Theorem ogrpgrp 20004

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 20003	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18812  oMndcomnd 19998  oGrpcogrp 19999

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-in 3910  df-ogrp 20001

This theorem is referenced by:  ogrpinv0le  20015  ogrpsub  20016  ogrpaddlt  20017  ogrpaddltbi  20018  ogrpaddltrbid  20020  ogrpsublt  20021  ogrpinv0lt  20022  ogrpinvlt  20023  isarchi3  33138  archirng  33139  archirngz  33140  archiabllem1a  33142  archiabllem1b  33143  archiabllem1  33144  archiabllem2a  33145  archiabllem2c  33146  archiabllem2b  33147  archiabllem2  33148