Theorem hvsubdistr1i

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

	Ref	Expression
Hypotheses	hvdistr1.1	|- A e. CC
	hvdistr1.2	|- B e. ~H
	hvdistr1.3	|- C e. ~H
Assertion	hvsubdistr1i	|- ( A .h ( B -h C ) ) = ( ( A .h B ) -h ( A .h C ) )

Proof

Step	Hyp	Ref	Expression
1		hvdistr1.1	|- A e. CC
2		hvdistr1.2	|- B e. ~H
3		hvdistr1.3	|- C e. ~H
4		hvsubdistr1	|- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( A .h ( B -h C ) ) = ( ( A .h B ) -h ( A .h C ) ) )
5	1 2 3 4	mp3an	|- ( A .h ( B -h C ) ) = ( ( A .h B ) -h ( A .h C ) )