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Theorem bcprod 35934

Description: A product identity for binomial coefficients. (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 7367	. . . . . . 7 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
2		1m1e0 12221	. . . . . . 7 ⊢ (1 − 1) = 0
3	1, 2	eqtrdi 2788	. . . . . 6 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
4	3	oveq2d 7376	. . . . 5 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = (1...0))
5		fz10 13465	. . . . 5 ⊢ (1...0) = ∅
6	4, 5	eqtrdi 2788	. . . 4 ⊢ (𝑚 = 1 → (1...(𝑚 − 1)) = ∅)
7	3	oveq1d 7375	. . . . 5 ⊢ (𝑚 = 1 → ((𝑚 − 1)C𝑘) = (0C𝑘))
8	7	adantr 480	. . . 4 ⊢ ((𝑚 = 1 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → ((𝑚 − 1)C𝑘) = (0C𝑘))
9	6, 8	prodeq12dv 15853	. . 3 ⊢ (𝑚 = 1 → ∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ ∅ (0C𝑘))
10		oveq2 7368	. . . . . 6 ⊢ (𝑚 = 1 → ((2 · 𝑘) − 𝑚) = ((2 · 𝑘) − 1))
11	10	oveq2d 7376	. . . . 5 ⊢ (𝑚 = 1 → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 1)))
12	11	adantr 480	. . . 4 ⊢ ((𝑚 = 1 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 1)))
13	6, 12	prodeq12dv 15853	. . 3 ⊢ (𝑚 = 1 → ∏𝑘 ∈ (1...(𝑚 − 1))(𝑘↑((2 · 𝑘) − 𝑚)) = ∏𝑘 ∈ ∅ (𝑘↑((2 · 𝑘) − 1)))
14	9, 13	eqeq12d 2753	. 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 7367	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
16	15	oveq2d 7376	. . . 4 ⊢ (𝑚 = 𝑛 → (1...(𝑚 − 1)) = (1...(𝑛 − 1)))
17	15	oveq1d 7375	. . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑚 − 1)C𝑘) = ((𝑛 − 1)C𝑘))
18	17	adantr 480	. . . 4 ⊢ ((𝑚 = 𝑛 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → ((𝑚 − 1)C𝑘) = ((𝑛 − 1)C𝑘))
19	16, 18	prodeq12dv 15853	. . 3 ⊢ (𝑚 = 𝑛 → ∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))((𝑛 − 1)C𝑘))
20		oveq2 7368	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((2 · 𝑘) − 𝑚) = ((2 · 𝑘) − 𝑛))
21	20	oveq2d 7376	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 𝑛)))
22	21	adantr 480	. . . 4 ⊢ ((𝑚 = 𝑛 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 𝑛)))
23	16, 22	prodeq12dv 15853	. . 3 ⊢ (𝑚 = 𝑛 → ∏𝑘 ∈ (1...(𝑚 − 1))(𝑘↑((2 · 𝑘) − 𝑚)) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)))
24	19, 23	eqeq12d 2753	. 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 7367	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
26	25	oveq2d 7376	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → (1...(𝑚 − 1)) = (1...((𝑛 + 1) − 1)))
27	25	oveq1d 7375	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → ((𝑚 − 1)C𝑘) = (((𝑛 + 1) − 1)C𝑘))
28	27	adantr 480	. . . 4 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑘 ∈ (1...(𝑚 − 1))) → ((𝑚 − 1)C𝑘) = (((𝑛 + 1) − 1)C𝑘))
29	26, 28	prodeq12dv 15853	. . 3 ⊢ (𝑚 = (𝑛 + 1) → ∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(((𝑛 + 1) − 1)C𝑘))
30		oveq2 7368	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((2 · 𝑘) − 𝑚) = ((2 · 𝑘) − (𝑛 + 1)))
31	30	oveq2d 7376	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − (𝑛 + 1))))
32	31	adantr 480	. . . 4 ⊢ ((𝑚 = (𝑛 + 1) ∧ 𝑘 ∈ (1...(𝑚 − 1))) → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − (𝑛 + 1))))
33	26, 32	prodeq12dv 15853	. . 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 2753	. 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 7367	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑚 − 1) = (𝑁 − 1))
36	35	oveq2d 7376	. . . 4 ⊢ (𝑚 = 𝑁 → (1...(𝑚 − 1)) = (1...(𝑁 − 1)))
37	35	oveq1d 7375	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝑚 − 1)C𝑘) = ((𝑁 − 1)C𝑘))
38	37	adantr 480	. . . 4 ⊢ ((𝑚 = 𝑁 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → ((𝑚 − 1)C𝑘) = ((𝑁 − 1)C𝑘))
39	36, 38	prodeq12dv 15853	. . 3 ⊢ (𝑚 = 𝑁 → ∏𝑘 ∈ (1...(𝑚 − 1))((𝑚 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘))
