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Theorem fourierdlem108 32200
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 32184 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 2457 . 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 6303 . . . . . 6  |-  ( w  =  y  ->  (
w  +  ( k  x.  T ) )  =  ( y  +  ( k  x.  T
) ) )
1514eleq1d 2526 . . . . 5  |-  ( w  =  y  ->  (
( w  +  ( k  x.  T ) )  e.  ran  Q  <->  ( y  +  ( k  x.  T ) )  e.  ran  Q ) )
1615rexbidv 2968 . . . 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 3108 . . 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 3651 . 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 6303 . . . . . . . . . 10  |-  ( l  =  k  ->  (
l  x.  T )  =  ( k  x.  T ) )
2019oveq2d 6312 . . . . . . . . 9  |-  ( l  =  k  ->  (
w  +  ( l  x.  T ) )  =  ( w  +  ( k  x.  T
) ) )
2120eleq1d 2526 . . . . . . . 8  |-  ( l  =  k  ->  (
( w  +  ( l  x.  T ) )  e.  ran  Q  <->  ( w  +  ( k  x.  T ) )  e.  ran  Q ) )
2221cbvrexv 3085 . . . . . . 7  |-  ( E. l  e.  ZZ  (
w  +  ( l  x.  T ) )  e.  ran  Q  <->  E. k  e.  ZZ  ( w  +  ( k  x.  T
) )  e.  ran  Q )
2322rgenw 2818 . . . . . 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 3036 . . . . . 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 208 . . . . 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 3651 . . . 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 5875 . . 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 6306 . 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 6220 . . . . 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 6216 . . . 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 257 . . 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 5563 . 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 6304 . . . . . . 7  |-  ( w  =  x  ->  ( B  -  w )  =  ( B  -  x ) )
3635oveq1d 6311 . . . . . 6  |-  ( w  =  x  ->  (
( B  -  w
)  /  T )  =  ( ( B  -  x )  /  T ) )
3736fveq2d 5876 . . . . 5  |-  ( w  =  x  ->  ( |_ `  ( ( B  -  w )  /  T ) )  =  ( |_ `  (
( B  -  x
)  /  T ) ) )
3837oveq1d 6311 . . . 4  |-  ( w  =  x  ->  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T )  =  ( ( |_
`  ( ( B  -  x )  /  T ) )  x.  T ) )
3934, 38oveq12d 6314 . . 3  |-  ( w  =  x  ->  (
w  +  ( ( |_ `  ( ( B  -  w )  /  T ) )  x.  T ) )  =  ( x  +  ( ( |_ `  ( ( B  -  x )  /  T
) )  x.  T
) ) )
4039cbvmptv 4548 . 2  |-  ( w  e.  RR  |->  ( w  +  ( ( |_
`  ( ( B  -  w )  /  T ) )  x.  T ) ) )  =  ( x  e.  RR  |->  ( x  +  ( ( |_ `  ( ( B  -  x )  /  T
) )  x.  T
) ) )
41 eqeq1 2461 . . . 4  |-  ( w  =  y  ->  (
w  =  B  <->  y  =  B ) )
42 id 22 . . . 4  |-  ( w  =  y  ->  w  =  y )
4341, 42ifbieq2d 3969 . . 3  |-  ( w  =  y  ->  if ( w  =  B ,  A ,  w )  =  if ( y  =  B ,  A ,  y ) )
4443cbvmptv 4548 . 2  |-  ( w  e.  ( A (,] B )  |->  if ( w  =  B ,  A ,  w )
)  =  ( y  e.  ( A (,] B )  |->  if ( y  =  B ,  A ,  y )
)
45 fveq2 5872 . . . . . . . 8  |-  ( z  =  x  ->  (
( w  e.  RR  |->  ( w  +  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T ) ) ) `  z
)  =  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T ) )  x.  T ) ) ) `  x ) )
4645fveq2d 5876 . . . . . . 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 4468 . . . . . 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 3101 . . . . 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 5872 . . . . . . 7  |-  ( j  =  i  ->  ( Q `  j )  =  ( Q `  i ) )
5049breq1d 4466 . . . . . 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 3108 . . . . 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 2514 . . . 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 7923 . . 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 4548 . 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 32199 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 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811    u. cun 3469   ifcif 3944   {cpr 4034   class class class wbr 4456    |-> cmpt 4515   ran crn 5009    |` cres 5010   iotacio 5555   -->wf 5590   ` cfv 5594    Isom wiso 5595  (class class class)co 6296    ^m cmap 7438   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   NNcn 10556   ZZcz 10885   RR+crp 11245   (,)cioo 11554   (,]cioc 11555   [,]cicc 11557   ...cfz 11697  ..^cfzo 11821   |_cfl 11930   #chash 12408   -cn->ccncf 21506   S.citg 22153   lim CC climc 22392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-cmp 20014  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-ovol 22002  df-vol 22003  df-mbf 22154  df-itg1 22155  df-itg2 22156  df-ibl 22157  df-itg 22158  df-0p 22203  df-ditg 22377  df-limc 22396  df-dv 22397
This theorem is referenced by:  fourierdlem109  32201
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