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Theorem nnaass 6064
 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 5520 . . . . . 6 (𝑥 = 𝐶 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 𝐶))
2 oveq2 5520 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
32oveq2d 5528 . . . . . 6 (𝑥 = 𝐶 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))
41, 3eqeq12d 2054 . . . . 5 (𝑥 = 𝐶 → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))
54imbi2d 219 . . . 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 5520 . . . . . 6 (𝑥 = ∅ → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 ∅))
7 oveq2 5520 . . . . . . 7 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
87oveq2d 5528 . . . . . 6 (𝑥 = ∅ → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 ∅)))
96, 8eqeq12d 2054 . . . . 5 (𝑥 = ∅ → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 (𝐵 +𝑜 ∅))))
10 oveq2 5520 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
11 oveq2 5520 . . . . . . 7 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
1211oveq2d 5528 . . . . . 6 (𝑥 = 𝑦 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
1310, 12eqeq12d 2054 . . . . 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 5520 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦))
15 oveq2 5520 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1615oveq2d 5528 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)))
1714, 16eqeq12d 2054 . . . . 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 6059 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω)
19 nna0 6053 . . . . . . 7 ((𝐴 +𝑜 𝐵) ∈ ω → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 𝐵))
2018, 19syl 14 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 𝐵))
21 nna0 6053 . . . . . . . 8 (𝐵 ∈ ω → (𝐵 +𝑜 ∅) = 𝐵)
2221oveq2d 5528 . . . . . . 7 (𝐵 ∈ ω → (𝐴 +𝑜 (𝐵 +𝑜 ∅)) = (𝐴 +𝑜 𝐵))
2322adantl 262 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 (𝐵 +𝑜 ∅)) = (𝐴 +𝑜 𝐵))
2420, 23eqtr4d 2075 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 (𝐵 +𝑜 ∅)))
25 suceq 4139 . . . . . . 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 6055 . . . . . . . . 9 (((𝐴 +𝑜 𝐵) ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
2718, 26sylan 267 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
28 nnasuc 6055 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
2928oveq2d 5528 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)))
3029adantl 262 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)))
31 nnacl 6059 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
32 nnasuc 6055 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ (𝐵 +𝑜 𝑦) ∈ ω) → (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3331, 32sylan2 270 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3430, 33eqtrd 2072 . . . . . . . . 9 ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3534anassrs 380 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3627, 35eqeq12d 2054 . . . . . . 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 145 . . . . . 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 109 . . . . 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 4324 . . . 4 (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥))))
405, 39vtoclga 2619 . . 3 (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))
4140com12 27 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ∈ ω → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))
42413impia 1101 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∅c0 3224  suc csuc 4102  ωcom 4313  (class class class)co 5512   +𝑜 coa 5998 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005 This theorem is referenced by:  nndi  6065  nnmsucr  6067  addasspig  6428  addassnq0  6560  prarloclemlo  6592
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