etransclem38 - Mathbox for Glauco Siliprandi

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Theorem etransclem38 39165

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 39143	. . 3 ⊢ (𝜑 → (𝐶‘𝐼) ∈ Fin)
4		etransclem38.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
5	4	nnzd 11357	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℤ)
6	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑃 ∈ ℕ)
7		etransclem38.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑀 ∈ ℕ₀)
9	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝐼 ∈ ℕ₀)
10		etransclem11 39138	. . . . . 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 39138	. . . . . 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 2636	. . . . 5 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑛) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑑‘𝑗) = 𝑛})
13		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑐 ∈ (𝐶‘𝐼))
14		etransclem38.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
15	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝐽 ∈ (0...𝑀))
16		eqid 2610	. . . . 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 39155	. . . 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 11211	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
19	4, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
20	19	faccld 12933	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
21	20	nnzd 11357	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (!‘(𝑃 − 1)) ∈ ℤ)
23	20	nnne0d 10942	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (!‘(𝑃 − 1)) ≠ 0)
25	14	elfzelzd 38471	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
26	25	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝐽 ∈ ℤ)
27	6, 8, 9, 26, 12, 13	etransclem26 39153	. . . . 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 14824	. . . . 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 1318	. . . 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 221	. . 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		pm3.22 464	. . . . . . . 8 ⊢ ((𝐽 = 0 ∧ 𝐼 = (𝑃 − 1)) → (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))
32	31	adantll 746	. . . . . . 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 762	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 = (𝑃 − 1)) → ¬ (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))
35	32, 34	pm2.65da 598	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) → ¬ 𝐼 = (𝑃 − 1))
36	35	neqned 2789	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) → 𝐼 ≠ (𝑃 − 1))
37	4	ad3antrrr 762	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝑃 ∈ ℕ)
38	7	ad3antrrr 762	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝑀 ∈ ℕ₀)
39	2	ad3antrrr 762	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝐼 ∈ ℕ₀)
40		simpr 476	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝐼 ≠ (𝑃 − 1))
41		simplr 788	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝐽 = 0)
42	13	ad2antrr 758	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ 𝐽 = 0) ∧ 𝐼 ≠ (𝑃 − 1)) → 𝑐 ∈ (𝐶‘𝐼))
43	37, 38, 39, 40, 41, 12, 42	etransclem24 39151	. . . . 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 699	. . . 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 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑃 ∈ ℕ)
46	7	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑀 ∈ ℕ₀)
47	2	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝐼 ∈ ℕ₀)
48	1, 2	etransclem12 39139	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶‘𝐼) = {𝑐 ∈ ((0...𝐼) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼})
49	48	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (𝐶‘𝐼) = {𝑐 ∈ ((0...𝐼) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼})
50	13, 49	eleqtrd 2690	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐼) ↑_𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼})
51		rabid 3095	. . . . . . . . . . 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 207	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (𝑐 ∈ ((0...𝐼) ↑_𝑚 (0...𝑀)) ∧ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼))
53	52	simpld 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → 𝑐 ∈ ((0...𝐼) ↑_𝑚 (0...𝑀)))
54		elmapi 7765	. . . . . . . . 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 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑐:(0...𝑀)⟶(0...𝐼))
57	52	simprd 478	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼)
58	57	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝐼)
59		1zzd 11285	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ∈ ℤ)
60	7	nn0zd 11356	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
61	60	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝑀 ∈ ℤ)
62	25	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℤ)
63	59, 61, 62	3jca 1235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐽 ∈ ℤ))
64		elfznn0 12302	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ∈ ℕ₀)
65	14, 64	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
