Theorem etransclem30

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

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

Proof

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