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Theorem omopthlem1 7732
Description: Lemma for omopthi 7734. (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 7083 . . . . 5 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
31, 2ax-mp 5 . . . 4 suc 𝐴 ∈ ω
4 omopthlem1.2 . . . 4 𝐶 ∈ ω
5 nnmwordi 7712 . . . 4 ((suc 𝐴 ∈ ω ∧ 𝐶 ∈ ω ∧ suc 𝐴 ∈ ω) → (suc 𝐴𝐶 → (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (suc 𝐴 ·𝑜 𝐶)))
63, 4, 3, 5mp3an 1423 . . 3 (suc 𝐴𝐶 → (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (suc 𝐴 ·𝑜 𝐶))
7 nnmwordri 7713 . . . 4 ((suc 𝐴 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐶 ∈ ω) → (suc 𝐴𝐶 → (suc 𝐴 ·𝑜 𝐶) ⊆ (𝐶 ·𝑜 𝐶)))
83, 4, 4, 7mp3an 1423 . . 3 (suc 𝐴𝐶 → (suc 𝐴 ·𝑜 𝐶) ⊆ (𝐶 ·𝑜 𝐶))
96, 8sstrd 3611 . 2 (suc 𝐴𝐶 → (suc 𝐴 ·𝑜 suc 𝐴) ⊆ (𝐶 ·𝑜 𝐶))
101nnoni 7069 . . 3 𝐴 ∈ On
114nnoni 7069 . . 3 𝐶 ∈ On
1210, 11onsucssi 7038 . 2 (𝐴𝐶 ↔ suc 𝐴𝐶)
131, 1nnmcli 7692 . . . . . 6 (𝐴 ·𝑜 𝐴) ∈ ω
14 2onn 7717 . . . . . . 7 2𝑜 ∈ ω
151, 14nnmcli 7692 . . . . . 6 (𝐴 ·𝑜 2𝑜) ∈ ω
1613, 15nnacli 7691 . . . . 5 ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ ω
1716nnoni 7069 . . . 4 ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ On
184, 4nnmcli 7692 . . . . 5 (𝐶 ·𝑜 𝐶) ∈ ω
1918nnoni 7069 . . . 4 (𝐶 ·𝑜 𝐶) ∈ On
2017, 19onsucssi 7038 . . 3 (((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ∈ (𝐶 ·𝑜 𝐶) ↔ suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) ⊆ (𝐶 ·𝑜 𝐶))
213, 1nnmcli 7692 . . . . . 6 (suc 𝐴 ·𝑜 𝐴) ∈ ω
22 nnasuc 7683 . . . . . 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 7684 . . . . . 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 7699 . . . . . . . 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 1423 . . . . . . 7 (((𝐴 ·𝑜 𝐴) +𝑜 𝐴) +𝑜 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 +𝑜 𝐴))
28 nnmcom 7703 . . . . . . . . . 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 7684 . . . . . . . . . 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 2643 . . . . . . . 8 (suc 𝐴 ·𝑜 𝐴) = ((𝐴 ·𝑜 𝐴) +𝑜 𝐴)
3332oveq1i 6657 . . . . . . 7 ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴) = (((𝐴 ·𝑜 𝐴) +𝑜 𝐴) +𝑜 𝐴)
34 nnm2 7726 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 ·𝑜 2𝑜) = (𝐴 +𝑜 𝐴))
351, 34ax-mp 5 . . . . . . . 8 (𝐴 ·𝑜 2𝑜) = (𝐴 +𝑜 𝐴)
3635oveq2i 6658 . . . . . . 7 ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 +𝑜 𝐴))
3727, 33, 363eqtr4ri 2654 . . . . . 6 ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = ((suc 𝐴 ·𝑜 𝐴) +𝑜 𝐴)
38 suceq 5788 . . . . . 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 2654 . . . 4 suc ((𝐴 ·𝑜 𝐴) +𝑜 (𝐴 ·𝑜 2𝑜)) = (suc 𝐴 ·𝑜 suc 𝐴)
4140sseq1i 3627 . . 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 1482  wcel 1989  wss 3572  suc csuc 5723  (class class class)co 6647  ωcom 7062  2𝑜c2o 7551   +𝑜 coa 7554   ·𝑜 comu 7555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-omul 7562
This theorem is referenced by:  omopthlem2  7733
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