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Theorem binom 14762
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 14761. 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 6799 . . . . . 6 (𝑥 = 0 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑0))
2 oveq2 6799 . . . . . . 7 (𝑥 = 0 → (0...𝑥) = (0...0))
3 oveq1 6798 . . . . . . . . 9 (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4 oveq1 6798 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥𝑘) = (0 − 𝑘))
54oveq2d 6807 . . . . . . . . . 10 (𝑥 = 0 → (𝐴↑(𝑥𝑘)) = (𝐴↑(0 − 𝑘)))
65oveq1d 6806 . . . . . . . . 9 (𝑥 = 0 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(0 − 𝑘)) · (𝐵𝑘)))
73, 6oveq12d 6809 . . . . . . . 8 (𝑥 = 0 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
87adantr 466 . . . . . . 7 ((𝑥 = 0 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
92, 8sumeq12dv 14638 . . . . . 6 (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
101, 9eqeq12d 2786 . . . . 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 329 . . . 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 6799 . . . . . 6 (𝑥 = 𝑛 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑛))
13 oveq2 6799 . . . . . . 7 (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14 oveq1 6798 . . . . . . . . 9 (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15 oveq1 6798 . . . . . . . . . . 11 (𝑥 = 𝑛 → (𝑥𝑘) = (𝑛𝑘))
1615oveq2d 6807 . . . . . . . . . 10 (𝑥 = 𝑛 → (𝐴↑(𝑥𝑘)) = (𝐴↑(𝑛𝑘)))
1716oveq1d 6806 . . . . . . . . 9 (𝑥 = 𝑛 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(𝑛𝑘)) · (𝐵𝑘)))
1814, 17oveq12d 6809 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
1918adantr 466 . . . . . . 7 ((𝑥 = 𝑛𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
2013, 19sumeq12dv 14638 . . . . . 6 (𝑥 = 𝑛 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
2112, 20eqeq12d 2786 . . . . 5 (𝑥 = 𝑛 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) ↔ ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘)))))
2221imbi2d 329 . . . 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 6799 . . . . . 6 (𝑥 = (𝑛 + 1) → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑(𝑛 + 1)))
24 oveq2 6799 . . . . . . 7 (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25 oveq1 6798 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26 oveq1 6798 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝑥𝑘) = ((𝑛 + 1) − 𝑘))
2726oveq2d 6807 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (𝐴↑(𝑥𝑘)) = (𝐴↑((𝑛 + 1) − 𝑘)))
2827oveq1d 6806 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘)))
2925, 28oveq12d 6809 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))
3029adantr 466 . . . . . . 7 ((𝑥 = (𝑛 + 1) ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))
3124, 30sumeq12dv 14638 . . . . . 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 2786 . . . . 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 329 . . . 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 6799 . . . . . 6 (𝑥 = 𝑁 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑁))
35 oveq2 6799 . . . . . . 7 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36 oveq1 6798 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37 oveq1 6798 . . . . . . . . . . 11 (𝑥 = 𝑁 → (𝑥𝑘) = (𝑁𝑘))
3837oveq2d 6807 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝐴↑(𝑥𝑘)) = (𝐴↑(𝑁𝑘)))
3938oveq1d 6806 . . . . . . . . 9 (𝑥 = 𝑁 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))
4036, 39oveq12d 6809 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4140adantr 466 . . . . . . 7 ((𝑥 = 𝑁𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4235, 41sumeq12dv 14638 . . . . . 6 (𝑥 = 𝑁 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4334, 42eqeq12d 2786 . . . . 5 (𝑥 = 𝑁 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) ↔ ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))))
4443imbi2d 329 . . . 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 13064 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝐴↑0) = 1)
46 exp0 13064 . . . . . . . . 9 (𝐵 ∈ ℂ → (𝐵↑0) = 1)
4745, 46oveqan12d 6810 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = (1 · 1))
48 1t1e1 11375 . . . . . . . 8 (1 · 1) = 1
4947, 48syl6eq 2821 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = 1)
5049oveq2d 6807 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = (1 · 1))
5150, 48syl6eq 2821 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = 1)
52 0z 11588 . . . . . 6 0 ∈ ℤ
53 ax-1cn 10194 . . . . . . 7 1 ∈ ℂ
5451, 53syl6eqel 2858 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) ∈ ℂ)
55 oveq2 6799 . . . . . . . . 9 (𝑘 = 0 → (0C𝑘) = (0C0))
56 0nn0 11507 . . . . . . . . . 10 0 ∈ ℕ0
57 bcn0 13294 . . . . . . . . . 10 (0 ∈ ℕ0 → (0C0) = 1)
5856, 57ax-mp 5 . . . . . . . . 9 (0C0) = 1
5955, 58syl6eq 2821 . . . . . . . 8 (𝑘 = 0 → (0C𝑘) = 1)
60 oveq2 6799 . . . . . . . . . . 11 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
61 0m0e0 11330 . . . . . . . . . . 11 (0 − 0) = 0
6260, 61syl6eq 2821 . . . . . . . . . 10 (𝑘 = 0 → (0 − 𝑘) = 0)
6362oveq2d 6807 . . . . . . . . 9 (𝑘 = 0 → (𝐴↑(0 − 𝑘)) = (𝐴↑0))
64 oveq2 6799 . . . . . . . . 9 (𝑘 = 0 → (𝐵𝑘) = (𝐵↑0))
6563, 64oveq12d 6809 . . . . . . . 8 (𝑘 = 0 → ((𝐴↑(0 − 𝑘)) · (𝐵𝑘)) = ((𝐴↑0) · (𝐵↑0)))
6659, 65oveq12d 6809 . . . . . . 7 (𝑘 = 0 → ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
6766fsum1 14677 . . . . . 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 575 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
69 addcl 10218 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
7069exp0d 13202 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = 1)
7151, 68, 703eqtr4rd 2816 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
72 simprl 754 . . . . . . 7 ((𝑛 ∈ ℕ0 ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ∈ ℂ)
73 simprr 756 . . . . . . 7 ((𝑛 ∈ ℕ0 ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐵 ∈ ℂ)
74 simpl 468 . . . . . . 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 14761 . . . . . 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 406 . . . . 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 11672 . . 3 (𝑁 ∈ ℕ0 → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))))
8079impcom 394 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
81803impa 1100 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 382  w3a 1071   = wceq 1631  wcel 2145  (class class class)co 6791  cc 10134  0cc0 10136  1c1 10137   + caddc 10139   · cmul 10141  cmin 10466  0cn0 11492  cz 11577  ...cfz 12526  cexp 13060  Ccbc 13286  Σcsu 14617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-inf2 8700  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213  ax-pre-sup 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-isom 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-sup 8502  df-oi 8569  df-card 8963  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-div 10885  df-nn 11221  df-2 11279  df-3 11280  df-n0 11493  df-z 11578  df-uz 11887  df-rp 12029  df-fz 12527  df-fzo 12667  df-seq 13002  df-exp 13061  df-fac 13258  df-bc 13287  df-hash 13315  df-cj 14040  df-re 14041  df-im 14042  df-sqrt 14176  df-abs 14177  df-clim 14420  df-sum 14618
This theorem is referenced by:  binom1p  14763  efaddlem  15022  basellem3  25023  jm2.22  38081  binomcxplemnn0  39067  altgsumbc  42651
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