Theorem simpggrp 20008

Description: A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Assertion

Ref	Expression
simpggrp	⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)

Proof of Theorem simpggrp

Step	Hyp	Ref	Expression
1		issimpg 20006	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2111   class class class wbr 5089  ‘cfv 6481  2_oc2o 8379   ≈ cen 8866  Grpcgrp 18846  NrmSGrpcnsg 19034  SimpGrpcsimpg 20004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-simpg 20005

This theorem is referenced by:  simpggrpd  20009  prmsimpcyc  33197