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Theorem binom 15852
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 15851. 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 7433 . . . . . 6 (𝑥 = 0 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑0))
2 oveq2 7433 . . . . . . 7 (𝑥 = 0 → (0...𝑥) = (0...0))
3 oveq1 7432 . . . . . . . . 9 (𝑥 = 0 → (𝑥C𝑘) = (0C𝑘))
4 oveq1 7432 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥𝑘) = (0 − 𝑘))
54oveq2d 7441 . . . . . . . . . 10 (𝑥 = 0 → (𝐴↑(𝑥𝑘)) = (𝐴↑(0 − 𝑘)))
65oveq1d 7440 . . . . . . . . 9 (𝑥 = 0 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(0 − 𝑘)) · (𝐵𝑘)))
73, 6oveq12d 7443 . . . . . . . 8 (𝑥 = 0 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
87adantr 480 . . . . . . 7 ((𝑥 = 0 ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
92, 8sumeq12dv 15728 . . . . . 6 (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
101, 9eqeq12d 2749 . . . . 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 340 . . . 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 7433 . . . . . 6 (𝑥 = 𝑛 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑛))
13 oveq2 7433 . . . . . . 7 (𝑥 = 𝑛 → (0...𝑥) = (0...𝑛))
14 oveq1 7432 . . . . . . . . 9 (𝑥 = 𝑛 → (𝑥C𝑘) = (𝑛C𝑘))
15 oveq1 7432 . . . . . . . . . . 11 (𝑥 = 𝑛 → (𝑥𝑘) = (𝑛𝑘))
1615oveq2d 7441 . . . . . . . . . 10 (𝑥 = 𝑛 → (𝐴↑(𝑥𝑘)) = (𝐴↑(𝑛𝑘)))
1716oveq1d 7440 . . . . . . . . 9 (𝑥 = 𝑛 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(𝑛𝑘)) · (𝐵𝑘)))
1814, 17oveq12d 7443 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
1918adantr 480 . . . . . . 7 ((𝑥 = 𝑛𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
2013, 19sumeq12dv 15728 . . . . . 6 (𝑥 = 𝑛 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
2112, 20eqeq12d 2749 . . . . 5 (𝑥 = 𝑛 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) ↔ ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘)))))
2221imbi2d 340 . . . 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 7433 . . . . . 6 (𝑥 = (𝑛 + 1) → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑(𝑛 + 1)))
24 oveq2 7433 . . . . . . 7 (𝑥 = (𝑛 + 1) → (0...𝑥) = (0...(𝑛 + 1)))
25 oveq1 7432 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → (𝑥C𝑘) = ((𝑛 + 1)C𝑘))
26 oveq1 7432 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝑥𝑘) = ((𝑛 + 1) − 𝑘))
2726oveq2d 7441 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → (𝐴↑(𝑥𝑘)) = (𝐴↑((𝑛 + 1) − 𝑘)))
2827oveq1d 7440 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘)))
2925, 28oveq12d 7443 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))
3029adantr 480 . . . . . . 7 ((𝑥 = (𝑛 + 1) ∧ 𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = (((𝑛 + 1)C𝑘) · ((𝐴↑((𝑛 + 1) − 𝑘)) · (𝐵𝑘))))
3124, 30sumeq12dv 15728 . . . . . 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 2749 . . . . 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 340 . . . 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 7433 . . . . . 6 (𝑥 = 𝑁 → ((𝐴 + 𝐵)↑𝑥) = ((𝐴 + 𝐵)↑𝑁))
35 oveq2 7433 . . . . . . 7 (𝑥 = 𝑁 → (0...𝑥) = (0...𝑁))
36 oveq1 7432 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑥C𝑘) = (𝑁C𝑘))
37 oveq1 7432 . . . . . . . . . . 11 (𝑥 = 𝑁 → (𝑥𝑘) = (𝑁𝑘))
3837oveq2d 7441 . . . . . . . . . 10 (𝑥 = 𝑁 → (𝐴↑(𝑥𝑘)) = (𝐴↑(𝑁𝑘)))
3938oveq1d 7440 . . . . . . . . 9 (𝑥 = 𝑁 → ((𝐴↑(𝑥𝑘)) · (𝐵𝑘)) = ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))
4036, 39oveq12d 7443 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4140adantr 480 . . . . . . 7 ((𝑥 = 𝑁𝑘 ∈ (0...𝑥)) → ((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = ((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4235, 41sumeq12dv 15728 . . . . . 6 (𝑥 = 𝑁 → Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
4334, 42eqeq12d 2749 . . . . 5 (𝑥 = 𝑁 → (((𝐴 + 𝐵)↑𝑥) = Σ𝑘 ∈ (0...𝑥)((𝑥C𝑘) · ((𝐴↑(𝑥𝑘)) · (𝐵𝑘))) ↔ ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))))
