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Theorem fourierdlem108 38190
Description: The integral of a piecewise continuous periodic function  F is unchanged if the domain is shifted by any positive value  X. This lemma generalizes fourierdlem92 38174 where the integral was shifted by the exact period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem108.a  |-  ( ph  ->  A  e.  RR )
fourierdlem108.b  |-  ( ph  ->  B  e.  RR )
fourierdlem108.t  |-  T  =  ( B  -  A
)
fourierdlem108.x  |-  ( ph  ->  X  e.  RR+ )
fourierdlem108.p  |-  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 ) ) ) } )
fourierdlem108.m  |-  ( ph  ->  M  e.  NN )
fourierdlem108.q  |-  ( ph  ->  Q  e.  ( P `
 M ) )
fourierdlem108.f  |-  ( ph  ->  F : RR --> CC )
fourierdlem108.fper  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
fourierdlem108.fcn  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
fourierdlem108.r  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 i ) ) )
fourierdlem108.l  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 ( i  +  1 ) ) ) )
Assertion
Ref Expression
fourierdlem108  |-  ( ph  ->  S. ( ( A  -  X ) [,] ( B  -  X
) ) ( F `
 x )  _d x  =  S. ( A [,] B ) ( F `  x
)  _d x )
Distinct variable groups:    A, i, x    A, m, p, i    B, i, x    B, m, p    i, F, x   
x, L    i, M, x    m, M, p    Q, i, x    Q, m, p   
x, R    T, i, x    T, m, p    i, X, x    m, X, p    ph, i, x
Allowed substitution hints:    ph( m, p)    P( x, i, m, p)    R( i, m, p)    F( m, p)    L( i, m, p)

Proof of Theorem fourierdlem108
Dummy variables  f 
g  k  w  y  j  z  l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem108.a . 2  |-  ( ph  ->  A  e.  RR )
2 fourierdlem108.b . 2  |-  ( ph  ->  B  e.  RR )
3 fourierdlem108.t . 2  |-  T  =  ( B  -  A
)
4 fourierdlem108.x . 2  |-  ( ph  ->  X  e.  RR+ )
5 fourierdlem108.p . 2  |-  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 ) ) ) } )
6 fourierdlem108.m . 2  |-  ( ph  ->  M  e.  NN )
7 fourierdlem108.q . 2  |-  ( ph  ->  Q  e.  ( P `
 M ) )
8 fourierdlem108.f . 2  |-  ( ph  ->  F : RR --> CC )
9 fourierdlem108.fper . 2  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
10 fourierdlem108.fcn . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
11 fourierdlem108.r . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 i ) ) )
12 fourierdlem108.l . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 ( i  +  1 ) ) ) )
13 eqid 2471 . 2  |-  ( m  e.  NN  |->  { p  e.  ( RR  ^m  (
0 ... m ) )  |  ( ( ( p `  0 )  =  ( A  -  X )  /\  (
p `  m )  =  A )  /\  A. i  e.  ( 0..^ m ) ( p `
 i )  < 
( p `  (
i  +  1 ) ) ) } )  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  (
0 ... m ) )  |  ( ( ( p `  0 )  =  ( A  -  X )  /\  (
p `  m )  =  A )  /\  A. i  e.  ( 0..^ m ) ( p `
 i )  < 
( p `  (
i  +  1 ) ) ) } )
14 oveq1 6315 . . . . . 6  |-  ( w  =  y  ->  (
w  +  ( k  x.  T ) )  =  ( y  +  ( k  x.  T
) ) )
1514eleq1d 2533 . . . . 5  |-  ( w  =  y  ->  (
( w  +  ( k  x.  T ) )  e.  ran  Q  <->  ( y  +  ( k  x.  T ) )  e.  ran  Q ) )
1615rexbidv 2892 . . . 4  |-  ( w  =  y  ->  ( E. k  e.  ZZ  ( w  +  (
k  x.  T ) )  e.  ran  Q  <->  E. k  e.  ZZ  (
y  +  ( k  x.  T ) )  e.  ran  Q ) )
1716cbvrabv 3030 . . 3  |-  { w  e.  ( ( A  -  X ) [,] A
)  |  E. k  e.  ZZ  ( w  +  ( k  x.  T
