etransclem37 - Mathbox for Glauco Siliprandi

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Theorem etransclem37 47417

Description: (𝑃 − 1) factorial divides the 𝑁-th derivative of 𝐹 applied to 𝐽. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem37.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem37.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
etransclem37.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem37.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem37.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
etransclem37.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
etransclem37.h	⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
etransclem37.c	⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
etransclem37.9	⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
etransclem37.j	⊢ (𝜑 → 𝐽 ∈ 𝑋)

**Assertion**
Ref	Expression
etransclem37	⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐽))

Distinct variable groups: 𝐶,𝑐,𝑗,𝑥 𝐻,𝑐,𝑗,𝑛,𝑥 𝐽,𝑐,𝑗,𝑥 𝑀,𝑐,𝑗,𝑛,𝑥 𝑁,𝑐,𝑗,𝑛,𝑥 𝑃,𝑐,𝑗,𝑥 𝑆,𝑐,𝑗,𝑛,𝑥 𝑗,𝑋,𝑛,𝑥 𝜑,𝑐,𝑗,𝑛,𝑥 Allowed substitution hints: 𝐶(𝑛) 𝑃(𝑛) 𝐹(𝑥, 𝑗, 𝑛, 𝑐) 𝐽(𝑛) 𝑋(𝑐)

Proof of Theorem etransclem37 Dummy variables 𝑘 𝑚 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem37.c	. . . 4 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
2		etransclem37.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
3	1, 2	etransclem16 47396	. . 3 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
4		etransclem37.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ)
5		nnm1nn0 12654	. . . . . 6 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
6	4, 5	syl 18	. . . . 5 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
7	6	faccld 14435	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
8	7	nnzd 12726	. . 3 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
9	1, 2	etransclem12 47392	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
10	9	eleq2d 2846	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑐 ∈ (𝐶‘𝑁) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}))
11	10	biimpa 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
12		rabid 3432	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ↔ (𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∧ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁))
13	12	biimpi 219	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → (𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∧ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁))
14	13	simprd 501	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁)
15	11, 14	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁)
16	15	eqcomd 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗))
17	16	fveq2d 6894	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) = (!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)))
18	17	oveq1d 7439	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) = ((!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))))
19		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑗𝑐
20		fzfid 14123	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
21		nn0ex 12619	. . . . . . . . . . 11 ⊢ ℕ₀ ∈ V
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → ℕ₀ ∈ V)
23		fzssnn0 46467	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
24		mapss 8925	. . . . . . . . . 10 ⊢ ((ℕ₀ ∈ V ∧ (0...𝑁) ⊆ ℕ₀) → ((0...𝑁) ↑_m (0...𝑀)) ⊆ (ℕ₀ ↑_m (0...𝑀)))
25	22, 23, 24	sylancl 598	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → ((0...𝑁) ↑_m (0...𝑀)) ⊆ (ℕ₀ ↑_m (0...𝑀)))
26	13	simpld 500	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)))
27	25, 26	sseldd 3910	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐 ∈ (ℕ₀ ↑_m (0...𝑀)))
28	11, 27	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (ℕ₀ ↑_m (0...𝑀)))
29	19, 20, 28	mccl 46746	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℕ)
30	18, 29	eqeltrd 2860	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℕ)
31	30	nnzd 12726	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℤ)
32	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑃 ∈ ℕ)
33		etransclem37.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
34	33	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑀 ∈ ℕ₀)
35		elmapi 8877	. . . . . . 7 ⊢ (𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
36	11, 26, 35	3syl 19	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐:(0...𝑀)⟶(0...𝑁))
37		etransclem37.9	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
38	37	elfzelzd 13664	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
39	38	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝐽 ∈ ℤ)
40	32, 34, 36, 39	etransclem10 47390	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) ∈ ℤ)
41		fzfid 14123	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (1...𝑀) ∈ Fin)
42	32	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑃 ∈ ℕ)
43	36	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
44		0z 12711	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
45		fzp1ss 13716	. . . . . . . . . . 11 ⊢ (0 ∈ ℤ → ((0 + 1)...𝑀) ⊆ (0...𝑀))
46	44, 45	ax-mp 5	. . . . . . . . . 10 ⊢ ((0 + 1)...𝑀) ⊆ (0...𝑀)
47	46	sseli 3905	. . . . . . . . 9 ⊢ (𝑗 ∈ ((0 + 1)...𝑀) → 𝑗 ∈ (0...𝑀))
48		1e0p1 12868	. . . . . . . . . 10 ⊢ 1 = (0 + 1)
49	48	oveq1i 7434	. . . . . . . . 9 ⊢ (1...𝑀) = ((0 + 1)...𝑀)
50	47, 49	eleq2s 2878	. . . . . . . 8 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ (0...𝑀))
