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Metamath Proof Explorer

Theorem coscld

Description: Closure of the cosine function. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypothesis sincld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion coscld ( 𝜑 → ( cos ‘ 𝐴 ) ∈ ℂ )

Proof

Step Hyp Ref Expression
1 sincld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 coscl ⊢ ( 𝐴 ∈ ℂ → ( cos ‘ 𝐴 ) ∈ ℂ )
3 1 2 syl ⊢ ( 𝜑 → ( cos ‘ 𝐴 ) ∈ ℂ )