etransclem28 - Mathbox for Glauco Siliprandi

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Theorem etransclem28 39155

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 11211	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
3	1, 2	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
4	3	faccld 12933	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
5	4	nnzd 11357	. . . . . . . . 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 39139	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
11	7, 10	eleqtrd 2690	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
12		fveq1 6102	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝐷 → (𝑐‘𝑗) = (𝐷‘𝑗))
13	12	sumeq2ad 38632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝐷 → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗))
14	13	eqeq1d 2612	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝐷 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁 ↔ Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗) = 𝑁))
15	14	elrab 3331	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . 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 2616	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 = Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗))
19	18	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘𝑁) = (!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)))
20	19	oveq1d 6564	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) = ((!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))))
21		nfcv 2751	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝐷
22		fzfid 12634	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
23		nn0ex 11175	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ ∈ V
24		fzssnn0 38474	. . . . . . . . . . . . . . . 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 7786	. . . . . . . . . . . . . . 15 ⊢ ((ℕ₀ ∈ V ∧ (0...𝑁) ⊆ ℕ₀) → ((0...𝑁) ↑_𝑚 (0...𝑀)) ⊆ (ℕ₀ ↑_𝑚 (0...𝑀)))
27	23, 25, 26	sylancr 694	. . . . . . . . . . . . . 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 3328	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ {𝑐 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} → 𝐷 ∈ ((0...𝑁) ↑_𝑚 (0...𝑀)))
29	27, 28	sseldd 3569	. . . . . . . . . . . . 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 38665	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘Σ𝑗 ∈ (0...𝑀)(𝐷‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℕ)
32	20, 31	eqeltrd 2688	. . . . . . . . . 10 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷‘𝑗))) ∈ ℕ)
33	32	nnzd 11357	. . . . . . . . 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		oveq1 6556	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 = 0 → (𝐽 − 𝑗) = (0 − 𝑗))
36		df-neg 10148	. . . . . . . . . . . . . . . 16 ⊢ -𝑗 = (0 − 𝑗)
37	35, 36	syl6reqr 2663	. . . . . . . . . . . . . . 15 ⊢ (𝐽 = 0 → -𝑗 = (𝐽 − 𝑗))
38	37	oveq1d 6564	. . . . . . . . . . . . . 14 ⊢ (𝐽 = 0 → (-𝑗↑(𝑃 − (𝐷‘𝑗))) = ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗))))
39	38	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (𝐽 = 0 → (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · (-𝑗↑(𝑃 − (𝐷‘𝑗)))) = (((!‘𝑃) / (!‘(𝑃 − (𝐷‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐷‘𝑗)))))
40	39	ifeq2d 4055	. . . . . . . . . . . 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 38659	. . . . . . . . . . 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 7765	. . . . . . . . . . . . 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 39134	. . . . . . . . . . 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 2688	. . . . . . . . 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 11364	. . . . . . . 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 1235	. . . . . . 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 14841	. . . . . . . 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 691	. . . . . . 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 14859	. . . . . . 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 63	. . . . . 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 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝑃 ∈ ℕ)
58		etransclem28.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
59	58	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝑀 ∈ ℕ₀)
60	45	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝐷:(0...𝑀)⟶(0...𝑁))
61		eqid 2610	. . . . . 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 788	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → 𝐽 = 0)
63		simpr 476	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ (𝐷‘0) = (𝑃 − 1)) → (𝐷‘0) = (𝑃 − 1))
64	57, 59, 60, 61, 62, 63	etransclem14 39141	. . . . 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 4611	. . . 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 14835	. . . . . . 7 ⊢ ((!‘(𝑃 − 1)) ∈ ℤ → (!‘(𝑃 − 1)) ∥ 0)
67	5, 66	syl 17	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ 0)
68	67	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → (!‘(𝑃 − 1)) ∥ 0)
69	1	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝑃 ∈ ℕ)
70	58	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝑀 ∈ ℕ₀)
71	9	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝑁 ∈ ℕ₀)
72	45	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝐷:(0...𝑀)⟶(0...𝑁))
73		simplr 788	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 = 0) ∧ ¬ (𝐷‘0) = (𝑃 − 1)) → 𝐽 = 0)
74		neqne 2790	. . . . . . 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 39142	. . . . 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 4611	. . . 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 828	. . 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 11357	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℤ)
80		elfznn0 12302	. . . . . . . . 9 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ∈ ℕ₀)
81	46, 80	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
82	81	nn0zd 11356	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
83	1, 58, 9, 82, 8, 7	etransclem26 39153	. . . . . 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 1235	. . . . 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 10913	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℂ)
87		1cnd 9935	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
88	86, 87	npcand 10275	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) + 1) = 𝑃)
89	88	eqcomd 2616	. . . . . . . 8 ⊢ (𝜑 → 𝑃 = ((𝑃 − 1) + 1))
90	89	fveq2d 6107	. . . . . . 7 ⊢ (𝜑 → (!‘𝑃) = (!‘((𝑃 − 1) + 1)))
91		facp1 12927	. . . . . . . 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 6565	. . . . . . 7 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · 𝑃))
94	90, 92, 93	3eqtrrd 2649	. . . . . 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 11285	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ∈ ℤ)
102	58	nn0zd 11356	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
103	102	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑀 ∈ ℤ)
104	82	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℤ)
105	101, 103, 104	3jca 1235	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐽 ∈ ℤ))
106	81	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℕ₀)
107		neqne 2790	. . . . . . . . . . 11 ⊢ (¬ 𝐽 = 0 → 𝐽 ≠ 0)
108	107	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ≠ 0)
109		elnnne0 11183	. . . . . . . . . 10 ⊢ (𝐽 ∈ ℕ ↔ (𝐽 ∈ ℕ₀ ∧ 𝐽 ≠ 0))
110	106, 108, 109	sylanbrc 695	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℕ)
111	110	nnge1d 10940	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ≤ 𝐽)
112		elfzle2 12216	. . . . . . . . . 10 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ≤ 𝑀)
113	46, 112	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ≤ 𝑀)
114	113	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ≤ 𝑀)
115	105, 111, 114	jca32 556	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑀)))
116		elfz2 12204	. . . . . . 7 ⊢ (𝐽 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑀)))
117	115, 116	sylibr 223	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ (1...𝑀))
118	96, 97, 98, 99, 100, 61, 117	etransclem25 39152	. . . . 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))))))))
119	95, 118	eqbrtrd 4605	. . . 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))))))))
120		muldvds1 14844	. . . 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)))))))))
121	85, 119, 120	sylc 63	. . 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))))))))
122	78, 121	pm2.61dan 828	. 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))))))))
123		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)))))))
124	122, 123	syl6breqr 4625	1 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ⊆ wss 3540 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑_𝑚 cmap 7744 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 ↑cexp 12722 !cfa 12922 Σcsu 14264 ∏cprod 14474 ∥ cdvds 14821

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-prod 14475 df-dvds 14822

This theorem is referenced by: etransclem37 39164 etransclem38 39165

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