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Theorem List for Metamath Proof Explorer - 37501-37600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwallispi2lem1 37501 An intermediate step between the first version of the Wallis' formula for π and the second version of Wallis' formula. This second version will then be used to prove Stirling's approximation formula for the factorial. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  ( N  e.  NN  ->  ( 
 seq 1 (  x. 
 ,  ( k  e. 
 NN  |->  ( ( ( 2  x.  k ) 
 /  ( ( 2  x.  k )  -  1 ) )  x.  ( ( 2  x.  k )  /  (
 ( 2  x.  k
 )  +  1 ) ) ) ) ) `
  N )  =  ( ( 1  /  ( ( 2  x.  N )  +  1 ) )  x.  (  seq 1 (  x.  ,  ( k  e.  NN  |->  ( ( ( 2  x.  k ) ^
 4 )  /  (
 ( ( 2  x.  k )  x.  (
 ( 2  x.  k
 )  -  1 ) ) ^ 2 ) ) ) ) `  N ) ) )
 
Theoremwallispi2lem2 37502 Two expressions are proven to be equal, and this is used to complete the proof of the second version of Wallis' formula for π . (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  ( N  e.  NN  ->  ( 
 seq 1 (  x. 
 ,  ( k  e. 
 NN  |->  ( ( ( 2  x.  k ) ^ 4 )  /  ( ( ( 2  x.  k )  x.  ( ( 2  x.  k )  -  1
 ) ) ^ 2
 ) ) ) ) `
  N )  =  ( ( ( 2 ^ ( 4  x.  N ) )  x.  ( ( ! `  N ) ^ 4
 ) )  /  (
 ( ! `  (
 2  x.  N ) ) ^ 2 ) ) )
 
Theoremwallispi2 37503 An alternative version of Wallis' formula for π ; this second formula uses factorials and it is later used to proof Stirling's approximation formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `
  n ) ^
 4 ) )  /  ( ( ! `  ( 2  x.  n ) ) ^ 2
 ) )  /  (
 ( 2  x.  n )  +  1 )
 ) )   =>    |-  V  ~~>  ( pi  / 
 2 )
 
21.30.14  Stirling's approximation formula for ` n ` factorial
 
Theoremstirlinglem1 37504 A simple limit of fractions is computed. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
 |-  H  =  ( n  e.  NN  |->  ( ( n ^
 2 )  /  ( n  x.  ( ( 2  x.  n )  +  1 ) ) ) )   &    |-  F  =  ( n  e.  NN  |->  ( 1  -  ( 1 
 /  ( ( 2  x.  n )  +  1 ) ) ) )   &    |-  G  =  ( n  e.  NN  |->  ( 1  /  ( ( 2  x.  n )  +  1 ) ) )   &    |-  L  =  ( n  e.  NN  |->  ( 1  /  n ) )   =>    |-  H  ~~>  ( 1  / 
 2 )
 
Theoremstirlinglem2 37505  A maps to positive reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   =>    |-  ( N  e.  NN  ->  ( A `  N )  e.  RR+ )
 
Theoremstirlinglem3 37506 Long but simple algebraic transformations are applied to show that  V, the Wallis formula for π , can be expressed in terms of  A, the Stirling's approximation formula for the factorial, up to a constant factor. This will allow (in a later theorem) to determine the right constant factor to be put into the  A, in order to get the exact Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  D  =  ( n  e.  NN  |->  ( A `  ( 2  x.  n ) ) )   &    |-  E  =  ( n  e.  NN  |->  ( ( sqr `  (
 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) )   &    |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `  n ) ^ 4
 ) )  /  (
 ( ! `  (
 2  x.  n ) ) ^ 2 ) )  /  ( ( 2  x.  n )  +  1 ) ) )   =>    |-  V  =  ( n  e.  NN  |->  ( ( ( ( A `  n ) ^ 4
 )  /  ( ( D `  n ) ^
 2 ) )  x.  ( ( n ^
 2 )  /  ( n  x.  ( ( 2  x.  n )  +  1 ) ) ) ) )
 
Theoremstirlinglem4 37507* Algebraic manipulation of  ( ( B n ) - ( B  ( n  +  1 ) ) ). It will be used in other theorems to show that  B is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  J  =  ( n  e.  NN  |->  ( ( ( ( 1  +  (
 2  x.  n ) )  /  2 )  x.  ( log `  (
 ( n  +  1 )  /  n ) ) )  -  1
 ) )   =>    |-  ( N  e.  NN  ->  ( ( B `  N )  -  ( B `  ( N  +  1 ) ) )  =  ( J `  N ) )
 
Theoremstirlinglem5 37508* If  T is between  0 and  1, then a series (without alternating negative and positive terms) is given that converges to log (1+T)/(1-T) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  D  =  ( j  e.  NN  |->  ( ( -u 1 ^ ( j  -  1 ) )  x.  ( ( T ^
 j )  /  j
 ) ) )   &    |-  E  =  ( j  e.  NN  |->  ( ( T ^
 j )  /  j
 ) )   &    |-  F  =  ( j  e.  NN  |->  ( ( ( -u 1 ^ ( j  -  1 ) )  x.  ( ( T ^
 j )  /  j
 ) )  +  (
 ( T ^ j
 )  /  j )
 ) )   &    |-  H  =  ( j  e.  NN0  |->  ( 2  x.  ( ( 1 
 /  ( ( 2  x.  j )  +  1 ) )  x.  ( T ^ (
 ( 2  x.  j
 )  +  1 ) ) ) ) )   &    |-  G  =  ( j  e.  NN0  |->  ( ( 2  x.  j )  +  1 ) )   &    |-  ( ph  ->  T  e.  RR+ )   &    |-  ( ph  ->  ( abs `  T )  < 
 1 )   =>    |-  ( ph  ->  seq 0
 (  +  ,  H ) 
 ~~>  ( log `  (
 ( 1  +  T )  /  ( 1  -  T ) ) ) )
 
Theoremstirlinglem6 37509* A series that converges to log (N+1)/N (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  H  =  ( j  e.  NN0  |->  ( 2  x.  (
 ( 1  /  (
 ( 2  x.  j
 )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ ( ( 2  x.  j )  +  1 ) ) ) ) )   =>    |-  ( N  e.  NN  ->  seq 0 (  +  ,  H )  ~~>  ( log `  ( ( N  +  1 )  /  N ) ) )
 
Theoremstirlinglem7 37510* Algebraic manipulation of the formula for J(n). (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  J  =  ( n  e.  NN  |->  ( ( ( ( 1  +  ( 2  x.  n ) ) 
 /  2 )  x.  ( log `  (
 ( n  +  1 )  /  n ) ) )  -  1
 ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1  /  (
 ( 2  x.  k
 )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ ( 2  x.  k ) ) ) )   &    |-  H  =  ( k  e.  NN0  |->  ( 2  x.  ( ( 1 
 /  ( ( 2  x.  k )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ (
 ( 2  x.  k
 )  +  1 ) ) ) ) )   =>    |-  ( N  e.  NN  ->  seq 1 (  +  ,  K )  ~~>  ( J `  N ) )
 
Theoremstirlinglem8 37511 If  A converges to  C, then  F converges to C^2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ n ph   &    |-  F/_ n A   &    |-  F/_ n D   &    |-  D  =  ( n  e.  NN  |->  ( A `
  ( 2  x.  n ) ) )   &    |-  ( ph  ->  A : NN
 --> RR+ )   &    |-  F  =  ( n  e.  NN  |->  ( ( ( A `  n ) ^ 4
 )  /  ( ( D `  n ) ^
 2 ) ) )   &    |-  L  =  ( n  e.  NN  |->  ( ( A `
  n ) ^
 4 ) )   &    |-  M  =  ( n  e.  NN  |->  ( ( D `  n ) ^ 2
 ) )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( D `  n )  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  ~~>  C )   =>    |-  ( ph  ->  F  ~~>  ( C ^ 2 ) )
 
Theoremstirlinglem9 37512*  ( ( B `  N )  -  ( B `  ( N  +  1
) ) ) is expressed as a limit of a series. This result will be used both to prove that  B is decreasing and to prove that  B is bounded (below). It will follow that  B converges in the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  J  =  ( n  e.  NN  |->  ( ( ( ( 1  +  (
 2  x.  n ) )  /  2 )  x.  ( log `  (
 ( n  +  1 )  /  n ) ) )  -  1
 ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1  /  (
 ( 2  x.  k
 )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ ( 2  x.  k ) ) ) )   =>    |-  ( N  e.  NN  ->  seq 1 (  +  ,  K )  ~~>  ( ( B `  N )  -  ( B `  ( N  +  1 ) ) ) )
 
