Theorem ogrpgrp 20369

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 20368	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 502	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Grpcgrp 19173  oMndcomnd 20363  oGrpcogrp 20364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-ogrp 20366

This theorem is used by:  ogrpinv0le  20380  ogrpsub  20381  ogrpaddlt  20382  ogrpaddltbi  20383  ogrpaddltrbid  20385  ogrpsublt  20386  ogrpinv0lt  20387  ogrpinvlt  20388  isarchi3  33818  archirng  33819  archirngz  33820  archiabllem1a  33822  archiabllem1b  33823  archiabllem1  33824  archiabllem2a  33825  archiabllem2c  33826  archiabllem2b  33827  archiabllem2  33828