40		oveq2 7368	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((2 · 𝑘) − 𝑚) = ((2 · 𝑘) − 𝑁))
41	40	oveq2d 7376	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 𝑁)))
42	41	adantr 480	. . . 4 ⊢ ((𝑚 = 𝑁 ∧ 𝑘 ∈ (1...(𝑚 − 1))) → (𝑘↑((2 · 𝑘) − 𝑚)) = (𝑘↑((2 · 𝑘) − 𝑁)))
43	36, 42	prodeq12dv 15853	. . 3 ⊢ (𝑚 = 𝑁 → ∏𝑘 ∈ (1...(𝑚 − 1))(𝑘↑((2 · 𝑘) − 𝑚)) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁)))
44	39, 43	eqeq12d 2753	. 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 15870	. . 3 ⊢ ∏𝑘 ∈ ∅ (0C𝑘) = 1
46		prod0 15870	. . 3 ⊢ ∏𝑘 ∈ ∅ (𝑘↑((2 · 𝑘) − 1)) = 1
47	45, 46	eqtr4i 2763	. 2 ⊢ ∏𝑘 ∈ ∅ (0C𝑘) = ∏𝑘 ∈ ∅ (𝑘↑((2 · 𝑘) − 1))
48		simpr 484	. . . . 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 7375	. . . 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 12157	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
51		1cnd 11131	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℂ)
52	50, 51	pncand 11497	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑛 + 1) − 1) = 𝑛)
53	52	oveq2d 7376	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (1...((𝑛 + 1) − 1)) = (1...𝑛))
54	52	oveq1d 7375	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (((𝑛 + 1) − 1)C𝑘) = (𝑛C𝑘))
55	54	adantr 480	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...((𝑛 + 1) − 1))) → (((𝑛 + 1) − 1)C𝑘) = (𝑛C𝑘))
56	53, 55	prodeq12dv 15853	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(((𝑛 + 1) − 1)C𝑘) = ∏𝑘 ∈ (1...𝑛)(𝑛C𝑘))
57		elnnuz 12795	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
58	57	biimpi 216	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ_≥‘1))
59		nnnn0 12412	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
60		elfzelz 13444	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℤ)
61		bccl 14249	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → (𝑛C𝑘) ∈ ℕ₀)
62	59, 60, 61	syl2an 597	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (𝑛C𝑘) ∈ ℕ₀)
63	62	nn0cnd 12468	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (𝑛C𝑘) ∈ ℂ)
64		oveq2 7368	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑛C𝑘) = (𝑛C𝑛))
65	58, 63, 64	fprodm1 15894	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...𝑛)(𝑛C𝑘) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) · (𝑛C𝑛)))
66		bcnn 14239	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (𝑛C𝑛) = 1)
67	59, 66	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛C𝑛) = 1)
68	67	oveq2d 7376	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) · (𝑛C𝑛)) = (∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) · 1))
69		fzfid 13900	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (1...(𝑛 − 1)) ∈ Fin)
70		elfzelz 13444	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑛 − 1)) → 𝑘 ∈ ℤ)
71	59, 70, 61	syl2an 597	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛C𝑘) ∈ ℕ₀)
72	71	nn0cnd 12468	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛C𝑘) ∈ ℂ)
73	69, 72	fprodcl 15879	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) ∈ ℂ)
74	73	mulridd 11153	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) · 1) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘))
75		fz1ssfz0 13543	. . . . . . . . . . 11 ⊢ (1...(𝑛 − 1)) ⊆ (0...(𝑛 − 1))
76	75	sseli 3930	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑛 − 1)) → 𝑘 ∈ (0...(𝑛 − 1)))
77		bcm1nt 35933	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑛 − 1))) → (𝑛C𝑘) = (((𝑛 − 1)C𝑘) · (𝑛 / (𝑛 − 𝑘))))
78	76, 77	sylan2 594	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛C𝑘) = (((𝑛 − 1)C𝑘) · (𝑛 / (𝑛 − 𝑘))))
79	78	prodeq2dv 15849	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛C𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(((𝑛 − 1)C𝑘) · (𝑛 / (𝑛 − 𝑘))))
