Theorem nnawordi 8616

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 7409	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2		oveq2 7409	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
3	1, 2	sseq12d 4007	. . . . . 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 7409	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
7		oveq2 7409	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
8	6, 7	sseq12d 4007	. . . . . 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 7409	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
12		oveq2 7409	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
13	11, 12	sseq12d 4007	. . . . . 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 7409	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 +o 𝑥) = (𝐴 +o 𝐶))
17		oveq2 7409	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
18	16, 17	sseq12d 4007	. . . . . 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 7854	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
22		nnon 7854	. . . . 5 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
23		oa0 8511	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
24	23	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o ∅) = 𝐴)
25		oa0 8511	. . . . . . . 8 ⊢ (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
26	25	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o ∅) = 𝐵)
27	24, 26	sseq12d 4007	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o ∅) ⊆ (𝐵 +o ∅) ↔ 𝐴 ⊆ 𝐵))
28	27	biimprd 247	. . . . 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 8606	. . . . . . . . . . . . . 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 714	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +o 𝑦) ∈ ω)
33		nnon 7854	. . . . . . . . . . . 12 ⊢ ((𝐴 +o 𝑦) ∈ ω → (𝐴 +o 𝑦) ∈ On)
34		eloni 6364	. . . . . . . . . . . 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 8606	. . . . . . . . . . . . . 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 713	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +o 𝑦) ∈ ω)
39		nnon 7854	. . . . . . . . . . . 12 ⊢ ((𝐵 +o 𝑦) ∈ ω → (𝐵 +o 𝑦) ∈ On)
40		eloni 6364	. . . . . . . . . . . 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 7804	. . . . . . . . . . 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 8601	. . . . . . . . . . . . 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 714	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
48		nnasuc 8601	. . . . . . . . . . . . 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 713	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
51	47, 50	sseq12d 4007	. . . . . . . . . 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 7882	. . 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 1114	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ⊆ wss 3940  ∅c0 4314  Ord word 6353  Oncon0 6354  suc csuc 6356  (class class class)co 7401  ωcom 7848   +_o coa 8458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-oadd 8465

This theorem is referenced by:  omopthlem2  8654