Theorem isogrp 20365

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 20363	. 2 ⊢ oGrp = (Grp ∩ oMnd)
2	1	elin2 4103	1 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  Grpcgrp 19170  oMndcomnd 20360  oGrpcogrp 20361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-ogrp 20363

This theorem is used by:  ogrpgrp  20366  ogrpinv0le  20377  ogrpsub  20378  ogrpaddlt  20379  orngsqr  21150  ornglmulle  21151  orngrmulle  21152  ofldtos  21157  suborng  21160  zsoring  28822  isarchi3  33815  archirng  33816  archirngz  33817  archiabllem1a  33819  archiabllem1b  33820  archiabllem2a  33822  archiabllem2c  33823  archiabllem2b  33824  archiabl  33826  reofld  33971  nn0omnd  33972