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Theorem isogrp 29830
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
StepHypRef Expression
1 df-ogrp 29828 . 2 oGrp = (Grp ∩ oMnd)
21elin2 3834 1 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2030  Grpcgrp 17469  oMndcomnd 29825  oGrpcogrp 29826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ogrp 29828
This theorem is referenced by:  ogrpgrp  29831  ogrpinvOLD  29843  ogrpinv0le  29844  ogrpsub  29845  ogrpaddlt  29846  isarchi3  29869  archirng  29870  archirngz  29871  archiabllem1a  29873  archiabllem1b  29874  archiabllem2a  29876  archiabllem2c  29877  archiabllem2b  29878  archiabl  29880  orngsqr  29932  ornglmulle  29933  orngrmulle  29934  ofldtos  29939  suborng  29943  reofld  29968  nn0omnd  29969
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