4443imbi2d 340 . . . 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 14092 . . . . . . . . 9 (𝐴 ∈ ℂ → (𝐴↑0) = 1)
46 exp0 14092 . . . . . . . . 9 (𝐵 ∈ ℂ → (𝐵↑0) = 1)
4745, 46oveqan12d 7444 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = (1 · 1))
48 1t1e1 12419 . . . . . . . 8 (1 · 1) = 1
4947, 48eqtrdi 2789 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑0) · (𝐵↑0)) = 1)
5049oveq2d 7441 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = (1 · 1))
5150, 48eqtrdi 2789 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) = 1)
52 0z 12615 . . . . . 6 0 ∈ ℤ
53 ax-1cn 11204 . . . . . . 7 1 ∈ ℂ
5451, 53eqeltrdi 2845 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · ((𝐴↑0) · (𝐵↑0))) ∈ ℂ)
55 oveq2 7433 . . . . . . . . 9 (𝑘 = 0 → (0C𝑘) = (0C0))
56 0nn0 12532 . . . . . . . . . 10 0 ∈ ℕ0
57 bcn0 14335 . . . . . . . . . 10 (0 ∈ ℕ0 → (0C0) = 1)
5856, 57ax-mp 5 . . . . . . . . 9 (0C0) = 1
5955, 58eqtrdi 2789 . . . . . . . 8 (𝑘 = 0 → (0C𝑘) = 1)
60 oveq2 7433 . . . . . . . . . . 11 (𝑘 = 0 → (0 − 𝑘) = (0 − 0))
61 0m0e0 12377 . . . . . . . . . . 11 (0 − 0) = 0
6260, 61eqtrdi 2789 . . . . . . . . . 10 (𝑘 = 0 → (0 − 𝑘) = 0)
6362oveq2d 7441 . . . . . . . . 9 (𝑘 = 0 → (𝐴↑(0 − 𝑘)) = (𝐴↑0))
64 oveq2 7433 . . . . . . . . 9 (𝑘 = 0 → (𝐵𝑘) = (𝐵↑0))
6563, 64oveq12d 7443 . . . . . . . 8 (𝑘 = 0 → ((𝐴↑(0 − 𝑘)) · (𝐵𝑘)) = ((𝐴↑0) · (𝐵↑0)))
6659, 65oveq12d 7443 . . . . . . 7 (𝑘 = 0 → ((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
6766fsum1 15769 . . . . . 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 586 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))) = (1 · ((𝐴↑0) · (𝐵↑0))))
69 addcl 11228 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
7069exp0d 14166 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = 1)
7151, 68, 703eqtr4rd 2784 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑0) = Σ𝑘 ∈ (0...0)((0C𝑘) · ((𝐴↑(0 − 𝑘)) · (𝐵𝑘))))
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↑(𝑛𝑘)) · (𝐵𝑘))) → ((𝐴 + 𝐵)↑𝑛) = Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((𝐴↑(𝑛𝑘)) · (𝐵𝑘))))
7672, 73, 74, 75binomlem 15851 . . . . . 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 419 . . . . 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 12704 . . 3 (𝑁 ∈ ℕ0 → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘)))))
8079impcom 407 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑁 ∈ ℕ0) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))
81803impa 1108 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 395  w3a 1085   = wceq 1535  wcel 2104  (class class class)co 7425  cc 11144  0cc0 11146  1c1 11147   + caddc 11149   · cmul 11151  cmin 11483  0cn0 12517  cz 12604  ...cfz 13537  cexp 14088  Ccbc 14327  Σcsu 15708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5366  ax-pr 5430  ax-un 7747  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-int 4954  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-se 5636  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6317  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7381  df-ov 7428  df-oprab 7429  df-mpo 7430  df-om 7881  df-1st 8007  df-2nd 8008  df-frecs 8299  df-wrecs 8330  df-recs 8404  df-rdg 8443  df-1o 8499  df-er 8738  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-sup 9473  df-oi 9541  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11485  df-neg 11486  df-div 11912  df-nn 12258  df-2 12320  df-3 12321  df-n0 12518  df-z 12605  df-uz 12870  df-rp 13026  df-fz 13538  df-fzo 13682  df-seq 14029  df-exp 14089  df-fac 14299  df-bc 14328  df-hash 14356  df-cj 15124  df-re 15125  df-im 15126  df-sqrt 15260  df-abs 15261  df-clim 15510  df-sum 15709
This theorem is referenced by:  binom1p  15853  efaddlem  16115  basellem3  27122  lcmineqlem1  41972  jm2.22  42938  binomcxplemnn0  44304  altgsumbc  48115
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