Theorem nnawordi 7701

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 6658	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
2		oveq2 6658	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
3	1, 2	sseq12d 3634	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
4	3	imbi2d 330	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅))))
5	4	imbi2d 330	. . . 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 6658	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
7		oveq2 6658	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
8	6, 7	sseq12d 3634	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)))
9	8	imbi2d 330	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦))))
10	9	imbi2d 330	. . . 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 6658	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
12		oveq2 6658	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
13	11, 12	sseq12d 3634	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
14	13	imbi2d 330	. . . . 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 330	. . . 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 6658	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐶))
17		oveq2 6658	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
18	16, 17	sseq12d 3634	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))
19	18	imbi2d 330	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶))))
20	19	imbi2d 330	. . . 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 7071	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
22		nnon 7071	. . . . 5 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
23		oa0 7596	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
24	23	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 ∅) = 𝐴)
25		oa0 7596	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
26	25	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 ∅) = 𝐵)
27	24, 26	sseq12d 3634	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅) ↔ 𝐴 ⊆ 𝐵))
28	27	biimprd 238	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
29	21, 22, 28	syl2an 494	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
30		nnacl 7691	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) ∈ ω)
31	30	ancoms 469	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 𝑦) ∈ ω)
32	31	adantrr 753	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +𝑜 𝑦) ∈ ω)
33		nnon 7071	. . . . . . . . . . . 12 ⊢ ((𝐴 +𝑜 𝑦) ∈ ω → (𝐴 +𝑜 𝑦) ∈ On)
34		eloni 5733	. . . . . . . . . . . 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 7691	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
37	36	ancoms 469	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
38	37	adantrl 752	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +𝑜 𝑦) ∈ ω)
39		nnon 7071	. . . . . . . . . . . 12 ⊢ ((𝐵 +𝑜 𝑦) ∈ ω → (𝐵 +𝑜 𝑦) ∈ On)
40		eloni 5733	. . . . . . . . . . . 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 7023	. . . . . . . . . . 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 693	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
44	43	biimpa 501	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)) → suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦))
45		nnasuc 7686	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
46	45	ancoms 469	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
47	46	adantrr 753	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
48		nnasuc 7686	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
49	48	ancoms 469	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
50	49	adantrl 752	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
51	47, 50	sseq12d 3634	. . . . . . . . . 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 481	. . . . . . . . 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 247	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦))
54	53	ex 450	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
55	54	imim2d 57	. . . . . 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 450	. . . . 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 7092	. . 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 1261	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ∅c0 3915  Ord word 5722  Oncon0 5723  suc csuc 5725  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  omopthlem2  7736