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Theorem fourierdlem109 38191
Description: The integral of a piecewise continuous periodic function  F is unchanged if the domain is shifted by any 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
fourierdlem109.a  |-  ( ph  ->  A  e.  RR )
fourierdlem109.b  |-  ( ph  ->  B  e.  RR )
fourierdlem109.t  |-  T  =  ( B  -  A
)
fourierdlem109.x  |-  ( ph  ->  X  e.  RR )
fourierdlem109.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 ) ) ) } )
fourierdlem109.m  |-  ( ph  ->  M  e.  NN )
fourierdlem109.q  |-  ( ph  ->  Q  e.  ( P `
 M ) )
fourierdlem109.f  |-  ( ph  ->  F : RR --> CC )
fourierdlem109.fper  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
fourierdlem109.fcn  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
fourierdlem109.r  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 i ) ) )
fourierdlem109.l  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 ( i  +  1 ) ) ) )
fourierdlem109.o  |-  O  =  ( 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 ) ) ) } )
fourierdlem109.h  |-  H  =  ( { ( A  -  X ) ,  ( B  -  X
) }  u.  {
x  e.  ( ( A  -  X ) [,] ( B  -  X ) )  |  E. k  e.  ZZ  ( x  +  (
k  x.  T ) )  e.  ran  Q } )
fourierdlem109.n  |-  N  =  ( ( # `  H
)  -  1 )
fourierdlem109.16  |-  S  =  ( iota f f 
Isom  <  ,  <  (
( 0 ... N
) ,  H ) )
fourierdlem109.17  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( B  -  x
)  /  T ) )  x.  T ) ) )
fourierdlem109.18  |-  J  =  ( y  e.  ( A (,] B ) 
|->  if ( y  =  B ,  A , 
y ) )
fourierdlem109.19  |-  I  =  ( x  e.  RR  |->  sup ( { j  e.  ( 0..^ M )  |  ( Q `  j )  <_  ( J `  ( E `  x ) ) } ,  RR ,  <  ) )
Assertion
Ref Expression
fourierdlem109  |-  ( ph  ->  S. ( ( A  -  X ) [,] ( B  -  X
) ) ( F `
 x )  _d x  =  S. ( A [,] B ) ( F `  x
)  _d x )
Distinct variable groups:    A, f,
j, k, y    A, i, x, j, k, y    A, m, p, i, j    B, f, j, k, y    B, i, x    B, m, p    f, E, j, k, y    i, E, x    i, F, j, x, y    f, H, y    x, H    f, I, k, y    i, I, x    i, J, j, x, y    x, L, y    i, M, x, y, j    m, M, p    f, N, j, k, y    i, N, x    m, N, p    Q, f, j, k, y    Q, i, x    Q, m, p    x, R, y    S, f, j, k, y    S, i, x    S, m, p    T, f, j, k, y    T, i, x    T, m, p    f, X, j, y    i, X, m, p    x, X    ph, f,
j, k, y    ph, i, x
Allowed substitution hints:    ph( m, p)    P( x, y, f, i, j, k, m, p)    R( f, i, j, k, m, p)    E( m, p)    F( f, k, m, p)    H( i, j, k, m, p)    I( j, m, p)    J( f, k, m, p)    L( f,
i, j, k, m, p)    M( f, k)    O( x, y, f, i, j, k, m, p)    X( k)

Proof of Theorem fourierdlem109
StepHypRef Expression
1 fourierdlem109.a . . . 4  |-  ( ph  ->  A  e.  RR )
21adantr 472 . . 3  |-  ( (
ph  /\  0  <  X )  ->  A  e.  RR )
3 fourierdlem109.b . . . 4  |-  ( ph  ->  B  e.  RR )
43adantr 472 . . 3  |-  ( (
ph  /\  0  <  X )  ->  B  e.  RR )
5 fourierdlem109.t . . 3  |-  T  =  ( B  -  A
)
6 fourierdlem109.x . . . . 5  |-  ( ph  ->  X  e.  RR )
