Theorem nnawordi 8677

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 7456	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2		oveq2 7456	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
3	1, 2	sseq12d 4042	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))
4	3	imbi2d 340	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ⊆ 𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +o ∅) ⊆ (𝐵 +o ∅))))
5	4	imbi2d 340	. . . 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 7456	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
7		oveq2 7456	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
8	6, 7	sseq12d 4042	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)))
9	8	imbi2d 340	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ 𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦))))
10	9	imbi2d 340	. . . 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 7456	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
12		oveq2 7456	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
13	11, 12	sseq12d 4042	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))
14	13	imbi2d 340	. . . . 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 340	. . . 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 7456	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 +o 𝑥) = (𝐴 +o 𝐶))
17		oveq2 7456	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
18	16, 17	sseq12d 4042	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))
19	18	imbi2d 340	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊆ 𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶))))
20	19	imbi2d 340	. . . 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 7909	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
22		nnon 7909	. . . . 5 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
23		oa0 8572	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
24	23	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o ∅) = 𝐴)
25		oa0 8572	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
26	25	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o ∅) = 𝐵)
27	24, 26	sseq12d 4042	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o ∅) ⊆ (𝐵 +o ∅) ↔ 𝐴 ⊆ 𝐵))
28	27	biimprd 248	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))
29	21, 22, 28	syl2an 595	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))
30		nnacl 8667	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) ∈ ω)
31	30	ancoms 458	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +o 𝑦) ∈ ω)
32	31	adantrr 716	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +o 𝑦) ∈ ω)
33		nnon 7909	. . . . . . . . . . . 12 ⊢ ((𝐴 +o 𝑦) ∈ ω → (𝐴 +o 𝑦) ∈ On)
34		eloni 6405	. . . . . . . . . . . 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 8667	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
37	36	ancoms 458	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
38	37	adantrl 715	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +o 𝑦) ∈ ω)
39		nnon 7909	. . . . . . . . . . . 12 ⊢ ((𝐵 +o 𝑦) ∈ ω → (𝐵 +o 𝑦) ∈ On)
40		eloni 6405	. . . . . . . . . . . 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 7859	. . . . . . . . . . 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 583	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
44	43	biimpa 476	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)) → suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦))
45		nnasuc 8662	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
46	45	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
47	46	adantrr 716	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
48		nnasuc 8662	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
49	48	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
50	49	adantrl 715	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
51	47, 50	sseq12d 4042	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)) → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦))
54	53	ex 412	. . . . . . 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 412	. . . . 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 7936	. . 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 1117	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  ∅c0 4352  Ord word 6394  Oncon0 6395  suc csuc 6397  (class class class)co 7448  ωcom 7903   +_o coa 8519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-oadd 8526

This theorem is referenced by:  omopthlem2  8716