Theorem cotcl

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

	Ref	Expression
Assertion	cotcl	$$⊢\left(A\in ℂ\wedge \sin A\ne 0\right)\to \cot \left(A\right)\in ℂ$$

Proof

Step	Hyp	Ref	Expression
1		cotval	$$⊢\left(A\in ℂ\wedge \sin A\ne 0\right)\to \cot \left(A\right)=\frac{\cos A}{\sin A}$$
2		coscl	$$⊢A\in ℂ\to \cos A\in ℂ$$
3	2	adantr	$$⊢\left(A\in ℂ\wedge \sin A\ne 0\right)\to \cos A\in ℂ$$
4		sincl	$$⊢A\in ℂ\to \sin A\in ℂ$$
5	4	adantr	$$⊢\left(A\in ℂ\wedge \sin A\ne 0\right)\to \sin A\in ℂ$$
6		simpr	$$⊢\left(A\in ℂ\wedge \sin A\ne 0\right)\to \sin A\ne 0$$
7	3 5 6	divcld	$$⊢\left(A\in ℂ\wedge \sin A\ne 0\right)\to \frac{\cos A}{\sin A}\in ℂ$$
8	1 7	eqeltrd	$$⊢\left(A\in ℂ\wedge \sin A\ne 0\right)\to \cot \left(A\right)\in ℂ$$