Theorem issimpg 20160

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

Assertion

Ref	Expression
issimpg	⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))

Proof of Theorem issimpg

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6879	. . 3 ⊢ (𝑔 = 𝐺 → (NrmSGrp‘𝑔) = (NrmSGrp‘𝐺))
2	1	breq1d 5120	. 2 ⊢ (𝑔 = 𝐺 → ((NrmSGrp‘𝑔) ≈ 2o ↔ (NrmSGrp‘𝐺) ≈ 2o))
3		df-simpg 20159	. 2 ⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o}
4	2, 3	elrab2 3663	1 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   class class class wbr 5110  ‘cfv 6534  2_oc2o 8443   ≈ cen 8936  Grpcgrp 18996  NrmSGrpcnsg 19183  SimpGrpcsimpg 20158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6490  df-fv 6542  df-simpg 20159

This theorem is referenced by:  issimpgd  20161  simpggrp  20162  simpg2nsg  20164