Theorem ogrpgrp 20100

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 20099	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 496	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  Grpcgrp 18909  oMndcomnd 20094  oGrpcogrp 20095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-in 3897  df-ogrp 20097

This theorem is referenced by:  ogrpinv0le  20111  ogrpsub  20112  ogrpaddlt  20113  ogrpaddltbi  20114  ogrpaddltrbid  20116  ogrpsublt  20117  ogrpinv0lt  20118  ogrpinvlt  20119  isarchi3  33248  archirng  33249  archirngz  33250  archiabllem1a  33252  archiabllem1b  33253  archiabllem1  33254  archiabllem2a  33255  archiabllem2c  33256  archiabllem2b  33257  archiabllem2  33258