Theorem isogrp 30218

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 30216	. 2 ⊢ oGrp = (Grp ∩ oMnd)
2	1	elin2 3999	1 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Syntax hints:   ↔ wb 198   ∧ wa 385   ∈ wcel 2157  Grpcgrp 17738  oMndcomnd 30213  oGrpcogrp 30214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-in 3776  df-ogrp 30216

This theorem is referenced by:  ogrpgrp  30219  ogrpinvOLD  30231  ogrpinv0le  30232  ogrpsub  30233  ogrpaddlt  30234  isarchi3  30257  archirng  30258  archirngz  30259  archiabllem1a  30261  archiabllem1b  30262  archiabllem2a  30264  archiabllem2c  30265  archiabllem2b  30266  archiabl  30268  orngsqr  30320  ornglmulle  30321  orngrmulle  30322  ofldtos  30327  suborng  30331  reofld  30356  nn0omnd  30357