Theorem spansnmul

Description: A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	spansnmul	|- ( ( A e. ~H /\ B e. CC ) -> ( B .h A ) e. ( span ` { A } ) )

Proof

Step	Hyp	Ref	Expression
1		spansnsh	|- ( A e. ~H -> ( span ` { A } ) e. SH )
2		spansnid	|- ( A e. ~H -> A e. ( span ` { A } ) )
3	1 2	jca	|- ( A e. ~H -> ( ( span ` { A } ) e. SH /\ A e. ( span ` { A } ) ) )
4		shmulcl	|- ( ( ( span ` { A } ) e. SH /\ B e. CC /\ A e. ( span ` { A } ) ) -> ( B .h A ) e. ( span ` { A } ) )
5	4	3com12	|- ( ( B e. CC /\ ( span ` { A } ) e. SH /\ A e. ( span ` { A } ) ) -> ( B .h A ) e. ( span ` { A } ) )
6	5	3expb	|- ( ( B e. CC /\ ( ( span ` { A } ) e. SH /\ A e. ( span ` { A } ) ) ) -> ( B .h A ) e. ( span ` { A } ) )
7	3 6	sylan2	|- ( ( B e. CC /\ A e. ~H ) -> ( B .h A ) e. ( span ` { A } ) )
8	7	ancoms	|- ( ( A e. ~H /\ B e. CC ) -> ( B .h A ) e. ( span ` { A } ) )