Theorem isogrp 29033

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 29031	. 2 ⊢ oGrp = (Grp ∩ oMnd)
2	1	elin2 3763	1 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Grpcgrp 17245  oMndcomnd 29028  oGrpcogrp 29029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ogrp 29031

This theorem is referenced by:  ogrpgrp  29034  ogrpinvOLD  29046  ogrpinv0le  29047  ogrpsub  29048  ogrpaddlt  29049  isarchi3  29072  archirng  29073  archirngz  29074  archiabllem1a  29076  archiabllem1b  29077  archiabllem2a  29079  archiabllem2c  29080  archiabllem2b  29081  archiabl  29083  orngsqr  29135  ornglmulle  29136  orngrmulle  29137  ofldtos  29142  suborng  29146  reofld  29171  nn0omnd  29172