Theorem nvadd4

Description: Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvgcl.1	|- X = ( BaseSet ` U )
	nvgcl.2	|- G = ( +v ` U )
Assertion	nvadd4	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( ( A G B ) G ( C G D ) ) = ( ( A G C ) G ( B G D ) ) )

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	ablo4	|- ( ( G e. AbelOp /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( ( A G B ) G ( C G D ) ) = ( ( A G C ) G ( B G D ) ) )
6	3 5	syl3an1	|- ( ( U e. NrmCVec /\ ( A e. X /\ B e. X ) /\ ( C e. X /\ D e. X ) ) -> ( ( A G B ) G ( C G D ) ) = ( ( A G C ) G ( B G D ) ) )