Theorem issimpgd

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

	Ref	Expression
Hypotheses	issimpgd.1	⊢ ( 𝜑 → 𝐺 ∈ Grp )
	issimpgd.2	⊢ ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) ≈ 2o )
Assertion	issimpgd	⊢ ( 𝜑 → 𝐺 ∈ SimpGrp )

Proof

Step	Hyp	Ref	Expression
1		issimpgd.1	⊢ ( 𝜑 → 𝐺 ∈ Grp )
2		issimpgd.2	⊢ ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) ≈ 2o )
3		issimpg	⊢ ( 𝐺 ∈ SimpGrp ↔ ( 𝐺 ∈ Grp ∧ ( NrmSGrp ‘ 𝐺 ) ≈ 2o ) )
4	1 2 3	sylanbrc	⊢ ( 𝜑 → 𝐺 ∈ SimpGrp )