Theorem isogrp 20065

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 20063	. 2 ⊢ oGrp = (Grp ∩ oMnd)
2	1	elin2 4157	1 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  Grpcgrp 18875  oMndcomnd 20060  oGrpcogrp 20061

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 3444  df-in 3910  df-ogrp 20063

This theorem is referenced by:  ogrpgrp  20066  ogrpinv0le  20077  ogrpsub  20078  ogrpaddlt  20079  orngsqr  20811  ornglmulle  20812  orngrmulle  20813  ofldtos  20818  suborng  20821  zsoring  28417  isarchi3  33281  archirng  33282  archirngz  33283  archiabllem1a  33285  archiabllem1b  33286  archiabllem2a  33288  archiabllem2c  33289  archiabllem2b  33290  archiabl  33292  reofld  33436  nn0omnd  33437