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Metamath Proof Explorer

Theorem psrridm

Description: The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014) (Proof shortened by AV, 8-Jul-2019)

Ref Expression
Hypotheses psrring.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
psrring.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
psrring.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
psr1cl.d ⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
psr1cl.z ⊢ 0 = ( 0g ‘ 𝑅 )
psr1cl.o ⊢ 1 = ( 1r ‘ 𝑅 )
psr1cl.u ⊢ 𝑈 = ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) )
psr1cl.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
psrlidm.t ⊢ · = ( .r ‘ 𝑆 )
psrlidm.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion psrridm ( 𝜑 → ( 𝑋 · 𝑈 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 psrring.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psrring.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
3 psrring.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
4 psr1cl.d ⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
5 psr1cl.z ⊢ 0 = ( 0g ‘ 𝑅 )
6 psr1cl.o ⊢ 1 = ( 1r ‘ 𝑅 )
7 psr1cl.u ⊢ 𝑈 = ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) )
8 psr1cl.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
9 psrlidm.t ⊢ · = ( .r ‘ 𝑆 )
10 psrlidm.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
11 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
12 1 2 3 4 5 6 7 8 psr1cl ⊢ ( 𝜑 → 𝑈 ∈ 𝐵 )
13 1 8 9 3 10 12 psrmulcl ⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) ∈ 𝐵 )
14 1 11 4 8 13 psrelbas ⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
15 14 ffnd ⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) Fn 𝐷 )
16 1 11 4 8 10 psrelbas ⊢ ( 𝜑 → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
17 16 ffnd ⊢ ( 𝜑 → 𝑋 Fn 𝐷 )
18 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
19 10 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑋 ∈ 𝐵 )
20 12 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑈 ∈ 𝐵 )
21 simpr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 ∈ 𝐷 )
22 1 8 18 9 4 19 20 21 psrmulval ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 · 𝑈 ) ‘ 𝑦 ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) )
23 breq1 ⊢ ( 𝑔 = 𝑦 → ( 𝑔 ∘r ≤ 𝑦 ↔ 𝑦 ∘r ≤ 𝑦 ) )
24 2 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝐼 ∈ 𝑉 )
25 4 psrbagf ⊢ ( 𝑦 ∈ 𝐷 → 𝑦 : 𝐼 ⟶ ℕ0 )
26 25 adantl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 : 𝐼 ⟶ ℕ0 )
27 nn0re ⊢ ( 𝑧 ∈ ℕ0 → 𝑧 ∈ ℝ )
28 27 leidd ⊢ ( 𝑧 ∈ ℕ0 → 𝑧 ≤ 𝑧 )
29 28 adantl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ℕ0 ) → 𝑧 ≤ 𝑧 )
30 24 26 29 caofref ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 ∘r ≤ 𝑦 )
31 23 21 30 elrabd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } )
32 31 snssd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → { 𝑦 } ⊆ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } )
33 32 resmptd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ↾ { 𝑦 } ) = ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) )
34 33 oveq2d ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ↾ { 𝑦 } ) ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) )
35 ringcmn ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ CMnd )
36 3 35 syl ⊢ ( 𝜑 → 𝑅 ∈ CMnd )
37 36 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑅 ∈ CMnd )
38 ovex ⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
39 4 38 rab2ex ⊢ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∈ V
40 39 a1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∈ V )
41 3 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑅 ∈ Ring )
42 16 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
43 breq1 ⊢ ( 𝑔 = 𝑧 → ( 𝑔 ∘r ≤ 𝑦 ↔ 𝑧 ∘r ≤ 𝑦 ) )
44 43 elrab ⊢ ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↔ ( 𝑧 ∈ 𝐷 ∧ 𝑧 ∘r ≤ 𝑦 ) )
45 44 bilani ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑧 ∈ 𝐷 ∧ 𝑧 ∘r ≤ 𝑦 ) )
46 45 simpld ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 ∈ 𝐷 )
47 42 46 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑋 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) )
