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Theorem nnawordi 8243
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 7157 . . . . . . 7 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2 oveq2 7157 . . . . . . 7 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
31, 2sseq12d 3986 . . . . . 6 (𝑥 = ∅ → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))
43imbi2d 344 . . . . 5 (𝑥 = ∅ → ((𝐴𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)) ↔ (𝐴𝐵 → (𝐴 +o ∅) ⊆ (𝐵 +o ∅))))
54imbi2d 344 . . . 4 (𝑥 = ∅ → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))))
6 oveq2 7157 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
7 oveq2 7157 . . . . . . 7 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
86, 7sseq12d 3986 . . . . . 6 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)))
98imbi2d 344 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)) ↔ (𝐴𝐵 → (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦))))
109imbi2d 344 . . . 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 7157 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
12 oveq2 7157 . . . . . . 7 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1311, 12sseq12d 3986 . . . . . 6 (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))
1413imbi2d 344 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)) ↔ (𝐴𝐵 → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦))))
1514imbi2d 344 . . . 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 7157 . . . . . . 7 (𝑥 = 𝐶 → (𝐴 +o 𝑥) = (𝐴 +o 𝐶))
17 oveq2 7157 . . . . . . 7 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
1816, 17sseq12d 3986 . . . . . 6 (𝑥 = 𝐶 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))
1918imbi2d 344 . . . . 5 (𝑥 = 𝐶 → ((𝐴𝐵 → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)) ↔ (𝐴𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶))))
2019imbi2d 344 . . . 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 7580 . . . . 5 (𝐴 ∈ ω → 𝐴 ∈ On)
22 nnon 7580 . . . . 5 (𝐵 ∈ ω → 𝐵 ∈ On)
23 oa0 8137 . . . . . . . 8 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
2423adantr 484 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o ∅) = 𝐴)
25 oa0 8137 . . . . . . . 8 (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
2625adantl 485 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o ∅) = 𝐵)
2724, 26sseq12d 3986 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o ∅) ⊆ (𝐵 +o ∅) ↔ 𝐴𝐵))
2827biimprd 251 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))
2921, 22, 28syl2an 598 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))
30 nnacl 8233 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) ∈ ω)
3130ancoms 462 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +o 𝑦) ∈ ω)
3231adantrr 716 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +o 𝑦) ∈ ω)
33 nnon 7580 . . . . . . . . . . . 12 ((𝐴 +o 𝑦) ∈ ω → (𝐴 +o 𝑦) ∈ On)
34 eloni 6188 . . . . . . . . . . . 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 8233 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
3736ancoms 462 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
3837adantrl 715 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +o 𝑦) ∈ ω)
39 nnon 7580 . . . . . . . . . . . 12 ((𝐵 +o 𝑦) ∈ ω → (𝐵 +o 𝑦) ∈ On)
40 eloni 6188 . . . . . . . . . . . 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 7532 . . . . . . . . . . 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 587 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
4443biimpa 480 . . . . . . . . 9 (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)) → suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦))
45 nnasuc 8228 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
4645ancoms 462 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
4746adantrr 716 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
48 nnasuc 8228 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
4948ancoms 462 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
5049adantrl 715 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
5147, 50sseq12d 3986 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
5251adantr 484 . . . . . . . . 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 260 . . . . . . . 8 (((𝑦 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ∧ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)) → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦))
5453ex 416 . . . . . . 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 416 . . . . 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 7603 . . 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 1114 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wss 3919  c0 4276  Ord word 6177  Oncon0 6178  suc csuc 6180  (class class class)co 7149  ωcom 7574   +o coa 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-oadd 8102
This theorem is referenced by:  omopthlem2  8279
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