Theorem ogrpgrp 20040

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 20039	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18848  oMndcomnd 20034  oGrpcogrp 20035

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 20037

This theorem is referenced by:  ogrpinv0le  20051  ogrpsub  20052  ogrpaddlt  20053  ogrpaddltbi  20054  ogrpaddltrbid  20056  ogrpsublt  20057  ogrpinv0lt  20058  ogrpinvlt  20059  isarchi3  33157  archirng  33158  archirngz  33159  archiabllem1a  33161  archiabllem1b  33162  archiabllem1  33163  archiabllem2a  33164  archiabllem2c  33165  archiabllem2b  33166  archiabllem2  33167