80		nnm1nn0 12446	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
81		bccl 14249	. . . . . . . . . . 11 ⊢ (((𝑛 − 1) ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → ((𝑛 − 1)C𝑘) ∈ ℕ₀)
82	80, 70, 81	syl2an 597	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → ((𝑛 − 1)C𝑘) ∈ ℕ₀)
83	82	nn0cnd 12468	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → ((𝑛 − 1)C𝑘) ∈ ℂ)
84	50	adantr 480	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑛 ∈ ℂ)
85		elfznn 13473	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...(𝑛 − 1)) → 𝑘 ∈ ℕ)
86	85	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑘 ∈ ℕ)
87	86	nnred 12164	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑘 ∈ ℝ)
88	80	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 1) ∈ ℕ₀)
89	88	nn0red 12467	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 1) ∈ ℝ)
90		nnre 12156	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
91	90	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑛 ∈ ℝ)
92		elfzle2 13448	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...(𝑛 − 1)) → 𝑘 ≤ (𝑛 − 1))
93	92	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑘 ≤ (𝑛 − 1))
94	91	ltm1d 12078	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 1) < 𝑛)
95	87, 89, 91, 93, 94	lelttrd 11295	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑘 < 𝑛)
96		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑛 ∈ ℕ)
97		nnsub 12193	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑘 < 𝑛 ↔ (𝑛 − 𝑘) ∈ ℕ))
98	86, 96, 97	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑘 < 𝑛 ↔ (𝑛 − 𝑘) ∈ ℕ))
99	95, 98	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 𝑘) ∈ ℕ)
100	99	nncnd 12165	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 𝑘) ∈ ℂ)
101	99	nnne0d 12199	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 − 𝑘) ≠ 0)
102	84, 100, 101	divcld 11921	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑛 / (𝑛 − 𝑘)) ∈ ℂ)
103	69, 83, 102	fprodmul 15887	. . . . . . . 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 15888	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 / (𝑛 − 𝑘)) = (∏𝑘 ∈ (1...(𝑛 − 1))𝑛 / ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 − 𝑘)))
105		fzfi 13899	. . . . . . . . . . . . 13 ⊢ (1...(𝑛 − 1)) ∈ Fin
106		fprodconst 15905	. . . . . . . . . . . . 13 ⊢ (((1...(𝑛 − 1)) ∈ Fin ∧ 𝑛 ∈ ℂ) → ∏𝑘 ∈ (1...(𝑛 − 1))𝑛 = (𝑛↑(♯‘(1...(𝑛 − 1)))))
107	105, 50, 106	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))𝑛 = (𝑛↑(♯‘(1...(𝑛 − 1)))))
108		hashfz1 14273	. . . . . . . . . . . . . 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 7376	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛↑(♯‘(1...(𝑛 − 1)))) = (𝑛↑(𝑛 − 1)))
111	107, 110	eqtr2d 2773	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛↑(𝑛 − 1)) = ∏𝑘 ∈ (1...(𝑛 − 1))𝑛)
112		fprodfac 15900	. . . . . . . . . . . . 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 12513	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
115		1zzd 12526	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℤ)
116	80	nn0zd 12517	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℤ)
117		elfznn 13473	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...(𝑛 − 1)) → 𝑗 ∈ ℕ)
118	117	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑛 − 1))) → 𝑗 ∈ ℕ)
119	118	nncnd 12165	. . . . . . . . . . . . 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 15904	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → ∏𝑗 ∈ (1...(𝑛 − 1))𝑗 = ∏𝑘 ∈ ((𝑛 − (𝑛 − 1))...(𝑛 − 1))(𝑛 − 𝑘))
122	50, 51	nncand 11501	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 − (𝑛 − 1)) = 1)
