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Theorem dirkerval 37521
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 458 . . . . . . 7  |-  ( ( m  =  N  /\  s  e.  RR )  ->  m  =  N )
21oveq2d 6321 . . . . . 6  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( 2  x.  m
)  =  ( 2  x.  N ) )
32oveq1d 6320 . . . . 5  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( ( 2  x.  m )  +  1 )  =  ( ( 2  x.  N )  +  1 ) )
43oveq1d 6320 . . . 4  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( ( ( 2  x.  m )  +  1 )  /  (
2  x.  pi ) )  =  ( ( ( 2  x.  N
)  +  1 )  /  ( 2  x.  pi ) ) )
51oveq1d 6320 . . . . . . 7  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( m  +  ( 1  /  2 ) )  =  ( N  +  ( 1  / 
2 ) ) )
65oveq1d 6320 . . . . . 6  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( ( m  +  ( 1  /  2
) )  x.  s
)  =  ( ( N  +  ( 1  /  2 ) )  x.  s ) )
76fveq2d 5885 . . . . 5  |-  ( ( m  =  N  /\  s  e.  RR )  ->  ( sin `  (
( m  +  ( 1  /  2 ) )  x.  s ) )  =  ( sin `  ( ( N  +  ( 1  /  2
) )  x.  s
) ) )
87oveq1d 6320 . . . 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 3935 . . 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 4512 . 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 simpl 458 . . . . . . . . 9  |-  ( ( n  =  m  /\  s  e.  RR )  ->  n  =  m )
1312oveq2d 6321 . . . . . . . 8  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( 2  x.  n
)  =  ( 2  x.  m ) )
1413oveq1d 6320 . . . . . . 7  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( ( 2  x.  n )  +  1 )  =  ( ( 2  x.  m )  +  1 ) )
1514oveq1d 6320 . . . . . 6  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( ( ( 2  x.  n )  +  1 )  /  (
2  x.  pi ) )  =  ( ( ( 2  x.  m
)  +  1 )  /  ( 2  x.  pi ) ) )
1612oveq1d 6320 . . . . . . . . 9  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( n  +  ( 1  /  2 ) )  =  ( m  +  ( 1  / 
2 ) ) )
1716oveq1d 6320 . . . . . . . 8  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( ( n  +  ( 1  /  2
) )  x.  s
)  =  ( ( m  +  ( 1  /  2 ) )  x.  s ) )
1817fveq2d 5885 . . . . . . 7  |-  ( ( n  =  m  /\  s  e.  RR )  ->  ( sin `  (
( n  +  ( 1  /  2 ) )  x.  s ) )  =  ( sin `  ( ( m  +  ( 1  /  2
) )  x.  s
) ) )
1918oveq1d 6320 . . . . . 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 ) ) ) ) )
2015, 19ifeq12d 3935 . . . . 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
) ) ) ) ) )
2120mpteq2dva 4512 . . . 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 ) ) ) ) ) ) )
2221cbvmptv 4518 . . 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 ) ) ) ) ) ) )
2311, 22eqtri 2458 . 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 ) ) ) ) ) ) )
24 reex 9629 . . 3  |-  RR  e.  _V
2524mptex 6151 . 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
2610, 23, 25fvmpt 5964 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 370    = wceq 1437    e. wcel 1870   ifcif 3915    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543    / cdiv 10268   NNcn 10609   2c2 10659    mod cmo 12093   sincsin 14094   picpi 14097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-cnex 9594  ax-resscn 9595
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308
This theorem is referenced by:  dirkerval2  37524  dirkerf  37527  dirkertrigeq  37531  dirkercncflem2  37534  dirkercncflem4  37536
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