etransclem37 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem37		Structured version Visualization version GIF version

Theorem etransclem37 46269

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 46248	. . 3 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
4		etransclem37.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ)
5		nnm1nn0 12483	. . . . . 6 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
7	6	faccld 14249	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
8	7	nnzd 12556	. . 3 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
9	1, 2	etransclem12 46244	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
10	9	eleq2d 2814	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑐 ∈ (𝐶‘𝑁) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}))
11	10	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
12		rabid 3427	. . . . . . . . . . . 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 216	. . . . . . . . . . 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 495	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁)
15	11, 14	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁)
16	15	eqcomd 2735	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗))
17	16	fveq2d 6862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) = (!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)))
18	17	oveq1d 7402	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) = ((!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))))
19		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑗𝑐
20		fzfid 13938	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
21		nn0ex 12448	. . . . . . . . . . 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 45314	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
24		mapss 8862	. . . . . . . . . 10 ⊢ ((ℕ₀ ∈ V ∧ (0...𝑁) ⊆ ℕ₀) → ((0...𝑁) ↑_m (0...𝑀)) ⊆ (ℕ₀ ↑_m (0...𝑀)))
25	22, 23, 24	sylancl 586	. . . . . . . . 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 494	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)))
27	25, 26	sseldd 3947	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐 ∈ (ℕ₀ ↑_m (0...𝑀)))
28	11, 27	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (ℕ₀ ↑_m (0...𝑀)))
29	19, 20, 28	mccl 45596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℕ)
30	18, 29	eqeltrd 2828	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℕ)
31	30	nnzd 12556	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℤ)
32	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑃 ∈ ℕ)
33		etransclem37.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
34	33	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑀 ∈ ℕ₀)
35		elmapi 8822	. . . . . . 7 ⊢ (𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
36	11, 26, 35	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐:(0...𝑀)⟶(0...𝑁))
37		etransclem37.9	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
38	37	elfzelzd 13486	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
39	38	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝐽 ∈ ℤ)
40	32, 34, 36, 39	etransclem10 46242	. . . . 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 13938	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (1...𝑀) ∈ Fin)
42	32	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑃 ∈ ℕ)
43	36	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
44		0z 12540	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
45		fzp1ss 13536	. . . . . . . . . . 11 ⊢ (0 ∈ ℤ → ((0 + 1)...𝑀) ⊆ (0...𝑀))
46	44, 45	ax-mp 5	. . . . . . . . . 10 ⊢ ((0 + 1)...𝑀) ⊆ (0...𝑀)
47	46	sseli 3942	. . . . . . . . 9 ⊢ (𝑗 ∈ ((0 + 1)...𝑀) → 𝑗 ∈ (0...𝑀))
48		1e0p1 12691	. . . . . . . . . 10 ⊢ 1 = (0 + 1)
49	48	oveq1i 7397	. . . . . . . . 9 ⊢ (1...𝑀) = ((0 + 1)...𝑀)
50	47, 49	eleq2s 2846	. . . . . . . 8 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ (0...𝑀))
51	50	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ (0...𝑀))
52	39	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝐽 ∈ ℤ)
53	42, 43, 51, 52	etransclem3 46235	. . . . . 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 15920	. . . . 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 12644	. . . 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 12644	. . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑁 ∈ ℕ₀)
58		etransclem11 46243	. . . . 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 2752	. . . 4 ⊢ 𝐶 = (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})
60		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (𝐶‘𝑁))
61	37	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝐽 ∈ (0...𝑀))
62		fveq2 6858	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑐‘𝑗) = (𝑐‘𝑘))
63	62	fveq2d 6862	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (!‘(𝑐‘𝑗)) = (!‘(𝑐‘𝑘)))
64	63	cbvprodv 15880	. . . . . 6 ⊢ ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) = ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))
65	64	oveq2i 7398	. . . . 5 ⊢ ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) = ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)))
66	62	breq2d 5119	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑃 < (𝑐‘𝑗) ↔ 𝑃 < (𝑐‘𝑘)))
67	62	oveq2d 7403	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑃 − (𝑐‘𝑗)) = (𝑃 − (𝑐‘𝑘)))
68	67	fveq2d 6862	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (!‘(𝑃 − (𝑐‘𝑗))) = (!‘(𝑃 − (𝑐‘𝑘))))
69	68	oveq2d 7403	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) = ((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑘)))))
70		oveq2 7395	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐽 − 𝑗) = (𝐽 − 𝑘))
71	70, 67	oveq12d 7405	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))) = ((𝐽 − 𝑘)↑(𝑃 − (𝑐‘𝑘))))
72	69, 71	oveq12d 7405	. . . . . . . 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 4515	. . . . . . 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 15880	. . . . . 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 7398	. . . . 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 7399	. . . 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 46260	. . 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 16278	. 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 46263	. 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 5135	1 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3405 Vcvv 3447 ⊆ wss 3914 ifcif 4488 {cpr 4591 class class class wbr 5107 ↦ cmpt 5188 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ↑_m cmap 8799 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 − cmin 11405 / cdiv 11835 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ...cfz 13468 ↑cexp 14026 !cfa 14238 Σcsu 15652 ∏cprod 15869 ∥ cdvds 16222 ↾_t crest 17383 TopOpenctopn 17384 ℂ_fldccnfld 21264 D^𝑛 cdvn 25765

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-prod 15870 df-dvds 16223 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-dvn 25769

This theorem is referenced by: etransclem44 46276 etransclem45 46277

W3C validator