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Theorem omopthlem2 7781
Description: Lemma for omopthi 7782. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
Hypotheses
Ref Expression
omopthlem2.1 𝐴 ∈ ω
omopthlem2.2 𝐵 ∈ ω
omopthlem2.3 𝐶 ∈ ω
omopthlem2.4 𝐷 ∈ ω
Assertion
Ref Expression
omopthlem2 ((𝐴 +𝑜 𝐵) ∈ 𝐶 → ¬ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵))

Proof of Theorem omopthlem2
StepHypRef Expression
1 omopthlem2.3 . . . . . . 7 𝐶 ∈ ω
21, 1nnmcli 7740 . . . . . 6 (𝐶 ·𝑜 𝐶) ∈ ω
3 omopthlem2.4 . . . . . 6 𝐷 ∈ ω
42, 3nnacli 7739 . . . . 5 ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ ω
54nnoni 7114 . . . 4 ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ On
65onirri 5872 . . 3 ¬ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷)
7 eleq1 2718 . . 3 (((𝐶 ·𝑜 𝐶) +𝑜 𝐷) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) → (((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ↔ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷)))
86, 7mtbii 315 . 2 (((𝐶 ·𝑜 𝐶) +𝑜 𝐷) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) → ¬ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷))
9 nnaword1 7754 . . . 4 (((𝐶 ·𝑜 𝐶) ∈ ω ∧ 𝐷 ∈ ω) → (𝐶 ·𝑜 𝐶) ⊆ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷))
102, 3, 9mp2an 708 . . 3 (𝐶 ·𝑜 𝐶) ⊆ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷)
11 omopthlem2.2 . . . . . . . . 9 𝐵 ∈ ω
12 omopthlem2.1 . . . . . . . . . . 11 𝐴 ∈ ω
1312, 11nnacli 7739 . . . . . . . . . 10 (𝐴 +𝑜 𝐵) ∈ ω
1413, 12nnacli 7739 . . . . . . . . 9 ((𝐴 +𝑜 𝐵) +𝑜 𝐴) ∈ ω
15 nnaword1 7754 . . . . . . . . 9 ((𝐵 ∈ ω ∧ ((𝐴 +𝑜 𝐵) +𝑜 𝐴) ∈ ω) → 𝐵 ⊆ (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴)))
1611, 14, 15mp2an 708 . . . . . . . 8 𝐵 ⊆ (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴))
17 nnacom 7742 . . . . . . . . 9 ((𝐵 ∈ ω ∧ ((𝐴 +𝑜 𝐵) +𝑜 𝐴) ∈ ω) → (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴)) = (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵))
1811, 14, 17mp2an 708 . . . . . . . 8 (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴)) = (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵)
1916, 18sseqtri 3670 . . . . . . 7 𝐵 ⊆ (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵)
20 nnaass 7747 . . . . . . . . 9 (((𝐴 +𝑜 𝐵) ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵) = ((𝐴 +𝑜 𝐵) +𝑜 (𝐴 +𝑜 𝐵)))
2113, 12, 11, 20mp3an 1464 . . . . . . . 8 (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵) = ((𝐴 +𝑜 𝐵) +𝑜 (𝐴 +𝑜 𝐵))
22 nnm2 7774 . . . . . . . . 9 ((𝐴 +𝑜 𝐵) ∈ ω → ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) = ((𝐴 +𝑜 𝐵) +𝑜 (𝐴 +𝑜 𝐵)))
2313, 22ax-mp 5 . . . . . . . 8 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) = ((𝐴 +𝑜 𝐵) +𝑜 (𝐴 +𝑜 𝐵))
2421, 23eqtr4i 2676 . . . . . . 7 (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵) = ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)
2519, 24sseqtri 3670 . . . . . 6 𝐵 ⊆ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)
26 2onn 7765 . . . . . . . 8 2𝑜 ∈ ω
2713, 26nnmcli 7740 . . . . . . 7 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) ∈ ω
2813, 13nnmcli 7740 . . . . . . 7 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω
29 nnawordi 7746 . . . . . . 7 ((𝐵 ∈ ω ∧ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) ∈ ω ∧ ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω) → (𝐵 ⊆ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) → (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)))))
3011, 27, 28, 29mp3an 1464 . . . . . 6 (𝐵 ⊆ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) → (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))))
3125, 30ax-mp 5 . . . . 5 (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)))
32 nnacom 7742 . . . . . 6 ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))))
3328, 11, 32mp2an 708 . . . . 5 (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)))
34 nnacom 7742 . . . . . 6 ((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω ∧ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) ∈ ω) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) = (((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵))))
3528, 27, 34mp2an 708 . . . . 5 (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) = (((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)))
3631, 33, 353sstr4i 3677 . . . 4 (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜))
3713, 1omopthlem1 7780 . . . 4 ((𝐴 +𝑜 𝐵) ∈ 𝐶 → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶))
3828, 11nnacli 7739 . . . . . 6 (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ ω
3938nnoni 7114 . . . . 5 (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ On
402nnoni 7114 . . . . 5 (𝐶 ·𝑜 𝐶) ∈ On
41 ontr2 5810 . . . . 5 (((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ On ∧ (𝐶 ·𝑜 𝐶) ∈ On) → (((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) ∧ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶)) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐶)))
4239, 40, 41mp2an 708 . . . 4 (((((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ⊆ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) ∧ (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶)) → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐶))
4336, 37, 42sylancr 696 . . 3 ((𝐴 +𝑜 𝐵) ∈ 𝐶 → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐶))
4410, 43sseldi 3634 . 2 ((𝐴 +𝑜 𝐵) ∈ 𝐶 → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷))
458, 44nsyl3 133 1 ((𝐴 +𝑜 𝐵) ∈ 𝐶 → ¬ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) = (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wss 3607  Oncon0 5761  (class class class)co 6690  ωcom 7107  2𝑜c2o 7599   +𝑜 coa 7602   ·𝑜 comu 7603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610
This theorem is referenced by:  omopthi  7782
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