# Metamath Proof Explorer

## Theorem etransclem36

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

Ref Expression
Hypotheses etransclem36.s ( 𝜑𝑆 ∈ { ℝ , ℂ } )
etransclem36.x ( 𝜑𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
etransclem36.p ( 𝜑𝑃 ∈ ℕ )
etransclem36.m ( 𝜑𝑀 ∈ ℕ0 )
etransclem36.f 𝐹 = ( 𝑥𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥𝑗 ) ↑ 𝑃 ) ) )
etransclem36.n ( 𝜑𝑁 ∈ ℕ0 )
etransclem36.h 𝐻 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑥𝑋 ↦ ( ( 𝑥𝑗 ) ↑ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) )
etransclem36.jx ( 𝜑𝐽𝑋 )
etransclem36.jz ( 𝜑𝐽 ∈ ℤ )
etransclem36.10 𝐶 = ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐𝑗 ) = 𝑛 } )
Assertion etransclem36 ( 𝜑 → ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ‘ 𝐽 ) ∈ ℤ )

### Proof

Step Hyp Ref Expression
1 etransclem36.s ( 𝜑𝑆 ∈ { ℝ , ℂ } )
2 etransclem36.x ( 𝜑𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) )
3 etransclem36.p ( 𝜑𝑃 ∈ ℕ )
4 etransclem36.m ( 𝜑𝑀 ∈ ℕ0 )
5 etransclem36.f 𝐹 = ( 𝑥𝑋 ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥𝑗 ) ↑ 𝑃 ) ) )
6 etransclem36.n ( 𝜑𝑁 ∈ ℕ0 )
7 etransclem36.h 𝐻 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑥𝑋 ↦ ( ( 𝑥𝑗 ) ↑ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) )
8 etransclem36.jx ( 𝜑𝐽𝑋 )
9 etransclem36.jz ( 𝜑𝐽 ∈ ℤ )
10 etransclem36.10 𝐶 = ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐𝑗 ) = 𝑛 } )
11 1 2 3 4 5 6 7 10 8 etransclem31 ( 𝜑 → ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ‘ 𝐽 ) = Σ 𝑐 ∈ ( 𝐶𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 𝐽 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐𝑗 ) ) ) ) · ( ( 𝐽𝑗 ) ↑ ( 𝑃 − ( 𝑐𝑗 ) ) ) ) ) ) ) )
12 10 6 etransclem16 ( 𝜑 → ( 𝐶𝑁 ) ∈ Fin )
13 3 adantr ( ( 𝜑𝑐 ∈ ( 𝐶𝑁 ) ) → 𝑃 ∈ ℕ )
14 4 adantr ( ( 𝜑𝑐 ∈ ( 𝐶𝑁 ) ) → 𝑀 ∈ ℕ0 )
15 6 adantr ( ( 𝜑𝑐 ∈ ( 𝐶𝑁 ) ) → 𝑁 ∈ ℕ0 )
16 9 adantr ( ( 𝜑𝑐 ∈ ( 𝐶𝑁 ) ) → 𝐽 ∈ ℤ )
17 etransclem11 ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐𝑗 ) = 𝑛 } ) = ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑𝑘 ) = 𝑚 } )
18 etransclem11 ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑𝑘 ) = 𝑚 } ) = ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑒𝑗 ) = 𝑛 } )
19 10 17 18 3eqtri 𝐶 = ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑒𝑗 ) = 𝑛 } )
20 simpr ( ( 𝜑𝑐 ∈ ( 𝐶𝑁 ) ) → 𝑐 ∈ ( 𝐶𝑁 ) )
21 13 14 15 16 19 20 etransclem26 ( ( 𝜑𝑐 ∈ ( 𝐶𝑁 ) ) → ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 𝐽 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐𝑗 ) ) ) ) · ( ( 𝐽𝑗 ) ↑ ( 𝑃 − ( 𝑐𝑗 ) ) ) ) ) ) ) ∈ ℤ )
22 12 21 fsumzcl ( 𝜑 → Σ 𝑐 ∈ ( 𝐶𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 𝐽 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐𝑗 ) ) ) ) · ( ( 𝐽𝑗 ) ↑ ( 𝑃 − ( 𝑐𝑗 ) ) ) ) ) ) ) ∈ ℤ )
23 11 22 eqeltrd ( 𝜑 → ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ‘ 𝐽 ) ∈ ℤ )