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Theorem binom 15180
Description: The binomial theorem: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴𝑘) · (𝐵↑(𝑁𝑘)). 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 15179. 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↑(𝑁𝑘)) · (𝐵𝑘))))
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.
StepHypRef Expression
1 oveq2 7158 . . . . . 6 (𝑥 = 0 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑0))
2 oveq2 7158 . . . . . . 7 (𝑥 = 0 → (0...𝑥) = (0...0))
3 oveq1 7157 . . . . . . . . 9 (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4 oveq1 7157 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥𝑘) = (0 − 𝑘))
54oveq2d 7166 . . . . . . . . . 10 (𝑥 = 0 → (𝐴↑(𝑥𝑘)) = (𝐴↑(0 − 𝑘)))
65oveq1d 7165 . . . . . . . . 9 (𝑥 = 0 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(0 − 𝑘)) · (𝐵𝑘)))
73, 6oveq12d 7168 . . . . . . . 8 (𝑥 = 0 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
87adantr 481 . . . . . . 7 ((𝑥 = 0 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
92, 8sumeq12dv 15058 . . . . . 6 (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
101, 9eqeq12d 2842 . . . . 5 (𝑥 = 0 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) ↔ ((𝐴 + 𝐵)↑0) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘)))))
1110imbi2d 342 . . . 4 (𝑥 = 0 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))))
12 oveq2 7158 . . . . . 6 (𝑥 = 𝑛 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑛))
13 oveq2 7158 . . . . . . 7 (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14 oveq1 7157 . . . . . . . . 9 (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15 oveq1 7157 . . . . . . . . . . 11 (𝑥 = 𝑛 → (𝑥𝑘) = (𝑛𝑘))
1615oveq2d 7166 . . . . . . . . . 10 (𝑥 = 𝑛 → (𝐴↑(𝑥𝑘)) = (𝐴↑(𝑛𝑘)))
1716oveq1d 7165 . . . . . . . . 9 (𝑥 = 𝑛 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(𝑛𝑘)) · (𝐵𝑘)))
1814, 17oveq12d 7168 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
1918adantr 481 . . . . . . 7 ((𝑥 = 𝑛𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
2013, 19sumeq12dv 15058 . . . . . 6 (𝑥 = 𝑛 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
2112, 20eqeq12d 2842 . . . . 5 (𝑥 = 𝑛 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) ↔ ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘)))))
2221imbi2d 342 . . . 4 (𝑥 = 𝑛 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))))
23 oveq2 7158 . . . . . 6 (𝑥 = (𝑛 + 1) → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑(𝑛 + 1)))
24 oveq2 7158 . . . . . . 7 (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25 oveq1 7157 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26 oveq1 7157 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝑥𝑘) = ((𝑛 + 1) − 𝑘))
2726oveq2d 7166 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (𝐴↑(𝑥𝑘)) = (𝐴↑((𝑛 + 1) − 𝑘)))
2827oveq1d 7165 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘)))
2925, 28oveq12d 7168 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))
3029adantr 481 . . . . . . 7 ((𝑥 = (𝑛 + 1) ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))
3124, 30sumeq12dv 15058 . . . . . 6 (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))
3223, 31eqeq12d 2842 . . . . 5 (𝑥 = (𝑛 + 1) → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) ↔ ((𝐴 + 𝐵)↑(𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘)))))
3332imbi2d 342 . . . 4 (𝑥 = (𝑛 + 1) → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑(𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))))
34 oveq2 7158 . . . . . 6 (𝑥 = 𝑁 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑁))
35 oveq2 7158 . . . . . . 7 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36 oveq1 7157 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37 oveq1 7157 . . . . . . . . . . 11 (𝑥 = 𝑁 → (𝑥𝑘) = (𝑁𝑘))
3837oveq2d 7166 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝐴↑(𝑥𝑘)) = (𝐴↑(𝑁𝑘)))
3938oveq1d 7165 . . . . . . . . 9 (𝑥 = 𝑁 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))
