Theorem srgacl

Description: Closure of the addition operation of a semiring. (Contributed by Mario Carneiro, 14-Jan-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

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

Proof

Step	Hyp	Ref	Expression
1		srgacl.b	|- B = ( Base ` R )
2		srgacl.p	|- .+ = ( +g ` R )
3		srgmnd	|- ( R e. SRing -> R e. Mnd )
4	1 2	mndcl	|- ( ( R e. Mnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
5	3 4	syl3an1	|- ( ( R e. SRing /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )