ( 𝑑 ‘ 𝑋 ) )
806 801 805 eqbrtrd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) )
807 806 adantr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑋 ) ≤ ( 𝑑 ‘ 𝑋 ) )
808 177 800 807 pm2.61ne ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) )
809 808 ralrimiva ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) )
810 67 adantr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑑 Fn 𝐼 )
811 810 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝑑 Fn 𝐼 )
812 eqidd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
813 773 811 778 778 74 779 812 ofrfval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → ( 𝑢 ∘r ≤ 𝑑 ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
814 809 813 mpbird ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) → 𝑢 ∘r ≤ 𝑑 )
815 814 ex ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑢 ‘ 𝑋 ) = 0 → 𝑢 ∘r ≤ 𝑑 ) )
816 767 simprd ⊢ ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } → ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) )
817 816 adantl ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) )
818 imnan ⊢ ( ( 𝑢 ∘r ≤ 𝑑 → ¬ ( 𝑢 ‘ 𝑋 ) = 0 ) ↔ ¬ ( 𝑢 ∘r ≤ 𝑑 ∧ ( 𝑢 ‘ 𝑋 ) = 0 ) )
819 817 818 sylibr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘r ≤ 𝑑 → ¬ ( 𝑢 ‘ 𝑋 ) = 0 ) )
820 819 con2d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑢 ‘ 𝑋 ) = 0 → ¬ 𝑢 ∘r ≤ 𝑑 ) )
821 815 820 pm2.65d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ¬ ( 𝑢 ‘ 𝑋 ) = 0 )
822 821 neqned ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ‘ 𝑋 ) ≠ 0 )
823 763 822 193 sylanbrc ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ‘ 𝑋 ) ∈ ℕ )
824 823 nnge1d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 1 ≤ ( 𝑢 ‘ 𝑋 ) )
825 824 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → 1 ≤ ( 𝑢 ‘ 𝑋 ) )
826 175 breq2d ⊢ ( 𝑖 = 𝑋 → ( 1 ≤ ( 𝑢 ‘ 𝑖 ) ↔ 1 ≤ ( 𝑢 ‘ 𝑋 ) ) )
827 825 826 syl5ibrcom ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑖 = 𝑋 → 1 ≤ ( 𝑢 ‘ 𝑖 ) ) )
828 827 imp ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑖 = 𝑋 ) → 1 ≤ ( 𝑢 ‘ 𝑖 ) )
829 761 ffvelcdmda ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℕ0 )
830 829 nn0ge0d ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → 0 ≤ ( 𝑢 ‘ 𝑖 ) )
831 830 adantr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) ∧ ¬ 𝑖 = 𝑋 ) → 0 ≤ ( 𝑢 ‘ 𝑖 ) )
832 828 831 ifpimpda ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → if- ( 𝑖 = 𝑋 , 1 ≤ ( 𝑢 ‘ 𝑖 ) , 0 ≤ ( 𝑢 ‘ 𝑖 ) ) )
833 brif1 ⊢ ( if ( 𝑖 = 𝑋 , 1 , 0 ) ≤ ( 𝑢 ‘ 𝑖 ) ↔ if- ( 𝑖 = 𝑋 , 1 ≤ ( 𝑢 ‘ 𝑖 ) , 0 ≤ ( 𝑢 ‘ 𝑖 ) ) )
834 832 833 sylibr ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ≤ ( 𝑢 ‘ 𝑖 ) )
835 834 ralrimiva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ∀ 𝑖 ∈ 𝐼 if ( 𝑖 = 𝑋 , 1 , 0 ) ≤ ( 𝑢 ‘ 𝑖 ) )
836 71 a1i ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) Fn 𝐼 )
837 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝐼 ∈ V )
838 80 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ‘ 𝑖 ) = if ( 𝑖 = 𝑋 , 1 , 0 ) )
839 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) = ( 𝑢 ‘ 𝑖 ) )
840 836 772 837 837 74 838 839 ofrfval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘r ≤ 𝑢 ↔ ∀ 𝑖 ∈ 𝐼 if ( 𝑖 = 𝑋 , 1 , 0 ) ≤ ( 𝑢 ‘ 𝑖 ) ) )
841 835 840 mpbird ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘r ≤ 𝑢 )
842 18 psrbagcon ⊢ ( ( 𝑢 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) : 𝐼 ⟶ ℕ0 ∧ ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∘r ≤ 𝑢 ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑢 ) )
843 758 759 841 842 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∧ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑢 ) )
844 843 simpld ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
845 eqidd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) = ( 𝑑 ‘ 𝑖 ) )
846 810 836 837 837 74 845 838 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
847 772 776 837 837 74 839 846 ofrfval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
848 769 847 mpbid ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ∀ 𝑖 ∈ 𝐼 ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
849 848 r19.21bi ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
850 829 nn0red ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℝ )
851 62 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℝ )
852 802 ffvelcdmda ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℕ0 )
853 852 nn0red ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑑 ‘ 𝑖 ) ∈ ℝ )
854 850 851 853 lesubaddd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ≤ ( 𝑑 ‘ 𝑖 ) ↔ ( 𝑢 ‘ 𝑖 ) ≤ ( ( 𝑑 ‘ 𝑖 ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
855 849 854 mpbird ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ≤ ( 𝑑 ‘ 𝑖 ) )
856 855 ralrimiva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ∀ 𝑖 ∈ 𝐼 ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ≤ ( 𝑑 ‘ 𝑖 ) )
857 772 836 837 837 74 offn ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) Fn 𝐼 )
858 772 836 837 837 74 839 838 ofval ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ‘ 𝑖 ) = ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) )
859 857 810 837 837 74 858 845 ofrfval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 ↔ ∀ 𝑖 ∈ 𝐼 ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) ≤ ( 𝑑 ‘ 𝑖 ) ) )
860 856 859 mpbird ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘r ≤ 𝑑 )
861 756 844 860 elrabd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } )
862 829 nn0cnd ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝑢 ‘ 𝑖 ) ∈ ℂ )
863 239 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → if ( 𝑖 = 𝑋 , 1 , 0 ) ∈ ℂ )
864 862 863 npcand ⊢ ( ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) = ( 𝑢 ‘ 𝑖 ) )
865 864 mpteq2dva ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( 𝑢 ‘ 𝑖 ) ) )
866 857 836 837 837 74 858 838 offval ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( ( ( 𝑢 ‘ 𝑖 ) − if ( 𝑖 = 𝑋 , 1 , 0 ) ) + if ( 𝑖 = 𝑋 , 1 , 0 ) ) ) )
867 761 feqmptd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 = ( 𝑖 ∈ 𝐼 ↦ ( 𝑢 ‘ 𝑖 ) ) )
868 865 866 867 3eqtr4rd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) → 𝑢 = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
