Theorem nnacom 8627

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 7410	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 +o 𝐵) = (𝐴 +o 𝐵))
2		oveq2 7411	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 +o 𝑥) = (𝐵 +o 𝐴))
3	1, 2	eqeq12d 2751	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
4	3	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 +o 𝐵) = (𝐵 +o 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))))
5		oveq1 7410	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐵) = (∅ +o 𝐵))
6		oveq2 7411	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
7	5, 6	eqeq12d 2751	. . . 4 ⊢ (𝑥 = ∅ → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (∅ +o 𝐵) = (𝐵 +o ∅)))
8		oveq1 7410	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 +o 𝐵) = (𝑦 +o 𝐵))
9		oveq2 7411	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
10	8, 9	eqeq12d 2751	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (𝑦 +o 𝐵) = (𝐵 +o 𝑦)))
11		oveq1 7410	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 +o 𝐵) = (suc 𝑦 +o 𝐵))
12		oveq2 7411	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
13	11, 12	eqeq12d 2751	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑥 +o 𝐵) = (𝐵 +o 𝑥) ↔ (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦)))
14		nna0r 8619	. . . . 5 ⊢ (𝐵 ∈ ω → (∅ +o 𝐵) = 𝐵)
15		nna0 8614	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
16	14, 15	eqtr4d 2773	. . . 4 ⊢ (𝐵 ∈ ω → (∅ +o 𝐵) = (𝐵 +o ∅))
17		suceq 6419	. . . . . 6 ⊢ ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → suc (𝑦 +o 𝐵) = suc (𝐵 +o 𝑦))
18		oveq2 7411	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o 𝐵))
19		oveq2 7411	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑦 +o 𝑥) = (𝑦 +o 𝐵))
20		suceq 6419	. . . . . . . . . . . 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 2751	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
23	22	imbi2d 340	. . . . . . . . 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 7411	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o ∅))
25		oveq2 7411	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑦 +o 𝑥) = (𝑦 +o ∅))
26		suceq 6419	. . . . . . . . . . . 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 2751	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o ∅) = suc (𝑦 +o ∅)))
29		oveq2 7411	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o 𝑧))
30		oveq2 7411	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑦 +o 𝑥) = (𝑦 +o 𝑧))
31		suceq 6419	. . . . . . . . . . . 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 2751	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧)))
34		oveq2 7411	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑧 → (suc 𝑦 +o 𝑥) = (suc 𝑦 +o suc 𝑧))
35		oveq2 7411	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑧 → (𝑦 +o 𝑥) = (𝑦 +o suc 𝑧))
36		suceq 6419	. . . . . . . . . . . 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 2751	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑧 → ((suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥) ↔ (suc 𝑦 +o suc 𝑧) = suc (𝑦 +o suc 𝑧)))
39		peano2 7884	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
40		nna0 8614	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc 𝑦)
41	39, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc 𝑦)
42		nna0 8614	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (𝑦 +o ∅) = 𝑦)
43		suceq 6419	. . . . . . . . . . . 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 2773	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (suc 𝑦 +o ∅) = suc (𝑦 +o ∅))
46		suceq 6419	. . . . . . . . . . . 12 ⊢ ((suc 𝑦 +o 𝑧) = suc (𝑦 +o 𝑧) → suc (suc 𝑦 +o 𝑧) = suc suc (𝑦 +o 𝑧))
47		nnasuc 8616	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +o suc 𝑧) = suc (suc 𝑦 +o 𝑧))
48	39, 47	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 +o suc 𝑧) = suc (suc 𝑦 +o 𝑧))
49		nnasuc 8616	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 +o suc 𝑧) = suc (𝑦 +o 𝑧))
50		suceq 6419	. . . . . . . . . . . . . 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 2751	. . . . . . . . . . . 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 246	. . . . . . . . . . 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 413	. . . . . . . . . 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 7892	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝑥) = suc (𝑦 +o 𝑥)))
56	23, 55	vtoclga 3556	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (𝑦 ∈ ω → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵)))
57	56	imp 406	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝑦 +o 𝐵) = suc (𝑦 +o 𝐵))
58		nnasuc 8616	. . . . . . 7 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
59	57, 58	eqeq12d 2751	. . . . . 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 246	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +o 𝐵) = (𝐵 +o 𝑦) → (suc 𝑦 +o 𝐵) = (𝐵 +o suc 𝑦)))
61	60	expcom 413	. . . 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 7892	. . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 +o 𝐵) = (𝐵 +o 𝑥)))
63	4, 62	vtoclga 3556	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)))
64	63	imp 406	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∅c0 4308  suc csuc 6354  (class class class)co 7403  ωcom 7859   +_o coa 8475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-oadd 8482

This theorem is referenced by:  nnaordr  8630  nnmsucr  8635  nnaword2  8640  omopthlem2  8670  omopthi  8671  eldifsucnn  8674  addcompi  10906  finxpreclem4  37358  oaabsb  43265  ofoacom  43332  naddcnfcom  43337