Theoremstirlinglem10 37513* A bound for any B(N)-B(N + 1) that will allow to find a lower bound for the whole  B sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1 
 /  ( ( 2  x.  k )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ (
 2  x.  k ) ) ) )   &    |-  L  =  ( k  e.  NN  |->  ( ( 1  /  ( ( ( 2  x.  N )  +  1 ) ^ 2
 ) ) ^ k
 ) )   =>    |-  ( N  e.  NN  ->  ( ( B `  N )  -  ( B `  ( N  +  1 ) ) ) 
 <_  ( ( 1  / 
 4 )  x.  (
 1  /  ( N  x.  ( N  +  1 ) ) ) ) )
 
Theoremstirlinglem11 37514*  B is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  K  =  ( k  e.  NN  |->  ( ( 1 
 /  ( ( 2  x.  k )  +  1 ) )  x.  ( ( 1  /  ( ( 2  x.  N )  +  1 ) ) ^ (
 2  x.  k ) ) ) )   =>    |-  ( N  e.  NN  ->  ( B `  ( N  +  1
 ) )  <  ( B `  N ) )
 
Theoremstirlinglem12 37515* The sequence  B is bounded below. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   &    |-  F  =  ( n  e.  NN  |->  ( 1  /  ( n  x.  ( n  +  1 )
 ) ) )   =>    |-  ( N  e.  NN  ->  ( ( B `
  1 )  -  ( 1  /  4
 ) )  <_  ( B `  N ) )
 
Theoremstirlinglem13 37516*  B is decreasing and has a lower bound, then it converges. Since  B is  log A, in another theorem it is proven that  A converges as well. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   =>    |-  E. d  e.  RR  B  ~~>  d
 
Theoremstirlinglem14 37517* The sequence  A converges to a positive real. This proves that the Stirling's formula converges to the factorial, up to a constant. In another theorem, using Wallis' formula for π& , such constant is exactly determined, thus proving the Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
 ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  B  =  ( n  e.  NN  |->  ( log `  ( A `  n ) ) )   =>    |-  E. c  e.  RR+  A  ~~>  c
 
Theoremstirlinglem15 37518* The Stirling's formula is proven using a number of local definitions. The main theorem stirling 37519 will use this final lemma, but it will not expose the local definitions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F/ n ph   &    |-  S  =  ( n  e.  NN0  |->  ( ( sqr `  ( (
 2  x.  pi )  x.  n ) )  x.  ( ( n 
 /  _e ) ^ n ) ) )   &    |-  A  =  ( n  e.  NN  |->  ( ( ! `
  n )  /  ( ( sqr `  (
 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )   &    |-  D  =  ( n  e.  NN  |->  ( A `  ( 2  x.  n ) ) )   &    |-  E  =  ( n  e.  NN  |->  ( ( sqr `  (
 2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) )   &    |-  V  =  ( n  e.  NN  |->  ( ( ( ( 2 ^ ( 4  x.  n ) )  x.  ( ( ! `  n ) ^ 4
 ) )  /  (
 ( ! `  (
 2  x.  n ) ) ^ 2 ) )  /  ( ( 2  x.  n )  +  1 ) ) )   &    |-  F  =  ( n  e.  NN  |->  ( ( ( A `  n ) ^ 4
 )  /  ( ( D `  n ) ^
 2 ) ) )   &    |-  H  =  ( n  e.  NN  |->  ( ( n ^ 2 )  /  ( n  x.  (
 ( 2  x.  n )  +  1 )
 ) ) )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  ~~>  C )   =>    |-  ( ph  ->  ( n  e.  NN  |->  ( ( ! `  n ) 
 /  ( S `  n ) ) )  ~~>  1 )
 
Theoremstirling 37519 Stirling's approximation formula for 
n factorial. The proof follows two major steps: first it is proven that  S and  n factorial are asymptotically equivalent, up to an unknown constant. Then, using Wallis' formula for π it is proven that the unknown constant is the square root of π and then the exact Stirling's formula is established. This is Metamath 100 proof #90. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  S  =  ( n  e.  NN0  |->  ( ( sqr `  (
 ( 2  x.  pi )  x.  n ) )  x.  ( ( n 
 /  _e ) ^ n ) ) )   =>    |-  ( n  e.  NN  |->  ( ( ! `  n )  /  ( S `  n ) ) )  ~~>  1
 
Theoremstirlingr 37520 Stirling's approximation formula for 
n factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 37519 is proven for convergence in the topology of complex numbers. The variable  R is used to denote convergence with respect to the standard topology on the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  S  =  ( n  e.  NN0  |->  ( ( sqr `  (
 ( 2  x.  pi )  x.  n ) )  x.  ( ( n 
 /  _e ) ^ n ) ) )   &    |-  R  =  ( ~~> t `  ( topGen `  ran  (,) )
 )   =>    |-  ( n  e.  NN  |->  ( ( ! `  n )  /  ( S `  n ) ) ) R 1
 
21.30.15  Dirichlet kernel
 
Theoremdirkerval 37521* The Nth Dirichlet Kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  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 )
 ) ) ) ) ) )   =>    |-  ( 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 )
 ) ) ) ) ) )
 
Theoremdirker2re 37522 The Dirchlet Kernel value is a real if the argument is not a multiple of π . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( ( N  e.  NN  /\  S  e.  RR )  /\  -.  ( S 
 mod  ( 2  x.  pi ) )  =  0 )  ->  (
 ( sin `  ( ( N  +  ( 1  /  2 ) )  x.  S ) ) 
 /  ( ( 2  x.  pi )  x.  ( sin `  ( S  /  2 ) ) ) )  e.  RR )
 
Theoremdirkerdenne0 37523 The Dirchlet Kernel denominator is never  0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( S  e.  RR  /\ 
 -.  ( S  mod  ( 2  x.  pi ) )  =  0
 )  ->  ( (
 2  x.  pi )  x.  ( sin `  ( S  /  2 ) ) )  =/=  0 )
 
Theoremdirkerval2 37524* The Nth Dirichlet Kernel evaluated at a specific point  S. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  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 )
 ) ) ) ) ) )   =>    |-  ( ( N  e.  NN  /\  S  e.  RR )  ->  ( ( D `
  N ) `  S )  =  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 ) ) ) ) ) )
 
Theoremdirkerre 37525* The Dirichlet Kernel at any point evaluates to a real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  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 )
 ) ) ) ) ) )   =>    |-  ( ( N  e.  NN  /\  S  e.  RR )  ->  ( ( D `
  N ) `  S )  e.  RR )
 
Theoremdirkerper 37526* the Dirichlet Kernel has period  2 pi. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  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 )
 ) ) ) ) ) )   &    |-  T  =  ( 2  x.  pi )   =>    |-  ( ( N  e.  NN  /\  x  e.  RR )  ->  ( ( D `
  N ) `  ( x  +  T ) )  =  (
 ( D `  N ) `  x ) )
 
Theoremdirkerf 37527* For any natural number  N, the Dirichlet Kernel  ( D `  N
) is a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  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 )
 ) ) ) ) ) )   =>    |-  ( N  e.  NN  ->  ( D `  N ) : RR --> RR )
 
Theoremdirkertrigeqlem1 37528* Sum of an even number of alternating cos values. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( K  e.  NN  ->  sum_
 n  e.  ( 1
 ... ( 2  x.  K ) ) ( cos `  ( n  x.  pi ) )  =  0 )
 
Theoremdirkertrigeqlem2 37529* Trigonomic equality lemma for the Dirichlet Kernel trigonomic equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  ( sin `  A )  =/=  0 )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  ( ( ( 1  / 
 2 )  +  sum_ n  e.  ( 1 ...
 N ) ( cos `  ( n  x.  A ) ) )  /  pi )  =  (
 ( sin `  ( ( N  +  ( 1  /  2 ) )  x.  A ) ) 
 /  ( ( 2  x.  pi )  x.  ( sin `  ( A  /  2 ) ) ) ) )
 
Theoremdirkertrigeqlem3 37530* Trigonometric equality lemma for the Dirichlet Kernel trigonometric equality. Here we handle the case for an angle that's an odd multiple of  pi. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  A  =  ( ( ( 2  x.  K )  +  1 )  x.  pi )   =>    |-  ( ph  ->  ( (
 ( 1  /  2
 )  +  sum_ n  e.  ( 1 ... N ) ( cos `  ( n  x.  A ) ) )  /  pi )  =  ( ( sin `  ( ( N  +  ( 1  /  2
 ) )  x.  A ) )  /  (
 ( 2  x.  pi )  x.  ( sin `  ( A  /  2 ) ) ) ) )
 
Theoremdirkertrigeq 37531* Trigonometric equality for the Dirichlet Kernel (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  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 )
 ) ) ) ) ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  F  =  ( D `  N )   &    |-  H  =  ( s  e.  RR  |->  ( ( ( 1  /  2
 )  +  sum_ k  e.  ( 1 ... N ) ( cos `  (
 k  x.  s ) ) )  /  pi ) )   =>    |-  ( ph  ->  F  =  H )
 
