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Theorem nndi 7590
 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 6557 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
21oveq2d 6565 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
3 oveq2 6557 . . . . . . 7 (𝑥 = 𝐶 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐶))
43oveq2d 6565 . . . . . 6 (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
52, 4eqeq12d 2625 . . . . 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 329 . . . 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 6557 . . . . . . 7 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
87oveq2d 6565 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 ∅)))
9 oveq2 6557 . . . . . . 7 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
109oveq2d 6565 . . . . . 6 (𝑥 = ∅ → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)))
118, 10eqeq12d 2625 . . . . 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 6557 . . . . . . 7 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
1312oveq2d 6565 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝑦)))
14 oveq2 6557 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
1514oveq2d 6565 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)))
1613, 15eqeq12d 2625 . . . . 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 6557 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1817oveq2d 6565 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)))
19 oveq2 6557 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
2019oveq2d 6565 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦)))
2118, 20eqeq12d 2625 . . . . 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 7571 . . . . . . . . 9 (𝐵 ∈ ω → (𝐵 +𝑜 ∅) = 𝐵)
2322adantl 481 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 ∅) = 𝐵)
2423oveq2d 6565 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = (𝐴 ·𝑜 𝐵))
25 nnmcl 7579 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)
26 nna0 7571 . . . . . . . 8 ((𝐴 ·𝑜 𝐵) ∈ ω → ((𝐴 ·𝑜 𝐵) +𝑜 ∅) = (𝐴 ·𝑜 𝐵))
2725, 26syl 17 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 ∅) = (𝐴 ·𝑜 𝐵))
2824, 27eqtr4d 2647 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 ∅))
29 nnm0 7572 . . . . . . . 8 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
3029adantr 480 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 ∅) = ∅)
3130oveq2d 6565 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 ∅))
3228, 31eqtr4d 2647 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)))
33 oveq1 6556 . . . . . . . . 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 7573 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
35343adant1 1072 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3635oveq2d 6565 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)))
37 nnacl 7578 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
38 nnmsuc 7574 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ (𝐵 +𝑜 𝑦) ∈ ω) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
3937, 38sylan2 490 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
40393impb 1252 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
4136, 40eqtrd 2644 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
42 nnmsuc 7574 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
43423adant2 1073 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
4443oveq2d 6565 . . . . . . . . . . 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 7579 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 𝑦) ∈ ω)
46 nnaass 7589 . . . . . . . . . . . . . . . . . . 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 1351 . . . . . . . . . . . . . . . . . 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 1352 . . . . . . . . . . . . . . . . 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 1256 . . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . . 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 52 . . . . . . . . . . . . . 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 92 . . . . . . . . . . . . 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 50 . . . . . . . . . . . 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 1249 . . . . . . . . . . 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 2647 . . . . . . . . . 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 2625 . . . . . . . . 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 235 . . . . . . . 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 1256 . . . . . . 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 85 . . . . . 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 446 . . . . 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 6986 . . . 4 (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥))))
626, 61vtoclga 3245 . . 3 (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶))))
6362expdcom 454 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐶 ∈ ω → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))))
64633imp 1249 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∅c0 3874  suc csuc 5642  (class class class)co 6549  ωcom 6957   +𝑜 coa 7444   ·𝑜 comu 7445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452 This theorem is referenced by:  nnmass  7591  distrpi  9599
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