Theorem nnacom 6086

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.

Step	Hyp	Ref	Expression
1		oveq1 5539	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 +𝑜 𝐵) = (𝐴 +𝑜 𝐵))
2		oveq2 5540	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐴))
3	1, 2	eqeq12d 2095	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴)))
4	3	imbi2d 228	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴))))
5		oveq1 5539	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 +𝑜 𝐵) = (∅ +𝑜 𝐵))
6		oveq2 5540	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
7	5, 6	eqeq12d 2095	. . . 4 ⊢ (𝑥 = ∅ → ((𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥) ↔ (∅ +𝑜 𝐵) = (𝐵 +𝑜 ∅)))
8		oveq1 5539	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 +𝑜 𝐵) = (𝑦 +𝑜 𝐵))
9		oveq2 5540	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
10	8, 9	eqeq12d 2095	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥) ↔ (𝑦 +𝑜 𝐵) = (𝐵 +𝑜 𝑦)))
11		oveq1 5539	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
12		oveq2 5540	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
13	11, 12	eqeq12d 2095	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥) ↔ (suc 𝑦 +𝑜 𝐵) = (𝐵 +𝑜 suc 𝑦)))
14		nna0r 6080	. . . . 5 ⊢ (𝐵 ∈ ω → (∅ +𝑜 𝐵) = 𝐵)
15		nna0 6076	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐵 +𝑜 ∅) = 𝐵)
16	14, 15	eqtr4d 2116	. . . 4 ⊢ (𝐵 ∈ ω → (∅ +𝑜 𝐵) = (𝐵 +𝑜 ∅))
17		suceq 4157	. . . . . 6 ⊢ ((𝑦 +𝑜 𝐵) = (𝐵 +𝑜 𝑦) → suc (𝑦 +𝑜 𝐵) = suc (𝐵 +𝑜 𝑦))
18		oveq2 5540	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (suc 𝑦 +𝑜 𝑥) = (suc 𝑦 +𝑜 𝐵))
19		oveq2 5540	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑦 +𝑜 𝑥) = (𝑦 +𝑜 𝐵))
20		suceq 4157	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 𝑥) = (𝑦 +𝑜 𝐵) → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝐵))
21	19, 20	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝐵))
22	18, 21	eqeq12d 2095	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((suc 𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑥) ↔ (suc 𝑦 +𝑜 𝐵) = suc (𝑦 +𝑜 𝐵)))
23	22	imbi2d 228	. . . . . . . . 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 5540	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (suc 𝑦 +𝑜 𝑥) = (suc 𝑦 +𝑜 ∅))
25		oveq2 5540	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑦 +𝑜 𝑥) = (𝑦 +𝑜 ∅))
26		suceq 4157	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 𝑥) = (𝑦 +𝑜 ∅) → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 ∅))
27	25, 26	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 ∅))
28	24, 27	eqeq12d 2095	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((suc 𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑥) ↔ (suc 𝑦 +𝑜 ∅) = suc (𝑦 +𝑜 ∅)))
29		oveq2 5540	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (suc 𝑦 +𝑜 𝑥) = (suc 𝑦 +𝑜 𝑧))
30		oveq2 5540	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑦 +𝑜 𝑥) = (𝑦 +𝑜 𝑧))
31		suceq 4157	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 𝑥) = (𝑦 +𝑜 𝑧) → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑧))
32	30, 31	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑧))
33	29, 32	eqeq12d 2095	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((suc 𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑥) ↔ (suc 𝑦 +𝑜 𝑧) = suc (𝑦 +𝑜 𝑧)))
34		oveq2 5540	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑧 → (suc 𝑦 +𝑜 𝑥) = (suc 𝑦 +𝑜 suc 𝑧))
35		oveq2 5540	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑧 → (𝑦 +𝑜 𝑥) = (𝑦 +𝑜 suc 𝑧))
