etransclem28 - Mathbox for Glauco Siliprandi

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Theorem etransclem28 44493

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

**Hypotheses**
Ref	Expression
etransclem28.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem28.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem28.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
etransclem28.c	⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
etransclem28.d	⊢ (𝜑 → 𝐷 ∈ (𝐶‘𝑁))
etransclem28.j	⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
etransclem28.t	⊢ 𝑇 = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗)))))))

**Assertion**
Ref	Expression
etransclem28	⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ 𝑇)

Distinct variable groups: 𝐷,𝑐,𝑗 𝑗,𝐽 𝑀,𝑐,𝑗,𝑛 𝑁,𝑐,𝑛 𝑃,𝑗 𝜑,𝑗,𝑛 Allowed substitution hints: 𝜑(𝑐) 𝐶(𝑗,𝑛,𝑐) 𝐷(𝑛) 𝑃(𝑛,𝑐) 𝑇(𝑗,𝑛,𝑐) 𝐽(𝑛,𝑐) 𝑁(𝑗)

**Proof of Theorem etransclem28**
Step	Hyp	Ref	Expression
1		etransclem28.p	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ)
2		nnm1nn0 12454	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
3	1, 2	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
4	3	faccld 14184	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
5	4	nnzd 12526	. . . . . . . . 9 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
6	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → (!‘(𝑃 − 1)) ∈ ℤ)
7		etransclem28.d	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷 ∈ (𝐶‘𝑁))
8		etransclem28.c	. . . . . . . . . . . . . . . . 17 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
9		etransclem28.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10	8, 9	etransclem12 44477	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
11	7, 10	eleqtrd 2840	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
12		fveq1 6841	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝐷 → (𝑐‘𝑗) = (𝐷‘𝑗))
13	12	sumeq2sdv 15589	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝐷 → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗))
14	13	eqeq1d 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝐷 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁 ↔ Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗) = 𝑁))
15	14	elrab 3645	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ↔ (𝐷 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∧ Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗) = 𝑁))
16	15	simprbi 497	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗) = 𝑁)
17	11, 16	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗) = 𝑁)
18	17	eqcomd 2742	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 = Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗))
19	18	fveq2d 6846	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘𝑁) = (!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)))
20	19	oveq1d 7372	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) = ((!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))))
21		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝐷
22		fzfid 13878	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
23		nn0ex 12419	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ ∈ V
24		fzssnn0 43541	. . . . . . . . . . . . . . . 16 ⊢ (0...𝑁) ⊆ ℕ₀
25	24	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → (0...𝑁) ⊆ ℕ₀)
26		mapss 8827	. . . . . . . . . . . . . . 15 ⊢ ((ℕ₀ ∈ V ∧ (0...𝑁) ⊆ ℕ₀) → ((0...𝑁) ↑_m (0...𝑀)) ⊆ (ℕ₀ ↑_m (0...𝑀)))
27	23, 25, 26	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → ((0...𝑁) ↑_m (0...𝑀)) ⊆ (ℕ₀ ↑_m (0...𝑀)))
28		elrabi 3639	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝐷 ∈ ((0...𝑁) ↑_m (0...𝑀)))
29	27, 28	sseldd 3945	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝐷 ∈ (ℕ₀ ↑_m (0...𝑀)))
30	11, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ (ℕ₀ ↑_m (0...𝑀)))
31	21, 22, 30	mccl 43829	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℕ)
32	20, 31	eqeltrd 2838	. . . . . . . . . 10 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℕ)
33	32	nnzd 12526	. . . . . . . . 9 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℤ)
34	33	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℤ)
35		df-neg 11388	. . . . . . . . . . . . . . . 16 ⊢ -𝑗 = (0 − 𝑗)
36		oveq1 7364	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 = 0 → (𝐽 − 𝑗) = (0 − 𝑗))
37	35, 36	eqtr4id 2795	. . . . . . . . . . . . . . 15 ⊢ (𝐽 = 0 → -𝑗 = (𝐽 − 𝑗))
38	37	oveq1d 7372	. . . . . . . . . . . . . 14 ⊢ (𝐽 = 0 → (-𝑗↑(𝑃 − (𝐷‘𝑗))) = ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))
