Theorem nnacom 8588

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 7404	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 +o 𝐵) = (𝐴 +o 𝐵))
2		oveq2 7405	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 +o 𝑥) = (𝐵 +o 𝐴))
3	1, 2	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
4	3	imbi2d 342	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 +o 𝐵) = (𝐵 +o 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))))
5		oveq1 7404	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐵) = (∅ +o 𝐵))
6		oveq2 7405	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
7	5, 6	eqeq12d 2779	. . . 4 ⊢ (𝑥 = ∅ → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (∅ +o 𝐵) = (𝐵 +o ∅)))
8		oveq1 7404	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 +o 𝐵) = (𝑦 +o 𝐵))
9		oveq2 7405	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
10	8, 9	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (𝑦 +o 𝐵) = (𝐵 +o 𝑦)))
11		oveq1 7404	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 +o 𝐵) = (suc 𝑦 +o 𝐵))
12		oveq2 7405	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
13	11, 12	eqeq12d 2779	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦)))
14		nna0r 8580	. . . . 5 ⊢ (𝐵 ∈ ω → (∅ +o 𝐵) = 𝐵)
15		nna0 8575	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
16	14, 15	eqtr4d 2801	. . . 4 ⊢ (𝐵 ∈ ω → (∅ +o 𝐵) = (𝐵 +o ∅))
17		suceq 6415	. . . . . 6 ⊢ ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → suc (𝑦 +o 𝐵) = suc (𝐵 +o 𝑦))
18		oveq2 7405	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o 𝐵))
19		oveq2 7405	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑦 +o 𝑥) = (𝑦 +o 𝐵))
20		suceq 6415	. . . . . . . . . . . 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 2779	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
23	22	imbi2d 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 7405	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o ∅))
25		oveq2 7405	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑦 +o 𝑥) = (𝑦 +o ∅))
26		suceq 6415	. . . . . . . . . . . 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 2779	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o ∅) = suc (𝑦 +o ∅)))
29		oveq2 7405	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o 𝑧))
30		oveq2 7405	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑦 +o 𝑥) = (𝑦 +o 𝑧))
31		suceq 6415	. . . . . . . . . . . 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 2779	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧)))
34		oveq2 7405	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑧 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o suc 𝑧))
35		oveq2 7405	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑧 → (𝑦 +o 𝑥) = (𝑦 +o suc 𝑧))
36		suceq 6415	. . . . . . . . . . . 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 2779	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑧 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o suc 𝑧) = suc (𝑦 +o suc 𝑧)))
39		peano2 7871	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
40		nna0 8575	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc 𝑦)
41	39, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc 𝑦)
42		nna0 8575	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (𝑦 +o ∅) = 𝑦)
43		suceq 6415	. . . . . . . . . . . 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 2801	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc (𝑦 +o ∅))
46		suceq 6415	. . . . . . . . . . . 12 ⊢ ((suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧) → suc (suc 𝑦 +o 𝑧) = suc suc (𝑦 +o 𝑧))
47		nnasuc 8577	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +o suc 𝑧) = suc (suc 𝑦 +o 𝑧))
48	39, 47	sylan 589	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +o suc 𝑧) = suc (suc 𝑦 +o 𝑧))
49		nnasuc 8577	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 +o suc 𝑧) = suc (𝑦 +o 𝑧))
50		suceq 6415	. . . . . . . . . . . . . 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 2779	. . . . . . . . . . . 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	imbitrrid 248	. . . . . . . . . . 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 417	. . . . . . . . . 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 7880	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥)))
56	23, 55	vtoclga 3542	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
57	56	imp 410	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵))
58		nnasuc 8577	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
59	57, 58	eqeq12d 2779	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦) ↔ suc (𝑦 +o 𝐵) = suc (𝐵 +o 𝑦)))
60	17, 59	imbitrrid 248	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦)))
61	60	expcom 417	. . . 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 7880	. . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 +o 𝐵) = (𝐵 +o 𝑥)))
63	4, 62	vtoclga 3542	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
64	63	imp 410	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  ∅c0 4286  suc csuc 6349  (class class class)co 7397  ωcom 7847   +_o coa 8435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-oadd 8442

This theorem is referenced by:  nnaordr  8591  nnmsucr  8596  nnaword2  8601  omopthlem2  8631  omopthi  8632  eldifsucnn  8635  addcompi  10853  fineqvnttrclselem3  35420  mh-inf3f1  36902  finxpreclem4  37889  oaabsb  43872  ofoacom  43939  naddcnfcom  43944