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Theorem binom 15776

Description: The binomial theorem: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)). Theorem 15-2.8 of [Gleason] p. 296. This part of the proof sets up the induction and does the base case, with the bulk of the work (the induction step) in binomlem 15775. This is Metamath 100 proof #44. (Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)

**Assertion**
Ref	Expression
binom	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝑁

Proof of Theorem binom Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7417	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑0))
2		oveq2 7417	. . . . . . 7 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3		oveq1 7416	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4		oveq1 7416	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 − 𝑘) = (0 − 𝑘))
5	4	oveq2d 7425	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(0 − 𝑘)))
6	5	oveq1d 7424	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘)))
7	3, 6	oveq12d 7427	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))))
8	7	adantr 482	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))))
9	2, 8	sumeq12dv 15652	. . . . . 6 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))))
10	1, 9	eqeq12d 2749	. . . . 5 ⊢ (𝑥 = 0 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) ↔ ((𝐴 + 𝐵)↑0) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘)))))
11	10	imbi2d 341	. . . 4 ⊢ (𝑥 = 0 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))))))
12		oveq2 7417	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑛))
13		oveq2 7417	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14		oveq1 7416	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15		oveq1 7416	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → (𝑥 − 𝑘) = (𝑛 − 𝑘))
16	15	oveq2d 7425	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(𝑛 − 𝑘)))
17	16	oveq1d 7424	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘)))
18	14, 17	oveq12d 7427	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘))))
19	18	adantr 482	. . . . . . 7 ⊢ ((𝑥 = 𝑛 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘))))
20	13, 19	sumeq12dv 15652	. . . . . 6 ⊢ (𝑥 = 𝑛 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘))))
21	12, 20	eqeq12d 2749	. . . . 5 ⊢ (𝑥 = 𝑛 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) ↔ ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘)))))
22	21	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝑛 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘))))))
23		oveq2 7417	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑(𝑛 + 1)))
24		oveq2 7417	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25		oveq1 7416	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26		oveq1 7416	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 − 𝑘) = ((𝑛 + 1) − 𝑘))
27	26	oveq2d 7425	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑((𝑛 + 1) − 𝑘)))
28	27	oveq1d 7424	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘)))
29	25, 28	oveq12d 7427	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘))))
30	29	adantr 482	. . . . . . 7 ⊢ ((𝑥 = (𝑛 + 1) ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘))))
31	24, 30	sumeq12dv 15652	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘))))
32	23, 31	eqeq12d 2749	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) ↔ ((𝐴 + 𝐵)↑(𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘)))))
33	32	imbi2d 341	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑(𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘))))))
34		oveq2 7417	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑁))
35		oveq2 7417	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36		oveq1 7416	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37		oveq1 7416	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → (𝑥 − 𝑘) = (𝑁 − 𝑘))
38	37	oveq2d 7425	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(𝑁 − 𝑘)))
39	38	oveq1d 7424	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))
40	36, 39	oveq12d 7427	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))
41	40	adantr 482	. . . . . . 7 ⊢ ((𝑥 = 𝑁 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))
42	35, 41	sumeq12dv 15652	. . . . . 6 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))
43	34, 42	eqeq12d 2749	. . . . 5 ⊢ (𝑥 = 𝑁 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) ↔ ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))))
44	43	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝑁 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))))
45		exp0 14031	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
46		exp0 14031	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵↑0) = 1)
47	45, 46	oveqan12d 7428	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = (1 · 1))
48		1t1e1 12374	. . . . . . . 8 ⊢ (1 · 1) = 1
49	47, 48	eqtrdi 2789	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = 1)
50	49	oveq2d 7425	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = (1 · 1))
51	50, 48	eqtrdi 2789	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = 1)
52		0z 12569	. . . . . 6 ⊢ 0 ∈ ℤ
53		ax-1cn 11168	. . . . . . 7 ⊢ 1 ∈ ℂ
54	51, 53	eqeltrdi 2842	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) ∈ ℂ)
55		oveq2 7417	. . . . . . . . 9 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
56		0nn0 12487	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
57		bcn0 14270	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
58	56, 57	ax-mp 5	. . . . . . . . 9 ⊢ (0C0) = 1
59	55, 58	eqtrdi 2789	. . . . . . . 8 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
60		oveq2 7417	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
61		0m0e0 12332	. . . . . . . . . . 11 ⊢ (0 − 0) = 0
62	60, 61	eqtrdi 2789	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
63	62	oveq2d 7425	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐴↑(0 − 𝑘)) = (𝐴↑0))
64		oveq2 7417	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐵↑𝑘) = (𝐵↑0))
65	63, 64	oveq12d 7427	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑0) · (𝐵↑0)))
66	59, 65	oveq12d 7427	. . . . . . 7 ⊢ (𝑘 = 0 → ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
67	66	fsum1 15693	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ (1 · ((𝐴↑0) · (𝐵↑0))) ∈ ℂ) → Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
68	52, 54, 67	sylancr 588	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
69		addcl 11192	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
70	69	exp0d 14105	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = 1)
71	51, 68, 70	3eqtr4rd 2784	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))))
72		simprl 770	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ∈ ℂ)
73		simprr 772	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐵 ∈ ℂ)
74		simpl 484	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝑛 ∈ ℕ₀)
75		id 22	. . . . . . 7 ⊢ (((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘))) → ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘))))
76	72, 73, 74, 75	binomlem 15775	. . . . . 6 ⊢ (((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) ∧ ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘)))) → ((𝐴 + 𝐵)↑(𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘))))
77	76	exp31 421	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘))) → ((𝐴 + 𝐵)↑(𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘))))))
78	77	a2d 29	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘)))) → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑(𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘))))))
79	11, 22, 33, 44, 71, 78	nn0ind 12657	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))))
80	79	impcom 409	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))
81	80	3impa 1111	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 (class class class)co 7409 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 − cmin 11444 ℕ₀cn0 12472 ℤcz 12558 ...cfz 13484 ↑cexp 14027 Ccbc 14262 Σcsu 15632

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633

This theorem is referenced by: binom1p 15777 efaddlem 16036 basellem3 26587 lcmineqlem1 40894 jm2.22 41734 binomcxplemnn0 43108 altgsumbc 47028

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