Theorem issimpg 19610

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 6756	. . 3 ⊢ (𝑔 = 𝐺 → (NrmSGrp‘𝑔) = (NrmSGrp‘𝐺))
2	1	breq1d 5080	. 2 ⊢ (𝑔 = 𝐺 → ((NrmSGrp‘𝑔) ≈ 2o ↔ (NrmSGrp‘𝐺) ≈ 2o))
3		df-simpg 19609	. 2 ⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o}
4	2, 3	elrab2 3620	1 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   class class class wbr 5070  ‘cfv 6418  2_oc2o 8261   ≈ cen 8688  Grpcgrp 18492  NrmSGrpcnsg 18665  SimpGrpcsimpg 19608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-simpg 19609

This theorem is referenced by:  issimpgd  19611  simpggrp  19612  simpg2nsg  19614