Theorem tancld

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

	Ref	Expression
Hypotheses	sincld.1	$$⊢\phi \to A\in ℂ$$
	tancld.2	$$⊢\phi \to \cos A\ne 0$$
Assertion	tancld	$$⊢\phi \to \tan A\in ℂ$$

Proof

Step	Hyp	Ref	Expression
1		sincld.1	$$⊢\phi \to A\in ℂ$$
2		tancld.2	$$⊢\phi \to \cos A\ne 0$$
3		tancl	$$⊢\left(A\in ℂ\wedge \cos A\ne 0\right)\to \tan A\in ℂ$$
4	1 2 3	syl2anc	$$⊢\phi \to \tan A\in ℂ$$