Theorem nnaword 6107

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 5539	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 +𝑜 𝐴) = (𝐶 +𝑜 𝐴))
2		oveq1 5539	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 +𝑜 𝐵) = (𝐶 +𝑜 𝐵))
3	1, 2	sseq12d 3028	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
4	3	bibi2d 230	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
5	4	imbi2d 228	. . . 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 5539	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 +𝑜 𝐴) = (∅ +𝑜 𝐴))
7		oveq1 5539	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 +𝑜 𝐵) = (∅ +𝑜 𝐵))
8	6, 7	sseq12d 3028	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵)))
9	8	bibi2d 230	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵))))
10		oveq1 5539	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +𝑜 𝐴) = (𝑦 +𝑜 𝐴))
11		oveq1 5539	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +𝑜 𝐵) = (𝑦 +𝑜 𝐵))
12	10, 11	sseq12d 3028	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)))
13	12	bibi2d 230	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵))))
14		oveq1 5539	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
15		oveq1 5539	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
16	14, 15	sseq12d 3028	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
17	16	bibi2d 230	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
18		nna0r 6080	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴)
19	18	eqcomd 2086	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝐴 = (∅ +𝑜 𝐴))
20	19	adantr 270	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 = (∅ +𝑜 𝐴))
21		nna0r 6080	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (∅ +𝑜 𝐵) = 𝐵)
22	21	eqcomd 2086	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 = (∅ +𝑜 𝐵))
23	22	adantl 271	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 = (∅ +𝑜 𝐵))
24	20, 23	sseq12d 3028	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵)))
25		nnacl 6082	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +𝑜 𝐴) ∈ ω)
26	25	3adant3 958	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐴) ∈ ω)
27		nnacl 6082	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐵) ∈ ω)
28	27	3adant2 957	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐵) ∈ ω)
29		nnsucsssuc 6094	. . . . . . . . . 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 403	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ suc (𝑦 +𝑜 𝐴) ⊆ suc (𝑦 +𝑜 𝐵)))
31		nnasuc 6078	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
32		peano2 4336	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
33		nnacom 6086	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐴))
34	32, 33	sylan2 280	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐴))
35		nnacom 6086	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴))
36		suceq 4157	. . . . . . . . . . . . . 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 2122	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
39	38	ancoms 264	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
40	39	3adant3 958	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
41		nnasuc 6078	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
42		nnacom 6086	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐵))
43	32, 42	sylan2 280	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐵))
44		nnacom 6086	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) = (𝑦 +𝑜 𝐵))
45		suceq 4157	. . . . . . . . . . . . . 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 2122	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
48	47	ancoms 264	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
49	48	3adant2 957	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
50	40, 49	sseq12d 3028	. . . . . . . . 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 186	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
52	51	bibi2d 230	. . . . . . 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 142	. . . . . 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 1141	. . . . 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 4342	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵))))
56	5, 55	vtoclga 2664	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
57	56	impcom 123	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
58	57	3impa 1133	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284   ∈ wcel 1433   ⊆ wss 2973  ∅c0 3251  suc csuc 4120  ωcom 4331  (class class class)co 5532   +_𝑜 coa 6021

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028

This theorem is referenced by:  nnacan  6108  nnawordi  6111