36		suceq 4157	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 𝑥) = (𝑦 +𝑜 suc 𝑧) → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 suc 𝑧))
37	35, 36	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑧 → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 suc 𝑧))
38	34, 37	eqeq12d 2095	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑧 → ((suc 𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑥) ↔ (suc 𝑦 +𝑜 suc 𝑧) = suc (𝑦 +𝑜 suc 𝑧)))
39		peano2 4336	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
40		nna0 6076	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ ω → (suc 𝑦 +𝑜 ∅) = suc 𝑦)
41	39, 40	syl 14	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (suc 𝑦 +𝑜 ∅) = suc 𝑦)
42		nna0 6076	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 ∅) = 𝑦)
43		suceq 4157	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 ∅) = 𝑦 → suc (𝑦 +𝑜 ∅) = suc 𝑦)
44	42, 43	syl 14	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → suc (𝑦 +𝑜 ∅) = suc 𝑦)
45	41, 44	eqtr4d 2116	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (suc 𝑦 +𝑜 ∅) = suc (𝑦 +𝑜 ∅))
46		suceq 4157	. . . . . . . . . . . 12 ⊢ ((suc 𝑦 +𝑜 𝑧) = suc (𝑦 +𝑜 𝑧) → suc (suc 𝑦 +𝑜 𝑧) = suc suc (𝑦 +𝑜 𝑧))
47		nnasuc 6078	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +𝑜 suc 𝑧) = suc (suc 𝑦 +𝑜 𝑧))
48	39, 47	sylan 277	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +𝑜 suc 𝑧) = suc (suc 𝑦 +𝑜 𝑧))
49		nnasuc 6078	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 +𝑜 suc 𝑧) = suc (𝑦 +𝑜 𝑧))
50		suceq 4157	. . . . . . . . . . . . . 14 ⊢ ((𝑦 +𝑜 suc 𝑧) = suc (𝑦 +𝑜 𝑧) → suc (𝑦 +𝑜 suc 𝑧) = suc suc (𝑦 +𝑜 𝑧))
51	49, 50	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → suc (𝑦 +𝑜 suc 𝑧) = suc suc (𝑦 +𝑜 𝑧))
52	48, 51	eqeq12d 2095	. . . . . . . . . . . 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)))
53	46, 52	syl5ibr 154	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((suc 𝑦 +𝑜 𝑧) = suc (𝑦 +𝑜 𝑧) → (suc 𝑦 +𝑜 suc 𝑧) = suc (𝑦 +𝑜 suc 𝑧)))
54	53	expcom 114	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (𝑦 ∈ ω → ((suc 𝑦 +𝑜 𝑧) = suc (𝑦 +𝑜 𝑧) → (suc 𝑦 +𝑜 suc 𝑧) = suc (𝑦 +𝑜 suc 𝑧))))
55	28, 33, 38, 45, 54	finds2 4342	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑥)))
56	23, 55	vtoclga 2664	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +𝑜 𝐵) = suc (𝑦 +𝑜 𝐵)))
57	56	imp 122	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝑦 +𝑜 𝐵) = suc (𝑦 +𝑜 𝐵))
58		nnasuc 6078	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
59	57, 58	eqeq12d 2095	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((suc 𝑦 +𝑜 𝐵) = (𝐵 +𝑜 suc 𝑦) ↔ suc (𝑦 +𝑜 𝐵) = suc (𝐵 +𝑜 𝑦)))
60	17, 59	syl5ibr 154	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 𝐵) = (𝐵 +𝑜 𝑦) → (suc 𝑦 +𝑜 𝐵) = (𝐵 +𝑜 suc 𝑦)))
61	60	expcom 114	. . . 4 ⊢ (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦 +𝑜 𝐵) = (𝐵 +𝑜 𝑦) → (suc 𝑦 +𝑜 𝐵) = (𝐵 +𝑜 suc 𝑦))))
62	7, 10, 13, 16, 61	finds2 4342	. . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥)))
63	4, 62	vtoclga 2664	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴)))
64	63	imp 122	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  ∅c0 3251  suc csuc 4120  ωcom 4331  (class class class)co 5532   +_𝑜 coa 6021

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028

This theorem is referenced by:  nnmsucr  6090  nnaordi  6104  nnaordr  6106  nnaword  6107  nnaword2  6110  nnawordi  6111  addcompig  6519  nqpnq0nq  6643  prarloclemlt  6683  prarloclemlo  6684