Theorem isogrp 32220

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 32218	. 2 ⊢ oGrp = (Grp ∩ oMnd)
2	1	elin2 4198	1 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∈ wcel 2107  Grpcgrp 18819  oMndcomnd 32215  oGrpcogrp 32216

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 32218

This theorem is referenced by:  ogrpgrp  32221  ogrpinv0le  32233  ogrpsub  32234  ogrpaddlt  32235  isarchi3  32333  archirng  32334  archirngz  32335  archiabllem1a  32337  archiabllem1b  32338  archiabllem2a  32340  archiabllem2c  32341  archiabllem2b  32342  archiabl  32344  orngsqr  32422  ornglmulle  32423  orngrmulle  32424  ofldtos  32429  suborng  32433  reofld  32459  nn0omnd  32460