4036, 39oveq12d 7168 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4140adantr 481 . . . . . . 7 ((𝑥 = 𝑁𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4235, 41sumeq12dv 15058 . . . . . 6 (𝑥 = 𝑁 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4334, 42eqeq12d 2842 . . . . 5 (𝑥 = 𝑁 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) ↔ ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))))
4443imbi2d 342 . . . 4 (𝑥 = 𝑁 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))))
45 exp0 13428 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝐴↑0) = 1)
46 exp0 13428 . . . . . . . . 9 (𝐵 ∈ ℂ → (𝐵↑0) = 1)
4745, 46oveqan12d 7169 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = (1 · 1))
48 1t1e1 11793 . . . . . . . 8 (1 · 1) = 1
4947, 48syl6eq 2877 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = 1)
5049oveq2d 7166 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = (1 · 1))
5150, 48syl6eq 2877 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = 1)
52 0z 11986 . . . . . 6 0 ∈ ℤ
53 ax-1cn 10589 . . . . . . 7 1 ∈ ℂ
5451, 53syl6eqel 2926 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) ∈ ℂ)
55 oveq2 7158 . . . . . . . . 9 (𝑘 = 0 → (0C𝑘) = (0C0))
56 0nn0 11906 . . . . . . . . . 10 0 ∈ ℕ0
57 bcn0 13665 . . . . . . . . . 10 (0 ∈ ℕ0 → (0C0) = 1)
5856, 57ax-mp 5 . . . . . . . . 9 (0C0) = 1
5955, 58syl6eq 2877 . . . . . . . 8 (𝑘 = 0 → (0C𝑘) = 1)
60 oveq2 7158 . . . . . . . . . . 11 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
61 0m0e0 11751 . . . . . . . . . . 11 (0 − 0) = 0
6260, 61syl6eq 2877 . . . . . . . . . 10 (𝑘 = 0 → (0 − 𝑘) = 0)
6362oveq2d 7166 . . . . . . . . 9 (𝑘 = 0 → (𝐴↑(0 − 𝑘)) = (𝐴↑0))
64 oveq2 7158 . . . . . . . . 9 (𝑘 = 0 → (𝐵𝑘) = (𝐵↑0))
6563, 64oveq12d 7168 . . . . . . . 8 (𝑘 = 0 → ((𝐴↑(0 − 𝑘)) · (𝐵𝑘)) = ((𝐴↑0) · (𝐵↑0)))
6659, 65oveq12d 7168 . . . . . . 7 (𝑘 = 0 → ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
6766fsum1 15097 . . . . . 6 ((0 ∈ ℤ ∧ (1 · ((𝐴↑0) · (𝐵↑0))) ∈ ℂ) → Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
6852, 54, 67sylancr 587 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
69 addcl 10613 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
7069exp0d 13499 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = 1)
7151, 68, 703eqtr4rd 2872 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
72 simprl 767 . . . . . . 7 ((𝑛 ∈ ℕ0 ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ∈ ℂ)
73 simprr 769 . . . . . . 7 ((𝑛 ∈ ℕ0 ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐵 ∈ ℂ)
74 simpl 483 . . . . . . 7 ((𝑛 ∈ ℕ0 ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝑛 ∈ ℕ0)
75 id 22 . . . . . . 7 (((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))) → ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
7672, 73, 74, 75binomlem 15179 . . . . . 6 (((𝑛 ∈ ℕ0 ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) ∧ ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘)))) → ((𝐴 + 𝐵)↑(𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))
7776exp31 420 . . . . 5 (𝑛 ∈ ℕ0 → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))) → ((𝐴 + 𝐵)↑(𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))))
7877a2d 29 . . . 4 (𝑛 ∈ ℕ0 → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘)))) → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑(𝑛 + 1)) = Σ𝑘 ∈ (0...(𝑛 + 1))(((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))))
7911, 22, 33, 44, 71, 78nn0ind 12071 . . 3 (𝑁 ∈ ℕ0 → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))))
8079impcom 408 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
81803impa 1104 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  (class class class)co 7150  cc 10529  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536  cmin 10864  0cn0 11891  cz 11975  ...cfz 12887  cexp 13424  Ccbc 13657  Σcsu 15037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12385  df-fz 12888  df-fzo 13029  df-seq 13365  df-exp 13425  df-fac 13629  df-bc 13658  df-hash 13686  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-sum 15038
This theorem is referenced by:  binom1p  15181  efaddlem  15441  basellem3  25593  jm2.22  39476  binomcxplemnn0  40565  altgsumbc  44302
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