Theorem simpggrp 19963

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 19961	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
2	1	simplbi 498	1 ⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2106   class class class wbr 5148  ‘cfv 6543  2_oc2o 8459   ≈ cen 8935  Grpcgrp 18818  NrmSGrpcnsg 19000  SimpGrpcsimpg 19959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-simpg 19960

This theorem is referenced by:  simpggrpd  19964  prmsimpcyc  32368