binom - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > binom		Structured version Visualization version GIF version

Theorem binom 15737

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 15736. 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 7357	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑0))
2		oveq2 7357	. . . . . . 7 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3		oveq1 7356	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4		oveq1 7356	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 − 𝑘) = (0 − 𝑘))
5	4	oveq2d 7365	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(0 − 𝑘)))
6	5	oveq1d 7364	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘)))
7	3, 6	oveq12d 7367	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))))
8	7	adantr 480	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))))
9	2, 8	sumeq12dv 15613	. . . . . 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 2745	. . . . 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 340	. . . 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 7357	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑛))
13		oveq2 7357	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14		oveq1 7356	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15		oveq1 7356	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → (𝑥 − 𝑘) = (𝑛 − 𝑘))
16	15	oveq2d 7365	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(𝑛 − 𝑘)))
17	16	oveq1d 7364	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘)))
18	14, 17	oveq12d 7367	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘))))
19	18	adantr 480	. . . . . . 7 ⊢ ((𝑥 = 𝑛 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘))))
20	13, 19	sumeq12dv 15613	. . . . . 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 2745	. . . . 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 340	. . . 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 7357	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑(𝑛 + 1)))
24		oveq2 7357	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25		oveq1 7356	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26		oveq1 7356	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 − 𝑘) = ((𝑛 + 1) − 𝑘))
27	26	oveq2d 7365	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑((𝑛 + 1) − 𝑘)))
28	27	oveq1d 7364	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘)))
29	25, 28	oveq12d 7367	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘))))
30	29	adantr 480	. . . . . . 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 15613	. . . . . 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 2745	. . . . 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 340	. . . 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 7357	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑁))
35		oveq2 7357	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36		oveq1 7356	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37		oveq1 7356	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → (𝑥 − 𝑘) = (𝑁 − 𝑘))
38	37	oveq2d 7365	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(𝑁 − 𝑘)))
39	38	oveq1d 7364	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))
40	36, 39	oveq12d 7367	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))
41	40	adantr 480	. . . . . . 7 ⊢ ((𝑥 = 𝑁 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))
42	35, 41	sumeq12dv 15613	. . . . . 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 2745	. . . . 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 340	. . . 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 13972	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
46		exp0 13972	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵↑0) = 1)
47	45, 46	oveqan12d 7368	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = (1 · 1))
48		1t1e1 12285	. . . . . . . 8 ⊢ (1 · 1) = 1
49	47, 48	eqtrdi 2780	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = 1)
50	49	oveq2d 7365	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = (1 · 1))
51	50, 48	eqtrdi 2780	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = 1)
52		0z 12482	. . . . . 6 ⊢ 0 ∈ ℤ
53		ax-1cn 11067	. . . . . . 7 ⊢ 1 ∈ ℂ
54	51, 53	eqeltrdi 2836	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) ∈ ℂ)
55		oveq2 7357	. . . . . . . . 9 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
56		0nn0 12399	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
57		bcn0 14217	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
58	56, 57	ax-mp 5	. . . . . . . . 9 ⊢ (0C0) = 1
59	55, 58	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
60		oveq2 7357	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
61		0m0e0 12243	. . . . . . . . . . 11 ⊢ (0 − 0) = 0
62	60, 61	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
63	62	oveq2d 7365	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐴↑(0 − 𝑘)) = (𝐴↑0))
64		oveq2 7357	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐵↑𝑘) = (𝐵↑0))
65	63, 64	oveq12d 7367	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑0) · (𝐵↑0)))
66	59, 65	oveq12d 7367	. . . . . . 7 ⊢ (𝑘 = 0 → ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
67	66	fsum1 15654	. . . . . 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 587	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
69		addcl 11091	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
70	69	exp0d 14047	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = 1)
71	51, 68, 70	3eqtr4rd 2775	. . . 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 482	. . . . . . 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 15736	. . . . . 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 419	. . . . 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 12571	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))))
80	79	impcom 407	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))
81	80	3impa 1109	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 (class class class)co 7349 ℂcc 11007 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 − cmin 11347 ℕ₀cn0 12384 ℤcz 12471 ...cfz 13410 ↑cexp 13968 Ccbc 14209 Σcsu 15593

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-n0 12385 df-z 12472 df-uz 12736 df-rp 12894 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594

This theorem is referenced by: binom1p 15738 efaddlem 16000 basellem3 26991 lcmineqlem1 42006 jm2.22 42972 binomcxplemnn0 44326 altgsumbc 48340

W3C validator