51	50	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ (0...𝑀))
52	39	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝐽 ∈ ℤ)
53	42, 43, 51, 52	etransclem3 47383	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))) ∈ ℤ)
54	41, 53	fprodzcl 16130	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))) ∈ ℤ)
55	40, 54	zmulcld 12816	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗)))))) ∈ ℤ)
56	31, 55	zmulcld 12816	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))) ∈ ℤ)
57	2	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑁 ∈ ℕ₀)
58		etransclem11 47391	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) = (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})
59	1, 58	eqtri 2783	. . . 4 ⊢ 𝐶 = (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})
60		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (𝐶‘𝑁))
61	37	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝐽 ∈ (0...𝑀))
62		fveq2 6890	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑐‘𝑗) = (𝑐‘𝑘))
63	62	fveq2d 6894	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (!‘(𝑐‘𝑗)) = (!‘(𝑐‘𝑘)))
64	63	cbvprodv 16092	. . . . . 6 ⊢ ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) = ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))
65	64	oveq2i 7435	. . . . 5 ⊢ ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) = ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)))
66	62	breq2d 5114	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑃 < (𝑐‘𝑗) ↔ 𝑃 < (𝑐‘𝑘)))
67	62	oveq2d 7440	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑃 − (𝑐‘𝑗)) = (𝑃 − (𝑐‘𝑘)))
68	67	fveq2d 6894	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (!‘(𝑃 − (𝑐‘𝑗))) = (!‘(𝑃 − (𝑐‘𝑘))))
69	68	oveq2d 7440	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) = ((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑘)))))
70		oveq2 7432	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐽 − 𝑗) = (𝐽 − 𝑘))
71	70, 67	oveq12d 7442	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))) = ((𝐽 − 𝑘)↑(𝑃 − (𝑐‘𝑘))))
72	69, 71	oveq12d 7442	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗)))) = (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑘)))) · ((𝐽 − 𝑘)↑(𝑃 − (𝑐‘𝑘)))))
73	66, 72	ifbieq2d 4508	. . . . . . 7 ⊢ (𝑗 = 𝑘 → if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))) = if(𝑃 < (𝑐‘𝑘), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑘)))) · ((𝐽 − 𝑘)↑(𝑃 − (𝑐‘𝑘))))))
74	73	cbvprodv 16092	. . . . . 6 ⊢ ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))) = ∏𝑘 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑘), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑘)))) · ((𝐽 − 𝑘)↑(𝑃 − (𝑐‘𝑘)))))
75	74	oveq2i 7435	. . . . 5 ⊢ (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗)))))) = (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑘 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑘), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑘)))) · ((𝐽 − 𝑘)↑(𝑃 − (𝑐‘𝑘))))))
76	65, 75	oveq12i 7436	. . . 4 ⊢ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))) = (((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑘 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑘), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑘)))) · ((𝐽 − 𝑘)↑(𝑃 − (𝑐‘𝑘)))))))
77	32, 34, 57, 59, 60, 61, 76	etransclem28 47408	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘(𝑃 − 1)) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))))
78	3, 8, 56, 77	fsumdvds 16487	. 2 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))))
79		etransclem37.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
80		etransclem37.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
81		etransclem37.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
82		etransclem37.h	. . 3 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
83		etransclem37.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑋)
84	79, 80, 4, 33, 81, 2, 82, 1, 83	etransclem31 47411	. 2 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐽) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝐽↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))))
85	78, 84	breqtrrd 5132	1 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐽))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 {crab 3412 Vcvv 3450 ⊆ wss 3896 ifcif 4481 {cpr 4586 class class class wbr 5102 ↦ cmpt 5185 ⟶wf 6539 ‘cfv 6543 (class class class)co 7424 ↑_m cmap 8855 ℂcc 11205 ℝcr 11206 0cc0 11207 1c1 11208 + caddc 11210 · cmul 11212 < clt 11350 − cmin 11548 / cdiv 11980 ℕcn 12342 ℕ₀cn0 12613 ℤcz 12700 ...cfz 13646 ↑cexp 14211 !cfa 14424 Σcsu 15862 ∏cprod 16081 ∥ cdvds 16431 ↾_t crest 17600 TopOpenctopn 17601 ℂ_fldccnfld 21687 D^𝑛 cdvn 26193

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 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 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-er 8725 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ico 13489 df-icc 13490 df-fz 13647 df-fzo 13796 df-seq 14152 df-exp 14212 df-fac 14425 df-bc 14454 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-clim 15664 df-sum 15863 df-prod 16082 df-dvds 16432 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-mulg 19287 df-cntz 19540 df-cmn 20005 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-lp 23463 df-perf 23464 df-cn 23554 df-cnp 23555 df-haus 23642 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-limc 26195 df-dv 26196 df-dvn 26197

This theorem is used by: etransclem44 47424 etransclem45 47425

W3C validator