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Theorem nnacom 8623
Description: Addition of natural numbers is commutative. Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 6-May-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnacom ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))

Proof of Theorem nnacom
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7419 . . . . 5 (𝑥 = 𝐴 → (𝑥 +o 𝐵) = (𝐴 +o 𝐵))
2 oveq2 7420 . . . . 5 (𝑥 = 𝐴 → (𝐵 +o 𝑥) = (𝐵 +o 𝐴))
31, 2eqeq12d 2747 . . . 4 (𝑥 = 𝐴 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
43imbi2d 340 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 +o 𝐵) = (𝐵 +o 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))))
5 oveq1 7419 . . . . 5 (𝑥 = ∅ → (𝑥 +o 𝐵) = (∅ +o 𝐵))
6 oveq2 7420 . . . . 5 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
75, 6eqeq12d 2747 . . . 4 (𝑥 = ∅ → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (∅ +o 𝐵) = (𝐵 +o ∅)))
8 oveq1 7419 . . . . 5 (𝑥 = 𝑦 → (𝑥 +o 𝐵) = (𝑦 +o 𝐵))
9 oveq2 7420 . . . . 5 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
108, 9eqeq12d 2747 . . . 4 (𝑥 = 𝑦 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (𝑦 +o 𝐵) = (𝐵 +o 𝑦)))
11 oveq1 7419 . . . . 5 (𝑥 = suc 𝑦 → (𝑥 +o 𝐵) = (suc 𝑦 +o 𝐵))
12 oveq2 7420 . . . . 5 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1311, 12eqeq12d 2747 . . . 4 (𝑥 = suc 𝑦 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦)))
14 nna0r 8615 . . . . 5 (𝐵 ∈ ω → (∅ +o 𝐵) = 𝐵)
15 nna0 8610 . . . . 5 (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
1614, 15eqtr4d 2774 . . . 4 (𝐵 ∈ ω → (∅ +o 𝐵) = (𝐵 +o ∅))
17 suceq 6430 . . . . . 6 ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → suc (𝑦 +o 𝐵) = suc (𝐵 +o 𝑦))
18 oveq2 7420 . . . . . . . . . . 11 (𝑥 = 𝐵 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o 𝐵))
19 oveq2 7420 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑦 +o 𝑥) = (𝑦 +o 𝐵))
20 suceq 6430 . . . . . . . . . . . 12 ((𝑦 +o 𝑥) = (𝑦 +o 𝐵) → suc (𝑦 +o 𝑥) = suc (𝑦 +o 𝐵))
2119, 20syl 17 . . . . . . . . . . 11 (𝑥 = 𝐵 → suc (𝑦 +o 𝑥) = suc (𝑦 +o 𝐵))
2218, 21eqeq12d 2747 . . . . . . . . . 10 (𝑥 = 𝐵 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
2322imbi2d 340 . . . . . . . . 9 (𝑥 = 𝐵 → ((𝑦 ∈ ω → (suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥)) ↔ (𝑦 ∈ ω → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵))))
24 oveq2 7420 . . . . . . . . . . 11 (𝑥 = ∅ → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o ∅))
25 oveq2 7420 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑦 +o 𝑥) = (𝑦 +o ∅))
26 suceq 6430 . . . . . . . . . . . 12 ((𝑦 +o 𝑥) = (𝑦 +o ∅) → suc (𝑦 +o 𝑥) = suc (𝑦 +o ∅))
2725, 26syl 17 . . . . . . . . . . 11 (𝑥 = ∅ → suc (𝑦 +o 𝑥) = suc (𝑦 +o ∅))
2824, 27eqeq12d 2747 . . . . . . . . . 10 (𝑥 = ∅ → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o ∅) = suc (𝑦 +o ∅)))
29 oveq2 7420 . . . . . . . . . . 11 (𝑥 = 𝑧 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o 𝑧))
30 oveq2 7420 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑦 +o 𝑥) = (𝑦 +o 𝑧))
31 suceq 6430 . . . . . . . . . . . 12 ((𝑦 +o 𝑥) = (𝑦 +o 𝑧) → suc (𝑦 +o 𝑥) = suc (𝑦 +o 𝑧))
3230, 31syl 17 . . . . . . . . . . 11 (𝑥 = 𝑧 → suc (𝑦 +o 𝑥) = suc (𝑦 +o 𝑧))
3329, 32eqeq12d 2747 . . . . . . . . . 10 (𝑥 = 𝑧 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧)))
34 oveq2 7420 . . . . . . . . . . 11 (𝑥 = suc 𝑧 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o suc 𝑧))
