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 15862

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 15861. 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 7437	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑0))
2		oveq2 7437	. . . . . . 7 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3		oveq1 7436	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4		oveq1 7436	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 − 𝑘) = (0 − 𝑘))
5	4	oveq2d 7445	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(0 − 𝑘)))
6	5	oveq1d 7444	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘)))
7	3, 6	oveq12d 7447	. . . . . . . 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 15738	. . . . . 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 2752	. . . . 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 7437	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑛))
13		oveq2 7437	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14		oveq1 7436	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15		oveq1 7436	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → (𝑥 − 𝑘) = (𝑛 − 𝑘))
16	15	oveq2d 7445	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(𝑛 − 𝑘)))
17	16	oveq1d 7444	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘)))
18	14, 17	oveq12d 7447	. . . . . . . 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 15738	. . . . . 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 2752	. . . . 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 7437	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑(𝑛 + 1)))
24		oveq2 7437	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25		oveq1 7436	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26		oveq1 7436	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 − 𝑘) = ((𝑛 + 1) − 𝑘))
27	26	oveq2d 7445	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑((𝑛 + 1) − 𝑘)))
28	27	oveq1d 7444	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘)))
29	25, 28	oveq12d 7447	. . . . . . . 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 15738	. . . . . 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 2752	. . . . 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 7437	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑁))
35		oveq2 7437	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36		oveq1 7436	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37		oveq1 7436	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → (𝑥 − 𝑘) = (𝑁 − 𝑘))
38	37	oveq2d 7445	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(𝑁 − 𝑘)))
39	38	oveq1d 7444	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))
40	36, 39	oveq12d 7447	. . . . . . . 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 15738	. . . . . 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 2752	. . . . 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 14102	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
46		exp0 14102	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵↑0) = 1)
47	45, 46	oveqan12d 7448	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = (1 · 1))
48		1t1e1 12424	. . . . . . . 8 ⊢ (1 · 1) = 1
49	47, 48	eqtrdi 2792	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = 1)
50	49	oveq2d 7445	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = (1 · 1))
51	50, 48	eqtrdi 2792	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = 1)
52		0z 12620	. . . . . 6 ⊢ 0 ∈ ℤ
53		ax-1cn 11209	. . . . . . 7 ⊢ 1 ∈ ℂ
54	51, 53	eqeltrdi 2848	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) ∈ ℂ)
55		oveq2 7437	. . . . . . . . 9 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
56		0nn0 12537	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
57		bcn0 14345	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
58	56, 57	ax-mp 5	. . . . . . . . 9 ⊢ (0C0) = 1
59	55, 58	eqtrdi 2792	. . . . . . . 8 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
60		oveq2 7437	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
61		0m0e0 12382	. . . . . . . . . . 11 ⊢ (0 − 0) = 0
62	60, 61	eqtrdi 2792	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
63	62	oveq2d 7445	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐴↑(0 − 𝑘)) = (𝐴↑0))
64		oveq2 7437	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐵↑𝑘) = (𝐵↑0))
65	63, 64	oveq12d 7447	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑0) · (𝐵↑0)))
66	59, 65	oveq12d 7447	. . . . . . 7 ⊢ (𝑘 = 0 → ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
67	66	fsum1 15779	. . . . . 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 11233	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
70	69	exp0d 14176	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = 1)
71	51, 68, 70	3eqtr4rd 2787	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))))
72		simprl 771	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ∈ ℂ)
73		simprr 773	. . . . . . 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 15861	. . . . . 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 12709	. . 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 1110	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 1087 = wceq 1540 ∈ wcel 2108 (class class class)co 7429 ℂcc 11149 0cc0 11151 1c1 11152 + caddc 11154 · cmul 11156 − cmin 11488 ℕ₀cn0 12522 ℤcz 12609 ...cfz 13543 ↑cexp 14098 Ccbc 14337 Σcsu 15718

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-inf2 9677 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-pre-sup 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-sup 9478 df-oi 9546 df-card 9975 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-div 11917 df-nn 12263 df-2 12325 df-3 12326 df-n0 12523 df-z 12610 df-uz 12875 df-rp 13031 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719

This theorem is referenced by: binom1p 15863 efaddlem 16125 basellem3 27116 lcmineqlem1 42008 jm2.22 42985 binomcxplemnn0 44346 altgsumbc 48241

W3C validator