76adantr 472 . . . 4  |-  ( (
ph  /\  0  <  X )  ->  X  e.  RR )
8 simpr 468 . . . 4  |-  ( (
ph  /\  0  <  X )  ->  0  <  X )
97, 8elrpd 11361 . . 3  |-  ( (
ph  /\  0  <  X )  ->  X  e.  RR+ )
10 fourierdlem109.p . . 3  |-  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 ) ) ) } )
11 fourierdlem109.m . . . 4  |-  ( ph  ->  M  e.  NN )
1211adantr 472 . . 3  |-  ( (
ph  /\  0  <  X )  ->  M  e.  NN )
13 fourierdlem109.q . . . 4  |-  ( ph  ->  Q  e.  ( P `
 M ) )
1413adantr 472 . . 3  |-  ( (
ph  /\  0  <  X )  ->  Q  e.  ( P `  M ) )
15 fourierdlem109.f . . . 4  |-  ( ph  ->  F : RR --> CC )
1615adantr 472 . . 3  |-  ( (
ph  /\  0  <  X )  ->  F : RR
--> CC )
17 fourierdlem109.fper . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
1817adantlr 729 . . 3  |-  ( ( ( ph  /\  0  <  X )  /\  x  e.  RR )  ->  ( F `  ( x  +  T ) )  =  ( F `  x
) )
19 fourierdlem109.fcn . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
2019adantlr 729 . . 3  |-  ( ( ( ph  /\  0  <  X )  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
21 fourierdlem109.r . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 i ) ) )
2221adantlr 729 . . 3  |-  ( ( ( ph  /\  0  <  X )  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
) )
23 fourierdlem109.l . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 ( i  +  1 ) ) ) )
2423adantlr 729 . . 3  |-  ( ( ( ph  /\  0  <  X )  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) ) )
252, 4, 5, 9, 10, 12, 14, 16, 18, 20, 22, 24fourierdlem108 38190 . 2  |-  ( (
ph  /\  0  <  X )  ->  S. (
( A  -  X
) [,] ( B  -  X ) ) ( F `  x
)  _d x  =  S. ( A [,] B ) ( F `
 x )  _d x )
26 oveq2 6316 . . . . . . 7  |-  ( X  =  0  ->  ( A  -  X )  =  ( A  - 
0 ) )
271recnd 9687 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
2827subid1d 9994 . . . . . . 7  |-  ( ph  ->  ( A  -  0 )  =  A )
2926, 28sylan9eqr 2527 . . . . . 6  |-  ( (
ph  /\  X  = 
0 )  ->  ( A  -  X )  =  A )
30 oveq2 6316 . . . . . . 7  |-  ( X  =  0  ->  ( B  -  X )  =  ( B  - 
0 ) )
313recnd 9687 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
3231subid1d 9994 . . . . . . 7  |-  ( ph  ->  ( B  -  0 )  =  B )
3330, 32sylan9eqr 2527 . . . . . 6  |-  ( (
ph  /\  X  = 
0 )  ->  ( B  -  X )  =  B )
3429, 33oveq12d 6326 . . . . 5  |-  ( (
ph  /\  X  = 
0 )  ->  (
( A  -  X
) [,] ( B  -  X ) )  =  ( A [,] B ) )
3534itgeq1d 37930 . . . 4  |-  ( (
ph  /\  X  = 
0 )  ->  S. ( ( A  -  X ) [,] ( B  -  X )
) ( F `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x )
3635adantlr 729 . . 3  |-  ( ( ( ph  /\  -.  0  <  X )  /\  X  =  0 )  ->  S. ( ( A  -  X ) [,] ( B  -  X ) ) ( F `  x )  _d x  =  S. ( A [,] B
) ( F `  x )  _d x )
37 simpll 768 . . . 4  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  ph )
3837, 6syl 17 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  X  e.  RR )
39 0red 9662 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  0  e.  RR )
40 simpr 468 . . . . . 6  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  -.  X  =  0 )