Theoremdirkeritg 37532* The definite integral of the Dirichlet Kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  D  =  ( n  e.  NN  |->  ( x  e.  RR  |->  if ( ( x  mod  ( 2  x.  pi ) )  =  0 ,  ( ( ( 2  x.  n )  +  1 )  /  (
 2  x.  pi ) ) ,  ( ( sin `  ( ( n  +  ( 1  /  2 ) )  x.  x ) ) 
 /  ( ( 2  x.  pi )  x.  ( sin `  ( x  /  2 ) ) ) ) ) ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  F  =  ( D `  N )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  G  =  ( x  e.  ( A [,] B )  |->  ( ( ( x  /  2 )  +  sum_ k  e.  (
 1 ... N ) ( ( sin `  (
 k  x.  x ) )  /  k ) )  /  pi ) )   =>    |-  ( ph  ->  S. ( A (,) B ) ( F `  x )  _d x  =  ( ( G `  B )  -  ( G `  A ) ) )
 
Theoremdirkercncflem1 37533* If  Y is a multiple of  pi then it belongs to an open inerval  ( A (,) B ) such that for any other point  y in the interval, cos y/2 and sin y/2 are non zero. Such an interval is needed to apply De L'Hopital theorem. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  A  =  ( Y  -  pi )   &    |-  B  =  ( Y  +  pi )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  ( Y  mod  ( 2  x.  pi ) )  =  0 )   =>    |-  ( ph  ->  ( Y  e.  ( A (,) B )  /\  A. y  e.  ( ( A (,) B )  \  { Y } ) ( ( sin `  (
 y  /  2 )
 )  =/=  0  /\  ( cos `  ( y  /  2 ) )  =/=  0 ) ) )
 
Theoremdirkercncflem2 37534* Lemma used to proof that the Dirichlet Kernel is continuous at  Y points that are multiples of 
( 2  x.  pi ). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  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 )
 ) ) ) ) ) )   &    |-  F  =  ( y  e.  ( ( A (,) B ) 
 \  { Y }
 )  |->  ( sin `  (
 ( N  +  (
 1  /  2 )
 )  x.  y ) ) )   &    |-  G  =  ( y  e.  ( ( A (,) B ) 
 \  { Y }
 )  |->  ( ( 2  x.  pi )  x.  ( sin `  (
 y  /  2 )
 ) ) )   &    |-  (
 ( ph  /\  y  e.  ( ( A (,) B )  \  { Y } ) )  ->  ( sin `  ( y  /  2 ) )  =/=  0 )   &    |-  H  =  ( y  e.  (
 ( A (,) B )  \  { Y }
 )  |->  ( ( N  +  ( 1  / 
 2 ) )  x.  ( cos `  (
 ( N  +  (
 1  /  2 )
 )  x.  y ) ) ) )   &    |-  I  =  ( y  e.  (
 ( A (,) B )  \  { Y }
 )  |->  ( pi  x.  ( cos `  ( y  /  2 ) ) ) )   &    |-  L  =  ( w  e.  ( A (,) B )  |->  ( ( ( N  +  ( 1  /  2
 ) )  x.  ( cos `  ( ( N  +  ( 1  / 
 2 ) )  x.  w ) ) ) 
 /  ( pi  x.  ( cos `  ( w  /  2 ) ) ) ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  Y  e.  ( A (,) B ) )   &    |-  ( ph  ->  ( Y  mod  ( 2  x.  pi ) )  =  0
 )   &    |-  ( ( ph  /\  y  e.  ( ( A (,) B )  \  { Y } ) )  ->  ( cos `  ( y  /  2 ) )  =/=  0 )   =>    |-  ( ph  ->  ( ( D `  N ) `  Y )  e.  ( ( ( D `
  N )  |`  ( ( A (,) B )  \  { Y } ) ) lim CC  Y ) )
 
Theoremdirkercncflem3 37535* The Dirichlet Kernel is continuous at  Y points that are multiples of  ( 2  x.  pi ). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  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 )
 ) ) ) ) ) )   &    |-  A  =  ( Y  -  pi )   &    |-  B  =  ( Y  +  pi )   &    |-  F  =  ( y  e.  ( A (,) B )  |->  ( ( sin `  (
 ( n  +  (
 1  /  2 )
 )  x.  y ) )  /  ( ( 2  x.  pi )  x.  ( sin `  (
 y  /  2 )
 ) ) ) )   &    |-  G  =  ( y  e.  ( A (,) B )  |->  ( ( 2  x.  pi )  x.  ( sin `  (
 y  /  2 )
 ) ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  ( Y  mod  ( 2  x.  pi ) )  =  0 )   =>    |-  ( ph  ->  ( ( D `  N ) `  Y )  e.  ( ( D `  N ) lim CC  Y ) )
 
Theoremdirkercncflem4 37536* The Dirichlet Kernel is continuos at points that are not multiple of 2 π . This is the easier condition, for the proof of the continuity of the Dirichlet kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  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 )
 ) ) ) ) ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  ( Y  mod  ( 2  x.  pi ) )  =/=  0 )   &    |-  A  =  ( |_ `  ( Y 
 /  ( 2  x.  pi ) ) )   &    |-  B  =  ( A  +  1 )   &    |-  C  =  ( A  x.  (
 2  x.  pi ) )   &    |-  E  =  ( B  x.  ( 2  x.  pi ) )   =>    |-  ( ph  ->  ( D `  N )  e.  (
 ( ( topGen `  ran  (,) )  CnP  ( topGen `  ran  (,) ) ) `  Y ) )
 
Theoremdirkercncf 37537* For any natural number  N, the Dirichlet Kernel  ( D `  N
) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  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 )
 ) ) ) ) ) )   =>    |-  ( N  e.  NN  ->  ( D `  N )  e.  ( RR -cn-> RR ) )
 
21.30.16  Fourier Series
 
Theoremfourierdlem1 37538 A partition interval is a subset of the partitioned interval (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  Q : ( 0 ...
 M ) --> ( A [,] B ) )   &    |-  ( ph  ->  I  e.  ( 0..^ M ) )   &    |-  ( ph  ->  X  e.  ( ( Q `  I ) [,] ( Q `  ( I  +  1 ) ) ) )   =>    |-  ( ph  ->  X  e.  ( A [,] B ) )
 
Theoremfourierdlem2 37539* Membership in a partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  A  /\  ( p `
  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   =>    |-  ( M  e.  NN  ->  ( Q  e.  ( P `
  M )  <->  ( Q  e.  ( RR  ^m  ( 0
 ... M ) ) 
 /\  ( ( ( Q `  0 )  =  A  /\  ( Q `  M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
  i )  < 
 ( Q `  (
 i  +  1 ) ) ) ) ) )
 
Theoremfourierdlem3 37540* Membership in a partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  P  =  ( m  e.  NN  |->  { p  e.  ( (
 -u pi [,] pi )  ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  -u pi  /\  ( p `
  m )  =  pi )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   =>    |-  ( M  e.  NN  ->  ( Q  e.  ( P `
  M )  <->  ( Q  e.  ( ( -u pi [,] pi )  ^m  (
 0 ... M ) ) 
 /\  ( ( ( Q `  0 )  =  -u pi  /\  ( Q `  M )  =  pi )  /\  A. i  e.  ( 0..^ M ) ( Q `
  i )  < 
 ( Q `  (
 i  +  1 ) ) ) ) ) )
 
Theoremfourierdlem4 37541*  E is a function that maps any point to a periodic corresponding point in  ( A ,  B ]. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  T  =  ( B  -  A )   &    |-  E  =  ( x  e.  RR  |->  ( x  +  ( ( |_ `  ( ( B  -  x )  /  T ) )  x.  T ) ) )   =>    |-  ( ph  ->  E : RR --> ( A (,] B ) )
 
Theoremfourierdlem5 37542*  S is a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  S  =  ( x  e.  ( -u pi [,] pi ) 
 |->  ( sin `  (
 ( X  +  (
 1  /  2 )
 )  x.  x ) ) )   =>    |-  ( X  e.  RR  ->  S : ( -u pi [,] pi ) --> RR )
 
Theoremfourierdlem6 37543  X is in the periodic partition, when the considered interval is centered at  X. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  T  =  ( B  -  A )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  I  e.  ZZ )   &    |-  ( ph  ->  J  e.  ZZ )   &    |-  ( ph  ->  I  <  J )   &    |-  ( ph  ->  ( X  +  ( I  x.  T ) )  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( X  +  ( J  x.  T ) )  e.  ( A [,] B ) )   =>    |-  ( ph  ->  J  =  ( I  +  1 ) )
 
