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Metamath Proof Explorer

Theorem hicli

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

Ref Expression
Hypotheses hicl.1 ⊢ 𝐴 ∈ ℋ
hicl.2 ⊢ 𝐵 ∈ ℋ
Assertion hicli ( 𝐴 ·ih 𝐵 ) ∈ ℂ

Proof

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