Theorem ogrpgrp 31231

Description: A left-ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.)

Assertion

Ref	Expression
ogrpgrp	⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Proof of Theorem ogrpgrp

Step	Hyp	Ref	Expression
1		isogrp 31230	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2108  Grpcgrp 18492  oMndcomnd 31225  oGrpcogrp 31226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ogrp 31228

This theorem is referenced by:  ogrpinv0le  31243  ogrpsub  31244  ogrpaddlt  31245  ogrpaddltbi  31246  ogrpaddltrbid  31248  ogrpsublt  31249  ogrpinv0lt  31250  ogrpinvlt  31251  isarchi3  31343  archirng  31344  archirngz  31345  archiabllem1a  31347  archiabllem1b  31348  archiabllem1  31349  archiabllem2a  31350  archiabllem2c  31351  archiabllem2b  31352  archiabllem2  31353