Theorem slmdvacl

Description: Closure of vector addition for a semiring left module. (Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	slmdvacl.v	|- V = ( Base ` W )
	slmdvacl.a	|- .+ = ( +g ` W )
Assertion	slmdvacl	|- ( ( W e. SLMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )

Proof

Step	Hyp	Ref	Expression
1		slmdvacl.v	|- V = ( Base ` W )
2		slmdvacl.a	|- .+ = ( +g ` W )
3		slmdmnd	|- ( W e. SLMod -> W e. Mnd )
4	1 2	mndcl	|- ( ( W e. Mnd /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )
5	3 4	syl3an1	|- ( ( W e. SLMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )