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Metamath Proof Explorer

Theorem tancld

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

Ref Expression
Hypotheses sincld.1 ⊢ φ → A ∈ ℂ
tancld.2 ⊢ φ → cos ⁡ A ≠ 0
Assertion tancld ⊢ φ → tan ⁡ A ∈ ℂ

Proof

Step Hyp Ref Expression
1 sincld.1 ⊢ φ → A ∈ ℂ
2 tancld.2 ⊢ φ → cos ⁡ A ≠ 0
3 tancl ⊢ A ∈ ℂ ∧ cos ⁡ A ≠ 0 → tan ⁡ A ∈ ℂ
4 1 2 3 syl2anc ⊢ φ → tan ⁡ A ∈ ℂ