4140neqned 2650 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  X  =/=  0 )
42 simplr 770 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  -.  0  <  X )
4338, 39, 41, 42lttri5d 37605 . . . 4  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  X  <  0 )
446recnd 9687 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  CC )
4527, 44subcld 10005 . . . . . . . . . . 11  |-  ( ph  ->  ( A  -  X
)  e.  CC )
4645, 44subnegd 10012 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  -  X )  -  -u X
)  =  ( ( A  -  X )  +  X ) )
4727, 44npcand 10009 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  -  X )  +  X
)  =  A )
4846, 47eqtrd 2505 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  X )  -  -u X
)  =  A )
4931, 44subcld 10005 . . . . . . . . . . 11  |-  ( ph  ->  ( B  -  X
)  e.  CC )
5049, 44subnegd 10012 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  -  X )  -  -u X
)  =  ( ( B  -  X )  +  X ) )
5131, 44npcand 10009 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  -  X )  +  X
)  =  B )
5250, 51eqtrd 2505 . . . . . . . . 9  |-  ( ph  ->  ( ( B  -  X )  -  -u X
)  =  B )
5348, 52oveq12d 6326 . . . . . . . 8  |-  ( ph  ->  ( ( ( A  -  X )  -  -u X ) [,] (
( B  -  X
)  -  -u X
) )  =  ( A [,] B ) )
5453eqcomd 2477 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  =  ( ( ( A  -  X
)  -  -u X
) [,] ( ( B  -  X )  -  -u X ) ) )
5554itgeq1d 37930 . . . . . 6  |-  ( ph  ->  S. ( A [,] B ) ( F `
 x )  _d x  =  S. ( ( ( A  -  X )  -  -u X
) [,] ( ( B  -  X )  -  -u X ) ) ( F `  x
)  _d x )
5655adantr 472 . . . . 5  |-  ( (
ph  /\  X  <  0 )  ->  S. ( A [,] B ) ( F `  x
)  _d x  =  S. ( ( ( A  -  X )  -  -u X ) [,] ( ( B  -  X )  -  -u X
) ) ( F `
 x )  _d x )
571, 6resubcld 10068 . . . . . . 7  |-  ( ph  ->  ( A  -  X
)  e.  RR )
5857adantr 472 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  ( A  -  X )  e.  RR )
593, 6resubcld 10068 . . . . . . 7  |-  ( ph  ->  ( B  -  X
)  e.  RR )
6059adantr 472 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  ( B  -  X )  e.  RR )
61 eqid 2471 . . . . . 6  |-  ( ( B  -  X )  -  ( A  -  X ) )  =  ( ( B  -  X )  -  ( A  -  X )
)
626renegcld 10067 . . . . . . . 8  |-  ( ph  -> 
-u X  e.  RR )
6362adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  <  0 )  ->  -u X  e.  RR )
646lt0neg1d 10204 . . . . . . . 8  |-  ( ph  ->  ( X  <  0  <->  0  <  -u X ) )
6564biimpa 492 . . . . . . 7  |-  ( (
ph  /\  X  <  0 )  ->  0  <  -u X )
6663, 65elrpd 11361 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  -u X  e.  RR+ )
67 fourierdlem109.o . . . . . . 7  |-  O  =  ( 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 ) ) ) } )
68 fveq2 5879 . . . . . . . . . . . . 13  |-  ( i  =  j  ->  (
p `  i )  =  ( p `  j ) )
69 oveq1 6315 . . . . . . . . . . . . . 14  |-  ( i  =  j  ->  (
i  +  1 )  =  ( j  +  1 ) )
7069fveq2d 5883 . . . . . . . . . . . . 13  |-  ( i  =  j  ->  (
p `  ( i  +  1 ) )  =  ( p `  ( j  +  1 ) ) )
