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Theorem binom 11494

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 11493. 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 5885	. . . . . 6 ⊢ (𝑥 = 0 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑0))
2		oveq2 5885	. . . . . . 7 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
3		oveq1 5884	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4		oveq1 5884	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 − 𝑘) = (0 − 𝑘))
5	4	oveq2d 5893	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(0 − 𝑘)))
6	5	oveq1d 5892	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘)))
7	3, 6	oveq12d 5895	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))))
8	7	adantr 276	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))))
9	2, 8	sumeq12dv 11382	. . . . . 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 2192	. . . . 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 230	. . . 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 5885	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑛))
13		oveq2 5885	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14		oveq1 5884	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15		oveq1 5884	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → (𝑥 − 𝑘) = (𝑛 − 𝑘))
16	15	oveq2d 5893	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(𝑛 − 𝑘)))
17	16	oveq1d 5892	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘)))
18	14, 17	oveq12d 5895	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘))))
19	18	adantr 276	. . . . . . 7 ⊢ ((𝑥 = 𝑛 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛 − 𝑘)) · (𝐵↑𝑘))))
20	13, 19	sumeq12dv 11382	. . . . . 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 2192	. . . . 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 230	. . . 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 5885	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑(𝑛 + 1)))
24		oveq2 5885	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25		oveq1 5884	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26		oveq1 5884	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 − 𝑘) = ((𝑛 + 1) − 𝑘))
27	26	oveq2d 5893	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑((𝑛 + 1) − 𝑘)))
28	27	oveq1d 5892	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘)))
29	25, 28	oveq12d 5895	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵↑𝑘))))
30	29	adantr 276	. . . . . . 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 11382	. . . . . 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 2192	. . . . 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 230	. . . 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 5885	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑁))
35		oveq2 5885	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36		oveq1 5884	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37		oveq1 5884	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → (𝑥 − 𝑘) = (𝑁 − 𝑘))
38	37	oveq2d 5893	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → (𝐴↑(𝑥 − 𝑘)) = (𝐴↑(𝑁 − 𝑘)))
39	38	oveq1d 5892	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))
40	36, 39	oveq12d 5895	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))
41	40	adantr 276	. . . . . . 7 ⊢ ((𝑥 = 𝑁 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥 − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))
42	35, 41	sumeq12dv 11382	. . . . . 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 2192	. . . . 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 230	. . . 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 10526	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
46		exp0 10526	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵↑0) = 1)
47	45, 46	oveqan12d 5896	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = (1 · 1))
48		1t1e1 9073	. . . . . . . 8 ⊢ (1 · 1) = 1
49	47, 48	eqtrdi 2226	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = 1)
50	49	oveq2d 5893	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = (1 · 1))
51	50, 48	eqtrdi 2226	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = 1)
52		0z 9266	. . . . . 6 ⊢ 0 ∈ ℤ
53		ax-1cn 7906	. . . . . . 7 ⊢ 1 ∈ ℂ
54	51, 53	eqeltrdi 2268	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) ∈ ℂ)
55		oveq2 5885	. . . . . . . . 9 ⊢ (𝑘 = 0 → (0C𝑘) = (0C0))
56		0nn0 9193	. . . . . . . . . 10 ⊢ 0 ∈ ℕ₀
57		bcn0 10737	. . . . . . . . . 10 ⊢ (0 ∈ ℕ₀ → (0C0) = 1)
58	56, 57	ax-mp 5	. . . . . . . . 9 ⊢ (0C0) = 1
59	55, 58	eqtrdi 2226	. . . . . . . 8 ⊢ (𝑘 = 0 → (0C𝑘) = 1)
60		oveq2 5885	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
61		0m0e0 9033	. . . . . . . . . . 11 ⊢ (0 − 0) = 0
62	60, 61	eqtrdi 2226	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0)
63	62	oveq2d 5893	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐴↑(0 − 𝑘)) = (𝐴↑0))
64		oveq2 5885	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝐵↑𝑘) = (𝐵↑0))
65	63, 64	oveq12d 5895	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑0) · (𝐵↑0)))
66	59, 65	oveq12d 5895	. . . . . . 7 ⊢ (𝑘 = 0 → ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
67	66	fsum1 11422	. . . . . 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 414	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
69		addcl 7938	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
70	69	exp0d 10650	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = 1)
71	51, 68, 70	3eqtr4rd 2221	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵↑𝑘))))
72		simprl 529	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ∈ ℂ)
73		simprr 531	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐵 ∈ ℂ)
74		simpl 109	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ₀ ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝑛 ∈ ℕ₀)
75		id 19	. . . . . . 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 11493	. . . . . 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 364	. . . . 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 26	. . . 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 9369	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))))
80	79	impcom 125	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))
81	80	3impa 1194	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 (class class class)co 5877 ℂcc 7811 0cc0 7813 1c1 7814 + caddc 7816 · cmul 7818 − cmin 8130 ℕ₀cn0 9178 ℤcz 9255 ...cfz 10010 ↑cexp 10521 Ccbc 10729 Σcsu 11363

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-en 6743 df-dom 6744 df-fin 6745 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-fz 10011 df-fzo 10145 df-seqfrec 10448 df-exp 10522 df-fac 10708 df-bc 10730 df-ihash 10758 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-sumdc 11364

This theorem is referenced by: binom1p 11495 efaddlem 11684

W3C validator