Theorem cscval

Description: Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014)

	Ref	Expression
Assertion	cscval	$$⊢\left(A\in ℂ\wedge \sin A\ne 0\right)\to \csc \left(A\right)=\frac{1}{\sin A}$$

Proof

Step	Hyp	Ref	Expression
1		fveq2	$$⊢y=A\to \sin y=\sin A$$
2	1	neeq1d	$$⊢y=A\to \left(\sin y\ne 0\leftrightarrow \sin A\ne 0\right)$$
3	2	elrab	$$⊢A\in \left\{y\in ℂ|\sin y\ne 0\right\}\leftrightarrow \left(A\in ℂ\wedge \sin A\ne 0\right)$$
4		fveq2	$$⊢x=A\to \sin x=\sin A$$
5	4	oveq2d	$$⊢x=A\to \frac{1}{\sin x}=\frac{1}{\sin A}$$
6		df-csc	$$⊢\csc =\left(x\in \left\{y\in ℂ|\sin y\ne 0\right\}⟼\frac{1}{\sin x}\right)$$
7		ovex	$$⊢\frac{1}{\sin A}\in \mathrm{V}$$
8	5 6 7	fvmpt	$$⊢A\in \left\{y\in ℂ|\sin y\ne 0\right\}\to \csc \left(A\right)=\frac{1}{\sin A}$$
9	3 8	sylbir	$$⊢\left(A\in ℂ\wedge \sin A\ne 0\right)\to \csc \left(A\right)=\frac{1}{\sin A}$$