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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) → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))

Proof of Theorem nnawordi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6557 . . . . . . 7 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
2 oveq2 6557 . . . . . . 7 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
31, 2sseq12d 3597 . . . . . 6 (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
43imbi2d 329 . . . . 5 (𝑥 = ∅ → ((𝐴𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)) ↔ (𝐴𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅))))
54imbi2d 329 . . . 4 (𝑥 = ∅ → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (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))
86, 7sseq12d 3597 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)))
98imbi2d 329 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)) ↔ (𝐴𝐵 → (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦))))
109imbi2d 329 . . . 4 (𝑥 = 𝑦 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (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))
1311, 12sseq12d 3597 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
1413imbi2d 329 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)) ↔ (𝐴𝐵 → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦))))
1514imbi2d 329 . . . 4 (𝑥 = suc 𝑦 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (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))
1816, 17sseq12d 3597 . . . . . 6 (𝑥 = 𝐶 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))
1918imbi2d 329 . . . . 5 (𝑥 = 𝐶 → ((𝐴𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)) ↔ (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶))))
2019imbi2d 329 . . . 4 (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (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)
2423adantr 480 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 ∅) = 𝐴)
25 oa0 7483 . . . . . . . 8 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
2625adantl 481 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 ∅) = 𝐵)
2724, 26sseq12d 3597 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅) ↔ 𝐴𝐵))
2827biimprd 237 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
2921, 22, 28syl2an 493 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
30 nnacl 7578 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) ∈ ω)
3130ancoms 468 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 𝑦) ∈ ω)
3231adantrr 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))
3532, 33, 343syl 18 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → Ord (𝐴 +𝑜 𝑦))
36 nnacl 7578 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
3736ancoms 468 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
3837adantrl 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))
4138, 39, 403syl 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)))
4335, 41, 42syl2anc 691 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
4443biimpa 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))
4645ancoms 468 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
4746adantrr 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))
4948ancoms 468 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
5049adantrl 748 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
5147, 50sseq12d 3597 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
5251adantr 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)))
5344, 52mpbird 246 . . . . . . . 8 (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦))
5453ex 449 . . . . . . 7 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
5554imim2d 55 . . . . . 6 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴𝐵 → (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)) → (𝐴𝐵 → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦))))
5655ex 449 . . . . 5 (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 → (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)) → (𝐴𝐵 → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))))
5756a2d 29 . . . 4 (𝑦 ∈ ω → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦))) → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))))
585, 10, 15, 20, 29, 57finds 6984 . . 3 (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶))))
5958com12 32 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ∈ ω → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶))))
60593impia 1253 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))
 Colors of variables: wff setvar class 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
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