Theorem etransclem27 46866

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 6882	. . . . 5 ⊢ (𝑥 = 𝐽 → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽))
3	2	prodeq2ad 46199	. . . 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 15753	. . 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 9291	. . . . 5 ⊢ (𝐶 ∈ Fin → dom 𝐶 ∈ Fin)
8	6, 7	syl 18	. . . 4 ⊢ (𝜑 → dom 𝐶 ∈ Fin)
9		fzfid 14008	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (0...𝑀) ∈ Fin)
10		etransclem27.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
11	10	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
12		etransclem27.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
13	12	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
14		etransclem27.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
15	14	ad2antrr 738	. . . . . . 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 46844	. . . . . . . 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 2792	. . . . . . 7 ⊢ 𝐻 = (𝑧 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
19		simpr 489	. . . . . . 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 7080	. . . . . . . . 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 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (𝐶‘𝑙):(0...𝑀)⟶ℕ0)
24	23	ffvelcdmda 7080	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℕ0)
25	11, 13, 15, 18, 19, 24	etransclem20 46859	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗)):𝑋⟶ℂ)
26	5	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝐽 ∈ 𝑋)
27	25, 26	ffvelcdmd 7081	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
28	9, 27	fprodcl 16005	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
29	8, 28	fsumcl 15783	. . 3 ⊢ (𝜑 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
30	1, 4, 5, 29	fvmptd3 7014	. 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 46860	. . . . 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 4498	. . . . . . . 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 12602	. . . . . . . 8 ⊢ (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗) → 0 ∈ ℤ)
34	32, 33	eqeltrd 2869	. . . . . . 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 486	. . . . . 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 12602	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 0 ∈ ℤ)
37		nnm1nn0 12544	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
38	14, 37	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ0)
39	14	nnnn0d 12564	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ0)
40	38, 39	ifcld 4539	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
41	40	nn0zd 12615	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
42	41	ad3antrrr 742	. . . . . . . . . . 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 12615	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℤ)
44	43	adantr 485	. . . . . . . . . . . 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 12704	. . . . . . . . . . 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 12699	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐶‘𝑙)‘𝑗) ∈ ℝ)
47	42	zred 12699	. . . . . . . . . . . . 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 489	. . . . . . . . . . . . 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 11359	. . . . . . . . . . . 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 11803	. . . . . . . . . . . 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 260	. . . . . . . . . . 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 11210	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ ℝ)
53	24	nn0red 12565	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℝ)
54	40	nn0red 12565	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
55	54	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
56	24	nn0ge0d 12567	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ≤ ((𝐶‘𝑙)‘𝑗))
57	52, 53, 55, 56	lesub2dd 11830	. . . . . . . . . . . . 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 11236	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
59	58	subid1d 11557	. . . . . . . . . . . . 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 5141	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 13542	. . . . . . . . . 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 14361	. . . . . . . . . 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 18	. . . . . . . . 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 12616	. . . . . . . 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 742	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 𝐽 ∈ ℤ)
68		elfzelz 13551	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
69	68	ad2antlr 739	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 𝑗 ∈ ℤ)
70	67, 69	zsubcld 12704	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (𝐽 − 𝑗) ∈ ℤ)
71		elnn0z 12603	. . . . . . . . . 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 594	. . . . . . . . 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 14111	. . . . . . . . 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 595	. . . . . . . 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 12705	. . . . . . 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 4539	. . . . . 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 824	. . . . 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 2869	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
79	9, 78	fprodzcl 16007	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
80	8, 79	fsumzcl 15785	. 2 ⊢ (𝜑 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
81	30, 80	eqeltrd 2869	1 ⊢ (𝜑 → (𝐺‘𝐽) ∈ ℤ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ifcif 4492  {cpr 4596   class class class wbr 5113   ↦ cmpt 5196  dom cdm 5662  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ↑_m cmap 8823  Fincfn 8942  ℂcc 11097  ℝcr 11098  0cc0 11099  1c1 11100   · cmul 11104   < clt 11242   ≤ cle 11243   − cmin 11440   / cdiv 11870  ℕcn 12232  ℕ₀cn0 12503  ℤcz 12590  ...cfz 13534  ↑cexp 14096  !cfa 14308  Σcsu 15736  ∏cprod 15956   ↾_t crest 17472  TopOpenctopn 17473  ℂ_fldccnfld 21490   D^𝑛 cdvn 25991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9609  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177  ax-addf 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-er 8693  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9321  df-fi 9370  df-sup 9401  df-inf 9402  df-oi 9471  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-q 12972  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-icc 13378  df-fz 13535  df-fzo 13682  df-seq 14037  df-exp 14097  df-fac 14309  df-bc 14338  df-hash 14366  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-clim 15538  df-sum 15737  df-prod 15957  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-starv 17324  df-sca 17325  df-vsca 17326  df-ip 17327  df-tset 17328  df-ple 17329  df-ds 17331  df-unif 17332  df-hom 17333  df-cco 17334  df-rest 17474  df-topn 17475  df-0g 17493  df-gsum 17494  df-topgen 17495  df-pt 17496  df-prds 17499  df-xrs 17555  df-qtop 17560  df-imas 17561  df-xps 17563  df-mre 17637  df-mrc 17638  df-acs 17640  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-submnd 18841  df-mulg 19133  df-cntz 19386  df-cmn 19851  df-psmet 21482  df-xmet 21483  df-met 21484  df-bl 21485  df-mopn 21486  df-fbas 21487  df-fg 21488  df-cnfld 21491  df-top 23019  df-topon 23036  df-topsp 23058  df-bases 23071  df-cld 23144  df-ntr 23145  df-cls 23146  df-nei 23223  df-lp 23261  df-perf 23262  df-cn 23352  df-cnp 23353  df-haus 23440  df-tx 23687  df-hmeo 23880  df-fil 23971  df-fm 24063  df-flim 24064  df-flf 24065  df-xms 24445  df-ms 24446  df-tms 24447  df-cncf 25005  df-limc 25993  df-dv 25994  df-dvn 25995

This theorem is referenced by: (None)