Theorem ogrpgrp 20059

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 20058	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  Grpcgrp 18868  oMndcomnd 20053  oGrpcogrp 20054

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 3443  df-in 3909  df-ogrp 20056

This theorem is referenced by:  ogrpinv0le  20070  ogrpsub  20071  ogrpaddlt  20072  ogrpaddltbi  20073  ogrpaddltrbid  20075  ogrpsublt  20076  ogrpinv0lt  20077  ogrpinvlt  20078  isarchi3  33273  archirng  33274  archirngz  33275  archiabllem1a  33277  archiabllem1b  33278  archiabllem1  33279  archiabllem2a  33280  archiabllem2c  33281  archiabllem2b  33282  archiabllem2  33283