Theorem ogrpgrp 29034

Description: An 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 29033	. 2 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
2	1	simplbi 475	1 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 1977  Grpcgrp 17245  oMndcomnd 29028  oGrpcogrp 29029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ogrp 29031

This theorem is referenced by:  ogrpinv0le  29047  ogrpsub  29048  ogrpaddlt  29049  ogrpaddltbi  29050  ogrpaddltrbid  29052  ogrpsublt  29053  ogrpinv0lt  29054  ogrpinvlt  29055  isarchi3  29072  archirng  29073  archirngz  29074  archiabllem1a  29076  archiabllem1b  29077  archiabllem1  29078  archiabllem2a  29079  archiabllem2c  29080  archiabllem2b  29081  archiabllem2  29082