Theoremfourierdlem7 37544* The difference between a point and it's periodic image in the interval, is decreasing. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  T  =  ( B  -  A )   &    |-  E  =  ( x  e.  RR  |->  ( x  +  ( ( |_ `  ( ( B  -  x )  /  T ) )  x.  T ) ) )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  X  <_  Y )   =>    |-  ( ph  ->  (
 ( E `  Y )  -  Y )  <_  ( ( E `  X )  -  X ) )
 
Theoremfourierdlem8 37545 A partition interval is a subset of the partitioned interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  Q : ( 0 ...
 M ) --> ( A [,] B ) )   &    |-  ( ph  ->  I  e.  ( 0..^ M ) )   =>    |-  ( ph  ->  ( ( Q `  I ) [,] ( Q `  ( I  +  1 )
 ) )  C_  ( A [,] B ) )
 
Theoremfourierdlem9 37546*  H is a complex function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  W  e.  RR )   &    |-  H  =  ( s  e.  ( -u pi [,] pi )  |->  if ( s  =  0 ,  0 ,  (
 ( ( F `  ( X  +  s
 ) )  -  if ( 0  <  s ,  Y ,  W ) )  /  s ) ) )   =>    |-  ( ph  ->  H : ( -u pi [,] pi ) --> RR )
 
Theoremfourierdlem10 37547 Condition on the bounds of a non empty subinterval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  C  <  D )   &    |-  ( ph  ->  ( C (,) D ) 
 C_  ( A (,) B ) )   =>    |-  ( ph  ->  ( A  <_  C  /\  D  <_  B ) )
 
Theoremfourierdlem11 37548* If there is a partition, than the lower bound is strictly less than the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  A  /\  ( p `
  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q  e.  ( P `  M ) )   =>    |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) )
 
Theoremfourierdlem12 37549* A point of a partition is not an element of any open interval determined by the partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  A  /\  ( p `
  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q  e.  ( P `  M ) )   &    |-  ( ph  ->  X  e.  ran  Q )   =>    |-  (
 ( ph  /\  i  e.  ( 0..^ M ) )  ->  -.  X  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )
 
Theoremfourierdlem13 37550* Value of  V in terms of value of  Q. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  X  e.  RR )   &    |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  ( A  +  X )  /\  ( p `  m )  =  ( B  +  X )
 )  /\  A. i  e.  ( 0..^ m ) ( p `  i
 )  <  ( p `  ( i  +  1 ) ) ) }
 )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  V  e.  ( P `  M ) )   &    |-  ( ph  ->  I  e.  (
 0 ... M ) )   &    |-  Q  =  ( i  e.  ( 0 ... M )  |->  ( ( V `
  i )  -  X ) )   =>    |-  ( ph  ->  ( ( Q `  I
 )  =  ( ( V `  I )  -  X )  /\  ( V `  I )  =  ( X  +  ( Q `  I ) ) ) )
 
Theoremfourierdlem14 37551* Given the partition  V,  Q is the partition shifted to the left by  X. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  X  e.  RR )   &    |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  ( A  +  X )  /\  ( p `  m )  =  ( B  +  X )
 )  /\  A. i  e.  ( 0..^ m ) ( p `  i
 )  <  ( p `  ( i  +  1 ) ) ) }
 )   &    |-  O  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  (
 0 ... m ) )  |  ( ( ( p `  0 )  =  A  /\  ( p `  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  V  e.  ( P `  M ) )   &    |-  Q  =  ( i  e.  ( 0
 ... M )  |->  ( ( V `  i
 )  -  X ) )   =>    |-  ( ph  ->  Q  e.  ( O `  M ) )
 
Theoremfourierdlem15 37552* The range of the partition is between its starting point and its ending point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  A  /\  ( p `
  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q  e.  ( P `  M ) )   =>    |-  ( ph  ->  Q : ( 0 ...
 M ) --> ( A [,] B ) )
 
Theoremfourierdlem16 37553* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  C  =  (
 -u pi (,) pi )   &    |-  ( ph  ->  ( F  |`  C )  e.  L^1 )   &    |-  A  =  ( n  e.  NN0  |->  ( S. C ( ( F `  x )  x.  ( cos `  ( n  x.  x ) ) )  _d x  /  pi ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( ( A `  N )  e.  RR  /\  ( x  e.  C  |->  ( F `  x ) )  e.  L^1 )  /\  S. C ( ( F `  x )  x.  ( cos `  ( N  x.  x ) ) )  _d x  e.  RR ) )
 
Theoremfourierdlem17 37554* The defined  L is actually a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  L  =  ( x  e.  ( A (,] B )  |->  if ( x  =  B ,  A ,  x ) )   =>    |-  ( ph  ->  L : ( A (,] B ) --> ( A [,] B ) )
 
Theoremfourierdlem18 37555* The function  S is continuous (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  N  e.  RR )   &    |-  S  =  ( s  e.  ( -u pi [,] pi )  |->  ( sin `  ( ( N  +  ( 1  /  2
 ) )  x.  s
 ) ) )   =>    |-  ( ph  ->  S  e.  ( ( -u pi [,] pi ) -cn-> RR ) )
 
Theoremfourierdlem19 37556* If two elements of  D have the same periodic image in  ( A (,] B
) then they are equal. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  X  e.  RR )   &    |-  D  =  { y  e.  ( ( A  +  X ) (,] ( B  +  X )
 )  |  E. k  e.  ZZ  ( y  +  ( k  x.  T ) )  e.  C }   &    |-  T  =  ( B  -  A )   &    |-  E  =  ( x  e.  RR  |->  ( x  +  (
 ( |_ `  (
 ( B  -  x )  /  T ) )  x.  T ) ) )   &    |-  ( ph  ->  W  e.  D )   &    |-  ( ph  ->  Z  e.  D )   &    |-  ( ph  ->  ( E `  Z )  =  ( E `  W ) )   =>    |-  ( ph  ->  -.  W  <  Z )
 
Theoremfourierdlem20 37557* Every interval in the partition  S is included in an interval of the partition  Q. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  Q : ( 0 ...
 M ) --> RR )   &    |-  ( ph  ->  ( Q `  0 )  <_  A )   &    |-  ( ph  ->  B  <_  ( Q `  M ) )   &    |-  ( ph  ->  J  e.  ( 0..^ N ) )   &    |-  T  =  ( { A ,  B }  u.  ( ran  Q  i^i  ( A (,) B ) ) )   &    |-  ( ph  ->  S  Isom  <  ,  <  ( ( 0
 ... N ) ,  T ) )   &    |-  I  =  sup ( { k  e.  ( 0..^ M )  |  ( Q `  k )  <_  ( S `
  J ) } ,  RR ,  <  )   =>    |-  ( ph  ->  E. i  e.  (
 0..^ M ) ( ( S `  J ) (,) ( S `  ( J  +  1
 ) ) )  C_  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )
 
Theoremfourierdlem21 37558* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  C  =  (
 -u pi (,) pi )   &    |-  ( ph  ->  ( F  |`  C )  e.  L^1 )   &    |-  B  =  ( n  e.  NN  |->  ( S. C ( ( F `  x )  x.  ( sin `  ( n  x.  x ) ) )  _d x  /  pi ) )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  ( ( ( B `  N )  e.  RR  /\  ( x  e.  C  |->  ( ( F `  x )  x.  ( sin `  ( N  x.  x ) ) ) )  e.  L^1 )  /\  S. C ( ( F `  x )  x.  ( sin `  ( N  x.  x ) ) )  _d x  e.  RR ) )
 
Theoremfourierdlem22 37559* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  C  =  (
 -u pi (,) pi )   &    |-  ( ph  ->  ( F  |`  C )  e.  L^1 )   &    |-  A  =  ( n  e.  NN0  |->  ( S. C ( ( F `  x )  x.  ( cos `  ( n  x.  x ) ) )  _d x  /  pi ) )   &    |-  B  =  ( n  e.  NN  |->  ( S. C ( ( F `  x )  x.  ( sin `  ( n  x.  x ) ) )  _d x  /  pi ) )   =>    |-  ( ph  ->  (
 ( n  e.  NN0  ->  ( A `  n )  e.  RR )  /\  ( n  e.  NN  ->  ( B `  n )  e.  RR )
 ) )
 
Theoremfourierdlem23 37560* If  F is continuous and  X is constant, then  ( F `  ( X  +  s
) ) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  F  e.  ( A -cn-> CC )
 )   &    |-  ( ph  ->  B  C_ 
 CC )   &    |-  ( ph  ->  X  e.  CC )   &    |-  (
 ( ph  /\  s  e.  B )  ->  ( X  +  s )  e.  A )   =>    |-  ( ph  ->  (
 s  e.  B  |->  ( F `  ( X  +  s ) ) )  e.  ( B
 -cn-> CC ) )
 
Theoremfourierdlem24 37561 A sufficient condition for module being nonzero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  ( ( -u pi [,] pi ) 
 \  { 0 } )  ->  ( A  mod  ( 2  x.  pi ) )  =/=  0
 )
 
Theoremfourierdlem25 37562* If  C is not in the range of the partition, then it is in an open interval induced by the partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q : ( 0 ...
 M ) --> RR )   &    |-  ( ph  ->  C  e.  (
 ( Q `  0
 ) [,] ( Q `  M ) ) )   &    |-  ( ph  ->  -.  C  e.  ran 
 Q )   &    |-  I  =  sup ( { k  e.  (
 0..^ M )  |  ( Q `  k
 )  <  C } ,  RR ,  <  )   =>    |-  ( ph  ->  E. j  e.  (
 0..^ M ) C  e.  ( ( Q `
  j ) (,) ( Q `  (
 j  +  1 ) ) ) )
 
Theoremfourierdlem26 37563* Periodic image of a point  Y that's in the period that begins with the point  X. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  T  =  ( B  -  A )   &    |-  E  =  ( x  e.  RR  |->  ( x  +  ( ( |_ `  ( ( B  -  x )  /  T ) )  x.  T ) ) )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  ( E `  X )  =  B )   &    |-  ( ph  ->  Y  e.  ( X (,] ( X  +  T )
 ) )   =>    |-  ( ph  ->  ( E `  Y )  =  ( A  +  ( Y  -  X ) ) )
 
Theoremfourierdlem27 37564 A partition open interval is a subset of the partitioned open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  Q : ( 0 ...
 M ) --> ( A [,] B ) )   &    |-  ( ph  ->  I  e.  ( 0..^ M ) )   =>    |-  ( ph  ->  ( ( Q `  I ) (,) ( Q `  ( I  +  1 )
 ) )  C_  ( A (,) B ) )
 
Theoremfourierdlem28 37565* Derivative of  ( F `  ( X  +  s
) ). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  D  =  ( RR  _D  ( F  |`  ( ( X  +  A ) (,) ( X  +  B )
 ) ) )   &    |-  ( ph  ->  D : ( ( X  +  A ) (,) ( X  +  B ) ) --> RR )   =>    |-  ( ph  ->  ( RR  _D  ( s  e.  ( A (,) B )  |->  ( F `  ( X  +  s ) ) ) )  =  ( s  e.  ( A (,) B )  |->  ( D `  ( X  +  s ) ) ) )
 
Theoremfourierdlem29 37566* Explicit function value for  K applied to  A. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  K  =  ( s  e.  ( -u pi [,] pi ) 
 |->  if ( s  =  0 ,  1 ,  ( s  /  (
 2  x.  ( sin `  ( s  /  2
 ) ) ) ) ) )   =>    |-  ( A  e.  ( -u pi [,] pi ) 
 ->  ( K `  A )  =  if ( A  =  0 , 
 1 ,  ( A 
 /  ( 2  x.  ( sin `  ( A  /  2 ) ) ) ) ) )
 
Theoremfourierdlem30 37567* Sum of three small pieces is less than ε (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ( x  e.  I  |->  ( F  x.  -u G ) )  e.  L^1 )   &    |-  (
 ( ph  /\  x  e.  I )  ->  F  e.  CC )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  G  e.  CC )   &    |-  ( ph  ->  A  e.  CC )   &    |-  X  =  ( abs `  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  Y  =  ( abs `  C )   &    |-  Z  =  ( abs `  S. I ( F  x.  -u G )  _d x )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  R  e.  RR )   &    |-  ( ph  ->  ( ( ( ( X  +  Y )  +  Z )  /  E )  +  1 )  <_  R )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( abs `  B )  <_ 
 1 )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  ( abs `  D )  <_  1 )   =>    |-  ( ph  ->  ( abs `  ( (
 ( A  x.  -u ( B  /  R ) )  -  ( C  x.  -u ( D  /  R ) ) )  -  S. I ( F  x.  -u ( G  /  R ) )  _d x ) )  <  E )
 
Theoremfourierdlem31 37568* If  A is finite and for any element in  A there is a number  m such that a property holds for all numbers larger than  m, then there is a number  n such that the property holds for all numbers larger than  n and for all elements in  A. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F/ i ph   &    |-  F/ r ph   &    |-  F/_ i V   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A. i  e.  A  E. m  e.  NN  A. r  e.  ( m (,) +oo ) ch )   &    |-  M  =  { m  e.  NN  |  A. r  e.  ( m (,) +oo ) ch }   &    |-  V  =  ( i  e.  A  |->  sup ( M ,  RR ,  `'  <  ) )   &    |-  N  =  sup ( ran  V ,  RR ,  <  )   =>    |-  ( ph  ->  E. n  e.  NN  A. r  e.  ( n (,) +oo ) A. i  e.  A  ch )
 
Theoremfourierdlem32 37569 Limit of a continuous function on an open subinterval. Lower bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F  e.  (
 ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  R  e.  ( F lim CC  A ) )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  C  <  D )   &    |-  ( ph  ->  ( C (,) D )  C_  ( A (,) B ) )   &    |-  Y  =  if ( C  =  A ,  R ,  ( F `
  C ) )   &    |-  J  =  ( ( TopOpen ` fld )t  ( A [,) B ) )   =>    |-  ( ph  ->  Y  e.  ( ( F  |`  ( C (,) D ) ) lim
 CC  C ) )
 
Theoremfourierdlem33 37570 Limit of a continuous function on an open subinterval. Upper bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F  e.  (
 ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  L  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  C  <  D )   &    |-  ( ph  ->  ( C (,) D )  C_  ( A (,) B ) )   &    |-  Y  =  if ( D  =  B ,  L ,  ( F `
  D ) )   &    |-  J  =  ( ( TopOpen ` fld )t  ( ( A (,) B )  u.  { B }
 ) )   =>    |-  ( ph  ->  Y  e.  ( ( F  |`  ( C (,) D ) ) lim
 CC  D ) )
 
Theoremfourierdlem34 37571* A partition is one to one. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  A  /\  ( p `
  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q  e.  ( P `  M ) )   =>    |-  ( ph  ->  Q : ( 0 ...
 M ) -1-1-> RR )
 
Theoremfourierdlem35 37572 There is a single point in  ( A (,] B
) that's distant from 
X a multiple integer of  T. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  T  =  ( B  -  A )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  I  e.  ZZ )   &    |-  ( ph  ->  J  e.  ZZ )   &    |-  ( ph  ->  ( X  +  ( I  x.  T ) )  e.  ( A (,] B ) )   &    |-  ( ph  ->  ( X  +  ( J  x.  T ) )  e.  ( A (,] B ) )   =>    |-  ( ph  ->  I  =  J )
 
Theoremfourierdlem36 37573*  F is an isomorphism. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_ 
 RR )   &    |-  F  =  (
 iota f f  Isom  <  ,  <  ( ( 0
 ... N ) ,  A ) )   &    |-  N  =  ( ( # `  A )  -  1 )   =>    |-  ( ph  ->  F 
 Isom  <  ,  <  (
 ( 0 ... N ) ,  A )
 )
 
Theoremfourierdlem37 37574*  I is a function that maps any real point to the point that in the partition that immediately precedes the corresponding periodic point in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  A  /\  ( p `
  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q  e.  ( P `  M ) )   &    |-  T  =  ( B  -  A )   &    |-  E  =  ( x  e.  RR  |->  ( x  +  ( ( |_ `  (
 ( B  -  x )  /  T ) )  x.  T ) ) )   &    |-  L  =  ( y  e.  ( A (,] B )  |->  if ( y  =  B ,  A ,  y ) )   &    |-  I  =  ( x  e.  RR  |->  sup ( { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `
  ( E `  x ) ) } ,  RR ,  <  )
 )   =>    |-  ( ph  ->  ( I : RR --> ( 0..^ M )  /\  ( x  e.  RR  ->  sup ( { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `
  ( E `  x ) ) } ,  RR ,  <  )  e.  { i  e.  (
 0..^ M )  |  ( Q `  i
 )  <_  ( L `  ( E `  x ) ) } )
 ) )
 
Theoremfourierdlem38 37575* The function  F is continuous on every interval induced by the partition  Q. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F  e.  ( dom  F -cn-> CC ) )   &    |-  P  =  ( n  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... n ) )  |  ( ( ( p `
  0 )  =  -u pi  /\  ( p `
  n )  =  pi )  /\  A. i  e.  ( 0..^ n ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q  e.  ( P `  M ) )   &    |-  H  =  ( A  u.  ( (
 -u pi [,] pi )  \  dom  F ) )   &    |-  ( ph  ->  ran 
 Q  =  H )   =>    |-  ( ( ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i
 ) (,) ( Q `  ( i  +  1
 ) ) ) -cn-> CC ) )
 
Theoremfourierdlem39 37576* Integration by parts of  S. ( A (,) B ) ( ( F `  x
)  x.  ( sin `  ( R  x.  x
) ) )  _d x (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> CC ) )   &    |-  G  =  ( RR  _D  F )   &    |-  ( ph  ->  G  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  E. y  e.  RR  A. x  e.  ( A (,) B ) ( abs `  ( G `  x ) )  <_  y )   &    |-  ( ph  ->  R  e.  RR+ )   =>    |-  ( ph  ->  S. ( A (,) B ) ( ( F `  x )  x.  ( sin `  ( R  x.  x ) ) )  _d x  =  ( ( ( ( F `
  B )  x.  -u ( ( cos `  ( R  x.  B ) ) 
 /  R ) )  -  ( ( F `
  A )  x.  -u ( ( cos `  ( R  x.  A ) ) 
 /  R ) ) )  -  S. ( A (,) B ) ( ( G `  x )  x.  -u ( ( cos `  ( R  x.  x ) )  /  R ) )  _d x ) )
 
Theoremfourierdlem40 37577*  H is a continuous function on any partition interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  A  e.  ( -u pi [,] pi ) )   &    |-  ( ph  ->  B  e.  ( -u pi [,] pi ) )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  -.  0  e.  ( A (,) B ) )   &    |-  ( ph  ->  ( F  |`  ( ( A  +  X ) (,) ( B  +  X ) ) )  e.  ( ( ( A  +  X ) (,) ( B  +  X ) ) -cn-> CC )
 )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  W  e.  RR )   &    |-  H  =  ( s  e.  ( -u pi [,] pi ) 
 |->  if ( s  =  0 ,  0 ,  ( ( ( F `
  ( X  +  s ) )  -  if ( 0  <  s ,  Y ,  W ) )  /  s ) ) )   =>    |-  ( ph  ->  ( H  |`  ( A (,) B ) )  e.  (
 ( A (,) B ) -cn-> CC ) )
 
Theoremfourierdlem41 37578* Lemma used to prove that every real is a limit point for the domain of the derivative of the periodic function to be approximated. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  T  =  ( B  -  A )   &    |-  ( ( ph  /\  x  e.  D  /\  k  e. 
 ZZ )  ->  ( x  +  ( k  x.  T ) )  e.  D )   &    |-  ( ph  ->  X  e.  RR )   &    |-  Z  =  ( x  e.  RR  |->  ( ( |_ `  (
 ( B  -  x )  /  T ) )  x.  T ) )   &    |-  E  =  ( x  e.  RR  |->  ( x  +  ( Z `  x ) ) )   &    |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  A  /\  ( p `
  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q  e.  ( P `  M ) )   &    |-  ( ( ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) 
 C_  D )   =>    |-  ( ph  ->  ( E. y  e.  RR  ( y  <  X  /\  ( y (,) X )  C_  D )  /\  E. y  e.  RR  ( X  <  y  /\  ( X (,) y )  C_  D ) ) )
 
Theoremfourierdlem42 37579* The set of points in a moved partition are finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <  C )   &    |-  T  =  ( C  -  B )   &    |-  ( ph  ->  A  C_  ( B [,] C ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  C  e.  A )   &    |-  D  =  ( abs  o.  -  )   &    |-  I  =  ( ( A  X.  A )  \  _I  )   &    |-  R  =  ran  ( D  |`  I )   &    |-  E  =  sup ( R ,  RR ,  `'  <  )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  J  =  ( topGen `  ran  (,) )   &    |-  K  =  ( Jt  ( X [,] Y ) )   &    |-  H  =  { x  e.  ( X [,] Y )  |  E. k  e.  ZZ  ( x  +  ( k  x.  T ) )  e.  A }   &    |-  ( ps  <->  ( ( ph  /\  ( a  e.  RR  /\  b  e.  RR  /\  a  <  b ) ) 
 /\  E. j  e.  ZZ  E. k  e.  ZZ  (
 ( a  +  (
 j  x.  T ) )  e.  A  /\  ( b  +  (
 k  x.  T ) )  e.  A ) ) )   =>    |-  ( ph  ->  H  e.  Fin )
 
Theoremfourierdlem43 37580  K is a real function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  K  =  ( s  e.  ( -u pi [,] pi ) 
 |->  if ( s  =  0 ,  1 ,  ( s  /  (
 2  x.  ( sin `  ( s  /  2
 ) ) ) ) ) )   =>    |-  K : ( -u pi [,] pi ) --> RR
 
Theoremfourierdlem44 37581 A condition for having  ( sin `  ( A  /  2 ) ) non zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  ( -u pi [,] pi ) 
 /\  A  =/=  0
 )  ->  ( sin `  ( A  /  2
 ) )  =/=  0
 )
 
Theoremfourierdlem45 37582* If  F has bounded derivative on  ( A (,) B ) and  R is a convergent sequence, with values in  ( A (,) B ), then the composition of  F and  R is a sequence converging to its  limsup. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): replace uses of fourierdlem45 37582 with ioodvbdlimc1lem1 37374 then delete fourierdlem45 37582.
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F  e.  (
 ( A (,) B ) -cn-> RR ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  E. y  e.  RR  A. x  e.  ( A (,) B ) ( abs `  ( ( RR  _D  F ) `  x ) )  <_  y )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  R : ( ZZ>= `  M )
 --> ( A (,) B ) )   &    |-  S  =  ( j  e.  ( ZZ>= `  M )  |->  ( F `
  ( R `  j ) ) )   &    |-  ( ph  ->  R  e.  dom  ~~>  )   &    |-  K  =  sup ( { k  e.  ( ZZ>=
 `  M )  | 
 A. i  e.  ( ZZ>=
 `  k ) ( abs `  ( ( R `  i )  -  ( R `  k ) ) )  <  ( x  /  ( sup ( ran  ( z  e.  ( A (,) B )  |->  ( abs `  ( ( RR  _D  F ) `  z ) ) ) ,  RR ,  <  )  +  1 ) ) } ,  RR ,  `'  <  )   =>    |-  ( ph  ->  S  ~~>  ( limsup `  S )
 )
 
Theoremfourierdlem46 37583* The function  F has a limit at the bounds of every interval induced by the partition  Q. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F  e.  ( dom  F -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( ( -u pi [,) pi )  \  dom  F ) )  ->  (
 ( F  |`  ( x (,) +oo ) ) lim CC  x )  =/=  (/) )   &    |-  (
 ( ph  /\  x  e.  ( ( -u pi (,] pi )  \  dom  F ) )  ->  (
 ( F  |`  ( -oo (,) x ) ) lim CC  x )  =/=  (/) )   &    |-  ( ph  ->  Q  Isom  <  ,  <  ( ( 0
 ... M ) ,  H ) )   &    |-  ( ph  ->  Q : ( 0 ... M ) --> H )   &    |-  ( ph  ->  I  e.  ( 0..^ M ) )   &    |-  ( ph  ->  ( Q `  I )  <  ( Q `  ( I  +  1
 ) ) )   &    |-  ( ph  ->  ( ( Q `
  I ) (,) ( Q `  ( I  +  1 )
 ) )  C_  ( -u pi (,) pi ) )   &    |-  ( ph  ->  C  e.  RR )   &    |-  H  =  ( { -u pi ,  pi ,  C }  u.  ( ( -u pi [,] pi )  \  dom  F ) )   &    |-  ( ph  ->  ran 
 Q  =  H )   =>    |-  ( ph  ->  ( (
 ( F  |`  ( ( Q `  I ) (,) ( Q `  ( I  +  1
 ) ) ) ) lim
 CC  ( Q `  I ) )  =/=  (/)  /\  ( ( F  |`  ( ( Q `  I ) (,) ( Q `  ( I  +  1 ) ) ) ) lim CC  ( Q `
  ( I  +  1 ) ) )  =/=  (/) ) )
 
Theoremfourierdlem47 37584* For  r large enough, the final expression is less than the given positive  E. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ( x  e.  I  |->  F )  e.  L^1 )   &    |-  (
 ( ph  /\  r  e. 
 RR )  ->  ( x  e.  I  |->  ( F  x.  -u G ) )  e.  L^1 )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  F  e.  CC )   &    |-  (
 ( ( ph  /\  x  e.  I )  /\  r  e.  CC )  ->  G  e.  CC )   &    |-  ( ( (
 ph  /\  x  e.  I )  /\  r  e. 
 RR )  ->  ( abs `  G )  <_ 
 1 )   &    |-  ( ph  ->  A  e.  CC )   &    |-  X  =  ( abs `  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  Y  =  ( abs `  C )   &    |-  Z  =  S. I ( abs `  F )  _d x   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ( ph  /\  r  e.  CC )  ->  B  e.  CC )   &    |-  ( ( ph  /\  r  e.  RR )  ->  ( abs `  B )  <_  1 )   &    |-  (
 ( ph  /\  r  e. 
 CC )  ->  D  e.  CC )   &    |-  ( ( ph  /\  r  e.  RR )  ->  ( abs `  D )  <_  1 )   &    |-  M  =  ( ( |_ `  (
 ( ( ( X  +  Y )  +  Z )  /  E )  +  1 ) )  +  1 )   =>    |-  ( ph  ->  E. m  e.  NN  A. r  e.  ( m (,) +oo ) ( abs `  ( ( ( A  x.  -u ( B  /  r ) )  -  ( C  x.  -u ( D  /  r ) ) )  -  S. I
 ( F  x.  -u ( G  /  r ) )  _d x ) )  <  E )
 
