etransclem38 - Mathbox for Glauco Siliprandi

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Theorem etransclem38 44923

Description: 𝑃 divides the I -th derivative of 𝐹 applied to 𝐽. if it is not the case that 𝐼 = 𝑃 − 1 and 𝐽 = 0. This is case 1 and the second part of case 2 proven in in [Juillerat] p. 13 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem38.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem38.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem38.f	⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
etransclem38.i	⊢ (𝜑 → 𝐼 ∈ ℕ₀)
etransclem38.j	⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
etransclem38.ij	⊢ (𝜑 → ¬ (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))
etransclem38.c	⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})

**Assertion**
Ref	Expression
etransclem38	⊢ (𝜑 → 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘𝐼)‘𝐽) / (!‘(𝑃 − 1))))

Distinct variable groups: 𝐶,𝑐,𝑗,𝑛,𝑥 𝐼,𝑐,𝑗,𝑛,𝑥 𝐽,𝑐,𝑗,𝑛,𝑥 𝑀,𝑐,𝑗,𝑛,𝑥 𝑃,𝑐,𝑗,𝑛,𝑥 𝜑,𝑐,𝑗,𝑛,𝑥 Allowed substitution hints: 𝐹(𝑥,𝑗,𝑛,𝑐)

Proof of Theorem etransclem38 Dummy variables 𝑑 𝑒 𝑘 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem38.c	. . . 4 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
2		etransclem38.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ ℕ₀)
3	1, 2	etransclem16 44901	. . 3 ⊢ (𝜑 → (𝐶‘𝐼) ∈ Fin)
4		etransclem38.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
5	4	nnzd 12581	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℤ)
6	4	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑃 ∈ ℕ)
7		etransclem38.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
8	7	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑀 ∈ ℕ₀)
9	2	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝐼 ∈ ℕ₀)
10		etransclem11 44896	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) = (𝑚 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑒‘𝑘) = 𝑚})
11		etransclem11 44896	. . . . . 6 ⊢ (𝑚 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑒‘𝑘) = 𝑚}) = (𝑛 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑑‘𝑗) = 𝑛})
12	1, 10, 11	3eqtri 2765	. . . . 5 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑑‘𝑗) = 𝑛})
13		simpr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑐 ∈ (𝐶‘𝐼))
14		etransclem38.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
15	14	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝐽 ∈ (0...𝑀))
16		eqid 2733	. . . . 5 ⊢ (((!‘𝐼) / ∏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))))))) = (((!‘𝐼) / ∏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)))))))
17	6, 8, 9, 12, 13, 15, 16	etransclem28 44913	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (!‘(𝑃 − 1)) ∥ (((!‘𝐼) / ∏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))))))))
18		nnm1nn0 12509	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
19	4, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
20	19	faccld 14240	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
21	20	nnzd 12581	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
22	21	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (!‘(𝑃 − 1)) ∈ ℤ)
23	20	nnne0d 12258	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
24	23	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (!‘(𝑃 − 1)) ≠ 0)
25	14	elfzelzd 13498	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
26	25	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝐽 ∈ ℤ)
27	6, 8, 9, 26, 12, 13	etransclem26 44911	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (((!‘𝐼) / ∏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)
28		dvdsval2 16196	. . . . 5 ⊢ (((!‘(𝑃 − 1)) ∈ ℤ ∧ (!‘(𝑃 − 1)) ≠ 0 ∧ (((!‘𝐼) / ∏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) → ((!‘(𝑃 − 1)) ∥ (((!‘𝐼) / ∏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))))))) ↔ ((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))) ∈ ℤ))
29	22, 24, 27, 28	syl3anc 1372	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → ((!‘(𝑃 − 1)) ∥ (((!‘𝐼) / ∏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))))))) ↔ ((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))) ∈ ℤ))
30	17, 29	mpbid 231	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → ((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))) ∈ ℤ)
31		pm3.22 461	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐼 = (𝑃 − 1)) → (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))
32	31	adantll 713	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 = (𝑃 − 1)) → (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))
33		etransclem38.ij	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))
34	33	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 = (𝑃 − 1)) → ¬ (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))
35	32, 34	pm2.65da 816	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) → ¬ 𝐼 = (𝑃 − 1))
36	35	neqned 2948	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) → 𝐼 ≠ (𝑃 − 1))
37	4	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝑃 ∈ ℕ)
