Theorem lmod4

Description: Commutative/associative law for left module vector sum. (Contributed by NM, 4-Feb-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmod4.v	|- V = ( Base ` W )
	lmod4.p	|- .+ = ( +g ` W )
Assertion	lmod4	|- ( ( W e. LMod /\ ( X e. V /\ Y e. V ) /\ ( Z e. V /\ U e. V ) ) -> ( ( X .+ Y ) .+ ( Z .+ U ) ) = ( ( X .+ Z ) .+ ( Y .+ U ) ) )

Proof

Step	Hyp	Ref	Expression
1		lmod4.v	|- V = ( Base ` W )
2		lmod4.p	|- .+ = ( +g ` W )
3		lmodcmn	|- ( W e. LMod -> W e. CMnd )
4	1 2	cmn4	|- ( ( W e. CMnd /\ ( X e. V /\ Y e. V ) /\ ( Z e. V /\ U e. V ) ) -> ( ( X .+ Y ) .+ ( Z .+ U ) ) = ( ( X .+ Z ) .+ ( Y .+ U ) ) )
5	3 4	syl3an1	|- ( ( W e. LMod /\ ( X e. V /\ Y e. V ) /\ ( Z e. V /\ U e. V ) ) -> ( ( X .+ Y ) .+ ( Z .+ U ) ) = ( ( X .+ Z ) .+ ( Y .+ U ) ) )