Theorem isogrp 33016

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 33014	. 2 ⊢ oGrp = (Grp ∩ oMnd)
2	1	elin2 4166	1 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Grpcgrp 18865  oMndcomnd 33011  oGrpcogrp 33012

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 3449  df-in 3921  df-ogrp 33014

This theorem is referenced by:  ogrpgrp  33017  ogrpinv0le  33029  ogrpsub  33030  ogrpaddlt  33031  isarchi3  33141  archirng  33142  archirngz  33143  archiabllem1a  33145  archiabllem1b  33146  archiabllem2a  33148  archiabllem2c  33149  archiabllem2b  33150  archiabl  33152  orngsqr  33282  ornglmulle  33283  orngrmulle  33284  ofldtos  33289  suborng  33293  reofld  33315  nn0omnd  33316