7168, 70breq12d 4408 . . . . . . . . . . . 12  |-  ( i  =  j  ->  (
( p `  i
)  <  ( p `  ( i  +  1 ) )  <->  ( p `  j )  <  (
p `  ( j  +  1 ) ) ) )
7271cbvralv 3005 . . . . . . . . . . 11  |-  ( A. i  e.  ( 0..^ m ) ( p `
 i )  < 
( p `  (
i  +  1 ) )  <->  A. j  e.  ( 0..^ m ) ( p `  j )  <  ( p `  ( j  +  1 ) ) )
7372anbi2i 708 . . . . . . . . . 10  |-  ( ( ( ( p ` 
0 )  =  ( A  -  X )  /\  ( p `  m )  =  ( B  -  X ) )  /\  A. i  e.  ( 0..^ m ) ( p `  i
)  <  ( p `  ( i  +  1 ) ) )  <->  ( (
( p `  0
)  =  ( A  -  X )  /\  ( p `  m
)  =  ( B  -  X ) )  /\  A. j  e.  ( 0..^ m ) ( p `  j
)  <  ( p `  ( j  +  1 ) ) ) )
7473a1i 11 . . . . . . . . 9  |-  ( p  e.  ( RR  ^m  ( 0 ... m
) )  ->  (
( ( ( p `
 0 )  =  ( A  -  X
)  /\  ( p `  m )  =  ( B  -  X ) )  /\  A. i  e.  ( 0..^ m ) ( p `  i
)  <  ( p `  ( i  +  1 ) ) )  <->  ( (
( p `  0
)  =  ( A  -  X )  /\  ( p `  m
)  =  ( B  -  X ) )  /\  A. j  e.  ( 0..^ m ) ( p `  j
)  <  ( p `  ( j  +  1 ) ) ) ) )
7574rabbiia 3019 . . . . . . . 8  |-  { p  e.  ( RR  ^m  (
0 ... m ) )  |  ( ( ( p `  0 )  =  ( A  -  X )  /\  (
p `  m )  =  ( B  -  X ) )  /\  A. i  e.  ( 0..^ m ) ( p `
 i )  < 
( p `  (
i  +  1 ) ) ) }  =  { p  e.  ( RR  ^m  ( 0 ... m ) )  |  ( ( ( p `
 0 )  =  ( A  -  X
)  /\  ( p `  m )  =  ( B  -  X ) )  /\  A. j  e.  ( 0..^ m ) ( p `  j
)  <  ( p `  ( j  +  1 ) ) ) }
7675mpteq2i 4479 . . . . . . 7  |-  ( 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 ) ) ) } )  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  (
0 ... m ) )  |  ( ( ( p `  0 )  =  ( A  -  X )  /\  (
p `  m )  =  ( B  -  X ) )  /\  A. j  e.  ( 0..^ m ) ( p `
 j )  < 
( p `  (
j  +  1 ) ) ) } )
7767, 76eqtri 2493 . . . . . 6  |-  O  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  ( 0 ... m ) )  |  ( ( ( p `
 0 )  =  ( A  -  X
)  /\  ( p `  m )  =  ( B  -  X ) )  /\  A. j  e.  ( 0..^ m ) ( p `  j
)  <  ( p `  ( j  +  1 ) ) ) } )
7810, 11, 13fourierdlem11 38092 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) )
7978simp3d 1044 . . . . . . . . . . 11  |-  ( ph  ->  A  <  B )
801, 3, 6, 79ltsub1dd 10246 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  X
)  <  ( B  -  X ) )
81 fourierdlem109.h . . . . . . . . . 10  |-  H  =  ( { ( A  -  X ) ,  ( B  -  X
) }  u.  {
x  e.  ( ( A  -  X ) [,] ( B  -  X ) )  |  E. k  e.  ZZ  ( x  +  (
k  x.  T ) )  e.  ran  Q } )
82 fourierdlem109.n . . . . . . . . . 10  |-  N  =  ( ( # `  H
)  -  1 )
83 fourierdlem109.16 . . . . . . . . . 10  |-  S  =  ( iota f f 
Isom  <  ,  <  (
( 0 ... N
) ,  H ) )
845, 10, 11, 13, 57, 59, 80, 67, 81, 82, 83fourierdlem54 38136 . . . . . . . . 9  |-  ( ph  ->  ( ( N  e.  NN  /\  S  e.  ( O `  N
) )  /\  S  Isom  <  ,  <  (
( 0 ... N
) ,  H ) ) )
8584simpld 466 . . . . . . . 8  |-  ( ph  ->  ( N  e.  NN  /\  S  e.  ( O `
 N ) ) )
8685simpld 466 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
8786adantr 472 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  N  e.  NN )
8885simprd 470 . . . . . . 7  |-  ( ph  ->  S  e.  ( O `