123	122	oveq1d 7375	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ((𝑛 − (𝑛 − 1))...(𝑛 − 1)) = (1...(𝑛 − 1)))
124	123	prodeq1d 15847	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ ((𝑛 − (𝑛 − 1))...(𝑛 − 1))(𝑛 − 𝑘) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 − 𝑘))
125	113, 121, 124	3eqtrd 2776	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (!‘(𝑛 − 1)) = ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 − 𝑘))
126	111, 125	oveq12d 7378	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1))) = (∏𝑘 ∈ (1...(𝑛 − 1))𝑛 / ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 − 𝑘)))
127	104, 126	eqtr4d 2775	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑛 / (𝑛 − 𝑘)) = ((𝑛↑(𝑛 − 1)) / (!‘(𝑛 − 1))))
128	127	oveq2d 7376	. . . . . . . 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 2776	. . . . . . 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 2776	. . . . . 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 2776	. . . . 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 480	. . . 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 15847	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...((𝑛 + 1) − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))) = ∏𝑘 ∈ (1...𝑛)(𝑘↑((2 · 𝑘) − (𝑛 + 1))))
134		elfznn 13473	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
135	134	adantl 481	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
136	135	nncnd 12165	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℂ)
137	135	nnne0d 12199	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ≠ 0)
138		2nn 12222	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
139	138	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → 2 ∈ ℕ)
140	139, 135	nnmulcld 12202	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (2 · 𝑘) ∈ ℕ)
141	140	nnzd 12518	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (2 · 𝑘) ∈ ℤ)
142		peano2nn 12161	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
143	142	adantr 480	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (𝑛 + 1) ∈ ℕ)
144	143	nnzd 12518	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (𝑛 + 1) ∈ ℤ)
145	141, 144	zsubcld 12605	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → ((2 · 𝑘) − (𝑛 + 1)) ∈ ℤ)
146	136, 137, 145	expclzd 14078	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...𝑛)) → (𝑘↑((2 · 𝑘) − (𝑛 + 1))) ∈ ℂ)
147		id 22	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
148		oveq2 7368	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (2 · 𝑘) = (2 · 𝑛))
149	148	oveq1d 7375	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((2 · 𝑘) − (𝑛 + 1)) = ((2 · 𝑛) − (𝑛 + 1)))
150	147, 149	oveq12d 7378	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑘↑((2 · 𝑘) − (𝑛 + 1))) = (𝑛↑((2 · 𝑛) − (𝑛 + 1))))
151	58, 146, 150	fprodm1 15894	. . . . . 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 12165	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑘 ∈ ℂ)
153	86	nnne0d 12199	. . . . . . . . . . 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 12202	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (2 · 𝑘) ∈ ℕ)
156	155	nnzd 12518	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (2 · 𝑘) ∈ ℤ)
157	114	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 𝑛 ∈ ℤ)
158	156, 157	zsubcld 12605	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → ((2 · 𝑘) − 𝑛) ∈ ℤ)
159	152, 153, 158	expclzd 14078	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑘↑((2 · 𝑘) − 𝑛)) ∈ ℂ)
160	69, 159, 152, 153	fproddiv 15888	. . . . . . . . 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 12165	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (2 · 𝑘) ∈ ℂ)
162		1cnd 11131	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → 1 ∈ ℂ)
