Theorem issimpg

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

	Ref	Expression
Assertion	issimpg	|- ( G e. SimpGrp <-> ( G e. Grp /\ ( NrmSGrp ` G ) ~~ 2o ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	|- ( g = G -> ( NrmSGrp ` g ) = ( NrmSGrp ` G ) )
2	1	breq1d	|- ( g = G -> ( ( NrmSGrp ` g ) ~~ 2o <-> ( NrmSGrp ` G ) ~~ 2o ) )
3		df-simpg	|- SimpGrp = { g e. Grp | ( NrmSGrp ` g ) ~~ 2o }
4	2 3	elrab2	|- ( G e. SimpGrp <-> ( G e. Grp /\ ( NrmSGrp ` G ) ~~ 2o ) )