869 oveq1 ⊢ ( 𝑚 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
870 869 eqeq2d ⊢ ( 𝑚 = ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) → ( 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑢 = ( ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
871 755 861 868 870 rspceb2dv ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∃ 𝑚 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } 𝑢 = ( 𝑚 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ↔ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) )
872 456 720 871 3bitrd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↔ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } ) )
873 872 eqrdv ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) } )
874 difrab ⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) = { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∧ ¬ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) ) }
875 873 874 eqtr4di ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) = ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) )
876 difssd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
877 875 876 eqsstrd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } )
878 704 877 115 fmptssfisupp ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )
879 difss ⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 }
880 disjdif ⊢ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) = ∅
881 ssdisj ⊢ ( ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ⊆ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∧ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) = ∅ ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) = ∅ )
882 879 880 881 mp2an ⊢ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ) = ∅
883 882 ineqcomi ⊢ ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) = ∅
884 883 a1i ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∩ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) = ∅ )
885 281 101 psdmullem ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∪ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) = ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) )
886 875 885 eqtr4d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) = ( ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ∪ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ) )
887 9 106 10 13 697 700 878 884 886 gsumsplit2 ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
888 693 887 eqtrd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( ( 𝑏 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑏 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( 𝑑 ∘f − 𝑏 ) ) ) ) ∘ ( 𝑢 ∈ ( ∘f + “ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } × { ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) } ) ) ↦ ( 𝑢 ∘f − ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
889 427 593 888 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) ‘ 𝑑 ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
890 417 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ∈ 𝐵 )
891 1 2 37 4 18 387 890 14 psrmulval ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ‘ 𝑑 ) = ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) ) ) ) )
892 8 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → 𝐺 ∈ 𝐵 )
893 1 2 18 287 892 249 psdcoef ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )
894 269 fveq2d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( 𝐺 ‘ ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) )
895 894 oveq2d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f − 𝑢 ) ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) )
896 893 895 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) )
897 896 oveq2d ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) ) = ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
898 311 nn0zd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℤ )
899 9 26 37 mulgass3 ⊢ ( ( 𝑅 ∈ Ring ∧ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ∈ ℤ ∧ ( 𝐹 ‘ 𝑢 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
900 226 898 230 273 899 syl13anc ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
901 897 900 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) ∧ 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) → ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) ) = ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) )
902 901 mpteq2dva ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) ) ) = ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) )
903 902 oveq2d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ‘ ( 𝑑 ∘f − 𝑢 ) ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) )
904 9 10 13 223 323 277 284 gsummptfidmsplit ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
905 891 903 904 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ‘ 𝑑 ) = ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) )
906 421 423 424 424 425 889 905 offval ⊢ ( 𝜑 → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) ∘f ( +g ‘ 𝑅 ) ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
907 419 906 eqtrd ⊢ ( 𝜑 → ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) + ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) = ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ ( ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( 𝑢 ‘ 𝑋 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( ( 𝑅 Σg ( 𝑢 ∈ ( { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ 𝑘 ∘r ≤ 𝑑 } ∖ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ) ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑢 ∈ { 𝑘 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ∣ ( 𝑘 ∘r ≤ 𝑑 ∧ ( 𝑘 ‘ 𝑋 ) = 0 ) } ↦ ( ( ( ( 𝑑 ∘f − 𝑢 ) ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑢 ) ( .r ‘ 𝑅 ) ( 𝐺 ‘ ( ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ∘f − 𝑢 ) ) ) ) ) ) ) ) ) )
908 411 413 907 3eqtr4d ⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐹 · 𝐺 ) ) = ( ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) · 𝐺 ) + ( 𝐹 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐺 ) ) ) )