Theorem nsgsubg

Description: A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015)

	Ref	Expression
Assertion	nsgsubg	⊢ ( 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
2		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
3	1 2	isnsg	⊢ ( 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ↔ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) ∀ 𝑦 ∈ ( Base ‘ 𝐺 ) ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ∈ 𝑆 ) ) )
4	3	simplbi	⊢ ( 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )