Theorem ogrpgrp 20039

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 20038	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18847  oMndcomnd 20033  oGrpcogrp 20034

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 3446  df-in 3918  df-ogrp 20036

This theorem is referenced by:  ogrpinv0le  20050  ogrpsub  20051  ogrpaddlt  20052  ogrpaddltbi  20053  ogrpaddltrbid  20055  ogrpsublt  20056  ogrpinv0lt  20057  ogrpinvlt  20058  isarchi3  33156  archirng  33157  archirngz  33158  archiabllem1a  33160  archiabllem1b  33161  archiabllem1  33162  archiabllem2a  33163  archiabllem2c  33164  archiabllem2b  33165  archiabllem2  33166