39	38	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ (𝐽 = 0 → (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗)))) = (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗)))))
40	39	ifeq2d 4506	. . . . . . . . . . . 12 ⊢ (𝐽 = 0 → if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗))))) = if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))
41	40	prodeq2ad 43823	. . . . . . . . . . 11 ⊢ (𝐽 = 0 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗))))) = ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))
42	41	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 = 0) → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗))))) = ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))
43	11, 28	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ((0...𝑁) ↑_m (0...𝑀)))
44		elmapi 8787	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ ((0...𝑁) ↑_m (0...𝑀)) → 𝐷:(0...𝑀)⟶(0...𝑁))
45	43, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷:(0...𝑀)⟶(0...𝑁))
46		etransclem28.j	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
47	1, 45, 46	etransclem7 44472	. . . . . . . . . . 11 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))) ∈ ℤ)
48	47	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 = 0) → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))) ∈ ℤ)
49	42, 48	eqeltrd 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 = 0) → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗))))) ∈ ℤ)
50	6, 49	zmulcld 12613	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗)))))) ∈ ℤ)
51	6, 34, 50	3jca 1128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 = 0) → ((!‘(𝑃 − 1)) ∈ ℤ ∧ ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℤ ∧ ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗)))))) ∈ ℤ))
52		dvdsmul1 16160	. . . . . . . 8 ⊢ (((!‘(𝑃 − 1)) ∈ ℤ ∧ ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗))))) ∈ ℤ) → (!‘(𝑃 − 1)) ∥ ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗)))))))
53	6, 49, 52	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 = 0) → (!‘(𝑃 − 1)) ∥ ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗)))))))
54		dvdsmultr2 16180	. . . . . . 7 ⊢ (((!‘(𝑃 − 1)) ∈ ℤ ∧ ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℤ ∧ ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗)))))) ∈ ℤ) → ((!‘(𝑃 − 1)) ∥ ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗)))))) → (!‘(𝑃 − 1)) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗)))))))))
55	51, 53, 54	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 = 0) → (!‘(𝑃 − 1)) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗))))))))
56	55	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → (!‘(𝑃 − 1)) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗))))))))
57	1	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝑃 ∈ ℕ)
58		etransclem28.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
59	58	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝑀 ∈ ℕ₀)
60	45	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝐷:(0...𝑀)⟶(0...𝑁))
61		eqid 2736	. . . . . 6 ⊢ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))) = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗)))))))
62		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝐽 = 0)
63		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → (𝐷‘0) = (𝑃 − 1))
64	57, 59, 60, 61, 62, 63	etransclem14 44479	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))) = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · ((!‘(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗))))))))
65	56, 64	breqtrrd 5133	. . . 4 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → (!‘(𝑃 − 1)) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))))
66		dvds0 16154	. . . . . . 7 ⊢ ((!‘(𝑃 − 1)) ∈ ℤ → (!‘(𝑃 − 1)) ∥ 0)
67	5, 66	syl 17	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ 0)
68	67	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → (!‘(𝑃 − 1)) ∥ 0)
69	1	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝑃 ∈ ℕ)
70	58	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝑀 ∈ ℕ₀)
71	9	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝑁 ∈ ℕ₀)
72	45	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝐷:(0...𝑀)⟶(0...𝑁))
73		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝐽 = 0)
74		neqne 2951	. . . . . . 7 ⊢ (¬ (𝐷‘0) = (𝑃 − 1) → (𝐷‘0) ≠ (𝑃 − 1))
75	74	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → (𝐷‘0) ≠ (𝑃 − 1))
76	69, 70, 71, 72, 61, 73, 75	etransclem15 44480	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))) = 0)
