etransclem37 - Mathbox for Glauco Siliprandi

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Theorem etransclem37 40488

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 40467	. . 3 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
4		etransclem37.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ)
5		nnm1nn0 11334	. . . . . 6 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
7	6	faccld 13071	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
8	7	nnzd 11481	. . 3 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
9	1, 2	etransclem12 40463	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
10	9	eleq2d 2687	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑐 ∈ (𝐶‘𝑁) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}))
11	10	biimpa 501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
12		rabid 3116	. . . . . . . . . . . 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 206	. . . . . . . . . . 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 479	. . . . . . . . . 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 2628	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗))
17	16	fveq2d 6195	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) = (!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)))
18	17	oveq1d 6665	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) = ((!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))))
19		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑗𝑐
20		fzfid 12772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
21		nn0ex 11298	. . . . . . . . . . 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 39533	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
24		mapss 7900	. . . . . . . . . 10 ⊢ ((ℕ₀ ∈ V ∧ (0...𝑁) ⊆ ℕ₀) → ((0...𝑁) ↑_𝑚 (0...𝑀)) ⊆ (ℕ₀ ↑_𝑚 (0...𝑀)))
25	22, 23, 24	sylancl 694	. . . . . . . . 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 475	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)))
27	25, 26	sseldd 3604	. . . . . . . 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 39830	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℕ)
30	18, 29	eqeltrd 2701	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℕ)
31	30	nnzd 11481	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℤ)
32	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑃 ∈ ℕ)
33		etransclem37.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
34	33	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑀 ∈ ℕ₀)
35		elmapi 7879	. . . . . . 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 39530	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
39	38	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝐽 ∈ ℤ)
40	32, 34, 36, 39	etransclem10 40461	. . . . 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 12772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (1...𝑀) ∈ Fin)
42	32	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑃 ∈ ℕ)
43	36	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
44		0z 11388	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
45		fzp1ss 12392	. . . . . . . . . . 11 ⊢ (0 ∈ ℤ → ((0 + 1)...𝑀) ⊆ (0...𝑀))
46	44, 45	ax-mp 5	. . . . . . . . . 10 ⊢ ((0 + 1)...𝑀) ⊆ (0...𝑀)
47	46	sseli 3599	. . . . . . . . 9 ⊢ (𝑗 ∈ ((0 + 1)...𝑀) → 𝑗 ∈ (0...𝑀))
48		1e0p1 11552	. . . . . . . . . 10 ⊢ 1 = (0 + 1)
49	48	oveq1i 6660	. . . . . . . . 9 ⊢ (1...𝑀) = ((0 + 1)...𝑀)
50	47, 49	eleq2s 2719	. . . . . . . 8 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ (0...𝑀))
51	50	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ (0...𝑀))
52	39	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (1...𝑀)) → 𝐽 ∈ ℤ)
53	42, 43, 51, 52	etransclem3 40454	. . . . . 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 14684	. . . . 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 11488	. . . 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 11488	. . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑁 ∈ ℕ₀)
58		etransclem11 40462	. . . . 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 2644	. . . 4 ⊢ 𝐶 = (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})
60		simpr 477	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (𝐶‘𝑁))
61	37	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝐽 ∈ (0...𝑀))
62		fveq2 6191	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑐‘𝑗) = (𝑐‘𝑘))
63	62	fveq2d 6195	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (!‘(𝑐‘𝑗)) = (!‘(𝑐‘𝑘)))
64	63	cbvprodv 14646	. . . . . 6 ⊢ ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) = ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))
65	64	oveq2i 6661	. . . . 5 ⊢ ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) = ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)))
66	62	breq2d 4665	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑃 < (𝑐‘𝑗) ↔ 𝑃 < (𝑐‘𝑘)))
67	62	oveq2d 6666	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑃 − (𝑐‘𝑗)) = (𝑃 − (𝑐‘𝑘)))
68	67	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (!‘(𝑃 − (𝑐‘𝑗))) = (!‘(𝑃 − (𝑐‘𝑘))))
69	68	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) = ((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑘)))))
70		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐽 − 𝑗) = (𝐽 − 𝑘))
71	70, 67	oveq12d 6668	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))) = ((𝐽 − 𝑘)↑(𝑃 − (𝑐‘𝑘))))
72	69, 71	oveq12d 6668	. . . . . . . 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 4111	. . . . . . 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 14646	. . . . . 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 6661	. . . . 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 6662	. . . 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 40479	. . 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 15030	. 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 40482	. 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 4681	1 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ⊆ wss 3574 ifcif 4086 {cpr 4179 class class class wbr 4653 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 − cmin 10266 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ...cfz 12326 ↑cexp 12860 !cfa 13060 Σcsu 14416 ∏cprod 14635 ∥ cdvds 14983 ↾_t crest 16081 TopOpenctopn 16082 ℂ_fldccnfld 19746 D^𝑛 cdvn 23628

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-prod 14636 df-dvds 14984 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-dvn 23632

This theorem is referenced by: etransclem44 40495 etransclem45 40496

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