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 7354	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑0))
2		oveq2 7354	. . . . . . 7 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3		oveq1 7353	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4		oveq1 7353	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 − 𝑘) = (0 − 𝑘))
5	4	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(0 − 𝑘)))
6	5	oveq1d 7361	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘)))
7	3, 6	oveq12d 7364	. . . . . . . 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 2747	. . . . 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 7354	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑛))
13		oveq2 7354	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14		oveq1 7353	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15		oveq1 7353	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → (𝑥 − 𝑘) = (𝑛 − 𝑘))
16	15	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(𝑛 − 𝑘)))
17	16	oveq1d 7361	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘)))
18	14, 17	oveq12d 7364	. . . . . . . 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 2747	. . . . 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 7354	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑(𝑛 + 1)))
24		oveq2 7354	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25		oveq1 7353	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26		oveq1 7353	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 − 𝑘) = ((𝑛 + 1) − 𝑘))
27	26	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑((𝑛 + 1) − 𝑘)))
28	27	oveq1d 7361	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘)))
29	25, 28	oveq12d 7364	. . . . . . . 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 2747	. . . . 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 7354	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑁))
35		oveq2 7354	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36		oveq1 7353	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37		oveq1 7353	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → (𝑥 − 𝑘) = (𝑁 − 𝑘))
38	37	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(𝑁 − 𝑘)))
39	38	oveq1d 7361	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))
40	36, 39	oveq12d 7364	. . . . . . . 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 2747	. . . . 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 7365	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = (1 · 1))
48		1t1e1 12282	. . . . . . . 8 ⊢ (1 · 1) = 1
49	47, 48	eqtrdi 2782	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = 1)
50	49	oveq2d 7362	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = (1 · 1))
51	50, 48	eqtrdi 2782	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = 1)
52		0z 12479	. . . . . 6 ⊢ 0 ∈ ℤ
53		ax-1cn 11064	. . . . . . 7 ⊢ 1 ∈ ℂ
54	51, 53	eqeltrdi 2839	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) ∈ ℂ)
55		oveq2 7354	. . . . . . . . 9 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
56		0nn0 12396	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
57		bcn0 14217	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
58	56, 57	ax-mp 5	. . . . . . . . 9 ⊢ (0C0) = 1
59	55, 58	eqtrdi 2782	. . . . . . . 8 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
60		oveq2 7354	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
61		0m0e0 12240	. . . . . . . . . . 11 ⊢ (0 − 0) = 0
62	60, 61	eqtrdi 2782	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
63	62	oveq2d 7362	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐴↑(0 − 𝑘)) = (𝐴↑0))
64		oveq2 7354	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐵↑𝑘) = (𝐵↑0))
65	63, 64	oveq12d 7364	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑0) · (𝐵↑0)))
66	59, 65	oveq12d 7364	. . . . . . 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 11088	. . . . . 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 2777	. . . 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 12568	. . 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 1541 ∈ wcel 2111 (class class class)co 7346 ℂcc 11004 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 − cmin 11344 ℕ₀cn0 12381 ℤcz 12468 ...cfz 13407 ↑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 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-rp 12891 df-fz 13408 df-fzo 13555 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 27020 lcmineqlem1 42132 jm2.22 43098 binomcxplemnn0 44452 altgsumbc 48462

W3C validator