48 1 11 4 8 20 psrelbas ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑈 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
49 48 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑈 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
50 21 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑦 ∈ 𝐷 )
51 4 psrbagf ⊢ ( 𝑧 ∈ 𝐷 → 𝑧 : 𝐼 ⟶ ℕ0 )
52 46 51 syl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 : 𝐼 ⟶ ℕ0 )
53 45 simprd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 ∘r ≤ 𝑦 )
54 4 psrbagcon ⊢ ( ( 𝑦 ∈ 𝐷 ∧ 𝑧 : 𝐼 ⟶ ℕ0 ∧ 𝑧 ∘r ≤ 𝑦 ) → ( ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 ∧ ( 𝑦 ∘f − 𝑧 ) ∘r ≤ 𝑦 ) )
55 50 52 53 54 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 ∧ ( 𝑦 ∘f − 𝑧 ) ∘r ≤ 𝑦 ) )
56 55 simpld ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 )
57 49 56 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ∈ ( Base ‘ 𝑅 ) )
58 11 18 ringcl ⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ∈ ( Base ‘ 𝑅 ) )
59 41 47 57 58 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ∈ ( Base ‘ 𝑅 ) )
60 59 fmpttd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) : { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ⟶ ( Base ‘ 𝑅 ) )
61 eldifi ⊢ ( 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) → 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } )
62 61 56 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 )
63 eqeq1 ⊢ ( 𝑥 = ( 𝑦 ∘f − 𝑧 ) → ( 𝑥 = ( 𝐼 × { 0 } ) ↔ ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ) )
64 63 ifbid ⊢ ( 𝑥 = ( 𝑦 ∘f − 𝑧 ) → if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) = if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) )
65 6 fvexi ⊢ 1 ∈ V
66 5 fvexi ⊢ 0 ∈ V
67 65 66 ifex ⊢ if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) ∈ V
68 64 7 67 fvmpt ⊢ ( ( 𝑦 ∘f − 𝑧 ) ∈ 𝐷 → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) )
69 62 68 syl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) )
70 eldifsni ⊢ ( 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) → 𝑧 ≠ 𝑦 )
71 70 adantl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → 𝑧 ≠ 𝑦 )
72 71 necomd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → 𝑦 ≠ 𝑧 )
73 24 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝐼 ∈ 𝑉 )
74 nn0sscn ⊢ ℕ0 ⊆ ℂ
75 fss ⊢ ( ( 𝑦 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑦 : 𝐼 ⟶ ℂ )
76 26 74 75 sylancl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑦 : 𝐼 ⟶ ℂ )
77 76 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑦 : 𝐼 ⟶ ℂ )
78 fss ⊢ ( ( 𝑧 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑧 : 𝐼 ⟶ ℂ )
79 52 74 78 sylancl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → 𝑧 : 𝐼 ⟶ ℂ )
80 ofsubeq0 ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑦 : 𝐼 ⟶ ℂ ∧ 𝑧 : 𝐼 ⟶ ℂ ) → ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) )
81 73 77 79 80 syl3anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) )
82 61 81 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑧 ) )
83 82 necon3bbid ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ¬ ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 ≠ 𝑧 ) )
84 72 83 mpbird ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ¬ ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) )
85 84 iffalsed ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → if ( ( 𝑦 ∘f − 𝑧 ) = ( 𝐼 × { 0 } ) , 1 , 0 ) = 0 )
86 69 85 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = 0 )
87 86 oveq2d ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) = ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) )
88 11 18 5 ringrz ⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ‘ 𝑧 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) = 0 )
89 41 47 88 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) = 0 )
90 61 89 sylan2 ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) 0 ) = 0 )
91 87 90 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑧 ∈ ( { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ∖ { 𝑦 } ) ) → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) = 0 )