163	161, 84, 162	subsub4d 11527	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (((2 · 𝑘) − 𝑛) − 1) = ((2 · 𝑘) − (𝑛 + 1)))
164	163	oveq2d 7376	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑘↑(((2 · 𝑘) − 𝑛) − 1)) = (𝑘↑((2 · 𝑘) − (𝑛 + 1))))
165	152, 153, 158	expm1d 14083	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑘↑(((2 · 𝑘) − 𝑛) − 1)) = ((𝑘↑((2 · 𝑘) − 𝑛)) / 𝑘))
166	164, 165	eqtr3d 2774	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑛 − 1))) → (𝑘↑((2 · 𝑘) − (𝑛 + 1))) = ((𝑘↑((2 · 𝑘) − 𝑛)) / 𝑘))
167	166	prodeq2dv 15849	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − (𝑛 + 1))) = ∏𝑘 ∈ (1...(𝑛 − 1))((𝑘↑((2 · 𝑘) − 𝑛)) / 𝑘))
168		fprodfac 15900	. . . . . . . . . . 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 7376	. . . . . . . . 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 2782	. . . . . . . 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 12202	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℕ)
175	174	nncnd 12165	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) ∈ ℂ)
176	175, 50, 51	subsub4d 11527	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (((2 · 𝑛) − 𝑛) − 1) = ((2 · 𝑛) − (𝑛 + 1)))
177	50	2timesd 12388	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (2 · 𝑛) = (𝑛 + 𝑛))
178	50, 50, 177	mvrladdd 11554	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((2 · 𝑛) − 𝑛) = 𝑛)
179	178	oveq1d 7375	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (((2 · 𝑛) − 𝑛) − 1) = (𝑛 − 1))
180	176, 179	eqtr3d 2774	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((2 · 𝑛) − (𝑛 + 1)) = (𝑛 − 1))
181	180	oveq2d 7376	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛↑((2 · 𝑛) − (𝑛 + 1))) = (𝑛↑(𝑛 − 1)))
182	171, 181	oveq12d 7378	. . . . . . 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 15879	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ∏𝑘 ∈ (1...(𝑛 − 1))(𝑘↑((2 · 𝑘) − 𝑛)) ∈ ℂ)
184		faccl 14210	. . . . . . . . . 10 ⊢ ((𝑛 − 1) ∈ ℕ₀ → (!‘(𝑛 − 1)) ∈ ℕ)
185	80, 184	syl 17	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (!‘(𝑛 − 1)) ∈ ℕ)
186	185	nncnd 12165	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (!‘(𝑛 − 1)) ∈ ℂ)
187	50, 80	expcld 14073	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛↑(𝑛 − 1)) ∈ ℂ)
188	185	nnne0d 12199	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (!‘(𝑛 − 1)) ≠ 0)
189	183, 186, 187, 188	div32d 11944	. . . . . . 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 2772	. . . . . 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 2776	. . . . 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 480	. . . 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 2782	. . 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 412	. 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 12167	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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∅c0 4286 class class class wbr 5099 ‘cfv 6493 (class class class)co 7360 Fincfn 8887 ℂcc 11028 ℝcr 11029 0cc0 11030 1c1 11031 + caddc 11033 · cmul 11035 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 ℕcn 12149 2c2 12204 ℕ₀cn0 12405 ℤcz 12492 ℤ_≥cuz 12755 ...cfz 13427 ↑cexp 13988 !cfa 14200 Ccbc 14229 ♯chash 14257 ∏cprod 15830

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-oi 9419 df-card 9855 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 12150 df-2 12212 df-3 12213 df-n0 12406 df-z 12493 df-uz 12756 df-rp 12910 df-fz 13428 df-fzo 13575 df-seq 13929 df-exp 13989 df-fac 14201 df-bc 14230 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-clim 15415 df-prod 15831

This theorem is referenced by: (None)

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