etransclem37 - Mathbox for Glauco Siliprandi

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Theorem etransclem37 46876

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 46855	. . 3 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
4		etransclem37.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ)
5		nnm1nn0 12544	. . . . . 6 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
6	4, 5	syl 18	. . . . 5 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
7	6	faccld 14319	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
8	7	nnzd 12616	. . 3 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
9	1, 2	etransclem12 46851	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
10	9	eleq2d 2855	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑐 ∈ (𝐶‘𝑁) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}))
11	10	biimpa 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
12		rabid 3444	. . . . . . . . . . . 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 500	. . . . . . . . . 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 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗))
17	16	fveq2d 6886	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) = (!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)))
18	17	oveq1d 7426	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) = ((!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))))
19		nfcv 2931	. . . . . . 7 ⊢ Ⅎ𝑗𝑐
20		fzfid 14008	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
21		nn0ex 12509	. . . . . . . . . . 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 45926	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
24		mapss 8886	. . . . . . . . . 10 ⊢ ((ℕ₀ ∈ V ∧ (0...𝑁) ⊆ ℕ₀) → ((0...𝑁) ↑_m (0...𝑀)) ⊆ (ℕ₀ ↑_m (0...𝑀)))
25	22, 23, 24	sylancl 597	. . . . . . . . 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 499	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)))
27	25, 26	sseldd 3946	. . . . . . . 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 46205	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℕ)
30	18, 29	eqeltrd 2869	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℕ)
31	30	nnzd 12616	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℤ)
32	4	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑃 ∈ ℕ)
33		etransclem37.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
34	33	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑀 ∈ ℕ₀)
35		elmapi 8845	. . . . . . 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 13552	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
39	38	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝐽 ∈ ℤ)
40	32, 34, 36, 39	etransclem10 46849	. . . . 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 14008	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (1...𝑀) ∈ Fin)
42	32	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑃 ∈ ℕ)
43	36	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
44		0z 12601	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
45		fzp1ss 13602	. . . . . . . . . . 11 ⊢ (0 ∈ ℤ → ((0 + 1)...𝑀) ⊆ (0...𝑀))
46	44, 45	ax-mp 5	. . . . . . . . . 10 ⊢ ((0 + 1)...𝑀) ⊆ (0...𝑀)
47	46	sseli 3941	. . . . . . . . 9 ⊢ (𝑗 ∈ ((0 + 1)...𝑀) → 𝑗 ∈ (0...𝑀))
48		1e0p1 12757	. . . . . . . . . 10 ⊢ 1 = (0 + 1)
49	48	oveq1i 7421	. . . . . . . . 9 ⊢ (1...𝑀) = ((0 + 1)...𝑀)
50	47, 49	eleq2s 2887	. . . . . . . 8 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ (0...𝑀))
51	50	adantl 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ (0...𝑀))
52	39	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝐽 ∈ ℤ)
53	42, 43, 51, 52	etransclem3 46842	. . . . . 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 16007	. . . . 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 12705	. . . 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 12705	. . 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 485	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑁 ∈ ℕ₀)
58		etransclem11 46850	. . . . 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 2792	. . . 4 ⊢ 𝐶 = (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})
60		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (𝐶‘𝑁))
61	37	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝐽 ∈ (0...𝑀))
62		fveq2 6882	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑐‘𝑗) = (𝑐‘𝑘))
63	62	fveq2d 6886	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (!‘(𝑐‘𝑗)) = (!‘(𝑐‘𝑘)))
64	63	cbvprodv 15967	. . . . . 6 ⊢ ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) = ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))
65	64	oveq2i 7422	. . . . 5 ⊢ ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) = ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)))
66	62	breq2d 5125	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑃 < (𝑐‘𝑗) ↔ 𝑃 < (𝑐‘𝑘)))
67	62	oveq2d 7427	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑃 − (𝑐‘𝑗)) = (𝑃 − (𝑐‘𝑘)))
68	67	fveq2d 6886	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (!‘(𝑃 − (𝑐‘𝑗))) = (!‘(𝑃 − (𝑐‘𝑘))))
69	68	oveq2d 7427	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) = ((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑘)))))
70		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐽 − 𝑗) = (𝐽 − 𝑘))
71	70, 67	oveq12d 7429	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))) = ((𝐽 − 𝑘)↑(𝑃 − (𝑐‘𝑘))))
72	69, 71	oveq12d 7429	. . . . . . . 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 4519	. . . . . . 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 15967	. . . . . 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 7422	. . . . 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 7423	. . . 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 46867	. . 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 16365	. 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 46870	. 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 5143	1 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 {crab 3423 Vcvv 3463 ⊆ wss 3913 ifcif 4492 {cpr 4596 class class class wbr 5113 ↦ cmpt 5196 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ↑_m cmap 8823 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 + caddc 11102 · cmul 11104 < clt 11242 − cmin 11440 / cdiv 11870 ℕcn 12232 ℕ₀cn0 12503 ℤcz 12590 ...cfz 13534 ↑cexp 14096 !cfa 14308 Σcsu 15736 ∏cprod 15956 ∥ cdvds 16309 ↾_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-ico 13377 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-dvds 16310 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: etransclem44 46883 etransclem45 46884

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