92 91 40 suppss2 ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) supp 0 ) ⊆ { 𝑦 } )
93 40 mptexd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ∈ V )
94 funmpt ⊢ Fun ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) )
95 94 a1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → Fun ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) )
96 66 a1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 0 ∈ V )
97 snfi ⊢ { 𝑦 } ∈ Fin
98 97 a1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → { 𝑦 } ∈ Fin )
99 suppssfifsupp ⊢ ( ( ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ∈ V ∧ Fun ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ∧ 0 ∈ V ) ∧ ( { 𝑦 } ∈ Fin ∧ ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) supp 0 ) ⊆ { 𝑦 } ) ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) finSupp 0 )
100 93 95 96 98 92 99 syl32anc ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) finSupp 0 )
101 11 5 37 40 60 92 100 gsumres ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ↾ { 𝑦 } ) ) = ( 𝑅 Σg ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) )
102 3 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑅 ∈ Ring )
103 ringmnd ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd )
104 102 103 syl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → 𝑅 ∈ Mnd )
105 eqid ⊢ 𝑦 = 𝑦
106 ofsubeq0 ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑦 : 𝐼 ⟶ ℂ ∧ 𝑦 : 𝐼 ⟶ ℂ ) → ( ( 𝑦 ∘f − 𝑦 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑦 ) )
107 24 76 76 106 syl3anc ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑦 ∘f − 𝑦 ) = ( 𝐼 × { 0 } ) ↔ 𝑦 = 𝑦 ) )
108 105 107 mpbiri ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑦 ∘f − 𝑦 ) = ( 𝐼 × { 0 } ) )
109 108 fveq2d ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) = ( 𝑈 ‘ ( 𝐼 × { 0 } ) ) )
110 fconstmpt ⊢ ( 𝐼 × { 0 } ) = ( 𝑤 ∈ 𝐼 ↦ 0 )
111 4 fczpsrbag ⊢ ( 𝐼 ∈ 𝑉 → ( 𝑤 ∈ 𝐼 ↦ 0 ) ∈ 𝐷 )
112 2 111 syl ⊢ ( 𝜑 → ( 𝑤 ∈ 𝐼 ↦ 0 ) ∈ 𝐷 )
113 110 112 eqeltrid ⊢ ( 𝜑 → ( 𝐼 × { 0 } ) ∈ 𝐷 )
114 113 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝐼 × { 0 } ) ∈ 𝐷 )
115 iftrue ⊢ ( 𝑥 = ( 𝐼 × { 0 } ) → if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) = 1 )
116 115 7 65 fvmpt ⊢ ( ( 𝐼 × { 0 } ) ∈ 𝐷 → ( 𝑈 ‘ ( 𝐼 × { 0 } ) ) = 1 )
117 114 116 syl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑈 ‘ ( 𝐼 × { 0 } ) ) = 1 )
118 109 117 eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) = 1 )
119 118 oveq2d ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) 1 ) )
120 16 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑋 ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) )
121 11 18 6 ringridm ⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ‘ 𝑦 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑋 ‘ 𝑦 ) )
122 102 120 121 syl2anc ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑋 ‘ 𝑦 ) )
123 119 122 eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) = ( 𝑋 ‘ 𝑦 ) )
124 123 120 eqeltrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) ∈ ( Base ‘ 𝑅 ) )
125 fveq2 ⊢ ( 𝑧 = 𝑦 → ( 𝑋 ‘ 𝑧 ) = ( 𝑋 ‘ 𝑦 ) )
126 oveq2 ⊢ ( 𝑧 = 𝑦 → ( 𝑦 ∘f − 𝑧 ) = ( 𝑦 ∘f − 𝑦 ) )
127 126 fveq2d ⊢ ( 𝑧 = 𝑦 → ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) = ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) )
128 125 127 oveq12d ⊢ ( 𝑧 = 𝑦 → ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) )
129 11 128 gsumsn ⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑦 ∈ 𝐷 ∧ ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) ∈ ( Base ‘ 𝑅 ) ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) )
130 104 21 124 129 syl3anc ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) )
131 34 101 130 3eqtr3d ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( 𝑅 Σg ( 𝑧 ∈ { 𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑦 } ↦ ( ( 𝑋 ‘ 𝑧 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑧 ) ) ) ) ) = ( ( 𝑋 ‘ 𝑦 ) ( .r ‘ 𝑅 ) ( 𝑈 ‘ ( 𝑦 ∘f − 𝑦 ) ) ) )
132 22 131 123 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐷 ) → ( ( 𝑋 · 𝑈 ) ‘ 𝑦 ) = ( 𝑋 ‘ 𝑦 ) )
133 15 17 132 eqfnfvd ⊢ ( 𝜑 → ( 𝑋 · 𝑈 ) = 𝑋 )