Theorem cotval

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

	Ref	Expression
Assertion	cotval	|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( cot ` A ) = ( ( cos ` A ) / ( 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 -> ( cos ` x ) = ( cos ` A ) )
5		fveq2	|- ( x = A -> ( sin ` x ) = ( sin ` A ) )
6	4 5	oveq12d	|- ( x = A -> ( ( cos ` x ) / ( sin ` x ) ) = ( ( cos ` A ) / ( sin ` A ) ) )
7		df-cot	|- cot = ( x e. { y e. CC | ( sin ` y ) =/= 0 } |-> ( ( cos ` x ) / ( sin ` x ) ) )
8		ovex	|- ( ( cos ` A ) / ( sin ` A ) ) e. _V
9	6 7 8	fvmpt	|- ( A e. { y e. CC | ( sin ` y ) =/= 0 } -> ( cot ` A ) = ( ( cos ` A ) / ( sin ` A ) ) )
10	3 9	sylbir	|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( cot ` A ) = ( ( cos ` A ) / ( sin ` A ) ) )