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Theorem nnacom 8233
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 7152 . . . . 5 (𝑥 = 𝐴 → (𝑥 +o 𝐵) = (𝐴 +o 𝐵))
2 oveq2 7153 . . . . 5 (𝑥 = 𝐴 → (𝐵 +o 𝑥) = (𝐵 +o 𝐴))
31, 2eqeq12d 2837 . . . 4 (𝑥 = 𝐴 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
43imbi2d 342 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 +o 𝐵) = (𝐵 +o 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))))
5 oveq1 7152 . . . . 5 (𝑥 = ∅ → (𝑥 +o 𝐵) = (∅ +o 𝐵))
6 oveq2 7153 . . . . 5 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
75, 6eqeq12d 2837 . . . 4 (𝑥 = ∅ → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (∅ +o 𝐵) = (𝐵 +o ∅)))
8 oveq1 7152 . . . . 5 (𝑥 = 𝑦 → (𝑥 +o 𝐵) = (𝑦 +o 𝐵))
9 oveq2 7153 . . . . 5 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
108, 9eqeq12d 2837 . . . 4 (𝑥 = 𝑦 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (𝑦 +o 𝐵) = (𝐵 +o 𝑦)))
11 oveq1 7152 . . . . 5 (𝑥 = suc 𝑦 → (𝑥 +o 𝐵) = (suc 𝑦 +o 𝐵))
12 oveq2 7153 . . . . 5 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1311, 12eqeq12d 2837 . . . 4 (𝑥 = suc 𝑦 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦)))
14 nna0r 8225 . . . . 5 (𝐵 ∈ ω → (∅ +o 𝐵) = 𝐵)
15 nna0 8220 . . . . 5 (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
1614, 15eqtr4d 2859 . . . 4 (𝐵 ∈ ω → (∅ +o 𝐵) = (𝐵 +o ∅))
17 suceq 6250 . . . . . 6 ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → suc (𝑦 +o 𝐵) = suc (𝐵 +o 𝑦))
18 oveq2 7153 . . . . . . . . . . 11 (𝑥 = 𝐵 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o 𝐵))
19 oveq2 7153 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑦 +o 𝑥) = (𝑦 +o 𝐵))
20 suceq 6250 . . . . . . . . . . . 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 2837 . . . . . . . . . 10 (𝑥 = 𝐵 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
2322imbi2d 342 . . . . . . . . 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 7153 . . . . . . . . . . 11 (𝑥 = ∅ → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o ∅))
25 oveq2 7153 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑦 +o 𝑥) = (𝑦 +o ∅))
26 suceq 6250 . . . . . . . . . . . 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 2837 . . . . . . . . . 10 (𝑥 = ∅ → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o ∅) = suc (𝑦 +o ∅)))
29 oveq2 7153 . . . . . . . . . . 11 (𝑥 = 𝑧 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o 𝑧))
30 oveq2 7153 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑦 +o 𝑥) = (𝑦 +o 𝑧))
31 suceq 6250 . . . . . . . . . . . 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 2837 . . . . . . . . . 10 (𝑥 = 𝑧 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧)))
34 oveq2 7153 . . . . . . . . . . 11 (𝑥 = suc 𝑧 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o suc 𝑧))
35 oveq2 7153 . . . . . . . . . . . 12 (𝑥 = suc 𝑧 → (𝑦 +o 𝑥) = (𝑦 +o suc 𝑧))
36 suceq 6250 . . . . . . . . . . . 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 2837 . . . . . . . . . 10 (𝑥 = suc 𝑧 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o suc 𝑧) = suc (𝑦 +o suc 𝑧)))
39 peano2 7590 . . . . . . . . . . . 12 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
40 nna0 8220 . . . . . . . . . . . 12 (suc 𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc 𝑦)
4139, 40syl 17 . . . . . . . . . . 11 (𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc 𝑦)
42 nna0 8220 . . . . . . . . . . . 12 (𝑦 ∈ ω → (𝑦 +o ∅) = 𝑦)
43 suceq 6250 . . . . . . . . . . . 12 ((𝑦 +o ∅) = 𝑦 → suc (𝑦 +o ∅) = suc 𝑦)
4442, 43syl 17 . . . . . . . . . . 11 (𝑦 ∈ ω → suc (𝑦 +o ∅) = suc 𝑦)
4541, 44eqtr4d 2859 . . . . . . . . . 10 (𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc (𝑦 +o ∅))
46 suceq 6250 . . . . . . . . . . . 12 ((suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧) → suc (suc 𝑦 +o 𝑧) = suc suc (𝑦 +o 𝑧))
47 nnasuc 8222 . . . . . . . . . . . . . 14 ((suc 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +o suc 𝑧) = suc (suc 𝑦 +o 𝑧))
4839, 47sylan 580 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +o suc 𝑧) = suc (suc 𝑦 +o 𝑧))
49 nnasuc 8222 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 +o suc 𝑧) = suc (𝑦 +o 𝑧))
50 suceq 6250 . . . . . . . . . . . . . 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 2837 . . . . . . . . . . . 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, 52syl5ibr 247 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧) → (suc 𝑦 +o suc 𝑧) = suc (𝑦 +o suc 𝑧)))
5453expcom 414 . . . . . . . . . 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 7598 . . . . . . . . 9 (𝑥 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥)))
5623, 55vtoclga 3574 . . . . . . . 8 (𝐵 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
5756imp 407 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵))
58 nnasuc 8222 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
5957, 58eqeq12d 2837 . . . . . 6 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦) ↔ suc (𝑦 +o 𝐵) = suc (𝐵 +o 𝑦)))
6017, 59syl5ibr 247 . . . . 5 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦)))
6160expcom 414 . . . 4 (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦))))
627, 10, 13, 16, 61finds2 7598 . . 3 (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 +o 𝐵) = (𝐵 +o 𝑥)))
634, 62vtoclga 3574 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
6463imp 407 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  c0 4290  suc csuc 6187  (class class class)co 7145  ωcom 7568   +o coa 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-oadd 8097
This theorem is referenced by:  nnaordr  8236  nnmsucr  8241  nnaword2  8246  omopthlem2  8273  omopthi  8274  addcompi  10305  finxpreclem4  34558
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