38	7	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝑀 ∈ ℕ₀)
39	2	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝐼 ∈ ℕ₀)
40		simpr 486	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝐼 ≠ (𝑃 − 1))
41		simplr 768	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝐽 = 0)
42	13	ad2antrr 725	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝑐 ∈ (𝐶‘𝐼))
43	37, 38, 39, 40, 41, 12, 42	etransclem24 44909	. . . . 5 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝑃 ∥ ((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))))
44	36, 43	mpdan 686	. . . 4 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) → 𝑃 ∥ ((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))))
45	4	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑃 ∈ ℕ)
46	7	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑀 ∈ ℕ₀)
47	2	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝐼 ∈ ℕ₀)
48	1, 2	etransclem12 44897	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶‘𝐼) = {𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼})
49	48	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (𝐶‘𝐼) = {𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼})
50	13, 49	eleqtrd 2836	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼})
51		rabid 3453	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼} ↔ (𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)) ∧ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼))
52	50, 51	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)) ∧ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼))
53	52	simpld 496	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)))
54		elmapi 8839	. . . . . . . . 9 ⊢ (𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝐼))
55	53, 54	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑐:(0...𝑀)⟶(0...𝐼))
56	55	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑐:(0...𝑀)⟶(0...𝐼))
57	52	simprd 497	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼)
58	57	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼)
59		1zzd 12589	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ∈ ℤ)
60	7	nn0zd 12580	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
61	60	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑀 ∈ ℤ)
62	25	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℤ)
63		elfznn0 13590	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ∈ ℕ₀)
64	14, 63	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
65		neqne 2949	. . . . . . . . . . . 12 ⊢ (¬ 𝐽 = 0 → 𝐽 ≠ 0)
66	64, 65	anim12i 614	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → (𝐽 ∈ ℕ₀ ∧ 𝐽 ≠ 0))
67		elnnne0 12482	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ℕ ↔ (𝐽 ∈ ℕ₀ ∧ 𝐽 ≠ 0))
68	66, 67	sylibr 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℕ)
69	68	nnge1d 12256	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ≤ 𝐽)
70		elfzle2 13501	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ≤ 𝑀)
71	14, 70	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ≤ 𝑀)
72	71	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ≤ 𝑀)
73	59, 61, 62, 69, 72	elfzd 13488	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ (1...𝑀))
74	73	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝐽 ∈ (1...𝑀))
75	45, 46, 47, 56, 58, 16, 74	etransclem25 44910	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (!‘𝑃) ∥ (((!‘𝐼) / ∏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))))))))
76	4	nncnd 12224	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
77		1cnd 11205	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
78	76, 77	npcand 11571	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 − 1) + 1) = 𝑃)
79	78	eqcomd 2739	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 = ((𝑃 − 1) + 1))
80	79	fveq2d 6892	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑃) = (!‘((𝑃 − 1) + 1)))
81		facp1 14234	. . . . . . . . 9 ⊢ ((𝑃 − 1) ∈ ℕ₀ → (!‘((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)))
82	19, 81	syl 17	. . . . . . . 8 ⊢ (𝜑 → (!‘((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)))
83	78	oveq2d 7420	. . . . . . . . 9 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · 𝑃))
84	20	nncnd 12224	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
85	84, 76	mulcomd 11231	. . . . . . . . 9 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · 𝑃) = (𝑃 · (!‘(𝑃 − 1))))
86	83, 85	eqtrd 2773	. . . . . . . 8 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)) = (𝑃 · (!‘(𝑃 − 1))))
87	80, 82, 86	3eqtrrd 2778	. . . . . . 7 ⊢ (𝜑 → (𝑃 · (!‘(𝑃 − 1))) = (!‘𝑃))
88	87	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (𝑃 · (!‘(𝑃 − 1))) = (!‘𝑃))
89	27	zcnd 12663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (((!‘𝐼) / ∏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))))))) ∈ ℂ)
90	84	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (!‘(𝑃 − 1)) ∈ ℂ)
91	89, 90, 24	divcan1d 11987	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))) · (!‘(𝑃 − 1))) = (((!‘𝐼) / ∏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))))))))
