Theorem hvmulcl

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

	Ref	Expression
Assertion	hvmulcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ )

Proof

Step	Hyp	Ref	Expression
1		ax-hfvmul	⊢ ·ℎ : ( ℂ × ℋ ) ⟶ ℋ
2	1	fovcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ℎ 𝐵 ) ∈ ℋ )