Theorem ogrpgrp 20037

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 20036	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2111  Grpcgrp 18846  oMndcomnd 20031  oGrpcogrp 20032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3904  df-ogrp 20034

This theorem is referenced by:  ogrpinv0le  20048  ogrpsub  20049  ogrpaddlt  20050  ogrpaddltbi  20051  ogrpaddltrbid  20053  ogrpsublt  20054  ogrpinv0lt  20055  ogrpinvlt  20056  isarchi3  33156  archirng  33157  archirngz  33158  archiabllem1a  33160  archiabllem1b  33161  archiabllem1  33162  archiabllem2a  33163  archiabllem2c  33164  archiabllem2b  33165  archiabllem2  33166