Theorem simpggrp 20065

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 20063	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
2	1	simplbi 496	1 ⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2098   class class class wbr 5152  ‘cfv 6553  2_oc2o 8489   ≈ cen 8969  Grpcgrp 18904  NrmSGrpcnsg 19090  SimpGrpcsimpg 20061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-simpg 20062

This theorem is referenced by:  simpggrpd  20066  prmsimpcyc  32964