etransclem25 - Mathbox for Glauco Siliprandi

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Theorem etransclem25 46702

Description: 𝑃 factorial divides the 𝑁-th derivative of 𝐹 applied to 𝐽. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem25.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem25.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem25.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
etransclem25.c	⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁))
etransclem25.sumc	⊢ (𝜑 → Σ𝑗 ∈ (0...𝑀)(𝐶‘𝑗) = 𝑁)
etransclem25.t	⊢ 𝑇 = (((!‘𝑁) / ∏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)))))))
etransclem25.j	⊢ (𝜑 → 𝐽 ∈ (1...𝑀))

**Assertion**
Ref	Expression
etransclem25	⊢ (𝜑 → (!‘𝑃) ∥ 𝑇)

Distinct variable groups: 𝐶,𝑗 𝑗,𝐽 𝑗,𝑀 𝑃,𝑗 𝜑,𝑗 Allowed substitution hints: 𝑇(𝑗) 𝑁(𝑗)

**Proof of Theorem etransclem25**
Step	Hyp	Ref	Expression
1		etransclem25.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
2	1	nnnn0d 12489	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
3	2	faccld 14237	. . . . 5 ⊢ (𝜑 → (!‘𝑃) ∈ ℕ)
4	3	nnzd 12541	. . . 4 ⊢ (𝜑 → (!‘𝑃) ∈ ℤ)
5		etransclem25.sumc	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑀)(𝐶‘𝑗) = 𝑁)
6	5	eqcomd 2745	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 = Σ𝑗 ∈ (0...𝑀)(𝐶‘𝑗))
7	6	fveq2d 6831	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) = (!‘Σ𝑗 ∈ (0...𝑀)(𝐶‘𝑗)))
8	7	oveq1d 7371	. . . . . . 7 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) = ((!‘Σ𝑗 ∈ (0...𝑀)(𝐶‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))))
9		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑗𝐶
10		fzfid 13926	. . . . . . . 8 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
11		etransclem25.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁))
12		nn0ex 12434	. . . . . . . . . . 11 ⊢ ℕ₀ ∈ V
13		fzssnn0 45764	. . . . . . . . . . 11 ⊢ (0...𝑁) ⊆ ℕ₀
14		mapss 8827	. . . . . . . . . . 11 ⊢ ((ℕ₀ ∈ V ∧ (0...𝑁) ⊆ ℕ₀) → ((0...𝑁) ↑_m (0...𝑀)) ⊆ (ℕ₀ ↑_m (0...𝑀)))
15	12, 13, 14	mp2an 698	. . . . . . . . . 10 ⊢ ((0...𝑁) ↑_m (0...𝑀)) ⊆ (ℕ₀ ↑_m (0...𝑀))
16		ovex 7389	. . . . . . . . . . . 12 ⊢ (0...𝑁) ∈ V
17		ovexd 7391	. . . . . . . . . . . 12 ⊢ (𝐶:(0...𝑀)⟶(0...𝑁) → (0...𝑀) ∈ V)
18		elmapg 8776	. . . . . . . . . . . 12 ⊢ (((0...𝑁) ∈ V ∧ (0...𝑀) ∈ V) → (𝐶 ∈ ((0...𝑁) ↑_m (0...𝑀)) ↔ 𝐶:(0...𝑀)⟶(0...𝑁)))
19	16, 17, 18	sylancr 593	. . . . . . . . . . 11 ⊢ (𝐶:(0...𝑀)⟶(0...𝑁) → (𝐶 ∈ ((0...𝑁) ↑_m (0...𝑀)) ↔ 𝐶:(0...𝑀)⟶(0...𝑁)))
20	19	ibir 269	. . . . . . . . . 10 ⊢ (𝐶:(0...𝑀)⟶(0...𝑁) → 𝐶 ∈ ((0...𝑁) ↑_m (0...𝑀)))
21	15, 20	sselid 3913	. . . . . . . . 9 ⊢ (𝐶:(0...𝑀)⟶(0...𝑁) → 𝐶 ∈ (ℕ₀ ↑_m (0...𝑀)))
22	11, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (ℕ₀ ↑_m (0...𝑀)))
23	9, 10, 22	mccl 46043	. . . . . . 7 ⊢ (𝜑 → ((!‘Σ𝑗 ∈ (0...𝑀)(𝐶‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) ∈ ℕ)
24	8, 23	eqeltrd 2839	. . . . . 6 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) ∈ ℕ)
25	24	nnzd 12541	. . . . 5 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) ∈ ℤ)
26		etransclem25.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
27		etransclem25.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (1...𝑀))
28	27	elfzelzd 13470	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℤ)
29	1, 26, 11, 28	etransclem10 46687	. . . . 5 ⊢ (𝜑 → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) ∈ ℤ)
30	25, 29	zmulcld 12630	. . . 4 ⊢ (𝜑 → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0)))))) ∈ ℤ)
31		fzfid 13926	. . . . 5 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
32	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑃 ∈ ℕ)
33	11	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝐶:(0...𝑀)⟶(0...𝑁))
34		0z 12526	. . . . . . . . 9 ⊢ 0 ∈ ℤ
35		fzp1ss 13520	. . . . . . . . 9 ⊢ (0 ∈ ℤ → ((0 + 1)...𝑀) ⊆ (0...𝑀))
