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Theorem bcprod 33704

Description: A product identity for binomial coefficents. (Contributed by Scott Fenton, 23-Jun-2020.)

**Assertion**
Ref	Expression
bcprod	⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁)))

Distinct variable group: 𝑘,𝑁

Proof of Theorem bcprod Dummy variables 𝑛 𝑚 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7282	. . . . . . 7 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
2		1m1e0 12045	. . . . . . 7 ⊢ (1 − 1) = 0
3	1, 2	eqtrdi 2794	. . . . . 6 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
4	3	oveq2d 7291	. . . . 5 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = (1...0))
5		fz10 13277	. . . . 5 ⊢ (1...0) = ∅
6	4, 5	eqtrdi 2794	. . . 4 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = ∅)
7	3	oveq1d 7290	. . . . 5 ⊢ (𝑚 = 1 → ((𝑚 − 1)C𝑘) = (0C𝑘))
8	7	adantr 481	. . . 4 ⊢ ((𝑚 = 1 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → ((𝑚 − 1)C𝑘) = (0C𝑘))
9	6, 8	prodeq12dv 15636	. . 3 ⊢ (𝑚 = 1 → ∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ ∅ (0C𝑘))
10		oveq2 7283	. . . . . 6 ⊢ (𝑚 = 1 → ((2 · 𝑘) − 𝑚) = ((2 · 𝑘) − 1))
11	10	oveq2d 7291	. . . . 5 ⊢ (𝑚 = 1 → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 1)))
12	11	adantr 481	. . . 4 ⊢ ((𝑚 = 1 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 1)))
13	6, 12	prodeq12dv 15636	. . 3 ⊢ (𝑚 = 1 → ∏𝑘 ∈ (1...(𝑚 − 1))(𝑘↑((2 · 𝑘) − 𝑚)) = ∏𝑘 ∈ ∅ (𝑘↑((2 · 𝑘) − 1)))
14	9, 13	eqeq12d 2754	. 2 ⊢ (𝑚 = 1 → (∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑚 − 1))(𝑘↑((2 · 𝑘) − 𝑚)) ↔ ∏𝑘 ∈ ∅ (0C𝑘) = ∏𝑘 ∈ ∅ (𝑘↑((2 · 𝑘) − 1))))
15		oveq1 7282	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
16	15	oveq2d 7291	. . . 4 ⊢ (𝑚 = 𝑛 → (1...(𝑚 − 1)) = (1...(𝑛 − 1)))
17	15	oveq1d 7290	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 − 1)C𝑘) = ((𝑛 − 1)C𝑘))
18	17	adantr 481	. . . 4 ⊢ ((𝑚 = 𝑛 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → ((𝑚 − 1)C𝑘) = ((𝑛 − 1)C𝑘))
19	16, 18	prodeq12dv 15636	. . 3 ⊢ (𝑚 = 𝑛 → ∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘))
20		oveq2 7283	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((2 · 𝑘) − 𝑚) = ((2 · 𝑘) − 𝑛))
21	20	oveq2d 7291	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 𝑛)))
22	21	adantr 481	. . . 4 ⊢ ((𝑚 = 𝑛 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 𝑛)))
23	16, 22	prodeq12dv 15636	. . 3 ⊢ (𝑚 = 𝑛 → ∏𝑘 ∈ (1...(𝑚 − 1))(𝑘↑((2 · 𝑘) − 𝑚)) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)))
24	19, 23	eqeq12d 2754	. 2 ⊢ (𝑚 = 𝑛 → (∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑚 − 1))(𝑘↑((2 · 𝑘) − 𝑚)) ↔ ∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛))))
25		oveq1 7282	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
26	25	oveq2d 7291	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → (1...(𝑚 − 1)) = (1...((𝑛 + 1) − 1)))
27	25	oveq1d 7290	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚 − 1)C𝑘) = (((𝑛 + 1) − 1)C𝑘))
28	27	adantr 481	. . . 4 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑘 ∈ (1...(𝑚 − 1))) → ((𝑚 − 1)C𝑘) = (((𝑛 + 1) − 1)C𝑘))
29	26, 28	prodeq12dv 15636	. . 3 ⊢ (𝑚 = (𝑛 + 1) → ∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(((𝑛 + 1) − 1)C𝑘))
30		oveq2 7283	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((2 · 𝑘) − 𝑚) = ((2 · 𝑘) − (𝑛 + 1)))
31	30	oveq2d 7291	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − (𝑛 + 1))))
32	31	adantr 481	. . . 4 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑘 ∈ (1...(𝑚 − 1))) → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − (𝑛 + 1))))
33	26, 32	prodeq12dv 15636	. . 3 ⊢ (𝑚 = (𝑛 + 1) → ∏𝑘 ∈ (1...(𝑚 − 1))(𝑘↑((2 · 𝑘) − 𝑚)) = ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))))
