Theorem ogrpgrp 32252

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 32251	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 499	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2107  Grpcgrp 18819  oMndcomnd 32246  oGrpcogrp 32247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ogrp 32249

This theorem is referenced by:  ogrpinv0le  32264  ogrpsub  32265  ogrpaddlt  32266  ogrpaddltbi  32267  ogrpaddltrbid  32269  ogrpsublt  32270  ogrpinv0lt  32271  ogrpinvlt  32272  isarchi3  32364  archirng  32365  archirngz  32366  archiabllem1a  32368  archiabllem1b  32369  archiabllem1  32370  archiabllem2a  32371  archiabllem2c  32372  archiabllem2b  32373  archiabllem2  32374