Metamath Proof Explorer


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 ( 𝜑𝑆 ∈ { ℝ , ℂ } )
etransclem42.x ( 𝜑𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
etransclem42.p ( 𝜑𝑃 ∈ ℕ )
etransclem42.m ( 𝜑𝑀 ∈ ℕ0 )
etransclem42.f 𝐹 = ( 𝑥𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥𝑗 ) ↑ 𝑃 ) ) )
etransclem42.n ( 𝜑𝑁 ∈ ℕ0 )
etransclem42.jx ( 𝜑𝐽𝑋 )
etransclem42.jz ( 𝜑𝐽 ∈ ℤ )
Assertion etransclem42 ( 𝜑 → ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ‘ 𝐽 ) ∈ ℤ )

Proof

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