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Theorem om1 7667
Description: Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 29-Oct-1995.)
Assertion
Ref Expression
om1 (𝐴 ∈ On → (𝐴 ·𝑜 1𝑜) = 𝐴)

Proof of Theorem om1
StepHypRef Expression
1 df-1o 7605 . . . 4 1𝑜 = suc ∅
21oveq2i 6701 . . 3 (𝐴 ·𝑜 1𝑜) = (𝐴 ·𝑜 suc ∅)
3 peano1 7127 . . . 4 ∅ ∈ ω
4 onmsuc 7654 . . . 4 ((𝐴 ∈ On ∧ ∅ ∈ ω) → (𝐴 ·𝑜 suc ∅) = ((𝐴 ·𝑜 ∅) +𝑜 𝐴))
53, 4mpan2 707 . . 3 (𝐴 ∈ On → (𝐴 ·𝑜 suc ∅) = ((𝐴 ·𝑜 ∅) +𝑜 𝐴))
62, 5syl5eq 2697 . 2 (𝐴 ∈ On → (𝐴 ·𝑜 1𝑜) = ((𝐴 ·𝑜 ∅) +𝑜 𝐴))
7 om0 7642 . . 3 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
87oveq1d 6705 . 2 (𝐴 ∈ On → ((𝐴 ·𝑜 ∅) +𝑜 𝐴) = (∅ +𝑜 𝐴))
9 oa0r 7663 . 2 (𝐴 ∈ On → (∅ +𝑜 𝐴) = 𝐴)
106, 8, 93eqtrd 2689 1 (𝐴 ∈ On → (𝐴 ·𝑜 1𝑜) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  c0 3948  Oncon0 5761  suc csuc 5763  (class class class)co 6690  ωcom 7107  1𝑜c1o 7598   +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-rep 4804  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-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610
This theorem is referenced by:  oe1m  7670  omword1  7698  oeordi  7712  oeoalem  7721  oeoa  7722  oeeui  7727  oaabs2  7770  infxpenc  8879
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