Theorem recoscl

Description: The cosine of a real number is real. (Contributed by NM, 30-Apr-2005)

	Ref	Expression
Assertion	recoscl	|- ( A e. RR -> ( cos ` A ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		recosval	|- ( A e. RR -> ( cos ` A ) = ( Re ` ( exp ` ( _i x. A ) ) ) )
2		ax-icn	|- _i e. CC
3		recn	|- ( A e. RR -> A e. CC )
4		mulcl	|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
5	2 3 4	sylancr	|- ( A e. RR -> ( _i x. A ) e. CC )
6		efcl	|- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) e. CC )
7	5 6	syl	|- ( A e. RR -> ( exp ` ( _i x. A ) ) e. CC )
8	7	recld	|- ( A e. RR -> ( Re ` ( exp ` ( _i x. A ) ) ) e. RR )
9	1 8	eqeltrd	|- ( A e. RR -> ( cos ` A ) e. RR )