Theorem lmodlcan

Description: Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodvacl.v	|- V = ( Base ` W )
	lmodvacl.a	|- .+ = ( +g ` W )
Assertion	lmodlcan	|- ( ( W e. LMod /\ ( X e. V /\ Y e. V /\ Z e. V ) ) -> ( ( Z .+ X ) = ( Z .+ Y ) <-> X = Y ) )

Proof

Step	Hyp	Ref	Expression
1		lmodvacl.v	|- V = ( Base ` W )
2		lmodvacl.a	|- .+ = ( +g ` W )
3		lmodgrp	|- ( W e. LMod -> W e. Grp )
4	1 2	grplcan	|- ( ( W e. Grp /\ ( X e. V /\ Y e. V /\ Z e. V ) ) -> ( ( Z .+ X ) = ( Z .+ Y ) <-> X = Y ) )
5	3 4	sylan	|- ( ( W e. LMod /\ ( X e. V /\ Y e. V /\ Z e. V ) ) -> ( ( Z .+ X ) = ( Z .+ Y ) <-> X = Y ) )