Theorem issimpg

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

	Ref	Expression
Assertion	issimpg	⊢ ( 𝐺 ∈ SimpGrp ↔ ( 𝐺 ∈ 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 ∈ Grp ∣ ( NrmSGrp ‘ 𝑔 ) ≈ 2o }
4	2 3	elrab2	⊢ ( 𝐺 ∈ SimpGrp ↔ ( 𝐺 ∈ Grp ∧ ( NrmSGrp ‘ 𝐺 ) ≈ 2o ) )