Theorem etransclem40

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

	Ref	Expression
Hypotheses	etransclem40.s	⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } )
	etransclem40.a	⊢ ( 𝜑 → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
	etransclem40.p	⊢ ( 𝜑 → 𝑃 ∈ ℕ )
	etransclem40.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
	etransclem40.f	⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑘 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑘 ) ↑ 𝑃 ) ) )
	etransclem40.6	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion	etransclem40	⊢ ( 𝜑 → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ∈ ( 𝑋 –cn→ ℂ ) )

Proof

Step	Hyp	Ref	Expression
1		etransclem40.s	⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } )
2		etransclem40.a	⊢ ( 𝜑 → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
3		etransclem40.p	⊢ ( 𝜑 → 𝑃 ∈ ℕ )
4		etransclem40.m	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
5		etransclem40.f	⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑘 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑘 ) ↑ 𝑃 ) ) )
6		etransclem40.6	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
7		etransclem5	⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ 𝑋 ↦ ( ( 𝑦 − 𝑗 ) ↑ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) = ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑥 ∈ 𝑋 ↦ ( ( 𝑥 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) )
8		etransclem11	⊢ ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑗 ) = 𝑚 } ) = ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) = 𝑛 } )
9	1 2 3 4 5 6 7 8	etransclem34	⊢ ( 𝜑 → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ∈ ( 𝑋 –cn→ ℂ ) )