34	29, 33	eqeq12d 2754	. 2 ⊢ (𝑚 = (𝑛 + 1) → (∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑚 − 1))(𝑘↑((2 · 𝑘) − 𝑚)) ↔ ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(((𝑛 + 1) − 1)C𝑘) = ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1)))))
35		oveq1 7282	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑚 − 1) = (𝑁 − 1))
36	35	oveq2d 7291	. . . 4 ⊢ (𝑚 = 𝑁 → (1...(𝑚 − 1)) = (1...(𝑁 − 1)))
37	35	oveq1d 7290	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝑚 − 1)C𝑘) = ((𝑁 − 1)C𝑘))
38	37	adantr 481	. . . 4 ⊢ ((𝑚 = 𝑁 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → ((𝑚 − 1)C𝑘) = ((𝑁 − 1)C𝑘))
39	36, 38	prodeq12dv 15636	. . 3 ⊢ (𝑚 = 𝑁 → ∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘))
40		oveq2 7283	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((2 · 𝑘) − 𝑚) = ((2 · 𝑘) − 𝑁))
41	40	oveq2d 7291	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 𝑁)))
42	41	adantr 481	. . . 4 ⊢ ((𝑚 = 𝑁 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 𝑁)))
43	36, 42	prodeq12dv 15636	. . 3 ⊢ (𝑚 = 𝑁 → ∏𝑘 ∈ (1...(𝑚 − 1))(𝑘↑((2 · 𝑘) − 𝑚)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁)))
44	39, 43	eqeq12d 2754	. 2 ⊢ (𝑚 = 𝑁 → (∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑚 − 1))(𝑘↑((2 · 𝑘) − 𝑚)) ↔ ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁))))
45		prod0 15653	. . 3 ⊢ ∏𝑘 ∈ ∅ (0C𝑘) = 1
46		prod0 15653	. . 3 ⊢ ∏𝑘 ∈ ∅ (𝑘↑((2 · 𝑘) − 1)) = 1
47	45, 46	eqtr4i 2769	. 2 ⊢ ∏𝑘 ∈ ∅ (0C𝑘) = ∏𝑘 ∈ ∅ (𝑘↑((2 · 𝑘) − 1))
48		simpr 485	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛))) → ∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)))
49	48	oveq1d 7290	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ ∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛))) → (∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) · ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1)))) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) · ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1)))))
50		nncn 11981	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
51		1cnd 10970	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℂ)
52	50, 51	pncand 11333	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑛 + 1) − 1) = 𝑛)
53	52	oveq2d 7291	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (1...((𝑛 + 1) − 1)) = (1...𝑛))
54	52	oveq1d 7290	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((𝑛 + 1) − 1)C𝑘) = (𝑛C𝑘))
55	54	adantr 481	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...((𝑛 + 1) − 1))) → (((𝑛 + 1) − 1)C𝑘) = (𝑛C𝑘))
56	53, 55	prodeq12dv 15636	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(((𝑛 + 1) − 1)C𝑘) = ∏𝑘 ∈ (1...𝑛)(𝑛C𝑘))
57		elnnuz 12622	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
58	57	biimpi 215	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘1))
59		nnnn0 12240	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
60		elfzelz 13256	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℤ)
61		bccl 14036	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → (𝑛C𝑘) ∈ ℕ₀)
62	59, 60, 61	syl2an 596	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (𝑛C𝑘) ∈ ℕ₀)
63	62	nn0cnd 12295	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (𝑛C𝑘) ∈ ℂ)
64		oveq2 7283	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑛C𝑘) = (𝑛C𝑛))
65	58, 63, 64	fprodm1 15677	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...𝑛)(𝑛C𝑘) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) · (𝑛C𝑛)))
66		bcnn 14026	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (𝑛C𝑛) = 1)
67	59, 66	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛C𝑛) = 1)
