Metamath Proof Explorer


Theorem etransclem30

Description: The N -th derivative of F . (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypotheses etransclem30.s ( 𝜑𝑆 ∈ { ℝ , ℂ } )
etransclem30.a ( 𝜑𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
etransclem30.p ( 𝜑𝑃 ∈ ℕ )
etransclem30.m ( 𝜑𝑀 ∈ ℕ0 )
etransclem30.f 𝐹 = ( 𝑥𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥𝑗 ) ↑ 𝑃 ) ) )
etransclem30.n ( 𝜑𝑁 ∈ ℕ0 )
etransclem30.h 𝐻 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑥𝑋 ↦ ( ( 𝑥𝑗 ) ↑ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) )
etransclem30.c 𝐶 = ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐𝑗 ) = 𝑛 } )
Assertion etransclem30 ( 𝜑 → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) = ( 𝑥𝑋 ↦ Σ 𝑐 ∈ ( 𝐶𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐𝑗 ) ) ) · ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( ( 𝑆 D𝑛 ( 𝐻𝑗 ) ) ‘ ( 𝑐𝑗 ) ) ‘ 𝑥 ) ) ) )

Proof

Step Hyp Ref Expression
1 etransclem30.s ( 𝜑𝑆 ∈ { ℝ , ℂ } )
2 etransclem30.a ( 𝜑𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
3 etransclem30.p ( 𝜑𝑃 ∈ ℕ )
4 etransclem30.m ( 𝜑𝑀 ∈ ℕ0 )
5 etransclem30.f 𝐹 = ( 𝑥𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥𝑗 ) ↑ 𝑃 ) ) )
6 etransclem30.n ( 𝜑𝑁 ∈ ℕ0 )
7 etransclem30.h 𝐻 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑥𝑋 ↦ ( ( 𝑥𝑗 ) ↑ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) )
8 etransclem30.c 𝐶 = ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐𝑗 ) = 𝑛 } )
9 eqid ( 𝑥𝑋 ↦ ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( 𝐻𝑗 ) ‘ 𝑥 ) ) = ( 𝑥𝑋 ↦ ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( 𝐻𝑗 ) ‘ 𝑥 ) )
10 1 2 3 4 5 6 7 8 9 etransclem29 ( 𝜑 → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) = ( 𝑥𝑋 ↦ Σ 𝑐 ∈ ( 𝐶𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐𝑗 ) ) ) · ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ( ( 𝑆 D𝑛 ( 𝐻𝑗 ) ) ‘ ( 𝑐𝑗 ) ) ‘ 𝑥 ) ) ) )