66		neqne 2790	. . . . . . . . . . . . 13 ⊢ (¬ 𝐽 = 0 → 𝐽 ≠ 0)
67	65, 66	anim12i 588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → (𝐽 ∈ ℕ₀ ∧ 𝐽 ≠ 0))
68		elnnne0 11183	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ ℕ ↔ (𝐽 ∈ ℕ₀ ∧ 𝐽 ≠ 0))
69	67, 68	sylibr 223	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ ℕ)
70	69	nnge1d 10940	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 1 ≤ 𝐽)
71		elfzle2 12216	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ≤ 𝑀)
72	14, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ≤ 𝑀)
73	72	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ≤ 𝑀)
74	63, 70, 73	jca32 556	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑀)))
75		elfz2 12204	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (1 ≤ 𝐽 ∧ 𝐽 ≤ 𝑀)))
76	74, 75	sylibr 223	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐽 = 0) → 𝐽 ∈ (1...𝑀))
77	76	adantlr 747	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝐽 ∈ (1...𝑀))
78	45, 46, 47, 56, 58, 16, 77	etransclem25 39152	. . . . . 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 10913	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
80		1cnd 9935	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
81	79, 80	npcand 10275	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 − 1) + 1) = 𝑃)
82	81	eqcomd 2616	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 = ((𝑃 − 1) + 1))
83	82	fveq2d 6107	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑃) = (!‘((𝑃 − 1) + 1)))
84		facp1 12927	. . . . . . . . 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 6565	. . . . . . . . 9 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)) = ((!‘(𝑃 − 1)) · 𝑃))
87	20	nncnd 10913	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
88	87, 79	mulcomd 9940	. . . . . . . . 9 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · 𝑃) = (𝑃 · (!‘(𝑃 − 1))))
89	86, 88	eqtrd 2644	. . . . . . . 8 ⊢ (𝜑 → ((!‘(𝑃 − 1)) · ((𝑃 − 1) + 1)) = (𝑃 · (!‘(𝑃 − 1))))
90	83, 85, 89	3eqtrrd 2649	. . . . . . 7 ⊢ (𝜑 → (𝑃 · (!‘(𝑃 − 1))) = (!‘𝑃))
91	90	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (𝑃 · (!‘(𝑃 − 1))) = (!‘𝑃))
92	27	zcnd 11359	. . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) → (!‘(𝑃 − 1)) ∈ ℂ)
94	92, 93, 24	divcan1d 10681	. . . . . . 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 480	. . . . . 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 4615	. . . . 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 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → 𝑃 ∈ ℤ)
98	30	adantr 480	. . . . . 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 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (!‘(𝑃 − 1)) ∈ ℤ)
100	23	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝐼)) ∧ ¬ 𝐽 = 0) → (!‘(𝑃 − 1)) ≠ 0)
101		dvdsmulcr 14849	. . . . . 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 1322	. . . . 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 221	. . . 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 828	. . 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 14868	. 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 9907	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
107	106	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
108		reopn 38442	. . . . . . 7 ⊢ ℝ ∈ (topGen‘ran (,))
109		eqid 2610	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
110	109	tgioo2 22414	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
111	108, 110	eleqtri 2686	. . . . . 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 39132	. . . . 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 38470	. . . . . 6 ⊢ (0...𝑀) ⊆ ℝ
116	115, 14	sseldi 3566	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℝ)
117	107, 112, 4, 7, 113, 2, 114, 1, 116	etransclem31 39158	. . . 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 6564	. . 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 14360	. . 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 2644	. 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 4611	1 ⊢ (𝜑 → 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘𝐼)‘𝐽) / (!‘(𝑃 − 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 ifcif 4036 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑_𝑚 cmap 7744 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 (,)cioo 12046 ...cfz 12197 ↑cexp 12722 !cfa 12922 Σcsu 14264 ∏cprod 14474 ∥ cdvds 14821 ↾_t crest 15904 TopOpenctopn 15905 topGenctg 15921 ℂ_fldccnfld 19567 D^𝑛 cdvn 23434

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 ax-addf 9894 ax-mulf 9895

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-iin 4458 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-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 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-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 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 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-dvn 23438

This theorem is referenced by: etransclem44 39171

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