36	34, 35	ax-mp 5	. . . . . . . 8 ⊢ ((0 + 1)...𝑀) ⊆ (0...𝑀)
37		id 22	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ (1...𝑀))
38		1e0p1 12677	. . . . . . . . . 10 ⊢ 1 = (0 + 1)
39	38	oveq1i 7366	. . . . . . . . 9 ⊢ (1...𝑀) = ((0 + 1)...𝑀)
40	37, 39	eleqtrdi 2849	. . . . . . . 8 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ((0 + 1)...𝑀))
41	36, 40	sselid 3913	. . . . . . 7 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ (0...𝑀))
42	41	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ (0...𝑀))
43	28	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝐽 ∈ ℤ)
44	32, 33, 42, 43	etransclem3 46680	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ)
45	31, 44	fprodzcl 15910	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ)
46	4, 30, 45	3jca 1134	. . 3 ⊢ (𝜑 → ((!‘𝑃) ∈ ℤ ∧ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0)))))) ∈ ℤ ∧ ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ))
47	28	zcnd 12625	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℂ)
48	47	subidd 11484	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 − 𝐽) = 0)
49	48	eqcomd 2745	. . . . . . . . 9 ⊢ (𝜑 → 0 = (𝐽 − 𝐽))
50	49	oveq1d 7371	. . . . . . . 8 ⊢ (𝜑 → (0↑(𝑃 − (𝐶‘𝐽))) = ((𝐽 − 𝐽)↑(𝑃 − (𝐶‘𝐽))))
51	50	oveq2d 7372	. . . . . . 7 ⊢ (𝜑 → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · ((𝐽 − 𝐽)↑(𝑃 − (𝐶‘𝐽)))))
52	51	ifeq2d 4475	. . . . . 6 ⊢ (𝜑 → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))) = if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · ((𝐽 − 𝐽)↑(𝑃 − (𝐶‘𝐽))))))
53		id 22	. . . . . . . . . 10 ⊢ (𝐽 ∈ (1...𝑀) → 𝐽 ∈ (1...𝑀))
54	53, 39	eleqtrdi 2849	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...𝑀) → 𝐽 ∈ ((0 + 1)...𝑀))
55	36, 54	sselid 3913	. . . . . . . 8 ⊢ (𝐽 ∈ (1...𝑀) → 𝐽 ∈ (0...𝑀))
56	27, 55	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
57	1, 11, 56, 28	etransclem3 46680	. . . . . 6 ⊢ (𝜑 → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · ((𝐽 − 𝐽)↑(𝑃 − (𝐶‘𝐽))))) ∈ ℤ)
58	52, 57	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))) ∈ ℤ)
59		fzfi 13925	. . . . . . 7 ⊢ (1...𝑀) ∈ Fin
60		diffi 9099	. . . . . . 7 ⊢ ((1...𝑀) ∈ Fin → ((1...𝑀) ∖ {𝐽}) ∈ Fin)
61	59, 60	mp1i 13	. . . . . 6 ⊢ (𝜑 → ((1...𝑀) ∖ {𝐽}) ∈ Fin)
62	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ((1...𝑀) ∖ {𝐽})) → 𝑃 ∈ ℕ)
63	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ((1...𝑀) ∖ {𝐽})) → 𝐶:(0...𝑀)⟶(0...𝑁))
64		eldifi 4061	. . . . . . . . 9 ⊢ (𝑗 ∈ ((1...𝑀) ∖ {𝐽}) → 𝑗 ∈ (1...𝑀))
65	64, 41	syl 17	. . . . . . . 8 ⊢ (𝑗 ∈ ((1...𝑀) ∖ {𝐽}) → 𝑗 ∈ (0...𝑀))
66	65	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ((1...𝑀) ∖ {𝐽})) → 𝑗 ∈ (0...𝑀))
67	28	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ((1...𝑀) ∖ {𝐽})) → 𝐽 ∈ ℤ)
68	62, 63, 66, 67	etransclem3 46680	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ((1...𝑀) ∖ {𝐽})) → if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ)
69	61, 68	fprodzcl 15910	. . . . 5 ⊢ (𝜑 → ∏𝑗 ∈ ((1...𝑀) ∖ {𝐽})if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ)
70		dvds0 16231	. . . . . . . . 9 ⊢ ((!‘𝑃) ∈ ℤ → (!‘𝑃) ∥ 0)
71	4, 70	syl 17	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑃) ∥ 0)
72	71	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 < (𝐶‘𝐽)) → (!‘𝑃) ∥ 0)
73		iftrue 4460	. . . . . . . . 9 ⊢ (𝑃 < (𝐶‘𝐽) → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))) = 0)
74	73	eqcomd 2745	. . . . . . . 8 ⊢ (𝑃 < (𝐶‘𝐽) → 0 = if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
75	74	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 < (𝐶‘𝐽)) → 0 = if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
76	72, 75	breqtrd 5098	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 < (𝐶‘𝐽)) → (!‘𝑃) ∥ if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
77		iddvds 16229	. . . . . . . . . 10 ⊢ ((!‘𝑃) ∈ ℤ → (!‘𝑃) ∥ (!‘𝑃))
78	4, 77	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (!‘𝑃) ∥ (!‘𝑃))
79	78	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ 𝑃 = (𝐶‘𝐽)) → (!‘𝑃) ∥ (!‘𝑃))
80		iffalse 4463	. . . . . . . . . 10 ⊢ (¬ 𝑃 < (𝐶‘𝐽) → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))) = (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))))
81	80	ad2antlr 733	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ 𝑃 = (𝐶‘𝐽)) → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))) = (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))))
82		oveq1 7363	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = (𝐶‘𝐽) → (𝑃 − (𝐶‘𝐽)) = ((𝐶‘𝐽) − (𝐶‘𝐽)))
83	82	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (𝑃 − (𝐶‘𝐽)) = ((𝐶‘𝐽) − (𝐶‘𝐽)))
84	11, 56	ffvelcdmd 7026	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐶‘𝐽) ∈ (0...𝑁))
85	84	elfzelzd 13470	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐶‘𝐽) ∈ ℤ)
86	85	zcnd 12625	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐶‘𝐽) ∈ ℂ)
87	86	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (𝐶‘𝐽) ∈ ℂ)
88	87	subidd 11484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → ((𝐶‘𝐽) − (𝐶‘𝐽)) = 0)
89	83, 88	eqtrd 2774	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (𝑃 − (𝐶‘𝐽)) = 0)
90	89	fveq2d 6831	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (!‘(𝑃 − (𝐶‘𝐽))) = (!‘0))
91		fac0 14229	. . . . . . . . . . . . . . 15 ⊢ (!‘0) = 1
92	90, 91	eqtrdi 2790	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (!‘(𝑃 − (𝐶‘𝐽))) = 1)
93	92	oveq2d 7372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) = ((!‘𝑃) / 1))
94	3	nncnd 12181	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (!‘𝑃) ∈ ℂ)
95	94	div1d 11914	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((!‘𝑃) / 1) = (!‘𝑃))
96	95	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → ((!‘𝑃) / 1) = (!‘𝑃))
97	93, 96	eqtrd 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) = (!‘𝑃))
98	89	oveq2d 7372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (0↑(𝑃 − (𝐶‘𝐽))) = (0↑0))
99		0cnd 11128	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → 0 ∈ ℂ)
100	99	exp0d 14093	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (0↑0) = 1)
101	98, 100	eqtrd 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (0↑(𝑃 − (𝐶‘𝐽))) = 1)
102	97, 101	oveq12d 7374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = ((!‘𝑃) · 1))
103	94	mulridd 11153	. . . . . . . . . . . 12 ⊢ (𝜑 → ((!‘𝑃) · 1) = (!‘𝑃))
104	103	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → ((!‘𝑃) · 1) = (!‘𝑃))
105	102, 104	eqtrd 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = (!‘𝑃))
106	105	adantlr 721	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ 𝑃 = (𝐶‘𝐽)) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = (!‘𝑃))
107	81, 106	eqtr2d 2775	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ 𝑃 = (𝐶‘𝐽)) → (!‘𝑃) = if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
108	79, 107	breqtrd 5098	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ 𝑃 = (𝐶‘𝐽)) → (!‘𝑃) ∥ if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
109	71	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → (!‘𝑃) ∥ 0)
110		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → ¬ 𝑃 < (𝐶‘𝐽))
111	110	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → ¬ 𝑃 < (𝐶‘𝐽))
112	111	iffalsed 4465	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))) = (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))))
113		simpll 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → 𝜑)
114	85	zred 12624	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶‘𝐽) ∈ ℝ)
115	114	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → (𝐶‘𝐽) ∈ ℝ)
116	1	nnred 12180	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℝ)
117	116	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → 𝑃 ∈ ℝ)
