Theorem simpggrp 19348

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 19346	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
2	1	simplbi 501	1 ⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114   class class class wbr 5040  ‘cfv 6350  2_oc2o 8138   ≈ cen 8565  Grpcgrp 18232  NrmSGrpcnsg 18405  SimpGrpcsimpg 19344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3402  df-un 3858  df-in 3860  df-ss 3870  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6308  df-fv 6358  df-simpg 19345

This theorem is referenced by:  simpggrpd  19349  prmsimpcyc  31071