Theorem resinhcl

Description: The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015)

	Ref	Expression
Assertion	resinhcl	|- ( A e. RR -> ( ( sin ` ( _i x. A ) ) / _i ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		recn	|- ( A e. RR -> A e. CC )
2		sinhval	|- ( A e. CC -> ( ( sin ` ( _i x. A ) ) / _i ) = ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) )
3	1 2	syl	|- ( A e. RR -> ( ( sin ` ( _i x. A ) ) / _i ) = ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) )
4		reefcl	|- ( A e. RR -> ( exp ` A ) e. RR )
5		renegcl	|- ( A e. RR -> -u A e. RR )
6	5	reefcld	|- ( A e. RR -> ( exp ` -u A ) e. RR )
7	4 6	resubcld	|- ( A e. RR -> ( ( exp ` A ) - ( exp ` -u A ) ) e. RR )
8	7	rehalfcld	|- ( A e. RR -> ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) e. RR )
9	3 8	eqeltrd	|- ( A e. RR -> ( ( sin ` ( _i x. A ) ) / _i ) e. RR )