Theorem subrgring

Description: A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014)

	Ref	Expression
Hypothesis	subrgring.1	|- S = ( R |`s A )
Assertion	subrgring	|- ( A e. ( SubRing ` R ) -> S e. Ring )

Proof

Step	Hyp	Ref	Expression
1		subrgring.1	|- S = ( R |`s A )
2		eqid	|- ( Base ` R ) = ( Base ` R )
3		eqid	|- ( 1r ` R ) = ( 1r ` R )
4	2 3	issubrg	|- ( A e. ( SubRing ` R ) <-> ( ( R e. Ring /\ ( R |`s A ) e. Ring ) /\ ( A C_ ( Base ` R ) /\ ( 1r ` R ) e. A ) ) )
5	4	simplbi	|- ( A e. ( SubRing ` R ) -> ( R e. Ring /\ ( R |`s A ) e. Ring ) )
6	5	simprd	|- ( A e. ( SubRing ` R ) -> ( R |`s A ) e. Ring )
7	1 6	eqeltrid	|- ( A e. ( SubRing ` R ) -> S e. Ring )