etransclem25 - Mathbox for Glauco Siliprandi

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Theorem etransclem25 42901

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 11943	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
3	2	faccld 13640	. . . . 5 ⊢ (𝜑 → (!‘𝑃) ∈ ℕ)
4	3	nnzd 12074	. . . 4 ⊢ (𝜑 → (!‘𝑃) ∈ ℤ)
5		etransclem25.sumc	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑗 ∈ (0...𝑀)(𝐶‘𝑗) = 𝑁)
6	5	eqcomd 2804	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 = Σ𝑗 ∈ (0...𝑀)(𝐶‘𝑗))
7	6	fveq2d 6649	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) = (!‘Σ𝑗 ∈ (0...𝑀)(𝐶‘𝑗)))
8	7	oveq1d 7150	. . . . . . 7 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) = ((!‘Σ𝑗 ∈ (0...𝑀)(𝐶‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))))
9		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑗𝐶
10		fzfid 13336	. . . . . . . 8 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
11		etransclem25.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁))
12		nn0ex 11891	. . . . . . . . . . 11 ⊢ ℕ₀ ∈ V
13		fzssnn0 41949	. . . . . . . . . . 11 ⊢ (0...𝑁) ⊆ ℕ₀
14		mapss 8436	. . . . . . . . . . 11 ⊢ ((ℕ₀ ∈ V ∧ (0...𝑁) ⊆ ℕ₀) → ((0...𝑁) ↑_m (0...𝑀)) ⊆ (ℕ₀ ↑_m (0...𝑀)))
15	12, 13, 14	mp2an 691	. . . . . . . . . 10 ⊢ ((0...𝑁) ↑_m (0...𝑀)) ⊆ (ℕ₀ ↑_m (0...𝑀))
16		ovex 7168	. . . . . . . . . . . 12 ⊢ (0...𝑁) ∈ V
17		ovexd 7170	. . . . . . . . . . . 12 ⊢ (𝐶:(0...𝑀)⟶(0...𝑁) → (0...𝑀) ∈ V)
18		elmapg 8402	. . . . . . . . . . . 12 ⊢ (((0...𝑁) ∈ V ∧ (0...𝑀) ∈ V) → (𝐶 ∈ ((0...𝑁) ↑_m (0...𝑀)) ↔ 𝐶:(0...𝑀)⟶(0...𝑁)))
19	16, 17, 18	sylancr 590	. . . . . . . . . . 11 ⊢ (𝐶:(0...𝑀)⟶(0...𝑁) → (𝐶 ∈ ((0...𝑁) ↑_m (0...𝑀)) ↔ 𝐶:(0...𝑀)⟶(0...𝑁)))
20	19	ibir 271	. . . . . . . . . 10 ⊢ (𝐶:(0...𝑀)⟶(0...𝑁) → 𝐶 ∈ ((0...𝑁) ↑_m (0...𝑀)))
21	15, 20	sseldi 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 42240	. . . . . . 7 ⊢ (𝜑 → ((!‘Σ𝑗 ∈ (0...𝑀)(𝐶‘𝑗)) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) ∈ ℕ)
24	8, 23	eqeltrd 2890	. . . . . 6 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) ∈ ℕ)
25	24	nnzd 12074	. . . . 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 12903	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℤ)
29	1, 26, 11, 28	etransclem10 42886	. . . . 5 ⊢ (𝜑 → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) ∈ ℤ)
30	25, 29	zmulcld 12081	. . . 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 13336	. . . . 5 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
32	1	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑃 ∈ ℕ)
33	11	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝐶:(0...𝑀)⟶(0...𝑁))
34		0z 11980	. . . . . . . . 9 ⊢ 0 ∈ ℤ
35		fzp1ss 12953	. . . . . . . . 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 12128	. . . . . . . . . 10 ⊢ 1 = (0 + 1)
39	38	oveq1i 7145	. . . . . . . . 9 ⊢ (1...𝑀) = ((0 + 1)...𝑀)
40	37, 39	eleqtrdi 2900	. . . . . . . 8 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ((0 + 1)...𝑀))
41	36, 40	sseldi 3913	. . . . . . 7 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ (0...𝑀))
42	41	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ (0...𝑀))
43	28	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → 𝐽 ∈ ℤ)
44	32, 33, 42, 43	etransclem3 42879	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ)
45	31, 44	fprodzcl 15300	. . . 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 1125	. . 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 12076	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℂ)
48	47	subidd 10974	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 − 𝐽) = 0)
49	48	eqcomd 2804	. . . . . . . . 9 ⊢ (𝜑 → 0 = (𝐽 − 𝐽))
50	49	oveq1d 7150	. . . . . . . 8 ⊢ (𝜑 → (0↑(𝑃 − (𝐶‘𝐽))) = ((𝐽 − 𝐽)↑(𝑃 − (𝐶‘𝐽))))
51	50	oveq2d 7151	. . . . . . 7 ⊢ (𝜑 → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · ((𝐽 − 𝐽)↑(𝑃 − (𝐶‘𝐽)))))
52	51	ifeq2d 4444	. . . . . 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 2900	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...𝑀) → 𝐽 ∈ ((0 + 1)...𝑀))
55	36, 54	sseldi 3913	. . . . . . . 8 ⊢ (𝐽 ∈ (1...𝑀) → 𝐽 ∈ (0...𝑀))
56	27, 55	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
57	1, 11, 56, 28	etransclem3 42879	. . . . . 6 ⊢ (𝜑 → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · ((𝐽 − 𝐽)↑(𝑃 − (𝐶‘𝐽))))) ∈ ℤ)
58	52, 57	eqeltrd 2890	. . . . 5 ⊢ (𝜑 → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))) ∈ ℤ)
59		fzfi 13335	. . . . . . 7 ⊢ (1...𝑀) ∈ Fin
60		diffi 8734	. . . . . . 7 ⊢ ((1...𝑀) ∈ Fin → ((1...𝑀) ∖ {𝐽}) ∈ Fin)
61	59, 60	mp1i 13	. . . . . 6 ⊢ (𝜑 → ((1...𝑀) ∖ {𝐽}) ∈ Fin)
62	1	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ((1...𝑀) ∖ {𝐽})) → 𝑃 ∈ ℕ)
63	11	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ((1...𝑀) ∖ {𝐽})) → 𝐶:(0...𝑀)⟶(0...𝑁))
64		eldifi 4054	. . . . . . . . 9 ⊢ (𝑗 ∈ ((1...𝑀) ∖ {𝐽}) → 𝑗 ∈ (1...𝑀))
65	64, 41	syl 17	. . . . . . . 8 ⊢ (𝑗 ∈ ((1...𝑀) ∖ {𝐽}) → 𝑗 ∈ (0...𝑀))
66	65	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ((1...𝑀) ∖ {𝐽})) → 𝑗 ∈ (0...𝑀))
67	28	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ((1...𝑀) ∖ {𝐽})) → 𝐽 ∈ ℤ)
68	62, 63, 66, 67	etransclem3 42879	. . . . . 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 15300	. . . . 5 ⊢ (𝜑 → ∏𝑗 ∈ ((1...𝑀) ∖ {𝐽})if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ)
70		dvds0 15617	. . . . . . . . 9 ⊢ ((!‘𝑃) ∈ ℤ → (!‘𝑃) ∥ 0)
71	4, 70	syl 17	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑃) ∥ 0)
72	71	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 < (𝐶‘𝐽)) → (!‘𝑃) ∥ 0)
73		iftrue 4431	. . . . . . . . 9 ⊢ (𝑃 < (𝐶‘𝐽) → if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))) = 0)
74	73	eqcomd 2804	. . . . . . . 8 ⊢ (𝑃 < (𝐶‘𝐽) → 0 = if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
75	74	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 < (𝐶‘𝐽)) → 0 = if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
76	72, 75	breqtrd 5056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 < (𝐶‘𝐽)) → (!‘𝑃) ∥ if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
77		iddvds 15615	. . . . . . . . . 10 ⊢ ((!‘𝑃) ∈ ℤ → (!‘𝑃) ∥ (!‘𝑃))
78	4, 77	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (!‘𝑃) ∥ (!‘𝑃))
79	78	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ 𝑃 = (𝐶‘𝐽)) → (!‘𝑃) ∥ (!‘𝑃))
80		iffalse 4434	. . . . . . . . . 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 726	. . . . . . . . 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 7142	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = (𝐶‘𝐽) → (𝑃 − (𝐶‘𝐽)) = ((𝐶‘𝐽) − (𝐶‘𝐽)))
83	82	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (𝑃 − (𝐶‘𝐽)) = ((𝐶‘𝐽) − (𝐶‘𝐽)))
84	11, 56	ffvelrnd 6829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐶‘𝐽) ∈ (0...𝑁))
85	84	elfzelzd 12903	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐶‘𝐽) ∈ ℤ)
86	85	zcnd 12076	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐶‘𝐽) ∈ ℂ)
87	86	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (𝐶‘𝐽) ∈ ℂ)
88	87	subidd 10974	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → ((𝐶‘𝐽) − (𝐶‘𝐽)) = 0)
89	83, 88	eqtrd 2833	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (𝑃 − (𝐶‘𝐽)) = 0)
90	89	fveq2d 6649	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (!‘(𝑃 − (𝐶‘𝐽))) = (!‘0))
91		fac0 13632	. . . . . . . . . . . . . . 15 ⊢ (!‘0) = 1
92	90, 91	eqtrdi 2849	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (!‘(𝑃 − (𝐶‘𝐽))) = 1)
93	92	oveq2d 7151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) = ((!‘𝑃) / 1))
94	3	nncnd 11641	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (!‘𝑃) ∈ ℂ)
95	94	div1d 11397	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((!‘𝑃) / 1) = (!‘𝑃))
96	95	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → ((!‘𝑃) / 1) = (!‘𝑃))
97	93, 96	eqtrd 2833	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) = (!‘𝑃))
98	89	oveq2d 7151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (0↑(𝑃 − (𝐶‘𝐽))) = (0↑0))
99		0cnd 10623	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → 0 ∈ ℂ)
100	99	exp0d 13500	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (0↑0) = 1)
101	98, 100	eqtrd 2833	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (0↑(𝑃 − (𝐶‘𝐽))) = 1)
102	97, 101	oveq12d 7153	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = ((!‘𝑃) · 1))
103	94	mulid1d 10647	. . . . . . . . . . . 12 ⊢ (𝜑 → ((!‘𝑃) · 1) = (!‘𝑃))
104	103	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → ((!‘𝑃) · 1) = (!‘𝑃))
105	102, 104	eqtrd 2833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = (𝐶‘𝐽)) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = (!‘𝑃))
106	105	adantlr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ 𝑃 = (𝐶‘𝐽)) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = (!‘𝑃))
107	81, 106	eqtr2d 2834	. . . . . . . 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 5056	. . . . . . 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 725	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → (!‘𝑃) ∥ 0)
110		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → ¬ 𝑃 < (𝐶‘𝐽))
111	110	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → ¬ 𝑃 < (𝐶‘𝐽))
112	111	iffalsed 4436	. . . . . . . . 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 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → 𝜑)
114	85	zred 12075	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶‘𝐽) ∈ ℝ)
115	114	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → (𝐶‘𝐽) ∈ ℝ)
116	1	nnred 11640	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℝ)
117	116	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → 𝑃 ∈ ℝ)
118	114	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (𝐶‘𝐽) ∈ ℝ)
119	116	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → 𝑃 ∈ ℝ)
120	118, 119, 110	nltled 10779	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (𝐶‘𝐽) ≤ 𝑃)
121	120	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → (𝐶‘𝐽) ≤ 𝑃)
122		neqne 2995	. . . . . . . . . . . 12 ⊢ (¬ 𝑃 = (𝐶‘𝐽) → 𝑃 ≠ (𝐶‘𝐽))
123	122	adantl 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → 𝑃 ≠ (𝐶‘𝐽))
124	115, 117, 121, 123	leneltd 10783	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → (𝐶‘𝐽) < 𝑃)
125	1	nnzd 12074	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ ℤ)
126	125	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → 𝑃 ∈ ℤ)
127	85	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝐶‘𝐽) ∈ ℤ)
128	126, 127	zsubcld 12080	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝑃 − (𝐶‘𝐽)) ∈ ℤ)
129		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝐶‘𝐽) < 𝑃)
130	114	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝐶‘𝐽) ∈ ℝ)
131	116	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → 𝑃 ∈ ℝ)
132	130, 131	posdifd 11216	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → ((𝐶‘𝐽) < 𝑃 ↔ 0 < (𝑃 − (𝐶‘𝐽))))
133	129, 132	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → 0 < (𝑃 − (𝐶‘𝐽)))
134		elnnz 11979	. . . . . . . . . . . . . 14 ⊢ ((𝑃 − (𝐶‘𝐽)) ∈ ℕ ↔ ((𝑃 − (𝐶‘𝐽)) ∈ ℤ ∧ 0 < (𝑃 − (𝐶‘𝐽))))
135	128, 133, 134	sylanbrc 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝑃 − (𝐶‘𝐽)) ∈ ℕ)
136	135	0expd 13499	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (0↑(𝑃 − (𝐶‘𝐽))) = 0)
137	136	oveq2d 7151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · 0))
138	94	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (!‘𝑃) ∈ ℂ)
139	135	nnnn0d 11943	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (𝑃 − (𝐶‘𝐽)) ∈ ℕ₀)
140	139	faccld 13640	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (!‘(𝑃 − (𝐶‘𝐽))) ∈ ℕ)
141	140	nncnd 11641	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (!‘(𝑃 − (𝐶‘𝐽))) ∈ ℂ)
142	140	nnne0d 11675	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (!‘(𝑃 − (𝐶‘𝐽))) ≠ 0)
143	138, 141, 142	divcld 11405	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) ∈ ℂ)
144	143	mul01d 10828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · 0) = 0)
145	137, 144	eqtrd 2833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶‘𝐽) < 𝑃) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = 0)
