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Theorem nnmass 7591
Description: Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnmass ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))

Proof of Theorem nnmass
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6557 . . . . . 6 (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶))
2 oveq2 6557 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐶))
32oveq2d 6565 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))
41, 3eqeq12d 2625 . . . . 5 (𝑥 = 𝐶 → (((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶))))
54imbi2d 329 . . . 4 (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))))
6 oveq2 6557 . . . . . 6 (𝑥 = ∅ → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 ∅))
7 oveq2 6557 . . . . . . 7 (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅))
87oveq2d 6565 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)))
96, 8eqeq12d 2625 . . . . 5 (𝑥 = ∅ → (((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = (𝐴 ·𝑜 (𝐵 ·𝑜 ∅))))
10 oveq2 6557 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦))
11 oveq2 6557 . . . . . . 7 (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦))
1211oveq2d 6565 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)))
1310, 12eqeq12d 2625 . . . . 5 (𝑥 = 𝑦 → (((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦))))
14 oveq2 6557 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦))
15 oveq2 6557 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦))
1615oveq2d 6565 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)))
1714, 16eqeq12d 2625 . . . . 5 (𝑥 = suc 𝑦 → (((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥)) ↔ ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦))))
18 nnmcl 7579 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)
19 nnm0 7572 . . . . . . 7 ((𝐴 ·𝑜 𝐵) ∈ ω → ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = ∅)
2018, 19syl 17 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = ∅)
21 nnm0 7572 . . . . . . . 8 (𝐵 ∈ ω → (𝐵 ·𝑜 ∅) = ∅)
2221oveq2d 6565 . . . . . . 7 (𝐵 ∈ ω → (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)) = (𝐴 ·𝑜 ∅))
23 nnm0 7572 . . . . . . 7 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
2422, 23sylan9eqr 2666 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)) = ∅)
2520, 24eqtr4d 2647 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 ∅) = (𝐴 ·𝑜 (𝐵 ·𝑜 ∅)))
26 oveq1 6556 . . . . . . . . 9 (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) +𝑜 (𝐴 ·𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))
27 nnmsuc 7574 . . . . . . . . . . 11 (((𝐴 ·𝑜 𝐵) ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) +𝑜 (𝐴 ·𝑜 𝐵)))
2818, 27stoic3 1692 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) +𝑜 (𝐴 ·𝑜 𝐵)))
29 nnmsuc 7574 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
30293adant1 1072 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
3130oveq2d 6565 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)) = (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
32 nnmcl 7579 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·𝑜 𝑦) ∈ ω)
33 nndi 7590 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ω ∧ (𝐵 ·𝑜 𝑦) ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))
3432, 33syl3an2 1352 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω) ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))
35343exp 1256 . . . . . . . . . . . . . . 15 (𝐴 ∈ ω → ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ∈ ω → (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))))
3635expd 451 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (𝐵 ∈ ω → (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵))))))
3736com34 89 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵))))))
3837pm2.43d 51 . . . . . . . . . . . 12 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))))
39383imp 1249 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))
4031, 39eqtrd 2644 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵)))
4128, 40eqeq12d 2625 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)) ↔ (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) +𝑜 (𝐴 ·𝑜 𝐵)) = ((𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) +𝑜 (𝐴 ·𝑜 𝐵))))
4226, 41syl5ibr 235 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦))))
43423exp 1256 . . . . . . 7 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦))))))
4443com3r 85 . . . . . 6 (𝑦 ∈ ω → (𝐴 ∈ ω → (𝐵 ∈ ω → (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦))))))
4544impd 446 . . . . 5 (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 ·𝑜 𝐵) ·𝑜 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝐵) ·𝑜 suc 𝑦) = (𝐴 ·𝑜 (𝐵 ·𝑜 suc 𝑦)))))
469, 13, 17, 25, 45finds2 6986 . . . 4 (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝑥) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝑥))))
475, 46vtoclga 3245 . . 3 (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶))))
4847expdcom 454 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐶 ∈ ω → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))))
49483imp 1249 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·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:  mulasspi  9598
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