68	67	oveq2d 7291	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) · (𝑛C𝑛)) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) · 1))
69		fzfid 13693	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (1...(𝑛 − 1)) ∈ Fin)
70		elfzelz 13256	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑛 − 1)) → 𝑘 ∈ ℤ)
71	59, 70, 61	syl2an 596	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛C𝑘) ∈ ℕ₀)
72	71	nn0cnd 12295	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛C𝑘) ∈ ℂ)
73	69, 72	fprodcl 15662	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) ∈ ℂ)
74	73	mulid1d 10992	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) · 1) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘))
75		fz1ssfz0 13352	. . . . . . . . . . 11 ⊢ (1...(𝑛 − 1)) ⊆ (0...(𝑛 − 1))
76	75	sseli 3917	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑛 − 1)) → 𝑘 ∈ (0...(𝑛 − 1)))
77		bcm1nt 33703	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑛 − 1))) → (𝑛C𝑘) = (((𝑛 − 1)C𝑘) · (𝑛 / (𝑛 − 𝑘))))
78	76, 77	sylan2 593	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛C𝑘) = (((𝑛 − 1)C𝑘) · (𝑛 / (𝑛 − 𝑘))))
79	78	prodeq2dv 15633	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(((𝑛 − 1)C𝑘) · (𝑛 / (𝑛 − 𝑘))))
80		nnm1nn0 12274	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
81		bccl 14036	. . . . . . . . . . 11 ⊢ (((𝑛 − 1) ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → ((𝑛 − 1)C𝑘) ∈ ℕ₀)
82	80, 70, 81	syl2an 596	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → ((𝑛 − 1)C𝑘) ∈ ℕ₀)
83	82	nn0cnd 12295	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → ((𝑛 − 1)C𝑘) ∈ ℂ)
84	50	adantr 481	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑛 ∈ ℂ)
85		elfznn 13285	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...(𝑛 − 1)) → 𝑘 ∈ ℕ)
86	85	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑘 ∈ ℕ)
87	86	nnred 11988	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑘 ∈ ℝ)
88	80	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 1) ∈ ℕ₀)
89	88	nn0red 12294	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 1) ∈ ℝ)
90		nnre 11980	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
91	90	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑛 ∈ ℝ)
92		elfzle2 13260	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...(𝑛 − 1)) → 𝑘 ≤ (𝑛 − 1))
93	92	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑘 ≤ (𝑛 − 1))
94	91	ltm1d 11907	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 1) < 𝑛)
95	87, 89, 91, 93, 94	lelttrd 11133	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑘 < 𝑛)
96		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑛 ∈ ℕ)
97		nnsub 12017	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑘 < 𝑛 ↔ (𝑛 − 𝑘) ∈ ℕ))
98	86, 96, 97	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑘 < 𝑛 ↔ (𝑛 − 𝑘) ∈ ℕ))
99	95, 98	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 𝑘) ∈ ℕ)
100	99	nncnd 11989	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 𝑘) ∈ ℂ)
101	99	nnne0d 12023	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 𝑘) ≠ 0)
102	84, 100, 101	divcld 11751	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 / (𝑛 − 𝑘)) ∈ ℂ)
103	69, 83, 102	fprodmul 15670	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(((𝑛 − 1)C𝑘) · (𝑛 / (𝑛 − 𝑘))) = (∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) · ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 / (𝑛 − 𝑘))))
104	69, 84, 100, 101	fproddiv 15671	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 / (𝑛 − 𝑘)) = (∏𝑘 ∈ (1...(𝑛 − 1))𝑛 / ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 − 𝑘)))
105		fzfi 13692	. . . . . . . . . . . . 13 ⊢ (1...(𝑛 − 1)) ∈ Fin
