Theorem cscval

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

	Ref	Expression
Assertion	cscval	|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( csc ` A ) = ( 1 / ( sin ` A ) ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	|- ( y = A -> ( sin ` y ) = ( sin ` A ) )
2	1	neeq1d	|- ( y = A -> ( ( sin ` y ) =/= 0 <-> ( sin ` A ) =/= 0 ) )
3	2	elrab	|- ( A e. { y e. CC | ( sin ` y ) =/= 0 } <-> ( A e. CC /\ ( sin ` A ) =/= 0 ) )
4		fveq2	|- ( x = A -> ( sin ` x ) = ( sin ` A ) )
5	4	oveq2d	|- ( x = A -> ( 1 / ( sin ` x ) ) = ( 1 / ( sin ` A ) ) )
6		df-csc	|- csc = ( x e. { y e. CC | ( sin ` y ) =/= 0 } |-> ( 1 / ( sin ` x ) ) )
7		ovex	|- ( 1 / ( sin ` A ) ) e. _V
8	5 6 7	fvmpt	|- ( A e. { y e. CC | ( sin ` y ) =/= 0 } -> ( csc ` A ) = ( 1 / ( sin ` A ) ) )
9	3 8	sylbir	|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( csc ` A ) = ( 1 / ( sin ` A ) ) )