Theorem etransclem27 45062

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

Hypotheses

Ref	Expression
etransclem27.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem27.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
etransclem27.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem27.h	⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
etransclem27.cfi	⊢ (𝜑 → 𝐶 ∈ Fin)
etransclem27.cf	⊢ (𝜑 → 𝐶:dom 𝐶⟶(ℕ0 ↑m (0...𝑀)))
etransclem27.g	⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥))
etransclem27.jx	⊢ (𝜑 → 𝐽 ∈ 𝑋)
etransclem27.jz	⊢ (𝜑 → 𝐽 ∈ ℤ)

Assertion

Ref	Expression
etransclem27	⊢ (𝜑 → (𝐺‘𝐽) ∈ ℤ)

Distinct variable groups:   𝐶,𝑗,𝑙,𝑥   𝑥,𝐻   𝑗,𝐽,𝑙,𝑥   𝑗,𝑀,𝑥   𝑃,𝑗,𝑥   𝑥,𝑆   𝑗,𝑋,𝑥   𝜑,𝑗,𝑙,𝑥

Allowed substitution hints:   𝑃(𝑙)   𝑆(𝑗,𝑙)   𝐺(𝑥,𝑗,𝑙)   𝐻(𝑗,𝑙)   𝑀(𝑙)   𝑋(𝑙)

