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Theorem bcxmas 15182
Description: Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
Assertion
Ref Expression
bcxmas ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
Distinct variable groups:   𝑗,𝑀   𝑗,𝑁

Proof of Theorem bcxmas
Dummy variables 𝑚 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bcxmaslem1 15181 . . 3 (𝑚 = 0 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 0)C0))
2 oveq2 7156 . . . 4 (𝑚 = 0 → (0...𝑚) = (0...0))
32sumeq1d 15050 . . 3 (𝑚 = 0 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
41, 3eqeq12d 2835 . 2 (𝑚 = 0 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗)))
5 bcxmaslem1 15181 . . 3 (𝑚 = 𝑘 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑘)C𝑘))
6 oveq2 7156 . . . 4 (𝑚 = 𝑘 → (0...𝑚) = (0...𝑘))
76sumeq1d 15050 . . 3 (𝑚 = 𝑘 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗))
85, 7eqeq12d 2835 . 2 (𝑚 = 𝑘 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)))
9 bcxmaslem1 15181 . . 3 (𝑚 = (𝑘 + 1) → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
10 oveq2 7156 . . . 4 (𝑚 = (𝑘 + 1) → (0...𝑚) = (0...(𝑘 + 1)))
1110sumeq1d 15050 . . 3 (𝑚 = (𝑘 + 1) → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
129, 11eqeq12d 2835 . 2 (𝑚 = (𝑘 + 1) → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗)))
13 bcxmaslem1 15181 . . 3 (𝑚 = 𝑀 → (((𝑁 + 1) + 𝑚)C𝑚) = (((𝑁 + 1) + 𝑀)C𝑀))
14 oveq2 7156 . . . 4 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
1514sumeq1d 15050 . . 3 (𝑚 = 𝑀 → Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
1613, 15eqeq12d 2835 . 2 (𝑚 = 𝑀 → ((((𝑁 + 1) + 𝑚)C𝑚) = Σ𝑗 ∈ (0...𝑚)((𝑁 + 𝑗)C𝑗) ↔ (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗)))
17 0nn0 11904 . . . 4 0 ∈ ℕ0
18 nn0addcl 11924 . . . . 5 ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → (𝑁 + 0) ∈ ℕ0)
19 bcn0 13662 . . . . 5 ((𝑁 + 0) ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
2018, 19syl 17 . . . 4 ((𝑁 ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 0)C0) = 1)
2117, 20mpan2 689 . . 3 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) = 1)
22 0z 11984 . . . 4 0 ∈ ℤ
23 1nn0 11905 . . . . . 6 1 ∈ ℕ0
2421, 23syl6eqel 2919 . . . . 5 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℕ0)
2524nn0cnd 11949 . . . 4 (𝑁 ∈ ℕ0 → ((𝑁 + 0)C0) ∈ ℂ)
26 bcxmaslem1 15181 . . . . 5 (𝑗 = 0 → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
2726fsum1 15094 . . . 4 ((0 ∈ ℤ ∧ ((𝑁 + 0)C0) ∈ ℂ) → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
2822, 25, 27sylancr 589 . . 3 (𝑁 ∈ ℕ0 → Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗) = ((𝑁 + 0)C0))
29 peano2nn0 11929 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
30 nn0addcl 11924 . . . . 5 (((𝑁 + 1) ∈ ℕ0 ∧ 0 ∈ ℕ0) → ((𝑁 + 1) + 0) ∈ ℕ0)
3129, 17, 30sylancl 588 . . . 4 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 0) ∈ ℕ0)
32 bcn0 13662 . . . 4 (((𝑁 + 1) + 0) ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
3331, 32syl 17 . . 3 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = 1)
3421, 28, 333eqtr4rd 2865 . 2 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 0)C0) = Σ𝑗 ∈ (0...0)((𝑁 + 𝑗)C𝑗))
35 simpr 487 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
36 elnn0uz 12275 . . . . . . 7 (𝑘 ∈ ℕ0𝑘 ∈ (ℤ‘0))
3735, 36sylib 220 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ‘0))
38 simpl 485 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
39 elfznn0 12992 . . . . . . . . 9 (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℕ0)
40 nn0addcl 11924 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑗 ∈ ℕ0) → (𝑁 + 𝑗) ∈ ℕ0)
4138, 39, 40syl2an 597 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → (𝑁 + 𝑗) ∈ ℕ0)
42 elfzelz 12900 . . . . . . . . 9 (𝑗 ∈ (0...(𝑘 + 1)) → 𝑗 ∈ ℤ)
4342adantl 484 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → 𝑗 ∈ ℤ)
44 bccl 13674 . . . . . . . 8 (((𝑁 + 𝑗) ∈ ℕ0𝑗 ∈ ℤ) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
4541, 43, 44syl2anc 586 . . . . . . 7 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℕ0)
4645nn0cnd 11949 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...(𝑘 + 1))) → ((𝑁 + 𝑗)C𝑗) ∈ ℂ)
47 bcxmaslem1 15181 . . . . . 6 (𝑗 = (𝑘 + 1) → ((𝑁 + 𝑗)C𝑗) = ((𝑁 + (𝑘 + 1))C(𝑘 + 1)))
4837, 46, 47fsump1 15103 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + ((𝑁 + (𝑘 + 1))C(𝑘 + 1))))
49 nn0cn 11899 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
5049adantr 483 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑁 ∈ ℂ)
51 nn0cn 11899 . . . . . . . . 9 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
5251adantl 484 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
53 1cnd 10628 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → 1 ∈ ℂ)
54 add32r 10851 . . . . . . . 8 ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
5550, 52, 53, 54syl3anc 1365 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑁 + (𝑘 + 1)) = ((𝑁 + 1) + 𝑘))