Theoremfourierdlem48 37585* The given periodic function  F has a right limit at every point in the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  A  /\  ( p `
  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  T  =  ( B  -  A )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q  e.  ( P `  M ) )   &    |-  ( ph  ->  F : D --> RR )   &    |-  ( ( ph  /\  x  e.  D  /\  k  e.  ZZ )  ->  ( x  +  (
 k  x.  T ) )  e.  D )   &    |-  ( ( ph  /\  x  e.  D  /\  k  e. 
 ZZ )  ->  ( F `  ( x  +  ( k  x.  T ) ) )  =  ( F `  x ) )   &    |-  ( ( ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1
 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1
 ) ) ) -cn-> CC ) )   &    |-  ( ( ph  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  ( ( Q `
  i ) (,) ( Q `  (
 i  +  1 ) ) ) ) lim CC  ( Q `  i ) ) )   &    |-  ( ph  ->  X  e.  RR )   &    |-  Z  =  ( x  e.  RR  |->  ( ( |_ `  (
 ( B  -  x )  /  T ) )  x.  T ) )   &    |-  E  =  ( x  e.  RR  |->  ( x  +  ( Z `  x ) ) )   &    |-  ( ch  <->  ( ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  y  e.  ( ( Q `  i ) [,) ( Q `  ( i  +  1 ) ) ) )  /\  k  e. 
 ZZ )  /\  y  =  ( X  +  (
 k  x.  T ) ) ) )   =>    |-  ( ph  ->  ( ( F  |`  ( X (,) +oo ) ) lim CC  X )  =/=  (/) )
 
Theoremfourierdlem49 37586* The given periodic function  F has a left limit at every point in the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  A  /\  ( p `
  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  T  =  ( B  -  A )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q  e.  ( P `  M ) )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F : D --> RR )   &    |-  (
 ( ph  /\  x  e.  D  /\  k  e. 
 ZZ )  ->  ( x  +  ( k  x.  T ) )  e.  D )   &    |-  ( ( ph  /\  x  e.  D  /\  k  e.  ZZ )  ->  ( F `  ( x  +  ( k  x.  T ) ) )  =  ( F `  x ) )   &    |-  (
 ( ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i
 ) (,) ( Q `  ( i  +  1
 ) ) ) -cn-> CC ) )   &    |-  ( ( ph  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( F  |`  ( ( Q `
  i ) (,) ( Q `  (
 i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) ) )   &    |-  ( ph  ->  X  e.  RR )   &    |-  Z  =  ( x  e.  RR  |->  ( ( |_ `  (
 ( B  -  x )  /  T ) )  x.  T ) )   &    |-  E  =  ( x  e.  RR  |->  ( x  +  ( Z `  x ) ) )   =>    |-  ( ph  ->  (
 ( F  |`  ( -oo (,)
 X ) ) lim CC  X )  =/=  (/) )
 
Theoremfourierdlem50 37587* Continuity of  O and its limits with respect to the  S partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  X  e.  RR )   &    |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  (
 0 ... m ) )  |  ( ( ( p `  0 )  =  ( -u pi  +  X )  /\  ( p `  m )  =  ( pi  +  X ) )  /\  A. i  e.  ( 0..^ m ) ( p `  i
 )  <  ( p `  ( i  +  1 ) ) ) }
 )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  V  e.  ( P `  M ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B )  C_  ( -u pi [,] pi ) )   &    |-  Q  =  ( i  e.  (
 0 ... M )  |->  ( ( V `  i
 )  -  X ) )   &    |-  T  =  ( { A ,  B }  u.  ( ran  Q  i^i  ( A (,) B ) ) )   &    |-  N  =  ( ( # `  T )  -  1 )   &    |-  S  =  ( iota f f 
 Isom  <  ,  <  (
 ( 0 ... N ) ,  T )
 )   &    |-  ( ph  ->  J  e.  ( 0..^ N ) )   &    |-  U  =  (
 iota_ i  e.  (
 0..^ M ) ( ( S `  J ) (,) ( S `  ( J  +  1
 ) ) )  C_  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )   &    |-  ( ch  <->  ( ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  ( ( S `  J ) (,) ( S `  ( J  +  1
 ) ) )  C_  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  /\  k  e.  ( 0..^ M ) )  /\  ( ( S `  J ) (,) ( S `  ( J  +  1
 ) ) )  C_  ( ( Q `  k ) (,) ( Q `  ( k  +  1 ) ) ) ) )   =>    |-  ( ph  ->  ( U  e.  ( 0..^ M )  /\  ( ( S `  J ) (,) ( S `  ( J  +  1
 ) ) )  C_  ( ( Q `  U ) (,) ( Q `  ( U  +  1 ) ) ) ) )
 
Theoremfourierdlem51 37588*  X is in the periodic partition, when the considered interval is centered at  X. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  0 )   &    |-  ( ph  ->  0  <  B )   &    |-  T  =  ( B  -  A )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  ( ph  ->  C  C_  ( A [,] B ) )   &    |-  ( ph  ->  B  e.  C )   &    |-  E  =  ( x  e.  RR  |->  ( x  +  (
 ( |_ `  (
 ( B  -  x )  /  T ) )  x.  T ) ) )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  ( E `  X )  e.  C )   &    |-  D  =  ( {
 ( A  +  X ) ,  ( B  +  X ) }  u.  { y  e.  ( ( A  +  X ) [,] ( B  +  X ) )  | 
 E. k  e.  ZZ  ( y  +  (
 k  x.  T ) )  e.  C }
 )   &    |-  F  =  ( iota
 f f  Isom  <  ,  <  ( ( 0
 ... ( ( # `  D )  -  1
 ) ) ,  D ) )   &    |-  H  =  {
 y  e.  ( ( A  +  X ) (,] ( B  +  X ) )  | 
 E. k  e.  ZZ  ( y  +  (
 k  x.  T ) )  e.  C }   =>    |-  ( ph  ->  X  e.  ran  F )
 
Theoremfourierdlem52 37589* d16:d17,d18:jca |- ( ph -> ( ( S  0 )  <_  A  /\  A  <_  ( S 0 ) ) ) (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  T  e.  Fin )   &    |-  N  =  ( ( # `  T )  -  1 )   &    |-  S  =  (
 iota f f  Isom  <  ,  <  ( ( 0
 ... N ) ,  T ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  T 
 C_  ( A [,] B ) )   &    |-  ( ph  ->  A  e.  T )   &    |-  ( ph  ->  B  e.  T )   =>    |-  ( ph  ->  (
 ( S : ( 0 ... N ) --> ( A [,] B )  /\  ( S `  0 )  =  A )  /\  ( S `  N )  =  B ) )
 
Theoremfourierdlem53 37590* The limit of  F ( s ) at  ( X  +  D ) is the limit of  F ( X  +  s ) at  D. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  A  C_  RR )   &    |-  G  =  ( s  e.  A  |->  ( F `
  ( X  +  s ) ) )   &    |-  ( ( ph  /\  s  e.  A )  ->  ( X  +  s )  e.  B )   &    |-  ( ph  ->  B 
 C_  RR )   &    |-  ( ( ph  /\  s  e.  A ) 
 ->  s  =/=  D )   &    |-  ( ph  ->  C  e.  ( ( F  |`  B ) lim
 CC  ( X  +  D ) ) )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  C  e.  ( G lim CC  D ) )
 
Theoremfourierdlem54 37591* Given a partition  Q and an arbitrary interval  [ C ,  D ], a partition  S on  [ C ,  D ] is built such that it preserves any periodic function piecewise continuous on  Q will be piecewise continuous on  S, with the same limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  T  =  ( B  -  A )   &    |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  (
 0 ... m ) )  |  ( ( ( p `  0 )  =  A  /\  ( p `  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q  e.  ( P `  M ) )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  C  <  D )   &    |-  O  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  C  /\  ( p `
  m )  =  D )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  H  =  ( { C ,  D }  u.  { x  e.  ( C [,] D )  |  E. k  e.  ZZ  ( x  +  ( k  x.  T ) )  e.  ran  Q } )   &    |-  N  =  ( ( # `  H )  -  1 )   &    |-  S  =  ( iota f f 
 Isom  <  ,  <  (
 ( 0 ... N ) ,  H )
 )   =>    |-  ( ph  ->  (
 ( N  e.  NN  /\  S  e.  ( O `
  N ) ) 
 /\  S  Isom  <  ,  <  ( ( 0
 ... N ) ,  H ) ) )
 