92	91	adantr 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))) · (!‘(𝑃 − 1))) = (((!‘𝐼) / ∏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))))))))
93	75, 88, 92	3brtr4d 5179	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (𝑃 · (!‘(𝑃 − 1))) ∥ (((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))) · (!‘(𝑃 − 1))))
94	5	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑃 ∈ ℤ)
95	30	adantr 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → ((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))) ∈ ℤ)
96	21	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (!‘(𝑃 − 1)) ∈ ℤ)
97	23	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (!‘(𝑃 − 1)) ≠ 0)
98		dvdsmulcr 16225	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ ((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))) ∈ ℤ ∧ ((!‘(𝑃 − 1)) ∈ ℤ ∧ (!‘(𝑃 − 1)) ≠ 0)) → ((𝑃 · (!‘(𝑃 − 1))) ∥ (((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))) · (!‘(𝑃 − 1))) ↔ 𝑃 ∥ ((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1)))))
99	94, 95, 96, 97, 98	syl112anc 1375	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → ((𝑃 · (!‘(𝑃 − 1))) ∥ (((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))) · (!‘(𝑃 − 1))) ↔ 𝑃 ∥ ((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1)))))
100	93, 99	mpbid 231	. . . 4 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑃 ∥ ((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))))
101	44, 100	pm2.61dan 812	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑃 ∥ ((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))))
102	3, 5, 30, 101	fsumdvds 16247	. 2 ⊢ (𝜑 → 𝑃 ∥ Σ𝑐 ∈ (𝐶‘𝐼)((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))))
103		reelprrecn 11198	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
104	103	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
105		reopn 43934	. . . . . . 7 ⊢ ℝ ∈ (topGen‘ran (,))
106		eqid 2733	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
107	106	tgioo2 24301	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
108	105, 107	eleqtri 2832	. . . . . 6 ⊢ ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
109	108	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
110		etransclem38.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
111		etransclem5 44890	. . . . 5 ⊢ (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ ℝ ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ ℝ ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
112		fzssre 43959	. . . . . 6 ⊢ (0...𝑀) ⊆ ℝ
113	112, 14	sselid 3979	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℝ)
114	104, 109, 4, 7, 110, 2, 111, 1, 113	etransclem31 44916	. . . 4 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘𝐼)‘𝐽) = Σ𝑐 ∈ (𝐶‘𝐼)(((!‘𝐼) / ∏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))))))))
115	114	oveq1d 7419	. . 3 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘𝐼)‘𝐽) / (!‘(𝑃 − 1))) = (Σ𝑐 ∈ (𝐶‘𝐼)(((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))))
116	3, 84, 89, 23	fsumdivc 15728	. . 3 ⊢ (𝜑 → (Σ𝑐 ∈ (𝐶‘𝐼)(((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))) = Σ𝑐 ∈ (𝐶‘𝐼)((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))))
117	115, 116	eqtrd 2773	. 2 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘𝐼)‘𝐽) / (!‘(𝑃 − 1))) = Σ𝑐 ∈ (𝐶‘𝐼)((((!‘𝐼) / ∏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))))))) / (!‘(𝑃 − 1))))
118	102, 117	breqtrrd 5175	1 ⊢ (𝜑 → 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘𝐼)‘𝐽) / (!‘(𝑃 − 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 {crab 3433 ifcif 4527 {cpr 4629 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ↑_m cmap 8816 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 (,)cioo 13320 ...cfz 13480 ↑cexp 14023 !cfa 14229 Σcsu 15628 ∏cprod 15845 ∥ cdvds 16193 ↾_t crest 17362 TopOpenctopn 17363 topGenctg 17379 ℂ_fldccnfld 20929 D^𝑛 cdvn 25363

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 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 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 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 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 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-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 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 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19643 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-fbas 20926 df-fg 20927 df-cnfld 20930 df-top 22378 df-topon 22395 df-topsp 22417 df-bases 22431 df-cld 22505 df-ntr 22506 df-cls 22507 df-nei 22584 df-lp 22622 df-perf 22623 df-cn 22713 df-cnp 22714 df-haus 22801 df-tx 23048 df-hmeo 23241 df-fil 23332 df-fm 23424 df-flim 23425 df-flf 23426 df-xms 23808 df-ms 23809 df-tms 23810 df-cncf 24376 df-limc 25365 df-dv 25366 df-dvn 25367

This theorem is referenced by: etransclem44 44929

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