Theorem hvmulcli

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

	Ref	Expression
Hypotheses	hvmulcl.1	⊢ 𝐴 ∈ ℂ
	hvmulcl.2	⊢ 𝐵 ∈ ℋ
Assertion	hvmulcli	⊢ ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ

Proof

Step	Hyp	Ref	Expression
1		hvmulcl.1	⊢ 𝐴 ∈ ℂ
2		hvmulcl.2	⊢ 𝐵 ∈ ℋ
3		hvmulcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ )
4	1 2 3	mp2an	⊢ ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