Theorem nnawordi 7588

Description: Adding to both sides of an inequality in ω. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)

Assertion

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

Proof of Theorem nnawordi

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

Step	Hyp	Ref	Expression
1		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
2		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
3	1, 2	sseq12d 3597	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
4	3	imbi2d 329	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅))))
5	4	imbi2d 329	. . . 4 ⊢ (𝑥 = ∅ → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))))
6		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
7		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
8	6, 7	sseq12d 3597	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)))
9	8	imbi2d 329	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦))))
10	9	imbi2d 329	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)))))
11		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
12		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
13	11, 12	sseq12d 3597	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
14	13	imbi2d 329	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦))))
15	14	imbi2d 329	. . . 4 ⊢ (𝑥 = suc 𝑦 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))))
16		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐶))
17		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
18	16, 17	sseq12d 3597	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))
19	18	imbi2d 329	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶))))
20	19	imbi2d 329	. . . 4 ⊢ (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))))
21		nnon 6963	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
22		nnon 6963	. . . . 5 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
23		oa0 7483	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
24	23	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 ∅) = 𝐴)
25		oa0 7483	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
26	25	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 ∅) = 𝐵)
27	24, 26	sseq12d 3597	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅) ↔ 𝐴 ⊆ 𝐵))
28	27	biimprd 237	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
29	21, 22, 28	syl2an 493	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
30		nnacl 7578	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) ∈ ω)
31	30	ancoms 468	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 𝑦) ∈ ω)
32	31	adantrr 749	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +𝑜 𝑦) ∈ ω)
33		nnon 6963	. . . . . . . . . . . 12 ⊢ ((𝐴 +𝑜 𝑦) ∈ ω → (𝐴 +𝑜 𝑦) ∈ On)
34		eloni 5650	. . . . . . . . . . . 12 ⊢ ((𝐴 +𝑜 𝑦) ∈ On → Ord (𝐴 +𝑜 𝑦))
35	32, 33, 34	3syl 18	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → Ord (𝐴 +𝑜 𝑦))
36		nnacl 7578	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
37	36	ancoms 468	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
38	37	adantrl 748	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +𝑜 𝑦) ∈ ω)
39		nnon 6963	. . . . . . . . . . . 12 ⊢ ((𝐵 +𝑜 𝑦) ∈ ω → (𝐵 +𝑜 𝑦) ∈ On)
40		eloni 5650	. . . . . . . . . . . 12 ⊢ ((𝐵 +𝑜 𝑦) ∈ On → Ord (𝐵 +𝑜 𝑦))
41	38, 39, 40	3syl 18	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → Ord (𝐵 +𝑜 𝑦))
42		ordsucsssuc 6915	. . . . . . . . . . 11 ⊢ ((Ord (𝐴 +𝑜 𝑦) ∧ Ord (𝐵 +𝑜 𝑦)) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
43	35, 41, 42	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
44	43	biimpa 500	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)) → suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦))
45		nnasuc 7573	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
46	45	ancoms 468	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
47	46	adantrr 749	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
48		nnasuc 7573	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
49	48	ancoms 468	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
50	49	adantrl 748	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
51	47, 50	sseq12d 3597	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
52	51	adantr 480	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)) → ((𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
53	44, 52	mpbird 246	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦))
54	53	ex 449	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
55	54	imim2d 55	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦))))
56	55	ex 449	. . . . 5 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))))
57	56	a2d 29	. . . 4 ⊢ (𝑦 ∈ ω → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦))) → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))))
58	5, 10, 15, 20, 29, 57	finds 6984	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶))))
59	58	com12 32	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ∈ ω → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶))))
60	59	3impia 1253	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∅c0 3874  Ord word 5639  Oncon0 5640  suc csuc 5642  (class class class)co 6549  ωcom 6957   +_𝑜 coa 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451

This theorem is referenced by:  omopthlem2  7623