Proof of Theorem etransclem27

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem27.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥))
2		fveq2 6891	. . . . 5 ⊢ (𝑥 = 𝐽 → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽))
3	2	prodeq2ad 44393	. . . 4 ⊢ (𝑥 = 𝐽 → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥) = ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽))
4	3	sumeq2sdv 15652	. . 3 ⊢ (𝑥 = 𝐽 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥) = Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽))
5		etransclem27.jx	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑋)
6		etransclem27.cfi	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Fin)
7		dmfi 9332	. . . . 5 ⊢ (𝐶 ∈ Fin → dom 𝐶 ∈ Fin)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → dom 𝐶 ∈ Fin)
9		fzfid 13940	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (0...𝑀) ∈ Fin)
10		etransclem27.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
11	10	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
12		etransclem27.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
13	12	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
14		etransclem27.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
15	14	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
16		etransclem27.h	. . . . . . . 8 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
17		etransclem5 45040	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑧 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
18	16, 17	eqtri 2760	. . . . . . 7 ⊢ 𝐻 = (𝑧 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
19		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
20		etransclem27.cf	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶:dom 𝐶⟶(ℕ0 ↑m (0...𝑀)))
21	20	ffvelcdmda 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (𝐶‘𝑙) ∈ (ℕ0 ↑m (0...𝑀)))
22		elmapi 8845	. . . . . . . . 9 ⊢ ((𝐶‘𝑙) ∈ (ℕ0 ↑m (0...𝑀)) → (𝐶‘𝑙):(0...𝑀)⟶ℕ0)
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (𝐶‘𝑙):(0...𝑀)⟶ℕ0)
24	23	ffvelcdmda 7086	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℕ0)
25	11, 13, 15, 18, 19, 24	etransclem20 45055	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗)):𝑋⟶ℂ)
26	5	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝐽 ∈ 𝑋)
27	25, 26	ffvelcdmd 7087	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
28	9, 27	fprodcl 15898	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
29	8, 28	fsumcl 15681	. . 3 ⊢ (𝜑 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
30	1, 4, 5, 29	fvmptd3 7021	. 2 ⊢ (𝜑 → (𝐺‘𝐽) = Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽))
31	11, 13, 15, 18, 19, 24, 26	etransclem21 45056	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) = if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))))
32		iftrue 4534	. . . . . . . 8 ⊢ (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) = 0)
33		0zd 12572	. . . . . . . 8 ⊢ (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗) → 0 ∈ ℤ)
34	32, 33	eqeltrd 2833	. . . . . . 7 ⊢ (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ)
35	34	adantl 482	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ)
36		0zd 12572	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 0 ∈ ℤ)
37		nnm1nn0 12515	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
38	14, 37	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ0)
39	14	nnnn0d 12534	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ0)
40	38, 39	ifcld 4574	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
41	40	nn0zd 12586	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
42	41	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
43	24	nn0zd 12586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℤ)
44	43	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐶‘𝑙)‘𝑗) ∈ ℤ)
45	42, 44	zsubcld 12673	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℤ)
46	44	zred 12668	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐶‘𝑙)‘𝑗) ∈ ℝ)
47	42	zred 12668	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
48		simpr 485	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗))
49	46, 47, 48	nltled 11366	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐶‘𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
50	47, 46	subge0d 11806	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ↔ ((𝐶‘𝑙)‘𝑗) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃)))
51	49, 50	mpbird 256	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))
52		0red 11219	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ ℝ)
53	24	nn0red 12535	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℝ)
54	40	nn0red 12535	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
55	54	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
56	24	nn0ge0d 12537	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ≤ ((𝐶‘𝑙)‘𝑗))
57	52, 53, 55, 56	lesub2dd 11833	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0))
58	55	recnd 11244	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
59	58	subid1d 11562	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − 0) = if(𝑗 = 0, (𝑃 − 1), 𝑃))
60	57, 59	breqtrd 5174	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
61	60	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ≤ if(𝑗 = 0, (𝑃 − 1), 𝑃))
62	36, 42, 45, 51, 61	elfzd 13494	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)))
63		permnn 14288	. . . . . . . . . 10 ⊢ ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ (0...if(𝑗 = 0, (𝑃 − 1), 𝑃)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℕ)
64	62, 63	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℕ)
65	64	nnzd 12587	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℤ)
66		etransclem27.jz	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℤ)
67	66	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 𝐽 ∈ ℤ)
68		elfzelz 13503	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
69	68	ad2antlr 725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 𝑗 ∈ ℤ)
70	67, 69	zsubcld 12673	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (𝐽 − 𝑗) ∈ ℤ)
71		elnn0z 12573	. . . . . . . . . 10 ⊢ ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℕ0 ↔ ((if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℤ ∧ 0 ≤ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))
72	45, 51, 71	sylanbrc 583	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℕ0)
73		zexpcl 14044	. . . . . . . . 9 ⊢ (((𝐽 − 𝑗) ∈ ℤ ∧ (if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)) ∈ ℕ0) → ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))) ∈ ℤ)
74	70, 72, 73	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))) ∈ ℤ)
75	65, 74	zmulcld 12674	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) ∈ ℤ)
76	36, 75	ifcld 4574	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ)
77	35, 76	pm2.61dan 811	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗)))) · ((𝐽 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − ((𝐶‘𝑙)‘𝑗))))) ∈ ℤ)
78	31, 77	eqeltrd 2833	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
79	9, 78	fprodzcl 15900	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
80	8, 79	fsumzcl 15683	. 2 ⊢ (𝜑 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
81	30, 80	eqeltrd 2833	1 ⊢ (𝜑 → (𝐺‘𝐽) ∈ ℤ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ifcif 4528  {cpr 4630   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ⟶wf 6539  ‘cfv 6543  (class class class)co 7411   ↑_m cmap 8822  Fincfn 8941  ℂcc 11110  ℝcr 11111  0cc0 11112  1c1 11113   · cmul 11117   < clt 11250   ≤ cle 11251   − cmin 11446   / cdiv 11873  ℕcn 12214  ℕ₀cn0 12474  ℤcz 12560  ...cfz 13486  ↑cexp 14029  !cfa 14235  Σcsu 15634  ∏cprod 15851   ↾_t crest 17368  TopOpenctopn 17369  ℂ_fldccnfld 20950   D^𝑛 cdvn 25388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12475  df-z 12561  df-dec 12680  df-uz 12825  df-q 12935  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-icc 13333  df-fz 13487  df-fzo 13630  df-seq 13969  df-exp 14030  df-fac 14236  df-bc 14265  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-sum 15635  df-prod 15852  df-struct 17082  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-mulr 17213  df-starv 17214  df-sca 17215  df-vsca 17216  df-ip 17217  df-tset 17218  df-ple 17219  df-ds 17221  df-unif 17222  df-hom 17223  df-cco 17224  df-rest 17370  df-topn 17371  df-0g 17389  df-gsum 17390  df-topgen 17391  df-pt 17392  df-prds 17395  df-xrs 17450  df-qtop 17455  df-imas 17456  df-xps 17458  df-mre 17532  df-mrc 17533  df-acs 17535  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-submnd 18674  df-mulg 18953  df-cntz 19183  df-cmn 19652  df-psmet 20942  df-xmet 20943  df-met 20944  df-bl 20945  df-mopn 20946  df-fbas 20947  df-fg 20948  df-cnfld 20951  df-top 22403  df-topon 22420  df-topsp 22442  df-bases 22456  df-cld 22530  df-ntr 22531  df-cls 22532  df-nei 22609  df-lp 22647  df-perf 22648  df-cn 22738  df-cnp 22739  df-haus 22826  df-tx 23073  df-hmeo 23266  df-fil 23357  df-fm 23449  df-flim 23450  df-flf 23451  df-xms 23833  df-ms 23834  df-tms 23835  df-cncf 24401  df-limc 25390  df-dv 25391  df-dvn 25392

This theorem is referenced by: (None)