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Theorem binom 11363
 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 11362. 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 5826 . . . . . 6 (𝑥 = 0 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑0))
2 oveq2 5826 . . . . . . 7 (𝑥 = 0 → (0...𝑥) = (0...0))
3 oveq1 5825 . . . . . . . . 9 (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4 oveq1 5825 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥𝑘) = (0 − 𝑘))
54oveq2d 5834 . . . . . . . . . 10 (𝑥 = 0 → (𝐴↑(𝑥𝑘)) = (𝐴↑(0 − 𝑘)))
65oveq1d 5833 . . . . . . . . 9 (𝑥 = 0 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(0 − 𝑘)) · (𝐵𝑘)))
73, 6oveq12d 5836 . . . . . . . 8 (𝑥 = 0 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
87adantr 274 . . . . . . 7 ((𝑥 = 0 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
92, 8sumeq12dv 11251 . . . . . 6 (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
101, 9eqeq12d 2172 . . . . 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 229 . . . 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 5826 . . . . . 6 (𝑥 = 𝑛 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑛))
13 oveq2 5826 . . . . . . 7 (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14 oveq1 5825 . . . . . . . . 9 (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15 oveq1 5825 . . . . . . . . . . 11 (𝑥 = 𝑛 → (𝑥𝑘) = (𝑛𝑘))
1615oveq2d 5834 . . . . . . . . . 10 (𝑥 = 𝑛 → (𝐴↑(𝑥𝑘)) = (𝐴↑(𝑛𝑘)))
1716oveq1d 5833 . . . . . . . . 9 (𝑥 = 𝑛 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(𝑛𝑘)) · (𝐵𝑘)))
1814, 17oveq12d 5836 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
1918adantr 274 . . . . . . 7 ((𝑥 = 𝑛𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
2013, 19sumeq12dv 11251 . . . . . 6 (𝑥 = 𝑛 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
2112, 20eqeq12d 2172 . . . . 5 (𝑥 = 𝑛 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) ↔ ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘)))))
2221imbi2d 229 . . . 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 5826 . . . . . 6 (𝑥 = (𝑛 + 1) → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑(𝑛 + 1)))
24 oveq2 5826 . . . . . . 7 (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25 oveq1 5825 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26 oveq1 5825 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝑥𝑘) = ((𝑛 + 1) − 𝑘))
2726oveq2d 5834 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (𝐴↑(𝑥𝑘)) = (𝐴↑((𝑛 + 1) − 𝑘)))
2827oveq1d 5833 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘)))
2925, 28oveq12d 5836 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))
3029adantr 274 . . . . . . 7 ((𝑥 = (𝑛 + 1) ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))
3124, 30sumeq12dv 11251 . . . . . 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 2172 . . . . 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 229 . . . 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 5826 . . . . . 6 (𝑥 = 𝑁 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑁))
35 oveq2 5826 . . . . . . 7 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36 oveq1 5825 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37 oveq1 5825 . . . . . . . . . . 11 (𝑥 = 𝑁 → (𝑥𝑘) = (𝑁𝑘))
3837oveq2d 5834 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝐴↑(𝑥𝑘)) = (𝐴↑(𝑁𝑘)))
3938oveq1d 5833 . . . . . . . . 9 (𝑥 = 𝑁 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))
4036, 39oveq12d 5836 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4140adantr 274 . . . . . . 7 ((𝑥 = 𝑁𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4235, 41sumeq12dv 11251 . . . . . 6 (𝑥 = 𝑁 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4334, 42eqeq12d 2172 . . . . 5 (𝑥 = 𝑁 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) ↔ ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))))
4443imbi2d 229 . . . 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 10405 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝐴↑0) = 1)
46 exp0 10405 . . . . . . . . 9 (𝐵 ∈ ℂ → (𝐵↑0) = 1)
4745, 46oveqan12d 5837 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = (1 · 1))
48 1t1e1 8968 . . . . . . . 8 (1 · 1) = 1
4947, 48eqtrdi 2206 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = 1)
5049oveq2d 5834 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = (1 · 1))
5150, 48eqtrdi 2206 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = 1)
52 0z 9161 . . . . . 6 0 ∈ ℤ
53 ax-1cn 7808 . . . . . . 7 1 ∈ ℂ
5451, 53eqeltrdi 2248 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) ∈ ℂ)
55 oveq2 5826 . . . . . . . . 9 (𝑘 = 0 → (0C𝑘) = (0C0))
56 0nn0 9088 . . . . . . . . . 10 0 ∈ ℕ0
57 bcn0 10611 . . . . . . . . . 10 (0 ∈ ℕ0 → (0C0) = 1)
5856, 57ax-mp 5 . . . . . . . . 9 (0C0) = 1
5955, 58eqtrdi 2206 . . . . . . . 8 (𝑘 = 0 → (0C𝑘) = 1)
60 oveq2 5826 . . . . . . . . . . 11 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
61 0m0e0 8928 . . . . . . . . . . 11 (0 − 0) = 0
6260, 61eqtrdi 2206 . . . . . . . . . 10 (𝑘 = 0 → (0 − 𝑘) = 0)
6362oveq2d 5834 . . . . . . . . 9 (𝑘 = 0 → (𝐴↑(0 − 𝑘)) = (𝐴↑0))
64 oveq2 5826 . . . . . . . . 9 (𝑘 = 0 → (𝐵𝑘) = (𝐵↑0))
6563, 64oveq12d 5836 . . . . . . . 8 (𝑘 = 0 → ((𝐴↑(0 − 𝑘)) · (𝐵𝑘)) = ((𝐴↑0) · (𝐵↑0)))
6659, 65oveq12d 5836 . . . . . . 7 (𝑘 = 0 → ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
6766fsum1 11291 . . . . . 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 411 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
69 addcl 7840 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
7069exp0d 10527 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = 1)
7151, 68, 703eqtr4rd 2201 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
72 simprl 521 . . . . . . 7 ((𝑛 ∈ ℕ0 ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ∈ ℂ)
73 simprr 522 . . . . . . 7 ((𝑛 ∈ ℕ0 ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐵 ∈ ℂ)
74 simpl 108 . . . . . . 7 ((𝑛 ∈ ℕ0 ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝑛 ∈ ℕ0)
75 id 19 . . . . . . 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 11362 . . . . . 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 362 . . . . 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 26 . . . 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 9261 . . 3 (𝑁 ∈ ℕ0 → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))))
8079impcom 124 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
81803impa 1177 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1335   ∈ wcel 2128  (class class class)co 5818  ℂcc 7713  0cc0 7715  1c1 7716   + caddc 7718   · cmul 7720   − cmin 8029  ℕ0cn0 9073  ℤcz 9150  ...cfz 9894  ↑cexp 10400  Ccbc 10603  Σcsu 11232 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834  ax-caucvg 7835 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-isom 5176  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-frec 6332  df-1o 6357  df-oadd 6361  df-er 6473  df-en 6679  df-dom 6680  df-fin 6681  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-3 8876  df-4 8877  df-n0 9074  df-z 9151  df-uz 9423  df-q 9511  df-rp 9543  df-fz 9895  df-fzo 10024  df-seqfrec 10327  df-exp 10401  df-fac 10582  df-bc 10604  df-ihash 10632  df-cj 10724  df-re 10725  df-im 10726  df-rsqrt 10880  df-abs 10881  df-clim 11158  df-sumdc 11233 This theorem is referenced by:  binom1p  11364  efaddlem  11553
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