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Metamath Proof Explorer

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