Theorem ogrpgrp 33017

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

Step	Hyp	Ref	Expression
1		isogrp 33016	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18865  oMndcomnd 33011  oGrpcogrp 33012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-in 3921  df-ogrp 33014

This theorem is referenced by:  ogrpinv0le  33029  ogrpsub  33030  ogrpaddlt  33031  ogrpaddltbi  33032  ogrpaddltrbid  33034  ogrpsublt  33035  ogrpinv0lt  33036  ogrpinvlt  33037  isarchi3  33141  archirng  33142  archirngz  33143  archiabllem1a  33145  archiabllem1b  33146  archiabllem1  33147  archiabllem2a  33148  archiabllem2c  33149  archiabllem2b  33150  archiabllem2  33151