Theorem nnaword 6084

Description: Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnaword	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))

Proof of Theorem nnaword

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 5519	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 +𝑜 𝐴) = (𝐶 +𝑜 𝐴))
2		oveq1 5519	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 +𝑜 𝐵) = (𝐶 +𝑜 𝐵))
3	1, 2	sseq12d 2974	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
4	3	bibi2d 221	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
5	4	imbi2d 219	. . . 4 ⊢ (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
6		oveq1 5519	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 +𝑜 𝐴) = (∅ +𝑜 𝐴))
7		oveq1 5519	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 +𝑜 𝐵) = (∅ +𝑜 𝐵))
8	6, 7	sseq12d 2974	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵)))
9	8	bibi2d 221	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵))))
10		oveq1 5519	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +𝑜 𝐴) = (𝑦 +𝑜 𝐴))
11		oveq1 5519	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +𝑜 𝐵) = (𝑦 +𝑜 𝐵))
12	10, 11	sseq12d 2974	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)))
13	12	bibi2d 221	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵))))
14		oveq1 5519	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
15		oveq1 5519	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
16	14, 15	sseq12d 2974	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
17	16	bibi2d 221	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
18		nna0r 6057	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴)
19	18	eqcomd 2045	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝐴 = (∅ +𝑜 𝐴))
20	19	adantr 261	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 = (∅ +𝑜 𝐴))
21		nna0r 6057	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (∅ +𝑜 𝐵) = 𝐵)
22	21	eqcomd 2045	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 = (∅ +𝑜 𝐵))
23	22	adantl 262	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 = (∅ +𝑜 𝐵))
24	20, 23	sseq12d 2974	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵)))
25		nnacl 6059	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +𝑜 𝐴) ∈ ω)
26	25	3adant3 924	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐴) ∈ ω)
27		nnacl 6059	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐵) ∈ ω)
28	27	3adant2 923	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐵) ∈ ω)
29		nnsucsssuc 6071	. . . . . . . . . 10 ⊢ (((𝑦 +𝑜 𝐴) ∈ ω ∧ (𝑦 +𝑜 𝐵) ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ suc (𝑦 +𝑜 𝐴) ⊆ suc (𝑦 +𝑜 𝐵)))
30	26, 28, 29	syl2anc 391	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ suc (𝑦 +𝑜 𝐴) ⊆ suc (𝑦 +𝑜 𝐵)))
31		nnasuc 6055	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
32		peano2 4318	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
33		nnacom 6063	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐴))
34	32, 33	sylan2 270	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐴))
35		nnacom 6063	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴))
36		suceq 4139	. . . . . . . . . . . . . 14 ⊢ ((𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴) → suc (𝐴 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐴))
37	35, 36	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐴 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐴))
38	31, 34, 37	3eqtr3rd 2081	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
39	38	ancoms 255	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
40	39	3adant3 924	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
41		nnasuc 6055	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
42		nnacom 6063	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐵))
43	32, 42	sylan2 270	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐵))
44		nnacom 6063	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) = (𝑦 +𝑜 𝐵))
45		suceq 4139	. . . . . . . . . . . . . 14 ⊢ ((𝐵 +𝑜 𝑦) = (𝑦 +𝑜 𝐵) → suc (𝐵 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐵))
46	44, 45	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐵 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐵))
47	41, 43, 46	3eqtr3rd 2081	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
48	47	ancoms 255	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
49	48	3adant2 923	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
50	40, 49	sseq12d 2974	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (𝑦 +𝑜 𝐴) ⊆ suc (𝑦 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
51	30, 50	bitrd 177	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
52	51	bibi2d 221	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ⊆ 𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
53	52	biimpd 132	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ⊆ 𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)) → (𝐴 ⊆ 𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
54	53	3expib 1107	. . . . 5 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ⊆ 𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)) → (𝐴 ⊆ 𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))))
55	9, 13, 17, 24, 54	finds2 4324	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵))))
56	5, 55	vtoclga 2619	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
57	56	impcom 116	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
58	57	3impa 1099	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393   ⊆ wss 2917  ∅c0 3224  suc csuc 4102  ωcom 4313  (class class class)co 5512   +_𝑜 coa 5998

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005

This theorem is referenced by:  nnacan  6085  nnawordi  6088