Theorem isogrp 20093

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 20091	. 2 ⊢ oGrp = (Grp ∩ oMnd)
2	1	elin2 4144	1 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  Grpcgrp 18903  oMndcomnd 20088  oGrpcogrp 20089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-in 3897  df-ogrp 20091

This theorem is referenced by:  ogrpgrp  20094  ogrpinv0le  20105  ogrpsub  20106  ogrpaddlt  20107  orngsqr  20837  ornglmulle  20838  orngrmulle  20839  ofldtos  20844  suborng  20847  zsoring  28418  isarchi3  33266  archirng  33267  archirngz  33268  archiabllem1a  33270  archiabllem1b  33271  archiabllem2a  33273  archiabllem2c  33274  archiabllem2b  33275  archiabl  33277  reofld  33421  nn0omnd  33422