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Theorem nnacom 6380
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 5781 . . . . 5 (𝑥 = 𝐴 → (𝑥 +o 𝐵) = (𝐴 +o 𝐵))
2 oveq2 5782 . . . . 5 (𝑥 = 𝐴 → (𝐵 +o 𝑥) = (𝐵 +o 𝐴))
31, 2eqeq12d 2154 . . . 4 (𝑥 = 𝐴 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
43imbi2d 229 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 +o 𝐵) = (𝐵 +o 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))))
5 oveq1 5781 . . . . 5 (𝑥 = ∅ → (𝑥 +o 𝐵) = (∅ +o 𝐵))
6 oveq2 5782 . . . . 5 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
75, 6eqeq12d 2154 . . . 4 (𝑥 = ∅ → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (∅ +o 𝐵) = (𝐵 +o ∅)))
8 oveq1 5781 . . . . 5 (𝑥 = 𝑦 → (𝑥 +o 𝐵) = (𝑦 +o 𝐵))
9 oveq2 5782 . . . . 5 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
108, 9eqeq12d 2154 . . . 4 (𝑥 = 𝑦 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (𝑦 +o 𝐵) = (𝐵 +o 𝑦)))
11 oveq1 5781 . . . . 5 (𝑥 = suc 𝑦 → (𝑥 +o 𝐵) = (suc 𝑦 +o 𝐵))
12 oveq2 5782 . . . . 5 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1311, 12eqeq12d 2154 . . . 4 (𝑥 = suc 𝑦 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦)))
14 nna0r 6374 . . . . 5 (𝐵 ∈ ω → (∅ +o 𝐵) = 𝐵)
15 nna0 6370 . . . . 5 (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
1614, 15eqtr4d 2175 . . . 4 (𝐵 ∈ ω → (∅ +o 𝐵) = (𝐵 +o ∅))
17 suceq 4324 . . . . . 6 ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → suc (𝑦 +o 𝐵) = suc (𝐵 +o 𝑦))
18 oveq2 5782 . . . . . . . . . . 11 (𝑥 = 𝐵 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o 𝐵))
19 oveq2 5782 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑦 +o 𝑥) = (𝑦 +o 𝐵))
20 suceq 4324 . . . . . . . . . . . 12 ((𝑦 +o 𝑥) = (𝑦 +o 𝐵) → suc (𝑦 +o 𝑥) = suc (𝑦 +o 𝐵))
2119, 20syl 14 . . . . . . . . . . 11 (𝑥 = 𝐵 → suc (𝑦 +o 𝑥) = suc (𝑦 +o 𝐵))
2218, 21eqeq12d 2154 . . . . . . . . . 10 (𝑥 = 𝐵 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
2322imbi2d 229 . . . . . . . . 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 5782 . . . . . . . . . . 11 (𝑥 = ∅ → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o ∅))
25 oveq2 5782 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑦 +o 𝑥) = (𝑦 +o ∅))
26 suceq 4324 . . . . . . . . . . . 12 ((𝑦 +o 𝑥) = (𝑦 +o ∅) → suc (𝑦 +o 𝑥) = suc (𝑦 +o ∅))
2725, 26syl 14 . . . . . . . . . . 11 (𝑥 = ∅ → suc (𝑦 +o 𝑥) = suc (𝑦 +o ∅))
2824, 27eqeq12d 2154 . . . . . . . . . 10 (𝑥 = ∅ → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o ∅) = suc (𝑦 +o ∅)))
29 oveq2 5782 . . . . . . . . . . 11 (𝑥 = 𝑧 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o 𝑧))
30 oveq2 5782 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑦 +o 𝑥) = (𝑦 +o 𝑧))
31 suceq 4324 . . . . . . . . . . . 12 ((𝑦 +o 𝑥) = (𝑦 +o 𝑧) → suc (𝑦 +o 𝑥) = suc (𝑦 +o 𝑧))
3230, 31syl 14 . . . . . . . . . . 11 (𝑥 = 𝑧 → suc (𝑦 +o 𝑥) = suc (𝑦 +o 𝑧))
3329, 32eqeq12d 2154 . . . . . . . . . 10 (𝑥 = 𝑧 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧)))
34 oveq2 5782 . . . . . . . . . . 11 (𝑥 = suc 𝑧 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o suc 𝑧))
35 oveq2 5782 . . . . . . . . . . . 12 (𝑥 = suc 𝑧 → (𝑦 +o 𝑥) = (𝑦 +o suc 𝑧))
