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Metamath Proof Explorer

Theorem psdvsca

Description: The derivative of a scaled power series is the scaled derivative. (Contributed by SN, 12-Apr-2025)

Ref Expression
Hypotheses psdvsca.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
psdvsca.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
psdvsca.m ⊢ · = ( ·𝑠 ‘ 𝑆 )
psdvsca.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
psdvsca.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
psdvsca.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
psdvsca.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
psdvsca.c ⊢ ( 𝜑 → 𝐶 ∈ 𝐾 )
Assertion psdvsca ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) = ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) )

Proof

Step Hyp Ref Expression
1 psdvsca.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psdvsca.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
3 psdvsca.m ⊢ · = ( ·𝑠 ‘ 𝑆 )
4 psdvsca.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
5 psdvsca.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
6 psdvsca.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
7 psdvsca.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
8 psdvsca.c ⊢ ( 𝜑 → 𝐶 ∈ 𝐾 )
9 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
10 eqid ⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
11 5 crngringd ⊢ ( 𝜑 → 𝑅 ∈ Ring )
12 ringmgm ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mgm )
13 11 12 syl ⊢ ( 𝜑 → 𝑅 ∈ Mgm )
14 1 3 4 2 11 8 7 psrvscacl ⊢ ( 𝜑 → ( 𝐶 · 𝐹 ) ∈ 𝐵 )
15 1 2 13 6 14 psdcl ⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) ∈ 𝐵 )
16 1 9 10 2 15 psrelbas ⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
17 16 ffnd ⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
18 1 2 13 6 7 psdcl ⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∈ 𝐵 )
19 1 3 4 2 11 8 18 psrvscacl ⊢ ( 𝜑 → ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) ∈ 𝐵 )
20 1 9 10 2 19 psrelbas ⊢ ( 𝜑 → ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝑅 ) )
21 20 ffnd ⊢ ( 𝜑 → ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
22 11 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑅 ∈ Ring )
23 10 psrbagf ⊢ ( 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } → 𝑑 : 𝐼 ⟶ ℕ0 )
24 23 adantl ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 : 𝐼 ⟶ ℕ0 )
25 6 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑋 ∈ 𝐼 )
26 24 25 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 )
27 peano2nn0 ⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℕ0 )
28 27 nn0zd ⊢ ( ( 𝑑 ‘ 𝑋 ) ∈ ℕ0 → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℤ )
29 26 28 syl ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℤ )
30 8 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐶 ∈ 𝐾 )
31 1 4 10 2 7 psrelbas ⊢ ( 𝜑 → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ 𝐾 )
32 31 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐹 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ⟶ 𝐾 )
33 simpr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
34 reldmpsr ⊢ Rel dom mPwSer
35 1 2 34 strov2rcl ⊢ ( 𝐹 ∈ 𝐵 → 𝐼 ∈ V )
36 7 35 syl ⊢ ( 𝜑 → 𝐼 ∈ V )
37 10 psrbagsn ⊢ ( 𝐼 ∈ V → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
38 36 37 syl ⊢ ( 𝜑 → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
39 38 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } )
40 10 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 } )
41 33 39 40 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 } )
42 32 41 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ∈ 𝐾 )
43 eqid ⊢ ( .g ‘ 𝑅 ) = ( .g ‘ 𝑅 )
44 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
45 4 43 44 mulgass3 ⊢ ( ( 𝑅 ∈ Ring ∧ ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ∈ ℤ ∧ 𝐶 ∈ 𝐾 ∧ ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ∈ 𝐾 ) ) → ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐶 ( .r ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) )
46 22 29 30 42 45 syl13anc ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐶 ( .r ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) )
47 7 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → 𝐹 ∈ 𝐵 )
48 1 2 10 25 47 33 psdcoef ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑑 ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )
49 48 oveq2d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑑 ) ) = ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) )
50 1 3 4 2 44 10 30 47 41 psrvscaval ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐶 · 𝐹 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝐶 ( .r ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )
51 50 oveq2d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐶 · 𝐹 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( 𝐶 ( .r ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) )
52 46 49 51 3eqtr4rd ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐶 · 𝐹 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑑 ) ) )
53 14 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( 𝐶 · 𝐹 ) ∈ 𝐵 )
54 1 2 10 25 53 33 psdcoef ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) ‘ 𝑑 ) = ( ( ( 𝑑 ‘ 𝑋 ) + 1 ) ( .g ‘ 𝑅 ) ( ( 𝐶 · 𝐹 ) ‘ ( 𝑑 ∘f + ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )
55 18 adantr ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ∈ 𝐵 )
56 1 3 4 2 44 10 30 55 33 psrvscaval ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) ‘ 𝑑 ) = ( 𝐶 ( .r ‘ 𝑅 ) ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝑑 ) ) )
57 52 54 56 3eqtr4d ⊢ ( ( 𝜑 ∧ 𝑑 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ) → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) ‘ 𝑑 ) = ( ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) ‘ 𝑑 ) )
58 17 21 57 eqfnfvd ⊢ ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ ( 𝐶 · 𝐹 ) ) = ( 𝐶 · ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ) )