Theorem slmdass

Description: Semiring left module vector sum is associative. (Contributed by NM, 10-Jan-2014) (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	slmdass	|- ( ( W e. SLMod /\ ( X e. V /\ Y e. V /\ Z e. V ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )

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	mndass	|- ( ( W e. Mnd /\ ( X e. V /\ Y e. V /\ Z e. V ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
5	3 4	sylan	|- ( ( W e. SLMod /\ ( X e. V /\ Y e. V /\ Z e. V ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )