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

Step	Hyp	Ref	Expression
1		isogrp 26330	. 2 |- ( G e. oGrp <-> ( G e. Grp /\ G e. oMnd ) )
2	1	simplbi 460	1 |- ( G e. oGrp -> G e. Grp )

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