) )  e.  ran  Q }  =  { y  e.  ( ( A  -  X ) [,] A )  |  E. k  e.  ZZ  (
y  +  ( k  x.  T ) )  e.  ran  Q }
1817uneq2i 3576 . 2  |-  ( { ( A  -  X
) ,  A }  u.  { w  e.  ( ( A  -  X
) [,] A )  |  E. k  e.  ZZ  ( w  +  ( k  x.  T
) )  e.  ran  Q } )  =  ( { ( A  -  X ) ,  A }  u.  { y  e.  ( ( A  -  X ) [,] A
)  |  E. k  e.  ZZ  ( y  +  ( k  x.  T
) )  e.  ran  Q } )
19 oveq1 6315 . . . . . . . . . 10  |-  ( l  =  k  ->  (
l  x.  T )  =  ( k  x.  T ) )
2019oveq2d 6324 . . . . . . . . 9  |-  ( l  =  k  ->  (
w  +  ( l  x.  T ) )  =  ( w  +  ( k  x.  T
) ) )
2120eleq1d 2533 . . . . . . . 8  |-  ( l  =  k  ->  (
( w  +  ( l  x.  T ) )  e.  ran  Q  <->  ( w  +  ( k  x.  T ) )  e.  ran  Q ) )
2221cbvrexv 3006 . . . . . . 7  |-  ( E. l  e.  ZZ  (
w  +  ( l  x.  T ) )  e.  ran  Q  <->  E. k  e.  ZZ  ( w  +  ( k  x.  T
) )  e.  ran  Q )
2322rgenw 2768 . . . . . 6  |-  A. w  e.  ( ( A  -  X ) [,] A
) ( E. l  e.  ZZ  ( w  +  ( l  x.  T
) )  e.  ran  Q  <->  E. k  e.  ZZ  ( w  +  (
k  x.  T ) )  e.  ran  Q
)
24 rabbi 2955 . . . . . 6  |-  ( A. w  e.  ( ( A  -  X ) [,] A ) ( E. l  e.  ZZ  (
w  +  ( l  x.  T ) )  e.  ran  Q  <->  E. k  e.  ZZ  ( w  +  ( k  x.  T
) )  e.  ran  Q )  <->  { w  e.  ( ( A  -  X
) [,] A )  |  E. l  e.  ZZ  ( w  +  ( l  x.  T
) )  e.  ran  Q }  =  { w  e.  ( ( A  -  X ) [,] A
)  |  E. k  e.  ZZ  ( w  +  ( k  x.  T
) )  e.  ran  Q } )
2523, 24mpbi 213 . . . . 5  |-  { w  e.  ( ( A  -  X ) [,] A
)  |  E. l  e.  ZZ  ( w  +  ( l  x.  T
) )  e.  ran  Q }  =  { w  e.  ( ( A  -  X ) [,] A
)  |  E. k  e.  ZZ  ( w  +  ( k  x.  T
) )  e.  ran  Q }
2625uneq2i 3576 . . . 4  |-  ( { ( A  -  X
) ,  A }  u.  { w  e.  ( ( A  -  X
) [,] A )  |  E. l  e.  ZZ  ( w  +  ( l  x.  T
) )  e.  ran  Q } )  =  ( { ( A  -  X ) ,  A }  u.  { w  e.  ( ( A  -  X ) [,] A
)  |  E. k  e.  ZZ  ( w  +  ( k  x.  T
) )  e.  ran  Q } )
2726fveq2i 5882 . . 3  |-  ( # `  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. l  e.  ZZ  ( w  +  (
l  x.  T ) )  e.  ran  Q } ) )  =  ( # `  ( { ( A  -  X ) ,  A }  u.  { w  e.  ( ( A  -  X ) [,] A
)  |  E. k  e.  ZZ  ( w  +  ( k  x.  T
) )  e.  ran  Q } ) )
2827oveq1i 6318 . 2  |-  ( (
# `  ( {
( A  -  X
) ,  A }  u.  { w  e.  ( ( A  -  X
) [,] A )  |  E. l  e.  ZZ  ( w  +  ( l  x.  T
) )  e.  ran  Q } ) )  - 
1 )  =  ( ( # `  ( { ( A  -  X ) ,  A }  u.  { w  e.  ( ( A  -  X ) [,] A
)  |  E. k  e.  ZZ  ( w  +  ( k  x.  T
) )  e.  ran  Q } ) )  - 
1 )
29 isoeq5 6232 . . . . 5  |-  ( ( { ( A  -  X ) ,  A }  u.  { w  e.  ( ( A  -  X ) [,] A
)  |  E. l  e.  ZZ  ( w  +  ( l  x.  T
) )  e.  ran  Q } )  =  ( { ( A  -  X ) ,  A }  u.  { w  e.  ( ( A  -  X ) [,] A
)  |  E. k  e.  ZZ  ( w  +  ( k  x.  T
) )  e.  ran  Q } )  ->  (
g  Isom  <  ,  <  ( ( 0 ... (
( # `  ( { ( A  -  X
) ,  A }  u.  { w  e.  ( ( A  -  X
) [,] A )  |  E. l  e.  ZZ  ( w  +  ( l  x.  T
) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. l  e.  ZZ  ( w  +  (
l  x.  T ) )  e.  ran  Q } ) )  <->  g  Isom  <  ,  <  ( ( 0 ... ( ( # `  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. l  e.  ZZ  ( w  +  (
l  x.  T ) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. k  e.  ZZ  ( w  +  (
k  x.  T ) )  e.  ran  Q } ) ) ) )
3026, 29ax-mp 5 . . . 4  |-  ( g 
Isom  <  ,  <  (
( 0 ... (
( # `  ( { ( A  -  X
) ,  A }  u.  { w  e.  ( ( A  -  X
) [,] A )  |  E. l  e.  ZZ  ( w  +  ( l  x.  T
) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. l  e.  ZZ  ( w  +  (
l  x.  T ) )  e.  ran  Q } ) )  <->  g  Isom  <  ,  <  ( ( 0 ... ( ( # `  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. l  e.  ZZ  ( w  +  (
l  x.  T ) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. k  e.  ZZ  ( w  +  (
k  x.  T ) )  e.  ran  Q } ) ) )
31 isoeq1 6228 . . . 4  |-  ( g  =  f  ->  (
g  Isom  <  ,  <  ( ( 0 ... (
( # `  ( { ( A  -  X
) ,  A }  u.  { w  e.  ( ( A  -  X
) [,] A )  |  E. l  e.  ZZ  ( w  +  ( l  x.  T
) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. k  e.  ZZ  ( w  +  (
k  x.  T ) )  e.  ran  Q } ) )  <->  f  Isom  <  ,  <  ( ( 0 ... ( ( # `  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. l  e.  ZZ  ( w  +  (
l  x.  T ) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. k  e.  ZZ  ( w  +  (
k  x.  T ) )  e.  ran  Q } ) ) ) )
3230, 31syl5bb 265 . . 3  |-  ( g  =  f  ->  (
g  Isom  <  ,  <  ( ( 0 ... (
( # `  ( { ( A  -  X
) ,  A }  u.  { w  e.  ( ( A  -  X
) [,] A )  |  E. l  e.  ZZ  ( w  +  ( l  x.  T
) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. l  e.  ZZ  ( w  +  (
l  x.  T ) )  e.  ran  Q } ) )  <->  f  Isom  <  ,  <  ( ( 0 ... ( ( # `  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. l  e.  ZZ  ( w  +  (
l  x.  T ) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. k  e.  ZZ  ( w  +  (
k  x.  T ) )  e.  ran  Q } ) ) ) )
3332cbviotav 5559 . 2  |-  ( iota g g  Isom  <  ,  <  ( ( 0 ... ( ( # `  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. l  e.  ZZ  ( w  +  (
l  x.  T ) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. l  e.  ZZ  ( w  +  (
l  x.  T ) )  e.  ran  Q } ) ) )  =  ( iota f
f  Isom  <  ,  <  ( ( 0 ... (
( # `  ( { ( A  -  X
) ,  A }  u.  { w  e.  ( ( A  -  X
) [,] A )  |  E. l  e.  ZZ  ( w  +  ( l  x.  T
) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { ( A  -  X ) ,  A }  u.  {
w  e.  ( ( A  -  X ) [,] A )  |  E. k  e.  ZZ  ( w  +  (
k  x.  T ) )  e.  ran  Q } ) ) )
34 id 22 . . . 4  |-  ( w  =  x  ->  w  =  x )
35 oveq2 6316 . . . . . . 7  |-  ( w  =  x  ->  ( B  -  w )  =  ( B  -  x ) )
3635oveq1d 6323 . . . . . 6  |-  ( w  =  x  ->  (
( B  -  w
)  /  T )  =  ( ( B  -  x )  /  T ) )
3736fveq2d 5883 . . . . 5  |-  ( w  =  x  ->  ( |_ `  ( ( B  -  w )  /  T ) )  =  ( |_ `  (
( B  -  x
)  /  T ) ) )
3837oveq1d 6323 . . . 4  |-  ( w  =  x  ->  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T )  =  ( ( |_
`  ( ( B  -  x )  /  T ) )  x.  T ) )
3934, 38oveq12d 6326 . . 3  |-  ( w  =  x  ->  (
w  +  ( ( |_ `  ( ( B  -  w )  /  T ) )  x.  T ) )  =  ( x  +  ( ( |_ `  ( ( B  -  x )  /  T
) )  x.  T
) ) )
4039cbvmptv 4488 . 2  |-  ( w  e.  RR  |->  ( w  +  ( ( |_
`  ( ( B  -  w )  /  T ) )  x.  T ) ) )  =  ( x  e.  RR  |->  ( x  +  ( ( |_ `  ( ( B  -  x )  /  T
) )  x.  T
) ) )
41 eqeq1 2475 . . . 4  |-  ( w  =  y  ->  (
w  =  B  <->  y  =  B ) )
42 id 22 . . . 4  |-  ( w  =  y  ->  w  =  y )
4341, 42ifbieq2d 3897 . . 3  |-  ( w  =  y  ->  if ( w  =  B ,  A ,  w )  =  if ( y  =  B ,  A ,  y ) )
4443cbvmptv 4488 . 2  |-  ( w  e.  ( A (,] B )  |->  if ( w  =  B ,  A ,  w )
)  =  ( y  e.  ( A (,] B )  |->  if ( y  =  B ,  A ,  y )
)
45 fveq2 5879 . . . . . . . 8  |-  ( z  =  x  ->  (
( w  e.  RR  |->  ( w  +  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T ) ) ) `  z