 N ) )
8988adantr 472 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  S  e.  ( O `  N
) )
9015adantr 472 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  F : RR --> CC )
9131, 27, 44nnncan2d 10040 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( B  -  X )  -  ( A  -  X )
)  =  ( B  -  A ) )
9291, 5syl6eqr 2523 . . . . . . . . . . 11  |-  ( ph  ->  ( ( B  -  X )  -  ( A  -  X )
)  =  T )
9392oveq2d 6324 . . . . . . . . . 10  |-  ( ph  ->  ( x  +  ( ( B  -  X
)  -  ( A  -  X ) ) )  =  ( x  +  T ) )
9493adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR )  ->  ( x  +  ( ( B  -  X )  -  ( A  -  X
) ) )  =  ( x  +  T
) )
9594fveq2d 5883 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  ( ( B  -  X )  -  ( A  -  X )
) ) )  =  ( F `  (
x  +  T ) ) )
9695, 17eqtrd 2505 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  ( ( B  -  X )  -  ( A  -  X )
) ) )  =  ( F `  x
) )
9796adantlr 729 . . . . . 6  |-  ( ( ( ph  /\  X  <  0 )  /\  x  e.  RR )  ->  ( F `  ( x  +  ( ( B  -  X )  -  ( A  -  X
) ) ) )  =  ( F `  x ) )
9811adantr 472 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  M  e.  NN )
9913adantr 472 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  Q  e.  ( P `  M ) )
10015adantr 472 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  F : RR --> CC )
10117adantlr 729 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 0..^ N ) )  /\  x  e.  RR )  ->  ( F `  ( x  +  T ) )  =  ( F `  x
) )
10219adantlr 729 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 0..^ N ) )  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
10357adantr 472 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  ( A  -  X )  e.  RR )
10457rexrd 9708 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  X
)  e.  RR* )
105 pnfxr 11435 . . . . . . . . . . 11  |- +oo  e.  RR*
106105a1i 11 . . . . . . . . . 10  |-  ( ph  -> +oo  e.  RR* )
10759ltpnfd 11446 . . . . . . . . . 10  |-  ( ph  ->  ( B  -  X
)  < +oo )
108104, 106, 59, 80, 107eliood 37691 . . . . . . . . 9  |-  ( ph  ->  ( B  -  X
)  e.  ( ( A  -  X ) (,) +oo ) )
109108adantr 472 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  ( B  -  X )  e.  ( ( A  -  X
) (,) +oo )
)
110 oveq1 6315 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
x  +  ( k  x.  T ) )  =  ( y  +  ( k  x.  T
) ) )
111110eleq1d 2533 . . . . . . . . . . . 12  |-  ( x  =  y  ->  (
( x  +  ( k  x.  T ) )  e.  ran  Q  <->  ( y  +  ( k  x.  T ) )  e.  ran  Q ) )
112111rexbidv 2892 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( E. k  e.  ZZ  ( x  +  (
k  x.  T ) )  e.  ran  Q  <->  E. k  e.  ZZ  (
y  +  ( k  x.  T ) )  e.  ran  Q ) )
113112cbvrabv 3030 . . . . . . . . . 10  |-  { x  e.  ( ( A  -  X ) [,] ( B  -  X )
)  |  E. k  e.  ZZ  ( x  +  ( k  x.  T
) )  e.  ran  Q }  =  { y  e.  ( ( A  -  X ) [,] ( B  -  X
) )  |  E. k  e.  ZZ  (
y  +  ( k  x.  T ) )  e.  ran  Q }
114113uneq2i 3576 . . . . . . . . 9  |-  ( { ( A  -  X
