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Metamath Proof Explorer

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 } ) )