106		fprodconst 15688	. . . . . . . . . . . . 13 ⊢ (((1...(𝑛 − 1)) ∈ Fin ∧ 𝑛 ∈ ℂ) → ∏𝑘 ∈ (1...(𝑛 − 1))𝑛 = (𝑛↑(♯‘(1...(𝑛 − 1)))))
107	105, 50, 106	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))𝑛 = (𝑛↑(♯‘(1...(𝑛 − 1)))))
108		hashfz1 14060	. . . . . . . . . . . . . 14 ⊢ ((𝑛 − 1) ∈ ℕ₀ → (♯‘(1...(𝑛 − 1))) = (𝑛 − 1))
109	80, 108	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (♯‘(1...(𝑛 − 1))) = (𝑛 − 1))
110	109	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛↑(♯‘(1...(𝑛 − 1)))) = (𝑛↑(𝑛 − 1)))
111	107, 110	eqtr2d 2779	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛↑(𝑛 − 1)) = ∏𝑘 ∈ (1...(𝑛 − 1))𝑛)
112		fprodfac 15683	. . . . . . . . . . . . 13 ⊢ ((𝑛 − 1) ∈ ℕ₀ → (!‘(𝑛 − 1)) = ∏𝑗 ∈ (1...(𝑛 − 1))𝑗)
113	80, 112	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (!‘(𝑛 − 1)) = ∏𝑗 ∈ (1...(𝑛 − 1))𝑗)
114		nnz 12342	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
115		1zzd 12351	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℤ)
116	80	nn0zd 12424	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
117		elfznn 13285	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...(𝑛 − 1)) → 𝑗 ∈ ℕ)
118	117	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑛 − 1))) → 𝑗 ∈ ℕ)
119	118	nncnd 11989	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑛 − 1))) → 𝑗 ∈ ℂ)
120		id 22	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑛 − 𝑘) → 𝑗 = (𝑛 − 𝑘))
121	114, 115, 116, 119, 120	fprodrev 15687	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → ∏𝑗 ∈ (1...(𝑛 − 1))𝑗 = ∏𝑘 ∈ ((𝑛 − (𝑛 − 1))...(𝑛 − 1))(𝑛 − 𝑘))
122	50, 51	nncand 11337	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 − (𝑛 − 1)) = 1)
123	122	oveq1d 7290	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((𝑛 − (𝑛 − 1))...(𝑛 − 1)) = (1...(𝑛 − 1)))
124	123	prodeq1d 15631	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ ((𝑛 − (𝑛 − 1))...(𝑛 − 1))(𝑛 − 𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 − 𝑘))
125	113, 121, 124	3eqtrd 2782	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (!‘(𝑛 − 1)) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 − 𝑘))
126	111, 125	oveq12d 7293	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1))) = (∏𝑘 ∈ (1...(𝑛 − 1))𝑛 / ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 − 𝑘)))
127	104, 126	eqtr4d 2781	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 / (𝑛 − 𝑘)) = ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1))))
128	127	oveq2d 7291	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) · ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 / (𝑛 − 𝑘))) = (∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) · ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1)))))
129	79, 103, 128	3eqtrd 2782	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) = (∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) · ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1)))))
130	68, 74, 129	3eqtrd 2782	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) · (𝑛C𝑛)) = (∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) · ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1)))))
131	56, 65, 130	3eqtrd 2782	. . . . 5 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(((𝑛 + 1) − 1)C𝑘) = (∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) · ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1)))))
132	131	adantr 481	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ ∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛))) → ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(((𝑛 + 1) − 1)C𝑘) = (∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) · ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1)))))
