Theorem ogrpgrp 20056

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 20055	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  Grpcgrp 18865  oMndcomnd 20050  oGrpcogrp 20051

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3441  df-in 3907  df-ogrp 20053

This theorem is referenced by:  ogrpinv0le  20067  ogrpsub  20068  ogrpaddlt  20069  ogrpaddltbi  20070  ogrpaddltrbid  20072  ogrpsublt  20073  ogrpinv0lt  20074  ogrpinvlt  20075  isarchi3  33248  archirng  33249  archirngz  33250  archiabllem1a  33252  archiabllem1b  33253  archiabllem1  33254  archiabllem2a  33255  archiabllem2c  33256  archiabllem2b  33257  archiabllem2  33258