Theorem ogrpgrp 33071

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 33070	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2108  Grpcgrp 18916  oMndcomnd 33065  oGrpcogrp 33066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-in 3933  df-ogrp 33068

This theorem is referenced by:  ogrpinv0le  33083  ogrpsub  33084  ogrpaddlt  33085  ogrpaddltbi  33086  ogrpaddltrbid  33088  ogrpsublt  33089  ogrpinv0lt  33090  ogrpinvlt  33091  isarchi3  33185  archirng  33186  archirngz  33187  archiabllem1a  33189  archiabllem1b  33190  archiabllem1  33191  archiabllem2a  33192  archiabllem2c  33193  archiabllem2b  33194  archiabllem2  33195