36 suceq 4324 . . . . . . . . . . . 12 ((𝑦 +o 𝑥) = (𝑦 +o suc 𝑧) → suc (𝑦 +o 𝑥) = suc (𝑦 +o suc 𝑧))
3735, 36syl 14 . . . . . . . . . . 11 (𝑥 = suc 𝑧 → suc (𝑦 +o 𝑥) = suc (𝑦 +o suc 𝑧))
3834, 37eqeq12d 2154 . . . . . . . . . 10 (𝑥 = suc 𝑧 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o suc 𝑧) = suc (𝑦 +o suc 𝑧)))
39 peano2 4509 . . . . . . . . . . . 12 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
40 nna0 6370 . . . . . . . . . . . 12 (suc 𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc 𝑦)
4139, 40syl 14 . . . . . . . . . . 11 (𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc 𝑦)
42 nna0 6370 . . . . . . . . . . . 12 (𝑦 ∈ ω → (𝑦 +o ∅) = 𝑦)
43 suceq 4324 . . . . . . . . . . . 12 ((𝑦 +o ∅) = 𝑦 → suc (𝑦 +o ∅) = suc 𝑦)
4442, 43syl 14 . . . . . . . . . . 11 (𝑦 ∈ ω → suc (𝑦 +o ∅) = suc 𝑦)
4541, 44eqtr4d 2175 . . . . . . . . . 10 (𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc (𝑦 +o ∅))
46 suceq 4324 . . . . . . . . . . . 12 ((suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧) → suc (suc 𝑦 +o 𝑧) = suc suc (𝑦 +o 𝑧))
47 nnasuc 6372 . . . . . . . . . . . . . 14 ((suc 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +o suc 𝑧) = suc (suc 𝑦 +o 𝑧))
4839, 47sylan 281 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +o suc 𝑧) = suc (suc 𝑦 +o 𝑧))
49 nnasuc 6372 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 +o suc 𝑧) = suc (𝑦 +o 𝑧))
50 suceq 4324 . . . . . . . . . . . . . 14 ((𝑦 +o suc 𝑧) = suc (𝑦 +o 𝑧) → suc (𝑦 +o suc 𝑧) = suc suc (𝑦 +o 𝑧))
5149, 50syl 14 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → suc (𝑦 +o suc 𝑧) = suc suc (𝑦 +o 𝑧))
5248, 51eqeq12d 2154 . . . . . . . . . . . 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 155 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧) → (suc 𝑦 +o suc 𝑧) = suc (𝑦 +o suc 𝑧)))
5453expcom 115 . . . . . . . . . 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 4515 . . . . . . . . 9 (𝑥 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥)))
5623, 55vtoclga 2752 . . . . . . . 8 (𝐵 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
5756imp 123 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵))
58 nnasuc 6372 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
5957, 58eqeq12d 2154 . . . . . 6 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦) ↔ suc (𝑦 +o 𝐵) = suc (𝐵 +o 𝑦)))
6017, 59syl5ibr 155 . . . . 5 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦)))
6160expcom 115 . . . 4 (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦))))
627, 10, 13, 16, 61finds2 4515 . . 3 (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 +o 𝐵) = (𝐵 +o 𝑥)))
634, 62vtoclga 2752 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
6463imp 123 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  c0 3363  suc csuc 4287  ωcom 4504  (class class class)co 5774   +o coa 6310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317
This theorem is referenced by:  nnmsucr  6384  nnaordi  6404  nnaordr  6406  nnaword  6407  nnaword2  6410  nnawordi  6411  addcompig  7144  nqpnq0nq  7268  prarloclemlt  7308  prarloclemlo  7309
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