Theorem nvrcan

Description: Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvgcl.1	|- X = ( BaseSet ` U )
	nvgcl.2	|- G = ( +v ` U )
Assertion	nvrcan	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G C ) = ( B G C ) <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		nvgcl.1	|- X = ( BaseSet ` U )
2		nvgcl.2	|- G = ( +v ` U )
3	2	nvgrp	|- ( U e. NrmCVec -> G e. GrpOp )
4	1 2	bafval	|- X = ran G
5	4	grporcan	|- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G C ) = ( B G C ) <-> A = B ) )
6	3 5	sylan	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G C ) = ( B G C ) <-> A = B ) )