Theorem nnaword 6310

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 5697	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 +o 𝐴) = (𝐶 +o 𝐴))
2		oveq1 5697	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 +o 𝐵) = (𝐶 +o 𝐵))
3	1, 2	sseq12d 3070	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
4	3	bibi2d 231	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))))
5	4	imbi2d 229	. . . 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 5697	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐴) = (∅ +o 𝐴))
7		oveq1 5697	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 +o 𝐵) = (∅ +o 𝐵))
8	6, 7	sseq12d 3070	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵)))
9	8	bibi2d 231	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵))))
10		oveq1 5697	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +o 𝐴) = (𝑦 +o 𝐴))
11		oveq1 5697	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +o 𝐵) = (𝑦 +o 𝐵))
12	10, 11	sseq12d 3070	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵)))
13	12	bibi2d 231	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵))))
14		oveq1 5697	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑥 +o 𝐴) = (suc 𝑦 +o 𝐴))
15		oveq1 5697	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝑥 +o 𝐵) = (suc 𝑦 +o 𝐵))
16	14, 15	sseq12d 3070	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵)))
17	16	bibi2d 231	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴 ⊆ 𝐵 ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵))))
18		nna0r 6279	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴)
19	18	eqcomd 2100	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝐴 = (∅ +o 𝐴))
20	19	adantr 271	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 = (∅ +o 𝐴))
21		nna0r 6279	. . . . . . . 8 ⊢ (𝐵 ∈ ω → (∅ +o 𝐵) = 𝐵)
22	21	eqcomd 2100	. . . . . . 7 ⊢ (𝐵 ∈ ω → 𝐵 = (∅ +o 𝐵))
23	22	adantl 272	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 = (∅ +o 𝐵))
24	20, 23	sseq12d 3070	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵)))
25		nnacl 6281	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o 𝐴) ∈ ω)
26	25	3adant3 966	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐴) ∈ ω)
27		nnacl 6281	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐵) ∈ ω)
28	27	3adant2 965	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐵) ∈ ω)
29		nnsucsssuc 6293	. . . . . . . . . 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 404	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵) ↔ suc (𝑦 +o 𝐴) ⊆ suc (𝑦 +o 𝐵)))
31		nnasuc 6277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
32		peano2 4438	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
33		nnacom 6285	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (suc 𝑦 +o 𝐴))
34	32, 33	sylan2 281	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (suc 𝑦 +o 𝐴))
35		nnacom 6285	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
36		suceq 4253	. . . . . . . . . . . . . 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 2136	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
39	38	ancoms 265	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
40	39	3adant3 966	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
41		nnasuc 6277	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
42		nnacom 6285	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = (suc 𝑦 +o 𝐵))
43	32, 42	sylan2 281	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = (suc 𝑦 +o 𝐵))
44		nnacom 6285	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) = (𝑦 +o 𝐵))
45		suceq 4253	. . . . . . . . . . . . . 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 2136	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
48	47	ancoms 265	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
49	48	3adant2 965	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
50	40, 49	sseq12d 3070	. . . . . . . . 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 187	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵) ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵)))
52	51	bibi2d 231	. . . . . . 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 143	. . . . . 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 1149	. . . . 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 4444	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵))))
56	5, 55	vtoclga 2699	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))))
57	56	impcom 124	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
58	57	3impa 1141	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 927   = wceq 1296   ∈ wcel 1445   ⊆ wss 3013  ∅c0 3302  suc csuc 4216  ωcom 4433  (class class class)co 5690   +_o coa 6216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-oadd 6223

This theorem is referenced by:  nnacan  6311  nnawordi  6314