Theorem ringacl

Description: Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014)

	Ref	Expression
Hypotheses	ringacl.b	|- B = ( Base ` R )
	ringacl.p	|- .+ = ( +g ` R )
Assertion	ringacl	|- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )

Proof

Step	Hyp	Ref	Expression
1		ringacl.b	|- B = ( Base ` R )
2		ringacl.p	|- .+ = ( +g ` R )
3		ringgrp	|- ( R e. Ring -> R e. Grp )
4	1 2	grpcl	|- ( ( R e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
5	3 4	syl3an1	|- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )