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Metamath Proof Explorer

Theorem recoscld

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

Ref Expression
Hypothesis resincld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
Assertion recoscld ( 𝜑 → ( cos ‘ 𝐴 ) ∈ ℝ )

Proof

Step Hyp Ref Expression
1 resincld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 recoscl ⊢ ( 𝐴 ∈ ℝ → ( cos ‘ 𝐴 ) ∈ ℝ )
3 1 2 syl ⊢ ( 𝜑 → ( cos ‘ 𝐴 ) ∈ ℝ )