Theorem issimpgd

Description: Deduce a simple group from its properties. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	issimpgd.1	|- ( ph -> G e. Grp )
	issimpgd.2	|- ( ph -> ( NrmSGrp ` G ) ~~ 2o )
Assertion	issimpgd	|- ( ph -> G e. SimpGrp )

Proof

Step	Hyp	Ref	Expression
1		issimpgd.1	|- ( ph -> G e. Grp )
2		issimpgd.2	|- ( ph -> ( NrmSGrp ` G ) ~~ 2o )
3		issimpg	|- ( G e. SimpGrp <-> ( G e. Grp /\ ( NrmSGrp ` G ) ~~ 2o ) )
4	1 2 3	sylanbrc	|- ( ph -> G e. SimpGrp )