Theorem nvcom

Description: The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007) (New usage is discouraged.)

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

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