5655oveq1d 7163 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + (𝑘 + 1))C(𝑘 + 1)) = (((𝑁 + 1) + 𝑘)C(𝑘 + 1)))
5756oveq2d 7164 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + ((𝑁 + (𝑘 + 1))C(𝑘 + 1))) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
5848, 57eqtrd 2854 . . . 4 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
5958adantr 483 . . 3 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
60 oveq1 7155 . . . 4 ((((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
6160adantl 484 . . 3 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
62 ax-1cn 10587 . . . . . . . . 9 1 ∈ ℂ
63 pncan 10884 . . . . . . . . 9 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
6452, 62, 63sylancl 588 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑘 + 1) − 1) = 𝑘)
6564oveq2d 7164 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1)) = (((𝑁 + 1) + 𝑘)C𝑘))
6665oveq2d 7164 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1))) = ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)))
67 nn0addcl 11924 . . . . . . . 8 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
6829, 67sylan 582 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
69 nn0p1nn 11928 . . . . . . . . 9 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
7069adantl 484 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ)
7170nnzd 12078 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℤ)
72 bcpasc 13673 . . . . . . 7 ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1))) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
7368, 71, 72syl2anc 586 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C((𝑘 + 1) − 1))) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
7466, 73eqtr3d 2856 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)))
75 nn0p1nn 11928 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
76 nnnn0addcl 11919 . . . . . . . . . 10 (((𝑁 + 1) ∈ ℕ ∧ 𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
7775, 76sylan 582 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ)
7877nnnn0d 11947 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((𝑁 + 1) + 𝑘) ∈ ℕ0)
79 bccl 13674 . . . . . . . 8 ((((𝑁 + 1) + 𝑘) ∈ ℕ0 ∧ (𝑘 + 1) ∈ ℤ) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
8078, 71, 79syl2anc 586 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℕ0)
8180nn0cnd 11949 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C(𝑘 + 1)) ∈ ℂ)
82 nn0z 11997 . . . . . . . . . 10 (𝑘 ∈ ℕ0𝑘 ∈ ℤ)
8382adantl 484 . . . . . . . . 9 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
84 bccl 13674 . . . . . . . . 9 ((((𝑁 + 1) + 𝑘) ∈ ℕ0𝑘 ∈ ℤ) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
8567, 83, 84syl2anc 586 . . . . . . . 8 (((𝑁 + 1) ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
8629, 85sylan 582 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℕ0)
8786nn0cnd 11949 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘)C𝑘) ∈ ℂ)
8881, 87addcomd 10834 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C(𝑘 + 1)) + (((𝑁 + 1) + 𝑘)C𝑘)) = ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))))
89 peano2cn 10804 . . . . . . . . 9 (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
9049, 89syl 17 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
9190adantr 483 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (𝑁 + 1) ∈ ℂ)
9291, 52, 53addassd 10655 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → (((𝑁 + 1) + 𝑘) + 1) = ((𝑁 + 1) + (𝑘 + 1)))
9392oveq1d 7163 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘) + 1)C(𝑘 + 1)) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
9474, 88, 933eqtr3d 2862 . . . 4 ((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
9594adantr 483 . . 3 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → ((((𝑁 + 1) + 𝑘)C𝑘) + (((𝑁 + 1) + 𝑘)C(𝑘 + 1))) = (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)))
9659, 61, 953eqtr2rd 2861 . 2 (((𝑁 ∈ ℕ0𝑘 ∈ ℕ0) ∧ (((𝑁 + 1) + 𝑘)C𝑘) = Σ𝑗 ∈ (0...𝑘)((𝑁 + 𝑗)C𝑗)) → (((𝑁 + 1) + (𝑘 + 1))C(𝑘 + 1)) = Σ𝑗 ∈ (0...(𝑘 + 1))((𝑁 + 𝑗)C𝑗))
974, 8, 12, 16, 34, 96nn0indd 12071 1 ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → (((𝑁 + 1) + 𝑀)C𝑀) = Σ𝑗 ∈ (0...𝑀)((𝑁 + 𝑗)C𝑗))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  cfv 6348  (class class class)co 7148  cc 10527  0cc0 10529  1c1 10530   + caddc 10532  cmin 10862  cn 11630  0cn0 11889  cz 11973  cuz 12235  ...cfz 12884  Ccbc 13654  Σcsu 15034
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 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 209  df-an 399  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 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12885  df-fzo 13026  df-seq 13362  df-exp 13422  df-fac 13626  df-bc 13655  df-hash 13683  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035
This theorem is referenced by:  arisum  15207
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