Theorem etransclem42

Description: The N -th derivative of F applied to J is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	etransclem42.s	⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } )
	etransclem42.x	⊢ ( 𝜑 → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
	etransclem42.p	⊢ ( 𝜑 → 𝑃 ∈ ℕ )
	etransclem42.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
	etransclem42.f	⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑗 ) ↑ 𝑃 ) ) )
	etransclem42.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
	etransclem42.jx	⊢ ( 𝜑 → 𝐽 ∈ 𝑋 )
	etransclem42.jz	⊢ ( 𝜑 → 𝐽 ∈ ℤ )
Assertion	etransclem42	⊢ ( 𝜑 → ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ‘ 𝐽 ) ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		etransclem42.s	⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } )
2		etransclem42.x	⊢ ( 𝜑 → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
3		etransclem42.p	⊢ ( 𝜑 → 𝑃 ∈ ℕ )
4		etransclem42.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
5		etransclem42.f	⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑗 ) ↑ 𝑃 ) ) )
6		etransclem42.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
7		etransclem42.jx	⊢ ( 𝜑 → 𝐽 ∈ 𝑋 )
8		etransclem42.jz	⊢ ( 𝜑 → 𝐽 ∈ ℤ )
9		etransclem5	⊢ ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 − 𝑗 ) ↑ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) )
10		etransclem11	⊢ ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) = ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = 𝑛 } )
11	1 2 3 4 5 6 9 7 8 10	etransclem36	⊢ ( 𝜑 → ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ‘ 𝐽 ) ∈ ℤ )