Theorem nnacom 7697

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 6657	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 +𝑜 𝐵) = (𝐴 +𝑜 𝐵))
2		oveq2 6658	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐴))
3	1, 2	eqeq12d 2637	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴)))
4	3	imbi2d 330	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴))))
5		oveq1 6657	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 +𝑜 𝐵) = (∅ +𝑜 𝐵))
6		oveq2 6658	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
7	5, 6	eqeq12d 2637	. . . 4 ⊢ (𝑥 = ∅ → ((𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥) ↔ (∅ +𝑜 𝐵) = (𝐵 +𝑜 ∅)))
8		oveq1 6657	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 +𝑜 𝐵) = (𝑦 +𝑜 𝐵))
9		oveq2 6658	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
10	8, 9	eqeq12d 2637	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥) ↔ (𝑦 +𝑜 𝐵) = (𝐵 +𝑜 𝑦)))
11		oveq1 6657	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
12		oveq2 6658	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
13	11, 12	eqeq12d 2637	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥) ↔ (suc 𝑦 +𝑜 𝐵) = (𝐵 +𝑜 suc 𝑦)))
14		nna0r 7689	. . . . 5 ⊢ (𝐵 ∈ ω → (∅ +𝑜 𝐵) = 𝐵)
15		nna0 7684	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐵 +𝑜 ∅) = 𝐵)
16	14, 15	eqtr4d 2659	. . . 4 ⊢ (𝐵 ∈ ω → (∅ +𝑜 𝐵) = (𝐵 +𝑜 ∅))
17		suceq 5790	. . . . . 6 ⊢ ((𝑦 +𝑜 𝐵) = (𝐵 +𝑜 𝑦) → suc (𝑦 +𝑜 𝐵) = suc (𝐵 +𝑜 𝑦))
18		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (suc 𝑦 +𝑜 𝑥) = (suc 𝑦 +𝑜 𝐵))
19		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑦 +𝑜 𝑥) = (𝑦 +𝑜 𝐵))
20		suceq 5790	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 𝑥) = (𝑦 +𝑜 𝐵) → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝐵))
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝐵))
22	18, 21	eqeq12d 2637	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((suc 𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑥) ↔ (suc 𝑦 +𝑜 𝐵) = suc (𝑦 +𝑜 𝐵)))
23	22	imbi2d 330	. . . . . . . . 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 6658	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (suc 𝑦 +𝑜 𝑥) = (suc 𝑦 +𝑜 ∅))
25		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑦 +𝑜 𝑥) = (𝑦 +𝑜 ∅))
26		suceq 5790	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 𝑥) = (𝑦 +𝑜 ∅) → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 ∅))
27	25, 26	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 ∅))
28	24, 27	eqeq12d 2637	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((suc 𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑥) ↔ (suc 𝑦 +𝑜 ∅) = suc (𝑦 +𝑜 ∅)))
29		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (suc 𝑦 +𝑜 𝑥) = (suc 𝑦 +𝑜 𝑧))
30		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑦 +𝑜 𝑥) = (𝑦 +𝑜 𝑧))
31		suceq 5790	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 𝑥) = (𝑦 +𝑜 𝑧) → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑧))
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑧))
33	29, 32	eqeq12d 2637	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((suc 𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑥) ↔ (suc 𝑦 +𝑜 𝑧) = suc (𝑦 +𝑜 𝑧)))
34		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑧 → (suc 𝑦 +𝑜 𝑥) = (suc 𝑦 +𝑜 suc 𝑧))
35		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑧 → (𝑦 +𝑜 𝑥) = (𝑦 +𝑜 suc 𝑧))
36		suceq 5790	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 𝑥) = (𝑦 +𝑜 suc 𝑧) → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 suc 𝑧))
37	35, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑧 → suc (𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 suc 𝑧))
38	34, 37	eqeq12d 2637	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑧 → ((suc 𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑥) ↔ (suc 𝑦 +𝑜 suc 𝑧) = suc (𝑦 +𝑜 suc 𝑧)))
39		peano2 7086	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
40		nna0 7684	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ ω → (suc 𝑦 +𝑜 ∅) = suc 𝑦)
41	39, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (suc 𝑦 +𝑜 ∅) = suc 𝑦)
42		nna0 7684	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 ∅) = 𝑦)
43		suceq 5790	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 ∅) = 𝑦 → suc (𝑦 +𝑜 ∅) = suc 𝑦)
44	42, 43	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → suc (𝑦 +𝑜 ∅) = suc 𝑦)
45	41, 44	eqtr4d 2659	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (suc 𝑦 +𝑜 ∅) = suc (𝑦 +𝑜 ∅))
46		suceq 5790	. . . . . . . . . . . 12 ⊢ ((suc 𝑦 +𝑜 𝑧) = suc (𝑦 +𝑜 𝑧) → suc (suc 𝑦 +𝑜 𝑧) = suc suc (𝑦 +𝑜 𝑧))
47		nnasuc 7686	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +𝑜 suc 𝑧) = suc (suc 𝑦 +𝑜 𝑧))
48	39, 47	sylan 488	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +𝑜 suc 𝑧) = suc (suc 𝑦 +𝑜 𝑧))
49		nnasuc 7686	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 +𝑜 suc 𝑧) = suc (𝑦 +𝑜 𝑧))
50		suceq 5790	. . . . . . . . . . . . . 14 ⊢ ((𝑦 +𝑜 suc 𝑧) = suc (𝑦 +𝑜 𝑧) → suc (𝑦 +𝑜 suc 𝑧) = suc suc (𝑦 +𝑜 𝑧))
51	49, 50	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → suc (𝑦 +𝑜 suc 𝑧) = suc suc (𝑦 +𝑜 𝑧))
52	48, 51	eqeq12d 2637	. . . . . . . . . . . 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 236	. . . . . . . . . . 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 451	. . . . . . . . . 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 7094	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +𝑜 𝑥) = suc (𝑦 +𝑜 𝑥)))
56	23, 55	vtoclga 3272	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +𝑜 𝐵) = suc (𝑦 +𝑜 𝐵)))
57	56	imp 445	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝑦 +𝑜 𝐵) = suc (𝑦 +𝑜 𝐵))
58		nnasuc 7686	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
59	57, 58	eqeq12d 2637	. . . . . 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 236	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 𝐵) = (𝐵 +𝑜 𝑦) → (suc 𝑦 +𝑜 𝐵) = (𝐵 +𝑜 suc 𝑦)))
61	60	expcom 451	. . . 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 7094	. . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 +𝑜 𝐵) = (𝐵 +𝑜 𝑥)))
63	4, 62	vtoclga 3272	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴)))
64	63	imp 445	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∅c0 3915  suc csuc 5725  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  nnaordr  7700  nnmsucr  7705  nnaword2  7710  omopthlem2  7736  omopthi  7737  addcompi  9716  finxpreclem4  33231