Theorem ogrpgrp 20058

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 20057	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 496	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  Grpcgrp 18867  oMndcomnd 20052  oGrpcogrp 20053

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 3432  df-in 3897  df-ogrp 20055

This theorem is referenced by:  ogrpinv0le  20069  ogrpsub  20070  ogrpaddlt  20071  ogrpaddltbi  20072  ogrpaddltrbid  20074  ogrpsublt  20075  ogrpinv0lt  20076  ogrpinvlt  20077  isarchi3  33253  archirng  33254  archirngz  33255  archiabllem1a  33257  archiabllem1b  33258  archiabllem1  33259  archiabllem2a  33260  archiabllem2c  33261  archiabllem2b  33262  archiabllem2  33263