133	53	prodeq1d 15631	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))) = ∏𝑘 ∈ (1...𝑛)(𝑘↑((2 · 𝑘) − (𝑛 + 1))))
134		elfznn 13285	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
135	134	adantl 482	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
136	135	nncnd 11989	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℂ)
137	135	nnne0d 12023	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ≠ 0)
138		2nn 12046	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
139	138	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → 2 ∈ ℕ)
140	139, 135	nnmulcld 12026	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (2 · 𝑘) ∈ ℕ)
141	140	nnzd 12425	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (2 · 𝑘) ∈ ℤ)
142		peano2nn 11985	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
143	142	adantr 481	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (𝑛 + 1) ∈ ℕ)
144	143	nnzd 12425	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (𝑛 + 1) ∈ ℤ)
145	141, 144	zsubcld 12431	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → ((2 · 𝑘) − (𝑛 + 1)) ∈ ℤ)
146	136, 137, 145	expclzd 13869	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (𝑘↑((2 · 𝑘) − (𝑛 + 1))) ∈ ℂ)
147		id 22	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
148		oveq2 7283	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (2 · 𝑘) = (2 · 𝑛))
149	148	oveq1d 7290	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((2 · 𝑘) − (𝑛 + 1)) = ((2 · 𝑛) − (𝑛 + 1)))
150	147, 149	oveq12d 7293	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑘↑((2 · 𝑘) − (𝑛 + 1))) = (𝑛↑((2 · 𝑛) − (𝑛 + 1))))
151	58, 146, 150	fprodm1 15677	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...𝑛)(𝑘↑((2 · 𝑘) − (𝑛 + 1))) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))) · (𝑛↑((2 · 𝑛) − (𝑛 + 1)))))
152	86	nncnd 11989	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑘 ∈ ℂ)
153	86	nnne0d 12023	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑘 ≠ 0)
154	138	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 2 ∈ ℕ)
155	154, 86	nnmulcld 12026	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (2 · 𝑘) ∈ ℕ)
156	155	nnzd 12425	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (2 · 𝑘) ∈ ℤ)
157	114	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑛 ∈ ℤ)
158	156, 157	zsubcld 12431	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → ((2 · 𝑘) − 𝑛) ∈ ℤ)
159	152, 153, 158	expclzd 13869	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑘↑((2 · 𝑘) − 𝑛)) ∈ ℂ)
160	69, 159, 152, 153	fproddiv 15671	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))((𝑘↑((2 · 𝑘) − 𝑛)) / 𝑘) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) / ∏𝑘 ∈ (1...(𝑛 − 1))𝑘))
161	155	nncnd 11989	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (2 · 𝑘) ∈ ℂ)
162		1cnd 10970	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 1 ∈ ℂ)
163	161, 84, 162	subsub4d 11363	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (((2 · 𝑘) − 𝑛) − 1) = ((2 · 𝑘) − (𝑛 + 1)))
164	163	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑘↑(((2 · 𝑘) − 𝑛) − 1)) = (𝑘↑((2 · 𝑘) − (𝑛 + 1))))
165	152, 153, 158	expm1d 13874	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑘↑(((2 · 𝑘) − 𝑛) − 1)) = ((𝑘↑((2 · 𝑘) − 𝑛)) / 𝑘))
166	164, 165	eqtr3d 2780	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑘↑((2 · 𝑘) − (𝑛 + 1))) = ((𝑘↑((2 · 𝑘) − 𝑛)) / 𝑘))
167	166	prodeq2dv 15633	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))) = ∏𝑘 ∈ (1...(𝑛 − 1))((𝑘↑((2 · 𝑘) − 𝑛)) / 𝑘))
168		fprodfac 15683	. . . . . . . . . . 11 ⊢ ((𝑛 − 1) ∈ ℕ₀ → (!‘(𝑛 − 1)) = ∏𝑘 ∈ (1...(𝑛 − 1))𝑘)
169	80, 168	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (!‘(𝑛 − 1)) = ∏𝑘 ∈ (1...(𝑛 − 1))𝑘)
