Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ogrpgrp Structured version   Unicode version

Theorem ogrpgrp 26331
Description: An left ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.)
Assertion
Ref Expression
ogrpgrp  |-  ( G  e. oGrp  ->  G  e.  Grp )

Proof of Theorem ogrpgrp
StepHypRef Expression
1 isogrp 26330 . 2  |-  ( G  e. oGrp 
<->  ( G  e.  Grp  /\  G  e. oMnd ) )
21simplbi 460 1  |-  ( G  e. oGrp  ->  G  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   Grpcgrp 15532  oMndcomnd 26325  oGrpcogrp 26326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-in 3446  df-ogrp 26328
This theorem is referenced by:  ogrpaddltbi  26347  ogrpaddltrbid  26349  ogrpsublt  26350  ogrpinv0lt  26351  ogrpinvlt  26352  isarchi3  26369
  Copyright terms: Public domain W3C validator