146	113, 124, 145	syl2anc 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) ∧ ¬ 𝑃 = (𝐶‘𝐽)) → (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽)))) = 0)
147	112, 146	eqtr2d 2834	. . . . . . . 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 5056	. . . . . . 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 812	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑃 < (𝐶‘𝐽)) → (!‘𝑃) ∥ if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
150	76, 149	pm2.61dan 812	. . . . 5 ⊢ (𝜑 → (!‘𝑃) ∥ if(𝑃 < (𝐶‘𝐽), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))) · (0↑(𝑃 − (𝐶‘𝐽))))))
151	4, 58, 69, 150	dvdsmultr1d 15640	. . . 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 12076	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℂ)
153		fveq2 6645	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (𝐶‘𝑗) = (𝐶‘𝐽))
154	153	breq2d 5042	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (𝑃 < (𝐶‘𝑗) ↔ 𝑃 < (𝐶‘𝐽)))
155	154	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → (𝑃 < (𝐶‘𝑗) ↔ 𝑃 < (𝐶‘𝐽)))
156	153	oveq2d 7151	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (𝑃 − (𝐶‘𝑗)) = (𝑃 − (𝐶‘𝐽)))
157	156	fveq2d 6649	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (!‘(𝑃 − (𝐶‘𝑗))) = (!‘(𝑃 − (𝐶‘𝐽))))
158	157	oveq2d 7151	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) = ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))))
159	158	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) = ((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝐽)))))
160		oveq2 7143	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (𝐽 − 𝑗) = (𝐽 − 𝐽))
161	160, 48	sylan9eqr 2855	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → (𝐽 − 𝑗) = 0)
162	156	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → (𝑃 − (𝐶‘𝑗)) = (𝑃 − (𝐶‘𝐽)))
163	161, 162	oveq12d 7153	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))) = (0↑(𝑃 − (𝐶‘𝐽))))
164	159, 163	oveq12d 7153	. . . . . 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 4450	. . . . 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 42235	. . . 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 5058	. . 3 ⊢ (𝜑 → (!‘𝑃) ∥ ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))))
168		dvdsmultr2 15641	. . 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 13640	. . . . . 6 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
172	171	nncnd 11641	. . . . 5 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
173	11	ffvelrnda 6828	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ (0...𝑁))
174	13, 173	sseldi 3913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ ℕ₀)
175	174	faccld 13640	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℕ)
176	175	nncnd 11641	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℂ)
177	10, 176	fprodcl 15298	. . . . 5 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ∈ ℂ)
178	175	nnne0d 11675	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ≠ 0)
179	10, 176, 178	fprodn0 15325	. . . . 5 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ≠ 0)
180	172, 177, 179	divcld 11405	. . . 4 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) ∈ ℂ)
181	29	zcnd 12076	. . . 4 ⊢ (𝜑 → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) ∈ ℂ)
182	31, 152	fprodcl 15298	. . . 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 10653	. . 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 2851	. 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 5056	1 ⊢ (𝜑 → (!‘𝑃) ∥ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ifcif 4425 {csn 4525 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑_m cmap 8389 Fincfn 8492 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 ℕ₀cn0 11885 ℤcz 11969 ...cfz 12885 ↑cexp 13425 !cfa 13629 Σcsu 15034 ∏cprod 15251 ∥ cdvds 15599

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-prod 15252 df-dvds 15600

This theorem is referenced by: etransclem28 42904 etransclem38 42914

W3C validator