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.

Step	Hyp	Ref	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))
3	1, 2	eqeq12d 2154	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
4	3	imbi2d 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 ∅))
7	5, 6	eqeq12d 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))
10	8, 9	eqeq12d 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))
13	11, 12	eqeq12d 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)
16	14, 15	eqtr4d 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))
21	19, 20	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → suc (𝑦 +o 𝑥) = suc (𝑦 +o 𝐵))
22	18, 21	eqeq12d 2154	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
23	22	imbi2d 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 ∅))
27	25, 26	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → suc (𝑦 +o 𝑥) = suc (𝑦 +o ∅))
28	24, 27	eqeq12d 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))
32	30, 31	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → suc (𝑦 +o 𝑥) = suc (𝑦 +o 𝑧))
33	29, 32	eqeq12d 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))
37	35, 36	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑧 → suc (𝑦 +o 𝑥) = suc (𝑦 +o suc 𝑧))
38	34, 37	eqeq12d 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)
41	39, 40	syl 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)
44	42, 43	syl 14	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → suc (𝑦 +o ∅) = suc 𝑦)
45	41, 44	eqtr4d 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))
48	39, 47	sylan 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))
51	49, 50	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → suc (𝑦 +o suc 𝑧) = suc suc (𝑦 +o 𝑧))
52	48, 51	eqeq12d 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)))
53	46, 52	syl5ibr 155	. . . . . . . . . . 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 115	. . . . . . . . . 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 4515	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥)))
56	23, 55	vtoclga 2752	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
57	56	imp 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))
59	57, 58	eqeq12d 2154	. . . . . 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 155	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦)))
61	60	expcom 115	. . . 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 4515	. . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 +o 𝐵) = (𝐵 +o 𝑥)))
63	4, 62	vtoclga 2752	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
64	63	imp 123	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))

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  7137  nqpnq0nq  7261  prarloclemlt  7301  prarloclemlo  7302