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 e. ( SubGrp ` G ) -> H e. Grp )

Proof

Step	Hyp	Ref	Expression
1		subggrp.h	|- H = ( G |`s S )
2		eqid	|- ( Base ` G ) = ( Base ` G )
3	2	issubg	|- ( S e. ( SubGrp ` G ) <-> ( G e. Grp /\ S C_ ( Base ` G ) /\ ( G |`s S ) e. Grp ) )
4	3	simp3bi	|- ( S e. ( SubGrp ` G ) -> ( G |`s S ) e. Grp )
5	1 4	eqeltrid	|- ( S e. ( SubGrp ` G ) -> H e. Grp )