118	114	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (𝐶‘𝐽) ∈ ℝ)
119	116	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → 𝑃 ∈ ℝ)
120	118, 119, 110	nltled 11287	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (𝐶‘𝐽) ≤ 𝑃)
121	120	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → (𝐶‘𝐽) ≤ 𝑃)
122		neqne 2942	. . . . . . . . . . . 12 ⊢ (¬ 𝑃 = (𝐶‘𝐽) → 𝑃 ≠ (𝐶‘𝐽))
123	122	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → 𝑃 ≠ (𝐶‘𝐽))
124	115, 117, 121, 123	leneltd 11291	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → (𝐶‘𝐽) < 𝑃)
125	1	nnzd 12541	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ ℤ)
126	125	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → 𝑃 ∈ ℤ)
127	85	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝐶‘𝐽) ∈ ℤ)
128	126, 127	zsubcld 12629	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝑃 − (𝐶‘𝐽)) ∈ ℤ)
129		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝐶‘𝐽) < 𝑃)
130	114	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝐶‘𝐽) ∈ ℝ)
131	116	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → 𝑃 ∈ ℝ)
132	130, 131	posdifd 11728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → ((𝐶‘𝐽) < 𝑃 ↔ 0 < (𝑃 − (𝐶‘𝐽))))
133	129, 132	mpbid 233	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → 0 < (𝑃 − (𝐶‘𝐽)))
134		elnnz 12525	. . . . . . . . . . . . . 14 ⊢ ((𝑃 − (𝐶‘𝐽)) ∈ ℕ ↔ ((𝑃 − (𝐶‘𝐽)) ∈ ℤ ∧ 0 < (𝑃 − (𝐶‘𝐽))))
135	128, 133, 134	sylanbrc 589	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝑃 − (𝐶‘𝐽)) ∈ ℕ)
136	135	0expd 14092	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (0↑(𝑃 − (𝐶‘𝐽))) = 0)
137	136	oveq2d 7372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · 0))
138	94	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (!‘𝑃) ∈ ℂ)
139	135	nnnn0d 12489	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝑃 − (𝐶‘𝐽)) ∈ ℕ₀)
140	139	faccld 14237	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (!‘(𝑃 − (𝐶‘𝐽))) ∈ ℕ)
141	140	nncnd 12181	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (!‘(𝑃 − (𝐶‘𝐽))) ∈ ℂ)
142	140	nnne0d 12218	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (!‘(𝑃 − (𝐶‘𝐽))) ≠ 0)
143	138, 141, 142	divcld 11922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) ∈ ℂ)
144	143	mul01d 11336	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · 0) = 0)
145	137, 144	eqtrd 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = 0)
146	113, 124, 145	syl2anc 590	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = 0)
147	112, 146	eqtr2d 2775	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → 0 = if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
148	109, 147	breqtrd 5098	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → (!‘𝑃) ∥ if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
149	108, 148	pm2.61dan 818	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (!‘𝑃) ∥ if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
150	76, 149	pm2.61dan 818	. . . . 5 ⊢ (𝜑 → (!‘𝑃) ∥ if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
151	4, 58, 69, 150	dvdsmultr1d 16257	. . . 4 ⊢ (𝜑 → (!‘𝑃) ∥ (if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))) · ∏𝑗 ∈ ((1...𝑀) ∖ {𝐽})if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))))))
152	44	zcnd 12625	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℂ)
153		fveq2 6827	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝐶‘𝑗) = (𝐶‘𝐽))
154	153	breq2d 5084	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (𝑃 < (𝐶‘𝑗) ↔ 𝑃 < (𝐶‘𝐽)))
155	154	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → (𝑃 < (𝐶‘𝑗) ↔ 𝑃 < (𝐶‘𝐽)))
156	153	oveq2d 7372	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (𝑃 − (𝐶‘𝑗)) = (𝑃 − (𝐶‘𝐽)))
157	156	fveq2d 6831	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (!‘(𝑃 − (𝐶‘𝑗))) = (!‘(𝑃 − (𝐶‘𝐽))))
158	157	oveq2d 7372	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) = ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))))
159	158	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) = ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))))
160		oveq2 7364	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (𝐽 − 𝑗) = (𝐽 − 𝐽))
161	160, 48	sylan9eqr 2796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → (𝐽 − 𝑗) = 0)
162	156	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → (𝑃 − (𝐶‘𝑗)) = (𝑃 − (𝐶‘𝐽)))
163	161, 162	oveq12d 7374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))) = (0↑(𝑃 − (𝐶‘𝐽))))
164	159, 163	oveq12d 7374	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))) = (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))))
165	155, 164	ifbieq2d 4481	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) = if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
166	31, 152, 27, 165	fprodsplit1 46038	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) = (if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))) · ∏𝑗 ∈ ((1...𝑀) ∖ {𝐽})if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))))))
167	151, 166	breqtrrd 5100	. . 3 ⊢ (𝜑 → (!‘𝑃) ∥ ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))))
168		dvdsmultr2 16258	. . 3 ⊢ (((!‘𝑃) ∈ ℤ ∧ (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0)))))) ∈ ℤ ∧ ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ) → ((!‘𝑃) ∥ ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) → (!‘𝑃) ∥ ((((!‘𝑁) / ∏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))))))))
169	46, 167, 168	sylc 65	. 2 ⊢ (𝜑 → (!‘𝑃) ∥ ((((!‘𝑁) / ∏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)))))))
170		etransclem25.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
171	170	faccld 14237	. . . . . 6 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
172	171	nncnd 12181	. . . . 5 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
173	11	ffvelcdmda 7025	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ (0...𝑁))
174	13, 173	sselid 3913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ ℕ₀)
175	174	faccld 14237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℕ)
176	175	nncnd 12181	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℂ)
177	10, 176	fprodcl 15908	. . . . 5 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ∈ ℂ)
178	175	nnne0d 12218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ≠ 0)
179	10, 176, 178	fprodn0 15935	. . . . 5 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ≠ 0)
180	172, 177, 179	divcld 11922	. . . 4 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) ∈ ℂ)
181	29	zcnd 12625	. . . 4 ⊢ (𝜑 → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) ∈ ℂ)
182	31, 152	fprodcl 15908	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℂ)
183	180, 181, 182	mulassd 11159	. . 3 ⊢ (𝜑 → ((((!‘𝑁) / ∏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)))))) = (((!‘𝑁) / ∏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))))))))
184		etransclem25.t	. . 3 ⊢ 𝑇 = (((!‘𝑁) / ∏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)))))))
185	183, 184	eqtr4di 2792	. 2 ⊢ (𝜑 → ((((!‘𝑁) / ∏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)))))) = 𝑇)
186	169, 185	breqtrd 5098	1 ⊢ (𝜑 → (!‘𝑃) ∥ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 Vcvv 3431 ∖ cdif 3880 ⊆ wss 3883 ifcif 4454 {csn 4555 class class class wbr 5072 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ↑_m cmap 8763 Fincfn 8883 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 ...cfz 13452 ↑cexp 14014 !cfa 14226 Σcsu 15639 ∏cprod 15859 ∥ cdvds 16212

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-sum 15640 df-prod 15860 df-dvds 16213

This theorem is referenced by: etransclem28 46705 etransclem38 46715

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