Theorem isogrp 20036

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 20034	. 2 ⊢ oGrp = (Grp ∩ oMnd)
2	1	elin2 4150	1 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2111  Grpcgrp 18846  oMndcomnd 20031  oGrpcogrp 20032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3904  df-ogrp 20034

This theorem is referenced by:  ogrpgrp  20037  ogrpinv0le  20048  ogrpsub  20049  ogrpaddlt  20050  orngsqr  20781  ornglmulle  20782  orngrmulle  20783  ofldtos  20788  suborng  20791  zsoring  28332  isarchi3  33156  archirng  33157  archirngz  33158  archiabllem1a  33160  archiabllem1b  33161  archiabllem2a  33163  archiabllem2c  33164  archiabllem2b  33165  archiabl  33167  reofld  33308  nn0omnd  33309