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Theorem dirkerval 31682
Description: The Nth Dirichlet Kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
dirkerval.1  |-  D  =  ( n  e.  NN  |->  ( s  e.  RR  |->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  n )  +  1 )  / 
( 2  x.  pi ) ) ,  ( ( sin `  (
( n  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) ) )
Assertion
Ref Expression
dirkerval  |-  ( N  e.  NN  ->  ( D `  N )  =  ( s  e.  RR  |->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  N )  +  1 )  /  ( 2  x.  pi ) ) ,  ( ( sin `  ( ( N  +  ( 1  /  2
) )  x.  s
) )  /  (
( 2  x.  pi )  x.  ( sin `  ( s  /  2
) ) ) ) ) ) )
Distinct variable groups:    N, s    n, s
Allowed substitution hints:    D( n, s)    N( n)

Proof of Theorem dirkerval
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . . . . 7  |-  ( ( m  =  N  /\  s  e.  RR )  ->  m  =  N )
21oveq2d 6310 . . . . . 6  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( 2  x.  m
)  =  ( 2  x.  N ) )
32oveq1d 6309 . . . . 5  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( ( 2  x.  m )  +  1 )  =  ( ( 2  x.  N )  +  1 ) )
43oveq1d 6309 . . . 4  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( ( ( 2  x.  m )  +  1 )  /  (
2  x.  pi ) )  =  ( ( ( 2  x.  N
)  +  1 )  /  ( 2  x.  pi ) ) )
51oveq1d 6309 . . . . . . 7  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( m  +  ( 1  /  2 ) )  =  ( N  +  ( 1  / 
2 ) ) )
65oveq1d 6309 . . . . . 6  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( ( m  +  ( 1  /  2
) )  x.  s
)  =  ( ( N  +  ( 1  /  2 ) )  x.  s ) )
76fveq2d 5875 . . . . 5  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( sin `  (
( m  +  ( 1  /  2 ) )  x.  s ) )  =  ( sin `  ( ( N  +  ( 1  /  2
) )  x.  s
) ) )
87oveq1d 6309 . . . 4  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( ( sin `  (
( m  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) )  =  ( ( sin `  (
( N  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) )
94, 8ifeq12d 3964 . . 3  |-  ( ( m  =  N  /\  s  e.  RR )  ->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  m
)  +  1 )  /  ( 2  x.  pi ) ) ,  ( ( sin `  (
( m  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) )  =  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  N )  +  1 )  /  ( 2  x.  pi ) ) ,  ( ( sin `  ( ( N  +  ( 1  /  2
) )  x.  s
) )  /  (
( 2  x.  pi )  x.  ( sin `  ( s  /  2
) ) ) ) ) )
109mpteq2dva 4538 . 2  |-  ( m  =  N  ->  (
s  e.  RR  |->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  m )  +  1 )  / 
( 2  x.  pi ) ) ,  ( ( sin `  (
( m  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) )  =  ( s  e.  RR  |->  if ( ( s  mod  (
2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  N )  +  1 )  /  (
2  x.  pi ) ) ,  ( ( sin `  ( ( N  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) ) )
11 dirkerval.1 . . 3  |-  D  =  ( n  e.  NN  |->  ( s  e.  RR  |->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  n )  +  1 )  / 
( 2  x.  pi ) ) ,  ( ( sin `  (
( n  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) ) )
12 nfcv 2629 . . . 4  |-  F/_ m
( s  e.  RR  |->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  n )  +  1 )  / 
( 2  x.  pi ) ) ,  ( ( sin `  (
( n  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) )
13 nfcv 2629 . . . 4  |-  F/_ n
( s  e.  RR  |->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  m )  +  1 )  / 
( 2  x.  pi ) ) ,  ( ( sin `  (
( m  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) )
14 simpl 457 . . . . . . . . 9  |-  ( ( n  =  m  /\  s  e.  RR )  ->  n  =  m )
1514oveq2d 6310 . . . . . . . 8  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( 2  x.  n
)  =  ( 2  x.  m ) )
