Theorem hvmulcl

Description: Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	hvmulcl	$$⊢\left(A\in ℂ\wedge B\in \mathscr{H}\right)\to A{\cdot }_{ℎ}B\in \mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		ax-hfvmul	$$⊢{\cdot }_{ℎ}:ℂ\times \mathscr{H}⟶\mathscr{H}$$
2	1	fovcl	$$⊢\left(A\in ℂ\wedge B\in \mathscr{H}\right)\to A{\cdot }_{ℎ}B\in \mathscr{H}$$