Theorem subggrp

Description: A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014)

	Ref	Expression
Hypothesis	subggrp.h	⊢ 𝐻 = ( 𝐺 ↾s 𝑆 )
Assertion	subggrp	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐻 ∈ Grp )

Proof

Step	Hyp	Ref	Expression
1		subggrp.h	⊢ 𝐻 = ( 𝐺 ↾s 𝑆 )
2		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
3	2	issubg	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝐺 ∈ Grp ∧ 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 𝐺 ↾s 𝑆 ) ∈ Grp ) )
4	3	simp3bi	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐺 ↾s 𝑆 ) ∈ Grp )
5	1 4	eqeltrid	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐻 ∈ Grp )