Theorem hvmulassi

Description: Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hvmulcom.1	|- A e. CC
	hvmulcom.2	|- B e. CC
	hvmulcom.3	|- C e. ~H
Assertion	hvmulassi	|- ( ( A x. B ) .h C ) = ( A .h ( B .h C ) )

Proof

Step	Hyp	Ref	Expression
1		hvmulcom.1	|- A e. CC
2		hvmulcom.2	|- B e. CC
3		hvmulcom.3	|- C e. ~H
4		ax-hvmulass	|- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A x. B ) .h C ) = ( A .h ( B .h C ) ) )
5	1 2 3 4	mp3an	|- ( ( A x. B ) .h C ) = ( A .h ( B .h C ) )