Theorem issimpg 20127

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 6907	. . 3 ⊢ (𝑔 = 𝐺 → (NrmSGrp‘𝑔) = (NrmSGrp‘𝐺))
2	1	breq1d 5158	. 2 ⊢ (𝑔 = 𝐺 → ((NrmSGrp‘𝑔) ≈ 2o ↔ (NrmSGrp‘𝐺) ≈ 2o))
3		df-simpg 20126	. 2 ⊢ SimpGrp = {𝑔 ∈ Grp ∣ (NrmSGrp‘𝑔) ≈ 2o}
4	2, 3	elrab2 3698	1 ⊢ (𝐺 ∈ SimpGrp ↔ (𝐺 ∈ Grp ∧ (NrmSGrp‘𝐺) ≈ 2o))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   class class class wbr 5148  ‘cfv 6563  2_oc2o 8499   ≈ cen 8981  Grpcgrp 18964  NrmSGrpcnsg 19152  SimpGrpcsimpg 20125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-simpg 20126

This theorem is referenced by:  issimpgd  20128  simpggrp  20129  simpg2nsg  20131