Theorem isogrp 20099

Description: A (left-)ordered group is a group with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)

Assertion

Ref	Expression
isogrp	⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Proof of Theorem isogrp

Step	Hyp	Ref	Expression
1		df-ogrp 20097	. 2 ⊢ oGrp = (Grp ∩ oMnd)
2	1	elin2 4143	1 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  Grpcgrp 18909  oMndcomnd 20094  oGrpcogrp 20095

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-in 3896  df-ogrp 20097

This theorem is referenced by:  ogrpgrp  20100  ogrpinv0le  20111  ogrpsub  20112  ogrpaddlt  20113  orngsqr  20843  ornglmulle  20844  orngrmulle  20845  ofldtos  20850  suborng  20853  zsoring  28401  isarchi3  33248  archirng  33249  archirngz  33250  archiabllem1a  33252  archiabllem1b  33253  archiabllem2a  33255  archiabllem2c  33256  archiabllem2b  33257  archiabl  33259  reofld  33403  nn0omnd  33404