Theorem simpggrp 20310

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 20308	. 2 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))
2	1	simplbi 502	1 ⊢ (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   class class class wbr 5103  ‘cfv 6538  2_oc2o 8470   ≈ cen 8970  Grpcgrp 19144  NrmSGrpcnsg 19331  SimpGrpcsimpg 20306

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-simpg 20307

This theorem is used by:  simpggrpd  20311  prmsimpcyc  33789