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Theorem ogrpgrp 31960
Description: A left-ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.)
Assertion
Ref Expression
ogrpgrp (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Proof of Theorem ogrpgrp
StepHypRef Expression
1 isogrp 31959 . 2 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simplbi 499 1 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Grpcgrp 18753  oMndcomnd 31954  oGrpcogrp 31955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-ogrp 31957
This theorem is referenced by:  ogrpinv0le  31972  ogrpsub  31973  ogrpaddlt  31974  ogrpaddltbi  31975  ogrpaddltrbid  31977  ogrpsublt  31978  ogrpinv0lt  31979  ogrpinvlt  31980  isarchi3  32072  archirng  32073  archirngz  32074  archiabllem1a  32076  archiabllem1b  32077  archiabllem1  32078  archiabllem2a  32079  archiabllem2c  32080  archiabllem2b  32081  archiabllem2  32082
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