etransclem28 - Mathbox for Glauco Siliprandi

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Theorem etransclem28 43693

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 12204	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
3	1, 2	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
4	3	faccld 13926	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
5	4	nnzd 12354	. . . . . . . . 9 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
6	5	adantr 480	. . . . . . . 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 43677	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
11	7, 10	eleqtrd 2841	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
12		fveq1 6755	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝐷 → (𝑐‘𝑗) = (𝐷‘𝑗))
13	12	sumeq2sdv 15344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝐷 → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗))
14	13	eqeq1d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝐷 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁 ↔ Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗) = 𝑁))
15	14	elrab 3617	. . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 = Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗))
19	18	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘𝑁) = (!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)))
20	19	oveq1d 7270	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) = ((!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))))
21		nfcv 2906	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝐷
22		fzfid 13621	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
23		nn0ex 12169	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ ∈ V
24		fzssnn0 42746	. . . . . . . . . . . . . . . 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 8635	. . . . . . . . . . . . . . 15 ⊢ ((ℕ₀ ∈ V ∧ (0...𝑁) ⊆ ℕ₀) → ((0...𝑁) ↑_m (0...𝑀)) ⊆ (ℕ₀ ↑_m (0...𝑀)))
27	23, 25, 26	sylancr 586	. . . . . . . . . . . . . 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 3611	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝐷 ∈ ((0...𝑁) ↑_m (0...𝑀)))
29	27, 28	sseldd 3918	. . . . . . . . . . . . 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 43029	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℕ)
32	20, 31	eqeltrd 2839	. . . . . . . . . 10 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℕ)
33	32	nnzd 12354	. . . . . . . . 9 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℤ)
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 = 0) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℤ)
35		df-neg 11138	. . . . . . . . . . . . . . . 16 ⊢ -𝑗 = (0 − 𝑗)
36		oveq1 7262	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 = 0 → (𝐽 − 𝑗) = (0 − 𝑗))
37	35, 36	eqtr4id 2798	. . . . . . . . . . . . . . 15 ⊢ (𝐽 = 0 → -𝑗 = (𝐽 − 𝑗))
38	37	oveq1d 7270	. . . . . . . . . . . . . 14 ⊢ (𝐽 = 0 → (-𝑗↑(𝑃 − (𝐷‘𝑗))) = ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))
39	38	oveq2d 7271	. . . . . . . . . . . . 13 ⊢ (𝐽 = 0 → (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗)))) = (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗)))))
40	39	ifeq2d 4476	. . . . . . . . . . . 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 43023	. . . . . . . . . . 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 481	. . . . . . . . . 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 8595	. . . . . . . . . . . . 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 43672	. . . . . . . . . . 11 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))) ∈ ℤ)
48	47	adantr 480	. . . . . . . . . 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 2839	. . . . . . . . 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 12361	. . . . . . . 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 1126	. . . . . . 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 15915	. . . . . . . 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 583	. . . . . . 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 15935	. . . . . . 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 480	. . . . 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 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝑃 ∈ ℕ)
58		etransclem28.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
59	58	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝑀 ∈ ℕ₀)
60	45	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝐷:(0...𝑀)⟶(0...𝑁))
61		eqid 2738	. . . . . 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 765	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝐽 = 0)
63		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → (𝐷‘0) = (𝑃 − 1))
64	57, 59, 60, 61, 62, 63	etransclem14 43679	. . . . 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 5098	. . . 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 15909	. . . . . . 7 ⊢ ((!‘(𝑃 − 1)) ∈ ℤ → (!‘(𝑃 − 1)) ∥ 0)
67	5, 66	syl 17	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ 0)
68	67	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → (!‘(𝑃 − 1)) ∥ 0)
69	1	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝑃 ∈ ℕ)
70	58	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝑀 ∈ ℕ₀)
71	9	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝑁 ∈ ℕ₀)
72	45	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝐷:(0...𝑀)⟶(0...𝑁))
73		simplr 765	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝐽 = 0)
74		neqne 2950	. . . . . . 7 ⊢ (¬ (𝐷‘0) = (𝑃 − 1) → (𝐷‘0) ≠ (𝑃 − 1))
75	74	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → (𝐷‘0) ≠ (𝑃 − 1))
76	69, 70, 71, 72, 61, 73, 75	etransclem15 43680	. . . . 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 5098	. . . 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 809	. . 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 12354	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ)
80		elfznn0 13278	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ∈ ℕ₀)
81	46, 80	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
82	81	nn0zd 12353	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
83	1, 58, 9, 82, 8, 7	etransclem26 43691	. . . . . 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 1126	. . . . 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 480	. . . 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 11919	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℂ)
87		1cnd 10901	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
88	86, 87	npcand 11266	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) + 1) = 𝑃)
89	88	eqcomd 2744	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = ((𝑃 − 1) + 1))
90	89	fveq2d 6760	. . . . . . 7 ⊢ (𝜑 → (!‘𝑃) = (!‘((𝑃 − 1) + 1)))
91		facp1 13920	. . . . . . . 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 7271	. . . . . . 7 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · 𝑃))
94	90, 92, 93	3eqtrrd 2783	. . . . . 6 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · 𝑃) = (!‘𝑃))
95	94	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → ((!‘(𝑃 − 1)) · 𝑃) = (!‘𝑃))
96	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑃 ∈ ℕ)
97	58	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑀 ∈ ℕ₀)
98	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑁 ∈ ℕ₀)
99	45	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐷:(0...𝑀)⟶(0...𝑁))
100	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗) = 𝑁)
101		1zzd 12281	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ∈ ℤ)
102	58	nn0zd 12353	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
103	102	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑀 ∈ ℤ)
104	82	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℤ)
105	81	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℕ₀)
106		neqne 2950	. . . . . . . . . 10 ⊢ (¬ 𝐽 = 0 → 𝐽 ≠ 0)
107	106	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ≠ 0)
108		elnnne0 12177	. . . . . . . . 9 ⊢ (𝐽 ∈ ℕ ↔ (𝐽 ∈ ℕ₀ ∧ 𝐽 ≠ 0))
109	105, 107, 108	sylanbrc 582	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℕ)
110	109	nnge1d 11951	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ≤ 𝐽)
111		elfzle2 13189	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ≤ 𝑀)
112	46, 111	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ≤ 𝑀)
113	112	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ≤ 𝑀)
114	101, 103, 104, 110, 113	elfzd 13176	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ (1...𝑀))
115	96, 97, 98, 99, 100, 61, 114	etransclem25 43690	. . . . 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 5092	. . . 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 15918	. . . 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 809	. 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 5112	1 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 {crab 3067 Vcvv 3422 ⊆ wss 3883 ifcif 4456 class class class wbr 5070 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ...cfz 13168 ↑cexp 13710 !cfa 13915 Σcsu 15325 ∏cprod 15543 ∥ cdvds 15891

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-prod 15544 df-dvds 15892

This theorem is referenced by: etransclem37 43702 etransclem38 43703

W3C validator