Theoremfourierdlem55 37592*  U is a real function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  W  e.  RR )   &    |-  H  =  ( s  e.  ( -u pi [,] pi )  |->  if ( s  =  0 ,  0 ,  (
 ( ( F `  ( X  +  s
 ) )  -  if ( 0  <  s ,  Y ,  W ) )  /  s ) ) )   &    |-  K  =  ( s  e.  ( -u pi [,] pi )  |->  if ( s  =  0 ,  1 ,  (
 s  /  ( 2  x.  ( sin `  (
 s  /  2 )
 ) ) ) ) )   &    |-  U  =  ( s  e.  ( -u pi [,] pi )  |->  ( ( H `  s
 )  x.  ( K `
  s ) ) )   =>    |-  ( ph  ->  U : ( -u pi [,] pi ) --> RR )
 
Theoremfourierdlem56 37593* Derivative of the  K function on an interval non containing ' 0 '. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  K  =  ( s  e.  ( -u pi [,] pi ) 
 |->  if ( s  =  0 ,  1 ,  ( s  /  (
 2  x.  ( sin `  ( s  /  2
 ) ) ) ) ) )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  ( ( -u pi [,] pi )  \  { 0 } )
 )   &    |-  ( ( ph  /\  s  e.  ( A (,) B ) )  ->  s  =/=  0 )   =>    |-  ( ph  ->  ( RR  _D  ( s  e.  ( A (,) B )  |->  ( K `  s ) ) )  =  ( s  e.  ( A (,) B )  |->  ( ( ( ( sin `  (
 s  /  2 )
 )  -  ( ( ( cos `  (
 s  /  2 )
 )  /  2 )  x.  s ) )  /  ( ( sin `  (
 s  /  2 )
 ) ^ 2 ) )  /  2 ) ) )
 
Theoremfourierdlem57 37594* The derivative of  O. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( X  +  A ) (,) ( X  +  B )
 ) ) ) : ( ( X  +  A ) (,) ( X  +  B )
 ) --> RR )   &    |-  ( ph  ->  ( A (,) B )  C_  ( -u pi [,] pi ) )   &    |-  ( ph  ->  -.  0  e.  ( A (,) B ) )   &    |-  ( ph  ->  C  e.  RR )   &    |-  O  =  ( s  e.  ( A (,) B )  |->  ( ( ( F `  ( X  +  s
 ) )  -  C )  /  ( 2  x.  ( sin `  (
 s  /  2 )
 ) ) ) )   =>    |-  ( ( ph  ->  ( ( RR  _D  O ) : ( A (,) B ) --> RR  /\  ( RR 
 _D  O )  =  ( s  e.  ( A (,) B )  |->  ( ( ( ( ( RR  _D  ( F  |`  ( ( X  +  A ) (,) ( X  +  B )
 ) ) ) `  ( X  +  s
 ) )  x.  (
 2  x.  ( sin `  ( s  /  2
 ) ) ) )  -  ( ( cos `  ( s  /  2
 ) )  x.  (
 ( F `  ( X  +  s )
 )  -  C ) ) )  /  (
 ( 2  x.  ( sin `  ( s  / 
 2 ) ) ) ^ 2 ) ) ) ) )  /\  ( RR  _D  (
 s  e.  ( A (,) B )  |->  ( 2  x.  ( sin `  ( s  /  2
 ) ) ) ) )  =  ( s  e.  ( A (,) B )  |->  ( cos `  (
 s  /  2 )
 ) ) )
 
Theoremfourierdlem58 37595* The derivative of  K is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  K  =  ( s  e.  A  |->  ( s  /  (
 2  x.  ( sin `  ( s  /  2
 ) ) ) ) )   &    |-  ( ph  ->  A 
 C_  ( -u pi [,] pi ) )   &    |-  ( ph  ->  -.  0  e.  A )   &    |-  ( ph  ->  A  e.  ( topGen `  ran  (,) ) )   =>    |-  ( ph  ->  ( RR  _D  K )  e.  ( A -cn-> RR )
 )
 
Theoremfourierdlem59 37596* The derivative of  H is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  -.  0  e.  ( A (,) B ) )   &    |-  ( ph  ->  ( RR  _D  ( F  |`  ( ( X  +  A ) (,) ( X  +  B ) ) ) )  e.  ( ( ( X  +  A ) (,) ( X  +  B ) ) -cn-> RR ) )   &    |-  ( ph  ->  C  e.  RR )   &    |-  H  =  ( s  e.  ( A (,) B )  |->  ( ( ( F `  ( X  +  s
 ) )  -  C )  /  s ) )   =>    |-  ( ph  ->  ( RR  _D  H )  e.  (
 ( A (,) B ) -cn-> RR ) )
 
Theoremfourierdlem60 37597* Given a differentiable function  F, with finite limit of the derivative at  A the derived function  H has a limit at  0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  Y  e.  ( F lim CC  B ) )   &    |-  G  =  ( RR  _D  F )   &    |-  ( ph  ->  dom  G  =  ( A (,) B ) )   &    |-  ( ph  ->  E  e.  ( G lim CC  B ) )   &    |-  H  =  ( s  e.  (
 ( A  -  B ) (,) 0 )  |->  ( ( ( F `  ( B  +  s
 ) )  -  Y )  /  s ) )   &    |-  N  =  ( s  e.  ( ( A  -  B ) (,) 0
 )  |->  ( ( F `
  ( B  +  s ) )  -  Y ) )   &    |-  D  =  ( s  e.  (
 ( A  -  B ) (,) 0 )  |->  s )   =>    |-  ( ph  ->  E  e.  ( H lim CC  0
 ) )
 
Theoremfourierdlem61 37598* Given a differentiable function  F, with finite limit of the derivative at  A the derived function  H has a limit at  0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  Y  e.  ( F lim CC  A ) )   &    |-  G  =  ( RR  _D  F )   &    |-  ( ph  ->  dom  G  =  ( A (,) B ) )   &    |-  ( ph  ->  E  e.  ( G lim CC  A ) )   &    |-  H  =  ( s  e.  (
 0 (,) ( B  -  A ) )  |->  ( ( ( F `  ( A  +  s
 ) )  -  Y )  /  s ) )   &    |-  N  =  ( s  e.  ( 0 (,) ( B  -  A ) ) 
 |->  ( ( F `  ( A  +  s
 ) )  -  Y ) )   &    |-  D  =  ( s  e.  ( 0 (,) ( B  -  A ) )  |->  s )   =>    |-  ( ph  ->  E  e.  ( H lim CC  0
 ) )
 
Theoremfourierdlem62 37599 The function  K is continuous (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  K  =  ( y  e.  ( -u pi [,] pi ) 
 |->  if ( y  =  0 ,  1 ,  ( y  /  (
 2  x.  ( sin `  ( y  /  2
 ) ) ) ) ) )   =>    |-  K  e.  ( (
 -u pi [,] pi ) -cn-> RR )
 
Theoremfourierdlem63 37600* The upper bound of intervals in the moved partition are mapped to points that are not greater than the corresponding upper bounds in the original partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  T  =  ( B  -  A )   &    |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  (
 0 ... m ) )  |  ( ( ( p `  0 )  =  A  /\  ( p `  m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Q  e.  ( P `  M ) )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  C  <  D )   &    |-  O  =  ( m  e.  NN  |->  { p  e.  ( RR 
 ^m  ( 0 ... m ) )  |  ( ( ( p `
  0 )  =  C  /\  ( p `
  m )  =  D )  /\  A. i  e.  ( 0..^ m ) ( p `
  i )  < 
 ( p `  (
 i  +  1 ) ) ) } )   &    |-  H  =  ( { C ,  D }  u.  { x  e.  ( C [,] D )  |  E. k  e.  ZZ  ( x  +  ( k  x.  T ) )  e.  ran  Q } )   &    |-  N  =  ( ( # `  H )  -  1 )   &    |-  S  =  ( iota f f 
 Isom  <  ,  <  (
 ( 0 ... N ) ,  H )
 )   &    |-  E  =  ( x  e.  RR  |->  ( x  +  ( ( |_ `  ( ( B  -  x )  /  T ) )  x.  T ) ) )   &    |-  ( ph  ->  K  e.  ( 0 ...
 M ) )   &    |-  ( ph  ->  J  e.  (
 0..^ N ) )   &    |-  ( ph  ->  Y  e.  ( ( S `  J ) [,) ( S `  ( J  +  1 ) ) ) )   &    |-  ( ph  ->  ( E `  Y )  <  ( Q `  K ) )   &    |-  X  =  ( ( Q `  K )  -  (
 ( E `  Y )  -  Y ) )   =>    |-  ( ph  ->  ( E `  ( S `  ( J  +  1 )
 ) )  <_  ( Q `  K ) )
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