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Theorem nnaass 7589
Description: Addition of natural numbers is associative. Theorem 4K(1) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnaass ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))

Proof of Theorem nnaass
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 nnacl 7578 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω)
19 nna0 7571 . . . . . . 7 ((𝐴 +𝑜 𝐵) ∈ ω → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 𝐵))
2018, 19syl 17 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 𝐵))
21 nna0 7571 . . . . . . . 8 (𝐵 ∈ ω → (𝐵 +𝑜 ∅) = 𝐵)
2221oveq2d 6565 . . . . . . 7 (𝐵 ∈ ω → (𝐴 +𝑜 (𝐵 +𝑜 ∅)) = (𝐴 +𝑜 𝐵))
2322adantl 481 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 (𝐵 +𝑜 ∅)) = (𝐴 +𝑜 𝐵))
2420, 23eqtr4d 2647 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 (𝐵 +𝑜 ∅)))
25 suceq 5707 . . . . . . 7 (((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
26 nnasuc 7573 . . . . . . . . 9 (((𝐴 +𝑜 𝐵) ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
2718, 26sylan 487 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
28 nnasuc 7573 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
2928oveq2d 6565 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)))
3029adantl 481 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)))
31 nnacl 7578 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
32 nnasuc 7573 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ (𝐵 +𝑜 𝑦) ∈ ω) → (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3331, 32sylan2 490 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3430, 33eqtrd 2644 . . . . . . . . 9 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3534anassrs 678 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3627, 35eqeq12d 2625 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → (((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) ↔ suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
3725, 36syl5ibr 235 . . . . . 6 (((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))))
3837expcom 450 . . . . 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)))))
399, 13, 17, 24, 38finds2 6986 . . . 4 (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥))))
405, 39vtoclga 3245 . . 3 (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))
4140com12 32 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ∈ ω → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))
42413impia 1253 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
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
This theorem is referenced by:  nndi  7590  nnmsucr  7592  omopthlem1  7622  omopthlem2  7623  addasspi  9596
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