Theorem ogrpgrp 20089

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 20088	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 496	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  Grpcgrp 18898  oMndcomnd 20083  oGrpcogrp 20084

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 3429  df-in 3892  df-ogrp 20086

This theorem is referenced by:  ogrpinv0le  20100  ogrpsub  20101  ogrpaddlt  20102  ogrpaddltbi  20103  ogrpaddltrbid  20105  ogrpsublt  20106  ogrpinv0lt  20107  ogrpinvlt  20108  isarchi3  33236  archirng  33237  archirngz  33238  archiabllem1a  33240  archiabllem1b  33241  archiabllem1  33242  archiabllem2a  33243  archiabllem2c  33244  archiabllem2b  33245  archiabllem2  33246