35 oveq2 7420 . . . . . . . . . . . 12 (𝑥 = suc 𝑧 → (𝑦 +o 𝑥) = (𝑦 +o suc 𝑧))
36 suceq 6430 . . . . . . . . . . . 12 ((𝑦 +o 𝑥) = (𝑦 +o suc 𝑧) → suc (𝑦 +o 𝑥) = suc (𝑦 +o suc 𝑧))
3735, 36syl 17 . . . . . . . . . . 11 (𝑥 = suc 𝑧 → suc (𝑦 +o 𝑥) = suc (𝑦 +o suc 𝑧))
3834, 37eqeq12d 2747 . . . . . . . . . 10 (𝑥 = suc 𝑧 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o suc 𝑧) = suc (𝑦 +o suc 𝑧)))
39 peano2 7885 . . . . . . . . . . . 12 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
40 nna0 8610 . . . . . . . . . . . 12 (suc 𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc 𝑦)
4139, 40syl 17 . . . . . . . . . . 11 (𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc 𝑦)
42 nna0 8610 . . . . . . . . . . . 12 (𝑦 ∈ ω → (𝑦 +o ∅) = 𝑦)
43 suceq 6430 . . . . . . . . . . . 12 ((𝑦 +o ∅) = 𝑦 → suc (𝑦 +o ∅) = suc 𝑦)
4442, 43syl 17 . . . . . . . . . . 11 (𝑦 ∈ ω → suc (𝑦 +o ∅) = suc 𝑦)
4541, 44eqtr4d 2774 . . . . . . . . . 10 (𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc (𝑦 +o ∅))
46 suceq 6430 . . . . . . . . . . . 12 ((suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧) → suc (suc 𝑦 +o 𝑧) = suc suc (𝑦 +o 𝑧))
47 nnasuc 8612 . . . . . . . . . . . . . 14 ((suc 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +o suc 𝑧) = suc (suc 𝑦 +o 𝑧))
4839, 47sylan 579 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +o suc 𝑧) = suc (suc 𝑦 +o 𝑧))
49 nnasuc 8612 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 +o suc 𝑧) = suc (𝑦 +o 𝑧))
50 suceq 6430 . . . . . . . . . . . . . 14 ((𝑦 +o suc 𝑧) = suc (𝑦 +o 𝑧) → suc (𝑦 +o suc 𝑧) = suc suc (𝑦 +o 𝑧))
5149, 50syl 17 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → suc (𝑦 +o suc 𝑧) = suc suc (𝑦 +o 𝑧))
5248, 51eqeq12d 2747 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((suc 𝑦 +o suc 𝑧) = suc (𝑦 +o suc 𝑧) ↔ suc (suc 𝑦 +o 𝑧) = suc suc (𝑦 +o 𝑧)))
5346, 52imbitrrid 245 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧) → (suc 𝑦 +o suc 𝑧) = suc (𝑦 +o suc 𝑧)))
5453expcom 413 . . . . . . . . . 10 (𝑧 ∈ ω → (𝑦 ∈ ω → ((suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧) → (suc 𝑦 +o suc 𝑧) = suc (𝑦 +o suc 𝑧))))
5528, 33, 38, 45, 54finds2 7895 . . . . . . . . 9 (𝑥 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥)))
5623, 55vtoclga 3566 . . . . . . . 8 (𝐵 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
5756imp 406 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵))
58 nnasuc 8612 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
5957, 58eqeq12d 2747 . . . . . 6 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦) ↔ suc (𝑦 +o 𝐵) = suc (𝐵 +o 𝑦)))
6017, 59imbitrrid 245 . . . . 5 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦)))
6160expcom 413 . . . 4 (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦))))
627, 10, 13, 16, 61finds2 7895 . . 3 (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 +o 𝐵) = (𝐵 +o 𝑥)))
634, 62vtoclga 3566 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
6463imp 406 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  c0 4322  suc csuc 6366  (class class class)co 7412  ωcom 7859   +o coa 8469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-oadd 8476
This theorem is referenced by:  nnaordr  8626  nnmsucr  8631  nnaword2  8636  omopthlem2  8665  omopthi  8666  eldifsucnn  8669  addcompi  10895  finxpreclem4  36741  oaabsb  42509  ofoacom  42576  naddcnfcom  42581
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