77	68, 76	breqtrrd 5133	. . . 4 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → (!‘(𝑃 − 1)) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))))
78	65, 77	pm2.61dan 811	. . 3 ⊢ ((𝜑 ∧ 𝐽 = 0) → (!‘(𝑃 − 1)) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))))
79	1	nnzd 12526	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ)
80		elfznn0 13534	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ∈ ℕ₀)
81	46, 80	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
82	81	nn0zd 12525	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
83	1, 58, 9, 82, 8, 7	etransclem26 44491	. . . . . 6 ⊢ (𝜑 → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))) ∈ ℤ)
84	5, 79, 83	3jca 1128	. . . . 5 ⊢ (𝜑 → ((!‘(𝑃 − 1)) ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))) ∈ ℤ))
85	84	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → ((!‘(𝑃 − 1)) ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))) ∈ ℤ))
86	1	nncnd 12169	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℂ)
87		1cnd 11150	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
88	86, 87	npcand 11516	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) + 1) = 𝑃)
89	88	eqcomd 2742	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = ((𝑃 − 1) + 1))
90	89	fveq2d 6846	. . . . . . 7 ⊢ (𝜑 → (!‘𝑃) = (!‘((𝑃 − 1) + 1)))
91		facp1 14178	. . . . . . . 8 ⊢ ((𝑃 − 1) ∈ ℕ₀ → (!‘((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)))
92	3, 91	syl 17	. . . . . . 7 ⊢ (𝜑 → (!‘((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)))
93	88	oveq2d 7373	. . . . . . 7 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · 𝑃))
94	90, 92, 93	3eqtrrd 2781	. . . . . 6 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · 𝑃) = (!‘𝑃))
95	94	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → ((!‘(𝑃 − 1)) · 𝑃) = (!‘𝑃))
96	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑃 ∈ ℕ)
97	58	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑀 ∈ ℕ₀)
98	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑁 ∈ ℕ₀)
99	45	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐷:(0...𝑀)⟶(0...𝑁))
100	17	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗) = 𝑁)
101		1zzd 12534	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ∈ ℤ)
102	58	nn0zd 12525	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
103	102	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑀 ∈ ℤ)
104	82	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℤ)
105	81	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℕ₀)
106		neqne 2951	. . . . . . . . . 10 ⊢ (¬ 𝐽 = 0 → 𝐽 ≠ 0)
107	106	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ≠ 0)
108		elnnne0 12427	. . . . . . . . 9 ⊢ (𝐽 ∈ ℕ ↔ (𝐽 ∈ ℕ₀ ∧ 𝐽 ≠ 0))
109	105, 107, 108	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℕ)
110	109	nnge1d 12201	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ≤ 𝐽)
111		elfzle2 13445	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ≤ 𝑀)
112	46, 111	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ≤ 𝑀)
113	112	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ≤ 𝑀)
114	101, 103, 104, 110, 113	elfzd 13432	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ (1...𝑀))
115	96, 97, 98, 99, 100, 61, 114	etransclem25 44490	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → (!‘𝑃) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))))
116	95, 115	eqbrtrd 5127	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → ((!‘(𝑃 − 1)) · 𝑃) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))))
117		muldvds1 16163	. . . 4 ⊢ (((!‘(𝑃 − 1)) ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))) ∈ ℤ) → (((!‘(𝑃 − 1)) · 𝑃) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))) → (!‘(𝑃 − 1)) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗)))))))))
118	85, 116, 117	sylc 65	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → (!‘(𝑃 − 1)) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))))
119	78, 118	pm2.61dan 811	. 2 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))))))
120		etransclem28.t	. 2 ⊢ 𝑇 = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗)))))))
121	119, 120	breqtrrdi 5147	1 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 {crab 3407 Vcvv 3445 ⊆ wss 3910 ifcif 4486 class class class wbr 5105 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 < clt 11189 ≤ cle 11190 − cmin 11385 -cneg 11386 / cdiv 11812 ℕcn 12153 ℕ₀cn0 12413 ℤcz 12499 ...cfz 13424 ↑cexp 13967 !cfa 14173 Σcsu 15570 ∏cprod 15788 ∥ cdvds 16136

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-prod 15789 df-dvds 16137

This theorem is referenced by: etransclem37 44502 etransclem38 44503

W3C validator