Theorem hvmulcli

Description: Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hvmulcl.1	$$⊢A\in ℂ$$
	hvmulcl.2	$$⊢B\in \mathscr{H}$$
Assertion	hvmulcli	$$⊢A{\cdot }_{ℎ}B\in \mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		hvmulcl.1	$$⊢A\in ℂ$$
2		hvmulcl.2	$$⊢B\in \mathscr{H}$$
3		hvmulcl	$$⊢\left(A\in ℂ\wedge B\in \mathscr{H}\right)\to A{\cdot }_{ℎ}B\in \mathscr{H}$$
4	1 2 3	mp2an	$$⊢A{\cdot }_{ℎ}B\in \mathscr{H}$$