)  =  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T ) )  x.  T ) ) ) `  x ) )
4645fveq2d 5883 . . . . . . 7  |-  ( z  =  x  ->  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  z ) )  =  ( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) )
4746breq2d 4407 . . . . . 6  |-  ( z  =  x  ->  (
( Q `  j
)  <_  ( (
w  e.  ( A (,] B )  |->  if ( w  =  B ,  A ,  w
) ) `  (
( w  e.  RR  |->  ( w  +  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T ) ) ) `  z
) )  <->  ( Q `  j )  <_  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) ) )
4847rabbidv 3022 . . . . 5  |-  ( z  =  x  ->  { j  e.  ( 0..^ M )  |  ( Q `
 j )  <_ 
( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  z ) ) }  =  { j  e.  ( 0..^ M )  |  ( Q `  j )  <_  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) } )
49 fveq2 5879 . . . . . . 7  |-  ( j  =  i  ->  ( Q `  j )  =  ( Q `  i ) )
5049breq1d 4405 . . . . . 6  |-  ( j  =  i  ->  (
( Q `  j
)  <_  ( (
w  e.  ( A (,] B )  |->  if ( w  =  B ,  A ,  w
) ) `  (
( w  e.  RR  |->  ( w  +  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T ) ) ) `  x
) )  <->  ( Q `  i )  <_  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) ) )
5150cbvrabv 3030 . . . . 5  |-  { j  e.  ( 0..^ M )  |  ( Q `
 j )  <_ 
( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) }  =  { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) }
5248, 51syl6eq 2521 . . . 4  |-  ( z  =  x  ->  { j  e.  ( 0..^ M )  |  ( Q `
 j )  <_ 
( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  z ) ) }  =  { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) } )
5352supeq1d 7978 . . 3  |-  ( z  =  x  ->  sup ( { j  e.  ( 0..^ M )  |  ( Q `  j
)  <_  ( (
w  e.  ( A (,] B )  |->  if ( w  =  B ,  A ,  w
) ) `  (
( w  e.  RR  |->  ( w  +  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T ) ) ) `  z
) ) } ,  RR ,  <  )  =  sup ( { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) } ,  RR ,  <  ) )
5453cbvmptv 4488 . 2  |-  ( z  e.  RR  |->  sup ( { j  e.  ( 0..^ M )  |  ( Q `  j
)  <_  ( (
w  e.  ( A (,] B )  |->  if ( w  =  B ,  A ,  w
) ) `  (
( w  e.  RR  |->  ( w  +  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T ) ) ) `  z
) ) } ,  RR ,  <  ) )  =  ( x  e.  RR  |->  sup ( { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) } ,  RR ,  <  ) )
551, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 28, 33, 40, 44, 54fourierdlem107 38189 1  |-  ( ph  ->  S. ( ( A  -  X ) [,] ( B  -  X
) ) ( F `
 x )  _d x  =  S. ( A [,] B ) ( F `  x
)  _d x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760    u. cun 3388   ifcif 3872   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   ran crn 4840    |` cres 4841   iotacio 5551   -->wf 5585   ` cfv 5589    Isom wiso 5590  (class class class)co 6308    ^m cmap 7490   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   ZZcz 10961   RR+crp 11325   (,)cioo 11660   (,]cioc 11661   [,]cicc 11663   ...cfz 11810  ..^cfzo 11942   |_cfl 12059   #chash 12553   -cn->ccncf 21986   S.citg 22655   lim CC climc 22896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-itg2 22658  df-ibl 22659  df-itg 22660  df-0p 22707  df-ditg 22881  df-limc 22900  df-dv 22901
This theorem is referenced by:  fourierdlem109  38191
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