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Theorem nndi 6229
Description: Distributive law for natural numbers (left-distributivity). Theorem 4K(3) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nndi ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))

Proof of Theorem nndi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5642 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
21oveq2d 5650 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
3 oveq2 5642 . . . . . . 7 (𝑥 = 𝐶 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐶))
43oveq2d 5650 . . . . . 6 (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
52, 4eqeq12d 2102 . . . . 5 (𝑥 = 𝐶 → ((𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) ↔ (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶))))
65imbi2d 228 . . . 4 (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))))
7 oveq2 5642 . . . . . . 7 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
87oveq2d 5650 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 ∅)))
9 oveq2 5642 . . . . . . 7 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
109oveq2d 5650 . . . . . 6 (𝑥 = ∅ → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)))
118, 10eqeq12d 2102 . . . . 5 (𝑥 = ∅ → ((𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) ↔ (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅))))
12 oveq2 5642 . . . . . . 7 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
1312oveq2d 5650 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝑦)))
14 oveq2 5642 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
1514oveq2d 5650 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)))
1613, 15eqeq12d 2102 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) ↔ (𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦))))
17 oveq2 5642 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1817oveq2d 5650 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)))
19 oveq2 5642 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
2019oveq2d 5650 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦)))
2118, 20eqeq12d 2102 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) ↔ (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦))))
22 nna0 6217 . . . . . . . . 9 (𝐵 ∈ ω → (𝐵 +𝑜 ∅) = 𝐵)
2322adantl 271 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 ∅) = 𝐵)
2423oveq2d 5650 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = (𝐴 ·𝑜 𝐵))
25 nnmcl 6224 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)
26 nna0 6217 . . . . . . . 8 ((𝐴 ·𝑜 𝐵) ∈ ω → ((𝐴 ·𝑜 𝐵) +𝑜 ∅) = (𝐴 ·𝑜 𝐵))
2725, 26syl 14 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 ∅) = (𝐴 ·𝑜 𝐵))
2824, 27eqtr4d 2123 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 ∅))
29 nnm0 6218 . . . . . . . 8 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
3029adantr 270 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 ∅) = ∅)
3130oveq2d 5650 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 ∅))
3228, 31eqtr4d 2123 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)))
33 oveq1 5641 . . . . . . . . 9 ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) → ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴) = (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴))
34 nnasuc 6219 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
35343adant1 961 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3635oveq2d 5650 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)))
37 nnacl 6223 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
38 nnmsuc 6220 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ (𝐵 +𝑜 𝑦) ∈ ω) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
3937, 38sylan2 280 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
40393impb 1139 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
4136, 40eqtrd 2120 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
42 nnmsuc 6220 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
43423adant2 962 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
4443oveq2d 5650 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 ((𝐴 ·𝑜 𝑦) +𝑜 𝐴)))
45 nnmcl 6224 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 𝑦) ∈ ω)
46 nnaass 6228 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ·𝑜 𝐵) ∈ ω ∧ (𝐴 ·𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 ((𝐴 ·𝑜 𝑦) +𝑜 𝐴)))
4725, 46syl3an1 1207 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 ((𝐴 ·𝑜 𝑦) +𝑜 𝐴)))
4845, 47syl3an2 1208 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ∈ ω ∧ 𝑦 ∈ ω) ∧ 𝐴 ∈ ω) → (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 ((𝐴 ·𝑜 𝑦) +𝑜 𝐴)))
49483exp 1142 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ∈ ω → (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 ((𝐴 ·𝑜 𝑦) +𝑜 𝐴)))))
5049exp4b 359 . . . . . . . . . . . . . . 15 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ∈ ω → (𝑦 ∈ ω → (𝐴 ∈ ω → (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 ((𝐴 ·𝑜 𝑦) +𝑜 𝐴)))))))
5150pm2.43a 50 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (𝐴 ∈ ω → (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))))))
5251com4r 85 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))))))
5352pm2.43i 48 . . . . . . . . . . . 12 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 ((𝐴 ·𝑜 𝑦) +𝑜 𝐴)))))
54533imp 1137 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 ((𝐴 ·𝑜 𝑦) +𝑜 𝐴)))
5544, 54eqtr4d 2123 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦)) = (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴))
5641, 55eqeq12d 2102 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦)) ↔ ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴) = (((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) +𝑜 𝐴)))
5733, 56syl5ibr 154 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦))))
58573exp 1142 . . . . . . 7 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦))))))
5958com3r 78 . . . . . 6 (𝑦 ∈ ω → (𝐴 ∈ ω → (𝐵 ∈ ω → ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦))))))
6059impd 251 . . . . 5 (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦)))))
6111, 16, 21, 32, 60finds2 4406 . . . 4 (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥))))
626, 61vtoclga 2685 . . 3 (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶))))
6362expdcom 1376 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐶 ∈ ω → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))))
64633imp 1137 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924   = wceq 1289  wcel 1438  c0 3284  suc csuc 4183  ωcom 4395  (class class class)co 5634   +𝑜 coa 6160   ·𝑜 comu 6161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-oadd 6167  df-omul 6168
This theorem is referenced by:  nnmass  6230  nndir  6233  distrpig  6871  addcmpblnq0  6981  nnanq0  6996  distrnq0  6997  addassnq0  7000
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