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 )