Theorem fourierdlem55

Description: U is a real function. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	fourierdlem55.f	|- ( ph -> F : RR --> RR )
	fourierdlem55.x	|- ( ph -> X e. RR )
	fourierdlem55.r	|- ( ph -> Y e. RR )
	fourierdlem55.w	|- ( ph -> W e. RR )
	fourierdlem55.h	|- H = ( s e. ( -u _pi [,] _pi ) |-> if ( s = 0 , 0 , ( ( ( F ` ( X + s ) ) - if ( 0 < s , Y , W ) ) / s ) ) )
	fourierdlem55.k	|- K = ( s e. ( -u _pi [,] _pi ) |-> if ( s = 0 , 1 , ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) )
	fourierdlem55.u	|- U = ( s e. ( -u _pi [,] _pi ) |-> ( ( H ` s ) x. ( K ` s ) ) )
Assertion	fourierdlem55	|- ( ph -> U : ( -u _pi [,] _pi ) --> RR )

Proof

Step	Hyp	Ref	Expression
1		fourierdlem55.f	|- ( ph -> F : RR --> RR )
2		fourierdlem55.x	|- ( ph -> X e. RR )
3		fourierdlem55.r	|- ( ph -> Y e. RR )
4		fourierdlem55.w	|- ( ph -> W e. RR )
5		fourierdlem55.h	|- H = ( s e. ( -u _pi [,] _pi ) |-> if ( s = 0 , 0 , ( ( ( F ` ( X + s ) ) - if ( 0 < s , Y , W ) ) / s ) ) )
6		fourierdlem55.k	|- K = ( s e. ( -u _pi [,] _pi ) |-> if ( s = 0 , 1 , ( s / ( 2 x. ( sin ` ( s / 2 ) ) ) ) ) )
7		fourierdlem55.u	|- U = ( s e. ( -u _pi [,] _pi ) |-> ( ( H ` s ) x. ( K ` s ) ) )
8	1 2 3 4 5	fourierdlem9	|- ( ph -> H : ( -u _pi [,] _pi ) --> RR )
9	8	ffvelrnda	|- ( ( ph /\ s e. ( -u _pi [,] _pi ) ) -> ( H ` s ) e. RR )
10	6	fourierdlem43	|- K : ( -u _pi [,] _pi ) --> RR
11	10	ffvelrni	|- ( s e. ( -u _pi [,] _pi ) -> ( K ` s ) e. RR )
12	11	adantl	|- ( ( ph /\ s e. ( -u _pi [,] _pi ) ) -> ( K ` s ) e. RR )
13	9 12	remulcld	|- ( ( ph /\ s e. ( -u _pi [,] _pi ) ) -> ( ( H ` s ) x. ( K ` s ) ) e. RR )
14	13 7	fmptd	|- ( ph -> U : ( -u _pi [,] _pi ) --> RR )