This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dirkerf

Description: For any natural number N , the Dirichlet Kernel ( DN ) is a function. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypothesis dirkerf.1 
|- D = ( n e. NN |-> ( y e. RR |-> if ( ( y mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. _pi ) x. ( sin ` ( y / 2 ) ) ) ) ) ) )
Assertion dirkerf 
|- ( N e. NN -> ( D ` N ) : RR --> RR )

Proof

Step Hyp Ref Expression
1 dirkerf.1 
 |-  D = ( n e. NN |-> ( y e. RR |-> if ( ( y mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. _pi ) x. ( sin ` ( y / 2 ) ) ) ) ) ) )
2 1 dirkerval2 
 |-  ( ( N e. NN /\ y e. RR ) -> ( ( D ` N ) ` y ) = if ( ( y mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. _pi ) x. ( sin ` ( y / 2 ) ) ) ) ) )
3 1 dirkerre 
 |-  ( ( N e. NN /\ y e. RR ) -> ( ( D ` N ) ` y ) e. RR )
4 2 3 eqeltrrd 
 |-  ( ( N e. NN /\ y e. RR ) -> if ( ( y mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. _pi ) x. ( sin ` ( y / 2 ) ) ) ) ) e. RR )
5 4 fmpttd 
 |-  ( N e. NN -> ( y e. RR |-> if ( ( y mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. _pi ) x. ( sin ` ( y / 2 ) ) ) ) ) ) : RR --> RR )
6 1 dirkerval 
 |-  ( N e. NN -> ( D ` N ) = ( y e. RR |-> if ( ( y mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. _pi ) x. ( sin ` ( y / 2 ) ) ) ) ) ) )
7 6 feq1d 
 |-  ( N e. NN -> ( ( D ` N ) : RR --> RR <-> ( y e. RR |-> if ( ( y mod ( 2 x. _pi ) ) = 0 , ( ( ( 2 x. N ) + 1 ) / ( 2 x. _pi ) ) , ( ( sin ` ( ( N + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. _pi ) x. ( sin ` ( y / 2 ) ) ) ) ) ) : RR --> RR ) )
8 5 7 mpbird 
 |-  ( N e. NN -> ( D ` N ) : RR --> RR )