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Theorem omopthlem1 7780
Description: Lemma for omopthi 7782. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
Hypotheses
Ref Expression
omopthlem1.1 𝐴 ∈ ω
omopthlem1.2 𝐶 ∈ ω
Assertion
Ref Expression
omopthlem1 (𝐴𝐶 → ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶))

Proof of Theorem omopthlem1
StepHypRef Expression
1 omopthlem1.1 . . . . 5 𝐴 ∈ ω
2 peano2 7128 . . . . 5 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
31, 2ax-mp 5 . . . 4 suc 𝐴 ∈ ω
4 omopthlem1.2 . . . 4 𝐶 ∈ ω
5 nnmwordi 7760 . . . 4 ((suc 𝐴 ∈ ω ∧ 𝐶 ∈ ω ∧ suc 𝐴 ∈ ω) → (suc 𝐴𝐶 → (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (suc 𝐴 ·𝑜 𝐶)))
63, 4, 3, 5mp3an 1464 . . 3 (suc 𝐴𝐶 → (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (suc 𝐴 ·𝑜 𝐶))
7 nnmwordri 7761 . . . 4 ((suc 𝐴 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐶 ∈ ω) → (suc 𝐴𝐶 → (suc 𝐴 ·𝑜 𝐶) ⊆ (𝐶 ·𝑜 𝐶)))
83, 4, 4, 7mp3an 1464 . . 3 (suc 𝐴𝐶 → (suc 𝐴 ·𝑜 𝐶) ⊆ (𝐶 ·𝑜 𝐶))
96, 8sstrd 3646 . 2 (suc 𝐴𝐶 → (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝐶))
101nnoni 7114 . . 3 𝐴 ∈ On
114nnoni 7114 . . 3 𝐶 ∈ On
1210, 11onsucssi 7083 . 2 (𝐴𝐶 ↔ suc 𝐴𝐶)
131, 1nnmcli 7740 . . . . . 6 (𝐴 ·𝑜 𝐴) ∈ ω
14 2onn 7765 . . . . . . 7 2𝑜 ∈ ω
151, 14nnmcli 7740 . . . . . 6 (𝐴 ·𝑜 2𝑜) ∈ ω
1613, 15nnacli 7739 . . . . 5 ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ ω
1716nnoni 7114 . . . 4 ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ On
184, 4nnmcli 7740 . . . . 5 (𝐶 ·𝑜 𝐶) ∈ ω
1918nnoni 7114 . . . 4 (𝐶 ·𝑜 𝐶) ∈ On
2017, 19onsucssi 7083 . . 3 (((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶) ↔ suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ⊆ (𝐶 ·𝑜 𝐶))
213, 1nnmcli 7740 . . . . . 6 (suc 𝐴 ·𝑜 𝐴) ∈ ω
22 nnasuc 7731 . . . . . 6 (((suc 𝐴 ·𝑜 𝐴) ∈ ω ∧ 𝐴 ∈ ω) → ((suc 𝐴 ·𝑜 𝐴) +𝑜 suc 𝐴) = suc ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴))
2321, 1, 22mp2an 708 . . . . 5 ((suc 𝐴 ·𝑜 𝐴) +𝑜 suc 𝐴) = suc ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴)
24 nnmsuc 7732 . . . . . 6 ((suc 𝐴 ∈ ω ∧ 𝐴 ∈ ω) → (suc 𝐴 ·𝑜 suc 𝐴) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 suc 𝐴))
253, 1, 24mp2an 708 . . . . 5 (suc 𝐴 ·𝑜 suc 𝐴) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 suc 𝐴)
26 nnaass 7747 . . . . . . . 8 (((𝐴 ·𝑜 𝐴) ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ·𝑜 𝐴) +𝑜 𝐴) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 +𝑜 𝐴)))
2713, 1, 1, 26mp3an 1464 . . . . . . 7 (((𝐴 ·𝑜 𝐴) +𝑜 𝐴) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 +𝑜 𝐴))
28 nnmcom 7751 . . . . . . . . . 10 ((suc 𝐴 ∈ ω ∧ 𝐴 ∈ ω) → (suc 𝐴 ·𝑜 𝐴) = (𝐴 ·𝑜 suc 𝐴))
293, 1, 28mp2an 708 . . . . . . . . 9 (suc 𝐴 ·𝑜 𝐴) = (𝐴 ·𝑜 suc 𝐴)
30 nnmsuc 7732 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐴 ∈ ω) → (𝐴 ·𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 𝐴))
311, 1, 30mp2an 708 . . . . . . . . 9 (𝐴 ·𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 𝐴)
3229, 31eqtri 2673 . . . . . . . 8 (suc 𝐴 ·𝑜 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 𝐴)
3332oveq1i 6700 . . . . . . 7 ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴) = (((𝐴 ·𝑜 𝐴) +𝑜 𝐴) +𝑜 𝐴)
34 nnm2 7774 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 ·𝑜 2𝑜) = (𝐴 +𝑜 𝐴))
351, 34ax-mp 5 . . . . . . . 8 (𝐴 ·𝑜 2𝑜) = (𝐴 +𝑜 𝐴)
3635oveq2i 6701 . . . . . . 7 ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 +𝑜 𝐴))
3727, 33, 363eqtr4ri 2684 . . . . . 6 ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴)
38 suceq 5828 . . . . . 6 (((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴) → suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = suc ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴))
3937, 38ax-mp 5 . . . . 5 suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = suc ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴)
4023, 25, 393eqtr4ri 2684 . . . 4 suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = (suc 𝐴 ·𝑜 suc 𝐴)
4140sseq1i 3662 . . 3 (suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ⊆ (𝐶 ·𝑜 𝐶) ↔ (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝐶))
4220, 41bitri 264 . 2 (((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶) ↔ (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝐶))
439, 12, 423imtr4i 281 1 (𝐴𝐶 → ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wss 3607  suc csuc 5763  (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:  omopthlem2  7781
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