Theorem isogrp 32490

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 32488	. 2 ⊢ oGrp = (Grp ∩ oMnd)
2	1	elin2 4196	1 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∈ wcel 2104  Grpcgrp 18855  oMndcomnd 32485  oGrpcogrp 32486

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ogrp 32488

This theorem is referenced by:  ogrpgrp  32491  ogrpinv0le  32503  ogrpsub  32504  ogrpaddlt  32505  isarchi3  32603  archirng  32604  archirngz  32605  archiabllem1a  32607  archiabllem1b  32608  archiabllem2a  32610  archiabllem2c  32611  archiabllem2b  32612  archiabl  32614  orngsqr  32692  ornglmulle  32693  orngrmulle  32694  ofldtos  32699  suborng  32703  reofld  32729  nn0omnd  32730