Theorem recsccl

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

	Ref	Expression
Assertion	recsccl	$$⊢\left(A\in ℝ\wedge \sin A\ne 0\right)\to \csc \left(A\right)\in ℝ$$

Proof

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