etransclem37 - Mathbox for Glauco Siliprandi

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Theorem etransclem37 42112

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 42091	. . 3 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
4		etransclem37.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ)
5		nnm1nn0 11788	. . . . . 6 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
7	6	faccld 13494	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
8	7	nnzd 11936	. . 3 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
9	1, 2	etransclem12 42087	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
10	9	eleq2d 2867	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑐 ∈ (𝐶‘𝑁) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}))
11	10	biimpa 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
12		rabid 3336	. . . . . . . . . . . 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 217	. . . . . . . . . . 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 496	. . . . . . . . . 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 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗))
17	16	fveq2d 6545	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) = (!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)))
18	17	oveq1d 7034	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) = ((!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))))
19		nfcv 2948	. . . . . . 7 ⊢ Ⅎ𝑗𝑐
20		fzfid 13191	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
21		nn0ex 11753	. . . . . . . . . . 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 41139	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
24		mapss 8305	. . . . . . . . . 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 495	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)))
27	25, 26	sseldd 3892	. . . . . . . 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 41434	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℕ)
30	18, 29	eqeltrd 2882	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℕ)
31	30	nnzd 11936	. . . 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 8281	. . . . . . 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 41136	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
39	38	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝐽 ∈ ℤ)
40	32, 34, 36, 39	etransclem10 42085	. . . . 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 13191	. . . . . 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 11842	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
45		fzp1ss 12808	. . . . . . . . . . 11 ⊢ (0 ∈ ℤ → ((0 + 1)...𝑀) ⊆ (0...𝑀))
46	44, 45	ax-mp 5	. . . . . . . . . 10 ⊢ ((0 + 1)...𝑀) ⊆ (0...𝑀)
47	46	sseli 3887	. . . . . . . . 9 ⊢ (𝑗 ∈ ((0 + 1)...𝑀) → 𝑗 ∈ (0...𝑀))
48		1e0p1 11990	. . . . . . . . . 10 ⊢ 1 = (0 + 1)
49	48	oveq1i 7029	. . . . . . . . 9 ⊢ (1...𝑀) = ((0 + 1)...𝑀)
50	47, 49	eleq2s 2900	. . . . . . . 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 42078	. . . . . 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 15141	. . . . 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 11943	. . . 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 11943	. . 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 42086	. . . . 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 2818	. . . 4 ⊢ 𝐶 = (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚})
60		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (𝐶‘𝑁))
61	37	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝐽 ∈ (0...𝑀))
62		fveq2 6541	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑐‘𝑗) = (𝑐‘𝑘))
63	62	fveq2d 6545	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (!‘(𝑐‘𝑗)) = (!‘(𝑐‘𝑘)))
64	63	cbvprodv 15103	. . . . . 6 ⊢ ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) = ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘))
65	64	oveq2i 7030	. . . . 5 ⊢ ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) = ((!‘𝑁) / ∏𝑘 ∈ (0...𝑀)(!‘(𝑐‘𝑘)))
66	62	breq2d 4976	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑃 < (𝑐‘𝑗) ↔ 𝑃 < (𝑐‘𝑘)))
67	62	oveq2d 7035	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑃 − (𝑐‘𝑗)) = (𝑃 − (𝑐‘𝑘)))
68	67	fveq2d 6545	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (!‘(𝑃 − (𝑐‘𝑗))) = (!‘(𝑃 − (𝑐‘𝑘))))
69	68	oveq2d 7035	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) = ((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑘)))))
70		oveq2 7027	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐽 − 𝑗) = (𝐽 − 𝑘))
71	70, 67	oveq12d 7037	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝐽 − 𝑗)↑(𝑃 − (𝑐‘𝑗))) = ((𝐽 − 𝑘)↑(𝑃 − (𝑐‘𝑘))))
72	69, 71	oveq12d 7037	. . . . . . . 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 4408	. . . . . . 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 15103	. . . . . 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 7030	. . . . 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 7031	. . . 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 42103	. . 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 15491	. 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 42106	. 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 4992	1 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2080 {crab 3108 Vcvv 3436 ⊆ wss 3861 ifcif 4383 {cpr 4476 class class class wbr 4964 ↦ cmpt 5043 ⟶wf 6224 ‘cfv 6228 (class class class)co 7019 ↑_𝑚 cmap 8259 ℂcc 10384 ℝcr 10385 0cc0 10386 1c1 10387 + caddc 10389 · cmul 10391 < clt 10524 − cmin 10719 / cdiv 11147 ℕcn 11488 ℕ₀cn0 11747 ℤcz 11831 ...cfz 12742 ↑cexp 13279 !cfa 13483 Σcsu 14876 ∏cprod 15092 ∥ cdvds 15440 ↾_t crest 16523 TopOpenctopn 16524 ℂ_fldccnfld 20227 D^𝑛 cdvn 24145

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-inf2 8953 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-pre-sup 10464 ax-addf 10465 ax-mulf 10466

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-iin 4830 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-se 5406 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-isom 6237 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-of 7270 df-om 7440 df-1st 7548 df-2nd 7549 df-supp 7685 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-2o 7957 df-oadd 7960 df-er 8142 df-map 8261 df-pm 8262 df-ixp 8314 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-fsupp 8683 df-fi 8724 df-sup 8755 df-inf 8756 df-oi 8823 df-card 9217 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-5 11553 df-6 11554 df-7 11555 df-8 11556 df-9 11557 df-n0 11748 df-z 11832 df-dec 11949 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-seq 13220 df-exp 13280 df-fac 13484 df-bc 13513 df-hash 13541 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-clim 14679 df-sum 14877 df-prod 15093 df-dvds 15441 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-mulg 17982 df-cntz 18188 df-cmn 18635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-ntr 21312 df-cls 21313 df-nei 21390 df-lp 21428 df-perf 21429 df-cn 21519 df-cnp 21520 df-haus 21607 df-tx 21854 df-hmeo 22047 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-xms 22613 df-ms 22614 df-tms 22615 df-cncf 23169 df-limc 24147 df-dv 24148 df-dvn 24149

This theorem is referenced by: etransclem44 42119 etransclem45 42120

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