1615oveq1d 6309 . . . . . . 7  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  m )  +  1 ) )
1716oveq1d 6309 . . . . . 6  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( ( ( 2  x.  n )  +  1 )  /  (
2  x.  pi ) )  =  ( ( ( 2  x.  m
)  +  1 )  /  ( 2  x.  pi ) ) )
1814oveq1d 6309 . . . . . . . . 9  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( n  +  ( 1  /  2 ) )  =  ( m  +  ( 1  / 
2 ) ) )
1918oveq1d 6309 . . . . . . . 8  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( ( n  +  ( 1  /  2
) )  x.  s
)  =  ( ( m  +  ( 1  /  2 ) )  x.  s ) )
2019fveq2d 5875 . . . . . . 7  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( sin `  (
( n  +  ( 1  /  2 ) )  x.  s ) )  =  ( sin `  ( ( m  +  ( 1  /  2
) )  x.  s
) ) )
2120oveq1d 6309 . . . . . 6  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( ( sin `  (
( n  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) )  =  ( ( sin `  (
( m  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) )
2217, 21ifeq12d 3964 . . . . 5  |-  ( ( n  =  m  /\  s  e.  RR )  ->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  n
)  +  1 )  /  ( 2  x.  pi ) ) ,  ( ( sin `  (
( n  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) )  =  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  m )  +  1 )  /  ( 2  x.  pi ) ) ,  ( ( sin `  ( ( m  +  ( 1  /  2
) )  x.  s
) )  /  (
( 2  x.  pi )  x.  ( sin `  ( s  /  2
) ) ) ) ) )
2322mpteq2dva 4538 . . . 4  |-  ( n  =  m  ->  (
s  e.  RR  |->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  n )  +  1 )  / 
( 2  x.  pi ) ) ,  ( ( sin `  (
( n  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) )  =  ( s  e.  RR  |->  if ( ( s  mod  (
2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  m )  +  1 )  /  (
2  x.  pi ) ) ,  ( ( sin `  ( ( m  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) ) )
2412, 13, 23cbvmpt 4542 . . 3  |-  ( n  e.  NN  |->  ( s  e.  RR  |->  if ( ( s  mod  (
2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  n )  +  1 )  /  (
2  x.  pi ) ) ,  ( ( sin `  ( ( n  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) ) )  =  ( m  e.  NN  |->  ( s  e.  RR  |->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  m )  +  1 )  / 
( 2  x.  pi ) ) ,  ( ( sin `  (
( m  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) ) )
2511, 24eqtri 2496 . 2  |-  D  =  ( m  e.  NN  |->  ( s  e.  RR  |->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  m )  +  1 )  / 
( 2  x.  pi ) ) ,  ( ( sin `  (
( m  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) ) )
26 reex 9593 . . 3  |-  RR  e.  _V
2726mptex 6141 . 2  |-  ( s  e.  RR  |->  if ( ( s  mod  (
2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  N )  +  1 )  /  (
2  x.  pi ) ) ,  ( ( sin `  ( ( N  +  ( 1  /  2 ) )  x.  s ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
s  /  2 ) ) ) ) ) )  e.  _V
2810, 25, 27fvmpt 5956 1  |-  ( N  e.  NN  ->  ( D `  N )  =  ( s  e.  RR  |->  if ( ( s  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  N )  +  1 )  /  ( 2  x.  pi ) ) ,  ( ( sin `  ( ( N  +  ( 1  /  2
) )  x.  s
) )  /  (
( 2  x.  pi )  x.  ( sin `  ( s  /  2
) ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ifcif 3944    |-> cmpt 4510   ` cfv 5593  (class class class)co 6294   RRcr 9501   0cc0 9502   1c1 9503    + caddc 9505    x. cmul 9507    / cdiv 10216   NNcn 10546   2c2 10595    mod cmo 11974   sincsin 13673   picpi 13676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-cnex 9558  ax-resscn 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297
This theorem is referenced by:  dirkerval2  31685  dirkerf  31688  dirkertrigeq  31692  dirkercncflem2  31695  dirkercncflem4  31697
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