Metamath Proof Explorer


Theorem psdcoef

Description: Coefficient of a term of the derivative of a power series. (Contributed by SN, 12-Apr-2025)

Ref Expression
Hypotheses psdval.s 𝑆 = ( 𝐼 mPwSer 𝑅 )
psdval.b 𝐵 = ( Base ‘ 𝑆 )
psdval.d 𝐷 = { ∈ ( ℕ0m 𝐼 ) ∣ ( “ ℕ ) ∈ Fin }
psdval.x ( 𝜑𝑋𝐼 )
psdval.f ( 𝜑𝐹𝐵 )
psdcoef.k ( 𝜑𝐾𝐷 )
Assertion psdcoef ( 𝜑 → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝐾 ) = ( ( ( 𝐾𝑋 ) + 1 ) ( .g𝑅 ) ( 𝐹 ‘ ( 𝐾f + ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 psdval.s 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psdval.b 𝐵 = ( Base ‘ 𝑆 )
3 psdval.d 𝐷 = { ∈ ( ℕ0m 𝐼 ) ∣ ( “ ℕ ) ∈ Fin }
4 psdval.x ( 𝜑𝑋𝐼 )
5 psdval.f ( 𝜑𝐹𝐵 )
6 psdcoef.k ( 𝜑𝐾𝐷 )
7 fveq1 ( 𝑘 = 𝐾 → ( 𝑘𝑋 ) = ( 𝐾𝑋 ) )
8 7 oveq1d ( 𝑘 = 𝐾 → ( ( 𝑘𝑋 ) + 1 ) = ( ( 𝐾𝑋 ) + 1 ) )
9 fvoveq1 ( 𝑘 = 𝐾 → ( 𝐹 ‘ ( 𝑘f + ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) = ( 𝐹 ‘ ( 𝐾f + ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) )
10 8 9 oveq12d ( 𝑘 = 𝐾 → ( ( ( 𝑘𝑋 ) + 1 ) ( .g𝑅 ) ( 𝐹 ‘ ( 𝑘f + ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) = ( ( ( 𝐾𝑋 ) + 1 ) ( .g𝑅 ) ( 𝐹 ‘ ( 𝐾f + ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )
11 1 2 3 4 5 psdval ( 𝜑 → ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) = ( 𝑘𝐷 ↦ ( ( ( 𝑘𝑋 ) + 1 ) ( .g𝑅 ) ( 𝐹 ‘ ( 𝑘f + ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ) )
12 ovexd ( 𝜑 → ( ( ( 𝐾𝑋 ) + 1 ) ( .g𝑅 ) ( 𝐹 ‘ ( 𝐾f + ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) ∈ V )
13 10 11 6 12 fvmptd4 ( 𝜑 → ( ( ( ( 𝐼 mPSDer 𝑅 ) ‘ 𝑋 ) ‘ 𝐹 ) ‘ 𝐾 ) = ( ( ( 𝐾𝑋 ) + 1 ) ( .g𝑅 ) ( 𝐹 ‘ ( 𝐾f + ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) ) )