Theorem nvadd32

Description: Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvgcl.1	|- X = ( BaseSet ` U )
	nvgcl.2	|- G = ( +v ` U )
Assertion	nvadd32	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( ( A G C ) G B ) )

Proof

Step	Hyp	Ref	Expression
1		nvgcl.1	|- X = ( BaseSet ` U )
2		nvgcl.2	|- G = ( +v ` U )
3	2	nvablo	|- ( U e. NrmCVec -> G e. AbelOp )
4	1 2	bafval	|- X = ran G
5	4	ablo32	|- ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( ( A G C ) G B ) )
6	3 5	sylan	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( ( A G C ) G B ) )