Theorem etransclem27 46505

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 6834	. . . . 5 ⊢ (𝑥 = 𝐽 → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽))
3	2	prodeq2ad 45838	. . . 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 15626	. . 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 9235	. . . . 5 ⊢ (𝐶 ∈ Fin → dom 𝐶 ∈ Fin)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → dom 𝐶 ∈ Fin)
9		fzfid 13896	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (0...𝑀) ∈ Fin)
10		etransclem27.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
11	10	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
12		etransclem27.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
13	12	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
14		etransclem27.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
15	14	ad2antrr 726	. . . . . . 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 46483	. . . . . . . 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 2759	. . . . . . 7 ⊢ 𝐻 = (𝑧 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑧)↑if(𝑧 = 0, (𝑃 − 1), 𝑃))))
19		simpr 484	. . . . . . 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 7029	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → (𝐶‘𝑙) ∈ (ℕ0 ↑m (0...𝑀)))
22		elmapi 8786	. . . . . . . . 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 7029	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℕ0)
25	11, 13, 15, 18, 19, 24	etransclem20 46498	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗)):𝑋⟶ℂ)
26	5	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 𝐽 ∈ 𝑋)
27	25, 26	ffvelcdmd 7030	. . . . 5 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
28	9, 27	fprodcl 15875	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
29	8, 28	fsumcl 15656	. . 3 ⊢ (𝜑 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℂ)
30	1, 4, 5, 29	fvmptd3 6964	. 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 46499	. . . . 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 4485	. . . . . . . 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 12500	. . . . . . . 8 ⊢ (if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗) → 0 ∈ ℤ)
34	32, 33	eqeltrd 2836	. . . . . . 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 481	. . . . . 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 12500	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 0 ∈ ℤ)
37		nnm1nn0 12442	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ0)
38	14, 37	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ0)
39	14	nnnn0d 12462	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ0)
40	38, 39	ifcld 4526	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℕ0)
41	40	nn0zd 12513	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
42	41	ad3antrrr 730	. . . . . . . . . . 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 12513	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℤ)
44	43	adantr 480	. . . . . . . . . . . 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 12601	. . . . . . . . . . 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 12596	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → ((𝐶‘𝑙)‘𝑗) ∈ ℝ)
47	42	zred 12596	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 11283	. . . . . . . . . . . 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 11727	. . . . . . . . . . . 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 257	. . . . . . . . . . 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 11135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ ℝ)
53	24	nn0red 12463	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → ((𝐶‘𝑙)‘𝑗) ∈ ℝ)
54	40	nn0red 12463	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
55	54	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
56	24	nn0ge0d 12465	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → 0 ≤ ((𝐶‘𝑙)‘𝑗))
57	52, 53, 55, 56	lesub2dd 11754	. . . . . . . . . . . . 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 11160	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → if(𝑗 = 0, (𝑃 − 1), 𝑃) ∈ ℂ)
59	58	subid1d 11481	. . . . . . . . . . . . 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 5124	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 13431	. . . . . . . . . 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 14249	. . . . . . . . . 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 12514	. . . . . . . 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 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 𝐽 ∈ ℤ)
68		elfzelz 13440	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
69	68	ad2antlr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → 𝑗 ∈ ℤ)
70	67, 69	zsubcld 12601	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ if(𝑗 = 0, (𝑃 − 1), 𝑃) < ((𝐶‘𝑙)‘𝑗)) → (𝐽 − 𝑗) ∈ ℤ)
71		elnn0z 12501	. . . . . . . . . 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 13999	. . . . . . . . 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 12602	. . . . . . 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 4526	. . . . . 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 812	. . . . 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 2836	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ dom 𝐶) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
79	9, 78	fprodzcl 15877	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ dom 𝐶) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
80	8, 79	fsumzcl 15658	. 2 ⊢ (𝜑 → Σ𝑙 ∈ dom 𝐶∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻‘𝑗))‘((𝐶‘𝑙)‘𝑗))‘𝐽) ∈ ℤ)
81	30, 80	eqeltrd 2836	1 ⊢ (𝜑 → (𝐺‘𝐽) ∈ ℤ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ifcif 4479  {cpr 4582   class class class wbr 5098   ↦ cmpt 5179  dom cdm 5624  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  Fincfn 8883  ℂcc 11024  ℝcr 11025  0cc0 11026  1c1 11027   · cmul 11031   < clt 11166   ≤ cle 11167   − cmin 11364   / cdiv 11794  ℕcn 12145  ℕ₀cn0 12401  ℤcz 12488  ...cfz 13423  ↑cexp 13984  !cfa 14196  Σcsu 15609  ∏cprod 15826   ↾_t crest 17340  TopOpenctopn 17341  ℂ_fldccnfld 21309   D^𝑛 cdvn 25821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104  ax-addf 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-q 12862  df-rp 12906  df-xneg 13026  df-xadd 13027  df-xmul 13028  df-icc 13268  df-fz 13424  df-fzo 13571  df-seq 13925  df-exp 13985  df-fac 14197  df-bc 14226  df-hash 14254  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-clim 15411  df-sum 15610  df-prod 15827  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-starv 17192  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-unif 17200  df-hom 17201  df-cco 17202  df-rest 17342  df-topn 17343  df-0g 17361  df-gsum 17362  df-topgen 17363  df-pt 17364  df-prds 17367  df-xrs 17423  df-qtop 17428  df-imas 17429  df-xps 17431  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-mulg 18998  df-cntz 19246  df-cmn 19711  df-psmet 21301  df-xmet 21302  df-met 21303  df-bl 21304  df-mopn 21305  df-fbas 21306  df-fg 21307  df-cnfld 21310  df-top 22838  df-topon 22855  df-topsp 22877  df-bases 22890  df-cld 22963  df-ntr 22964  df-cls 22965  df-nei 23042  df-lp 23080  df-perf 23081  df-cn 23171  df-cnp 23172  df-haus 23259  df-tx 23506  df-hmeo 23699  df-fil 23790  df-fm 23882  df-flim 23883  df-flf 23884  df-xms 24264  df-ms 24265  df-tms 24266  df-cncf 24827  df-limc 25823  df-dv 25824  df-dvn 25825

This theorem is referenced by: (None)