Theorem retancl

Description: The closure of the tangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014)

	Ref	Expression
Assertion	retancl	$$⊢\left(A\in ℝ\wedge \cos A\ne 0\right)\to \tan A\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		recn	$$⊢A\in ℝ\to A\in ℂ$$
2		tanval	$$⊢\left(A\in ℂ\wedge \cos A\ne 0\right)\to \tan A=\frac{\sin A}{\cos A}$$
3	1 2	sylan	$$⊢\left(A\in ℝ\wedge \cos A\ne 0\right)\to \tan A=\frac{\sin A}{\cos A}$$
4		recoscl	$$⊢A\in ℝ\to \cos A\in ℝ$$
5		resincl	$$⊢A\in ℝ\to \sin A\in ℝ$$
6		redivcl	$$⊢\left(\sin A\in ℝ\wedge \cos A\in ℝ\wedge \cos A\ne 0\right)\to \frac{\sin A}{\cos A}\in ℝ$$
7	5 6	syl3an1	$$⊢\left(A\in ℝ\wedge \cos A\in ℝ\wedge \cos A\ne 0\right)\to \frac{\sin A}{\cos A}\in ℝ$$
8	4 7	syl3an2	$$⊢\left(A\in ℝ\wedge A\in ℝ\wedge \cos A\ne 0\right)\to \frac{\sin A}{\cos A}\in ℝ$$
9	8	3anidm12	$$⊢\left(A\in ℝ\wedge \cos A\ne 0\right)\to \frac{\sin A}{\cos A}\in ℝ$$
10	3 9	eqeltrd	$$⊢\left(A\in ℝ\wedge \cos A\ne 0\right)\to \tan A\in ℝ$$