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Theorem omopthlem2 8385
Description: Lemma for omopthi 8386. (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 8343 . . . . . 6 (𝐶 ·o 𝐶) ∈ ω
3 omopthlem2.4 . . . . . 6 𝐷 ∈ ω
42, 3nnacli 8342 . . . . 5 ((𝐶 ·o 𝐶) +o 𝐷) ∈ ω
54nnoni 7651 . . . 4 ((𝐶 ·o 𝐶) +o 𝐷) ∈ On
65onirri 6320 . . 3 ¬ ((𝐶 ·o 𝐶) +o 𝐷) ∈ ((𝐶 ·o 𝐶) +o 𝐷)
7 eleq1 2825 . . 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 329 . 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 8357 . . . 4 (((𝐶 ·o 𝐶) ∈ ω ∧ 𝐷 ∈ ω) → (𝐶 ·o 𝐶) ⊆ ((𝐶 ·o 𝐶) +o 𝐷))
102, 3, 9mp2an 692 . . 3 (𝐶 ·o 𝐶) ⊆ ((𝐶 ·o 𝐶) +o 𝐷)
11 omopthlem2.2 . . . . . . . . 9 𝐵 ∈ ω
12 omopthlem2.1 . . . . . . . . . . 11 𝐴 ∈ ω
1312, 11nnacli 8342 . . . . . . . . . 10 (𝐴 +o 𝐵) ∈ ω
1413, 12nnacli 8342 . . . . . . . . 9 ((𝐴 +o 𝐵) +o 𝐴) ∈ ω
15 nnaword1 8357 . . . . . . . . 9 ((𝐵 ∈ ω ∧ ((𝐴 +o 𝐵) +o 𝐴) ∈ ω) → 𝐵 ⊆ (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴)))
1611, 14, 15mp2an 692 . . . . . . . 8 𝐵 ⊆ (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴))
17 nnacom 8345 . . . . . . . . 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 692 . . . . . . . 8 (𝐵 +o ((𝐴 +o 𝐵) +o 𝐴)) = (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵)
1916, 18sseqtri 3937 . . . . . . 7 𝐵 ⊆ (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵)
20 nnaass 8350 . . . . . . . . 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 1463 . . . . . . . 8 (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵) = ((𝐴 +o 𝐵) +o (𝐴 +o 𝐵))
22 nnm2 8378 . . . . . . . . 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 2768 . . . . . . 7 (((𝐴 +o 𝐵) +o 𝐴) +o 𝐵) = ((𝐴 +o 𝐵) ·o 2o)
2519, 24sseqtri 3937 . . . . . 6 𝐵 ⊆ ((𝐴 +o 𝐵) ·o 2o)
26 2onn 8368 . . . . . . . 8 2o ∈ ω
2713, 26nnmcli 8343 . . . . . . 7 ((𝐴 +o 𝐵) ·o 2o) ∈ ω
2813, 13nnmcli 8343 . . . . . . 7 ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) ∈ ω
29 nnawordi 8349 . . . . . . 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 1463 . . . . . 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 8345 . . . . . 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 692 . . . . 5 (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (𝐵 +o ((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)))
34 nnacom 8345 . . . . . 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 692 . . . . 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 3944 . . . 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 8384 . . . 4 ((𝐴 +o 𝐵) ∈ 𝐶 → (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o ((𝐴 +o 𝐵) ·o 2o)) ∈ (𝐶 ·o 𝐶))
3828, 11nnacli 8342 . . . . . 6 (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ ω
3938nnoni 7651 . . . . 5 (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ On
402nnoni 7651 . . . . 5 (𝐶 ·o 𝐶) ∈ On
41 ontr2 6260 . . . . 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 692 . . . 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 590 . . 3 ((𝐴 +o 𝐵) ∈ 𝐶 → (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ (𝐶 ·o 𝐶))
4410, 43sselid 3898 . 2 ((𝐴 +o 𝐵) ∈ 𝐶 → (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) ∈ ((𝐶 ·o 𝐶) +o 𝐷))
458, 44nsyl3 140 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 399   = wceq 1543  wcel 2110  wss 3866  Oncon0 6213  (class class class)co 7213  ωcom 7644  2oc2o 8196   +o coa 8199   ·o comu 8200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-oadd 8206  df-omul 8207
This theorem is referenced by:  omopthi  8386
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