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Theorem isogrp 31959
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 31957 . 2 oGrp = (Grp ∩ oMnd)
21elin2 4158 1 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wcel 2107  Grpcgrp 18753  oMndcomnd 31954  oGrpcogrp 31955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-ogrp 31957
This theorem is referenced by:  ogrpgrp  31960  ogrpinv0le  31972  ogrpsub  31973  ogrpaddlt  31974  isarchi3  32072  archirng  32073  archirngz  32074  archiabllem1a  32076  archiabllem1b  32077  archiabllem2a  32079  archiabllem2c  32080  archiabllem2b  32081  archiabl  32083  orngsqr  32146  ornglmulle  32147  orngrmulle  32148  ofldtos  32153  suborng  32157  reofld  32183  nn0omnd  32184
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