etransclem38 - Mathbox for Glauco Siliprandi

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Theorem etransclem38 42556

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 42534	. . 3 ⊢ (𝜑 → (𝐶‘𝐼) ∈ Fin)
4		etransclem38.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
5	4	nnzd 12085	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℤ)
6	4	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑃 ∈ ℕ)
7		etransclem38.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
8	7	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑀 ∈ ℕ₀)
9	2	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝐼 ∈ ℕ₀)
10		etransclem11 42529	. . . . . 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 42529	. . . . . 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 2848	. . . . 5 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑑‘𝑗) = 𝑛})
13		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑐 ∈ (𝐶‘𝐼))
14		etransclem38.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
15	14	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝐽 ∈ (0...𝑀))
16		eqid 2821	. . . . 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 42546	. . . 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 11937	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
19	4, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
20	19	faccld 13643	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
21	20	nnzd 12085	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
22	21	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (!‘(𝑃 − 1)) ∈ ℤ)
23	20	nnne0d 11686	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
24	23	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (!‘(𝑃 − 1)) ≠ 0)
25	14	elfzelzd 41580	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
26	25	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝐽 ∈ ℤ)
27	6, 8, 9, 26, 12, 13	etransclem26 42544	. . . . 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 15609	. . . . 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 1367	. . . 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 234	. . 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		pm3.22 462	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐼 = (𝑃 − 1)) → (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))
32	31	adantll 712	. . . . . . 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 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 = (𝑃 − 1)) → ¬ (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))
35	32, 34	pm2.65da 815	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) → ¬ 𝐼 = (𝑃 − 1))
36	35	neqned 3023	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) → 𝐼 ≠ (𝑃 − 1))
37	4	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝑃 ∈ ℕ)
38	7	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝑀 ∈ ℕ₀)
39	2	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝐼 ∈ ℕ₀)
40		simpr 487	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝐼 ≠ (𝑃 − 1))
41		simplr 767	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝐽 = 0)
42	13	ad2antrr 724	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝑐 ∈ (𝐶‘𝐼))
43	37, 38, 39, 40, 41, 12, 42	etransclem24 42542	. . . . 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 685	. . . 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 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑃 ∈ ℕ)
46	7	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑀 ∈ ℕ₀)
47	2	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝐼 ∈ ℕ₀)
48	1, 2	etransclem12 42530	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶‘𝐼) = {𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼})
49	48	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (𝐶‘𝐼) = {𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼})
50	13, 49	eleqtrd 2915	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼})
51		rabid 3378	. . . . . . . . . . 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 220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)) ∧ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼))
53	52	simpld 497	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑐 ∈ ((0...𝐼) ↑_m (0...𝑀)))
54		elmapi 8427	. . . . . . . . 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 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑐:(0...𝑀)⟶(0...𝐼))
57	52	simprd 498	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼)
58	57	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼)
59		1zzd 12012	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ∈ ℤ)
60	7	nn0zd 12084	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
61	60	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑀 ∈ ℤ)
62	25	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℤ)
63	59, 61, 62	3jca 1124	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐽 ∈ ℤ))
64		elfznn0 12999	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ∈ ℕ₀)
65	14, 64	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
66		neqne 3024	. . . . . . . . . . . . 13 ⊢ (¬ 𝐽 = 0 → 𝐽 ≠ 0)
67	65, 66	anim12i 614	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → (𝐽 ∈ ℕ₀ ∧ 𝐽 ≠ 0))
68		elnnne0 11910	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ ℕ ↔ (𝐽 ∈ ℕ₀ ∧ 𝐽 ≠ 0))
69	67, 68	sylibr 236	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℕ)
70	69	nnge1d 11684	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ≤ 𝐽)