170	169	oveq2d 7291	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) / (!‘(𝑛 − 1))) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) / ∏𝑘 ∈ (1...(𝑛 − 1))𝑘))
171	160, 167, 170	3eqtr4d 2788	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) / (!‘(𝑛 − 1))))
172	138	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 2 ∈ ℕ)
173		id 22	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
174	172, 173	nnmulcld 12026	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℕ)
175	174	nncnd 11989	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℂ)
176	175, 50, 51	subsub4d 11363	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (((2 · 𝑛) − 𝑛) − 1) = ((2 · 𝑛) − (𝑛 + 1)))
177	50	2timesd 12216	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) = (𝑛 + 𝑛))
178	50, 50, 177	mvrladdd 11388	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2 · 𝑛) − 𝑛) = 𝑛)
179	178	oveq1d 7290	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (((2 · 𝑛) − 𝑛) − 1) = (𝑛 − 1))
180	176, 179	eqtr3d 2780	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((2 · 𝑛) − (𝑛 + 1)) = (𝑛 − 1))
181	180	oveq2d 7291	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛↑((2 · 𝑛) − (𝑛 + 1))) = (𝑛↑(𝑛 − 1)))
182	171, 181	oveq12d 7293	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))) · (𝑛↑((2 · 𝑛) − (𝑛 + 1)))) = ((∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) / (!‘(𝑛 − 1))) · (𝑛↑(𝑛 − 1))))
183	69, 159	fprodcl 15662	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) ∈ ℂ)
184		faccl 13997	. . . . . . . . . 10 ⊢ ((𝑛 − 1) ∈ ℕ₀ → (!‘(𝑛 − 1)) ∈ ℕ)
185	80, 184	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (!‘(𝑛 − 1)) ∈ ℕ)
186	185	nncnd 11989	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (!‘(𝑛 − 1)) ∈ ℂ)
187	50, 80	expcld 13864	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛↑(𝑛 − 1)) ∈ ℂ)
188	185	nnne0d 12023	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (!‘(𝑛 − 1)) ≠ 0)
189	183, 186, 187, 188	div32d 11774	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) / (!‘(𝑛 − 1))) · (𝑛↑(𝑛 − 1))) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) · ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1)))))
190	182, 189	eqtrd 2778	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))) · (𝑛↑((2 · 𝑛) − (𝑛 + 1)))) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) · ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1)))))
191	133, 151, 190	3eqtrd 2782	. . . . 5 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) · ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1)))))
192	191	adantr 481	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ ∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛))) → ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) · ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1)))))
193	49, 132, 192	3eqtr4d 2788	. . 3 ⊢ ((𝑛 ∈ ℕ ∧ ∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛))) → ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(((𝑛 + 1) − 1)C𝑘) = ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))))
194	193	ex 413	. 2 ⊢ (𝑛 ∈ ℕ → (∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) → ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(((𝑛 + 1) − 1)C𝑘) = ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1)))))
195	14, 24, 34, 44, 47, 194	nnind 11991	1 ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∅c0 4256 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 ↑cexp 13782 !cfa 13987 Ccbc 14016 ♯chash 14044 ∏cprod 15615

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-prod 15616

This theorem is referenced by: (None)

W3C validator