etransclem28 - Mathbox for Glauco Siliprandi

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Theorem etransclem28 44964

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 12509	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
3	1, 2	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
4	3	faccld 14240	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
5	4	nnzd 12581	. . . . . . . . 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 44948	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
11	7, 10	eleqtrd 2835	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
12		fveq1 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝐷 → (𝑐‘𝑗) = (𝐷‘𝑗))
13	12	sumeq2sdv 15646	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝐷 → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗))
14	13	eqeq1d 2734	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝐷 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁 ↔ Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗) = 𝑁))
15	14	elrab 3682	. . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 = Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗))
19	18	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘𝑁) = (!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)))
20	19	oveq1d 7420	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) = ((!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))))
21		nfcv 2903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝐷
22		fzfid 13934	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
23		nn0ex 12474	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ ∈ V
24		fzssnn0 44013	. . . . . . . . . . . . . . . 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 8879	. . . . . . . . . . . . . . 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 3676	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝐷 ∈ ((0...𝑁) ↑_m (0...𝑀)))
29	27, 28	sseldd 3982	. . . . . . . . . . . . 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 44300	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℕ)
32	20, 31	eqeltrd 2833	. . . . . . . . . 10 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℕ)
33	32	nnzd 12581	. . . . . . . . 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 11443	. . . . . . . . . . . . . . . 16 ⊢ -𝑗 = (0 − 𝑗)
36		oveq1 7412	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 = 0 → (𝐽 − 𝑗) = (0 − 𝑗))
37	35, 36	eqtr4id 2791	. . . . . . . . . . . . . . 15 ⊢ (𝐽 = 0 → -𝑗 = (𝐽 − 𝑗))
38	37	oveq1d 7420	. . . . . . . . . . . . . 14 ⊢ (𝐽 = 0 → (-𝑗↑(𝑃 − (𝐷‘𝑗))) = ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))
39	38	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ (𝐽 = 0 → (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗)))) = (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗)))))
40	39	ifeq2d 4547	. . . . . . . . . . . 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 44294	. . . . . . . . . . 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 8839	. . . . . . . . . . . . 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 44943	. . . . . . . . . . 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 2833	. . . . . . . . 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 12668	. . . . . . . 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 16217	. . . . . . . 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 16237	. . . . . . 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 2732	. . . . . 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 44950	. . . . 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 5175	. . . 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 16211	. . . . . . 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 2948	. . . . . . 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 44951	. . . . 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 5175	. . . 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 12581	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ)
80		elfznn0 13590	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ∈ ℕ₀)
81	46, 80	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
82	81	nn0zd 12580	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
83	1, 58, 9, 82, 8, 7	etransclem26 44962	. . . . . 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 12224	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℂ)
87		1cnd 11205	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
88	86, 87	npcand 11571	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) + 1) = 𝑃)
89	88	eqcomd 2738	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = ((𝑃 − 1) + 1))
90	89	fveq2d 6892	. . . . . . 7 ⊢ (𝜑 → (!‘𝑃) = (!‘((𝑃 − 1) + 1)))
91		facp1 14234	. . . . . . . 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 7421	. . . . . . 7 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · 𝑃))
94	90, 92, 93	3eqtrrd 2777	. . . . . 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 12589	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ∈ ℤ)
102	58	nn0zd 12580	. . . . . . . 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 2948	. . . . . . . . . 10 ⊢ (¬ 𝐽 = 0 → 𝐽 ≠ 0)
107	106	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ≠ 0)
108		elnnne0 12482	. . . . . . . . 9 ⊢ (𝐽 ∈ ℕ ↔ (𝐽 ∈ ℕ₀ ∧ 𝐽 ≠ 0))
109	105, 107, 108	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℕ)
110	109	nnge1d 12256	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ≤ 𝐽)
111		elfzle2 13501	. . . . . . . . 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 13488	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ (1...𝑀))
115	96, 97, 98, 99, 100, 61, 114	etransclem25 44961	. . . . 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 5169	. . . 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 16220	. . . 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 5189	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 2940 {crab 3432 Vcvv 3474 ⊆ wss 3947 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 -cneg 11441 / cdiv 11867 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ...cfz 13480 ↑cexp 14023 !cfa 14229 Σcsu 15628 ∏cprod 15845 ∥ cdvds 16193

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-prod 15846 df-dvds 16194

This theorem is referenced by: etransclem37 44973 etransclem38 44974

W3C validator