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Theorem omopthlem2 7682
Description: Lemma for omopthi 7683. (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 7641 . . . . . 6 (𝐶 ·𝑜 𝐶) ∈ ω
3 omopthlem2.4 . . . . . 6 𝐷 ∈ ω
42, 3nnacli 7640 . . . . 5 ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ ω
54nnoni 7020 . . . 4 ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ On
65onirri 5796 . . 3 ¬ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷) ∈ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷)
7 eleq1 2692 . . 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 316 . 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 7655 . . . 4 (((𝐶 ·𝑜 𝐶) ∈ ω ∧ 𝐷 ∈ ω) → (𝐶 ·𝑜 𝐶) ⊆ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷))
102, 3, 9mp2an 707 . . 3 (𝐶 ·𝑜 𝐶) ⊆ ((𝐶 ·𝑜 𝐶) +𝑜 𝐷)
11 omopthlem2.2 . . . . . . . . 9 𝐵 ∈ ω
12 omopthlem2.1 . . . . . . . . . . 11 𝐴 ∈ ω
1312, 11nnacli 7640 . . . . . . . . . 10 (𝐴 +𝑜 𝐵) ∈ ω
1413, 12nnacli 7640 . . . . . . . . 9 ((𝐴 +𝑜 𝐵) +𝑜 𝐴) ∈ ω
15 nnaword1 7655 . . . . . . . . 9 ((𝐵 ∈ ω ∧ ((𝐴 +𝑜 𝐵) +𝑜 𝐴) ∈ ω) → 𝐵 ⊆ (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴)))
1611, 14, 15mp2an 707 . . . . . . . 8 𝐵 ⊆ (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴))
17 nnacom 7643 . . . . . . . . 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 707 . . . . . . . 8 (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) +𝑜 𝐴)) = (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵)
1916, 18sseqtri 3621 . . . . . . 7 𝐵 ⊆ (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵)
20 nnaass 7648 . . . . . . . . 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 1421 . . . . . . . 8 (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵) = ((𝐴 +𝑜 𝐵) +𝑜 (𝐴 +𝑜 𝐵))
22 nnm2 7675 . . . . . . . . 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 2651 . . . . . . 7 (((𝐴 +𝑜 𝐵) +𝑜 𝐴) +𝑜 𝐵) = ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)
2519, 24sseqtri 3621 . . . . . 6 𝐵 ⊆ ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)
26 2onn 7666 . . . . . . . 8 2𝑜 ∈ ω
2713, 26nnmcli 7641 . . . . . . 7 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜) ∈ ω
2813, 13nnmcli 7641 . . . . . . 7 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) ∈ ω
29 nnawordi 7647 . . . . . . 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 1421 . . . . . 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 7643 . . . . . 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 707 . . . . 5 (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) = (𝐵 +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)))
34 nnacom 7643 . . . . . 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 707 . . . . 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 3628 . . . 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 7681 . . . 4 ((𝐴 +𝑜 𝐵) ∈ 𝐶 → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 ((𝐴 +𝑜 𝐵) ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶))
3828, 11nnacli 7640 . . . . . 6 (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ ω
3938nnoni 7020 . . . . 5 (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ On
402nnoni 7020 . . . . 5 (𝐶 ·𝑜 𝐶) ∈ On
41 ontr2 5734 . . . . 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 707 . . . 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 694 . . 3 ((𝐴 +𝑜 𝐵) ∈ 𝐶 → (((𝐴 +𝑜 𝐵) ·𝑜 (𝐴 +𝑜 𝐵)) +𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐶))
4410, 43sseldi 3586 . 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 384   = wceq 1480  wcel 1992  wss 3560  Oncon0 5685  (class class class)co 6605  ωcom 7013  2𝑜c2o 7500   +𝑜 coa 7503   ·𝑜 comu 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-omul 7511
This theorem is referenced by:  omopthi  7683
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