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Theorem nnawordi 7870
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 6821 . . . . . . 7 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
2 oveq2 6821 . . . . . . 7 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
31, 2sseq12d 3775 . . . . . 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 6821 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
7 oveq2 6821 . . . . . . 7 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
86, 7sseq12d 3775 . . . . . 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 6821 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
12 oveq2 6821 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1311, 12sseq12d 3775 . . . . . 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 6821 . . . . . . 7 (𝑥 = 𝐶 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐶))
17 oveq2 6821 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
1816, 17sseq12d 3775 . . . . . 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 7236 . . . . 5 (𝐴 ∈ ω → 𝐴 ∈ On)
22 nnon 7236 . . . . 5 (𝐵 ∈ ω → 𝐵 ∈ On)
23 oa0 7765 . . . . . . . 8 (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
2423adantr 472 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 ∅) = 𝐴)
25 oa0 7765 . . . . . . . 8 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
2625adantl 473 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 ∅) = 𝐵)
2724, 26sseq12d 3775 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅) ↔ 𝐴𝐵))
2827biimprd 238 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
2921, 22, 28syl2an 495 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
30 nnacl 7860 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) ∈ ω)
3130ancoms 468 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 𝑦) ∈ ω)
3231adantrr 755 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +𝑜 𝑦) ∈ ω)
33 nnon 7236 . . . . . . . . . . . 12 ((𝐴 +𝑜 𝑦) ∈ ω → (𝐴 +𝑜 𝑦) ∈ On)
34 eloni 5894 . . . . . . . . . . . 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 7860 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
3736ancoms 468 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
3837adantrl 754 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +𝑜 𝑦) ∈ ω)
39 nnon 7236 . . . . . . . . . . . 12 ((𝐵 +𝑜 𝑦) ∈ ω → (𝐵 +𝑜 𝑦) ∈ On)
40 eloni 5894 . . . . . . . . . . . 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 7188 . . . . . . . . . . 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 696 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
4443biimpa 502 . . . . . . . . 9 (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)) → suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦))
45 nnasuc 7855 . . . . . . . . . . . . 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 755 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
48 nnasuc 7855 . . . . . . . . . . . . 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 754 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
5147, 50sseq12d 3775 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
5251adantr 472 . . . . . . . . 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 247 . . . . . . . 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 57 . . . . . 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 7257 . . 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 1110 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wss 3715  c0 4058  Ord word 5883  Oncon0 5884  suc csuc 5886  (class class class)co 6813  ωcom 7230   +𝑜 coa 7726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733
This theorem is referenced by:  omopthlem2  7905
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