71		elfzle2 12910	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ≤ 𝑀)
72	14, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ≤ 𝑀)
73	72	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ≤ 𝑀)
74	63, 70, 73	jca32 518	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑀)))
75		elfz2 12898	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑀)))
76	74, 75	sylibr 236	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ (1...𝑀))
77	76	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝐽 ∈ (1...𝑀))
78	45, 46, 47, 56, 58, 16, 77	etransclem25 42543	. . . . . 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))))))))
79	4	nncnd 11653	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
80		1cnd 10635	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
81	79, 80	npcand 11000	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 − 1) + 1) = 𝑃)
82	81	eqcomd 2827	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 = ((𝑃 − 1) + 1))
83	82	fveq2d 6673	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑃) = (!‘((𝑃 − 1) + 1)))
84		facp1 13637	. . . . . . . . 9 ⊢ ((𝑃 − 1) ∈ ℕ₀ → (!‘((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)))
85	19, 84	syl 17	. . . . . . . 8 ⊢ (𝜑 → (!‘((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)))
86	81	oveq2d 7171	. . . . . . . . 9 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · 𝑃))
87	20	nncnd 11653	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
88	87, 79	mulcomd 10661	. . . . . . . . 9 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · 𝑃) = (𝑃 · (!‘(𝑃 − 1))))
89	86, 88	eqtrd 2856	. . . . . . . 8 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)) = (𝑃 · (!‘(𝑃 − 1))))
90	83, 85, 89	3eqtrrd 2861	. . . . . . 7 ⊢ (𝜑 → (𝑃 · (!‘(𝑃 − 1))) = (!‘𝑃))
91	90	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (𝑃 · (!‘(𝑃 − 1))) = (!‘𝑃))
92	27	zcnd 12087	. . . . . . . 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)
93	87	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (!‘(𝑃 − 1)) ∈ ℂ)
94	92, 93, 24	divcan1d 11416	. . . . . . 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))))))))
95	94	adantr 483	. . . . . 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))))))))
96	78, 91, 95	3brtr4d 5097	. . . . 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))))
97	5	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑃 ∈ ℤ)
98	30	adantr 483	. . . . . 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)
99	21	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (!‘(𝑃 − 1)) ∈ ℤ)
100	23	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (!‘(𝑃 − 1)) ≠ 0)
101		dvdsmulcr 15638	. . . . . 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)))))
102	97, 98, 99, 100, 101	syl112anc 1370	. . . . 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)))))
103	96, 102	mpbid 234	. . . 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))))
104	44, 103	pm2.61dan 811	. . 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))))
105	3, 5, 30, 104	fsumdvds 15657	. 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))))
106		reelprrecn 10628	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
107	106	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
108		reopn 41553	. . . . . . 7 ⊢ ℝ ∈ (topGen‘ran (,))
109		eqid 2821	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
110	109	tgioo2 23410	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
111	108, 110	eleqtri 2911	. . . . . 6 ⊢ ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
112	111	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
113		etransclem38.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
114		etransclem5 42523	. . . . 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))))
115		fzssre 41579	. . . . . 6 ⊢ (0...𝑀) ⊆ ℝ
116	115, 14	sseldi 3964	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℝ)
117	107, 112, 4, 7, 113, 2, 114, 1, 116	etransclem31 42549	. . . 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))))))))
118	117	oveq1d 7170	. . 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))))
119	3, 87, 92, 23	fsumdivc 15140	. . 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))))
120	118, 119	eqtrd 2856	. 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))))
121	105, 120	breqtrrd 5093	1 ⊢ (𝜑 → 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘𝐼)‘𝐽) / (!‘(𝑃 − 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {crab 3142 ifcif 4466 {cpr 4568 class class class wbr 5065 ↦ cmpt 5145 ran crn 5555 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ↑_m cmap 8405 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 · cmul 10541 < clt 10674 ≤ cle 10675 − cmin 10869 / cdiv 11296 ℕcn 11637 ℕ₀cn0 11896 ℤcz 11980 (,)cioo 12737 ...cfz 12891 ↑cexp 13428 !cfa 13632 Σcsu 15041 ∏cprod 15258 ∥ cdvds 15606 ↾_t crest 16693 TopOpenctopn 16694 topGenctg 16710 ℂ_fldccnfld 20544 D^𝑛 cdvn 24461

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-fac 13633 df-bc 13662 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 df-prod 15259 df-dvds 15607 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cn 21834 df-cnp 21835 df-haus 21922 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-xms 22929 df-ms 22930 df-tms 22931 df-cncf 23485 df-limc 24463 df-dv 24464 df-dvn 24465

This theorem is referenced by: etransclem44 42562

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