) ,  ( B  -  X ) }  u.  { x  e.  ( ( A  -  X ) [,] ( B  -  X )
)  |  E. k  e.  ZZ  ( x  +  ( k  x.  T
) )  e.  ran  Q } )  =  ( { ( A  -  X ) ,  ( B  -  X ) }  u.  { y  e.  ( ( A  -  X ) [,] ( B  -  X
) )  |  E. k  e.  ZZ  (
y  +  ( k  x.  T ) )  e.  ran  Q }
)
11581, 114eqtri 2493 . . . . . . . 8  |-  H  =  ( { ( A  -  X ) ,  ( B  -  X
) }  u.  {
y  e.  ( ( A  -  X ) [,] ( B  -  X ) )  |  E. k  e.  ZZ  ( y  +  ( k  x.  T ) )  e.  ran  Q } )
116 fourierdlem109.17 . . . . . . . 8  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( B  -  x
)  /  T ) )  x.  T ) ) )
117 fourierdlem109.18 . . . . . . . 8  |-  J  =  ( y  e.  ( A (,] B ) 
|->  if ( y  =  B ,  A , 
y ) )
118 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  j  e.  ( 0..^ N ) )
119 eqid 2471 . . . . . . . 8  |-  ( ( S `  ( j  +  1 ) )  -  ( E `  ( S `  ( j  +  1 ) ) ) )  =  ( ( S `  (
j  +  1 ) )  -  ( E `
 ( S `  ( j  +  1 ) ) ) )
120 eqid 2471 . . . . . . . 8  |-  ( F  |`  ( ( J `  ( E `  ( S `
 j ) ) ) (,) ( E `
 ( S `  ( j  +  1 ) ) ) ) )  =  ( F  |`  ( ( J `  ( E `  ( S `
 j ) ) ) (,) ( E `
 ( S `  ( j  +  1 ) ) ) ) )
121 eqid 2471 . . . . . . . 8  |-  ( y  e.  ( ( ( J `  ( E `
 ( S `  j ) ) )  +  ( ( S `
 ( j  +  1 ) )  -  ( E `  ( S `
 ( j  +  1 ) ) ) ) ) (,) (
( E `  ( S `  ( j  +  1 ) ) )  +  ( ( S `  ( j  +  1 ) )  -  ( E `  ( S `  ( j  +  1 ) ) ) ) ) ) 
|->  ( ( F  |`  ( ( J `  ( E `  ( S `
 j ) ) ) (,) ( E `
 ( S `  ( j  +  1 ) ) ) ) ) `  ( y  -  ( ( S `
 ( j  +  1 ) )  -  ( E `  ( S `
 ( j  +  1 ) ) ) ) ) ) )  =  ( y  e.  ( ( ( J `
 ( E `  ( S `  j ) ) )  +  ( ( S `  (
j  +  1 ) )  -  ( E `
 ( S `  ( j  +  1 ) ) ) ) ) (,) ( ( E `  ( S `
 ( j  +  1 ) ) )  +  ( ( S `
 ( j  +  1 ) )  -  ( E `  ( S `
 ( j  +  1 ) ) ) ) ) )  |->  ( ( F  |`  (
( J `  ( E `  ( S `  j ) ) ) (,) ( E `  ( S `  ( j  +  1 ) ) ) ) ) `  ( y  -  (
( S `  (
j  +  1 ) )  -  ( E `
 ( S `  ( j  +  1 ) ) ) ) ) ) )
122 fourierdlem109.19 . . . . . . . . 9  |-  I  =  ( x  e.  RR  |->  sup ( { j  e.  ( 0..^ M )  |  ( Q `  j )  <_  ( J `  ( E `  x ) ) } ,  RR ,  <  ) )
123 fveq2 5879 . . . . . . . . . . . . 13  |-  ( j  =  i  ->  ( Q `  j )  =  ( Q `  i ) )
124123breq1d 4405 . . . . . . . . . . . 12  |-  ( j  =  i  ->  (
( Q `  j
)  <_  ( J `  ( E `  x
) )  <->  ( Q `  i )  <_  ( J `  ( E `  x ) ) ) )
125124cbvrabv 3030 . . . . . . . . . . 11  |-  { j  e.  ( 0..^ M )  |  ( Q `
 j )  <_ 
( J `  ( E `  x )
) }  =  {
i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( J `  ( E `  x ) ) }
126125supeq1i 7979 . . . . . . . . . 10  |-  sup ( { j  e.  ( 0..^ M )  |  ( Q `  j
)  <_  ( J `  ( E `  x
) ) } ,  RR ,  <  )  =  sup ( { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( J `  ( E `  x )
) } ,  RR ,  <  )
127126mpteq2i 4479 . . . . . . . . 9  |-  ( x  e.  RR  |->  sup ( { j  e.  ( 0..^ M )  |  ( Q `  j
)  <_  ( J `  ( E `  x
) ) } ,  RR ,  <  ) )  =  ( x  e.  RR  |->  sup ( { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( J `  ( E `  x )
) } ,  RR ,  <  ) )
128122, 127eqtri 2493 . . . . . . . 8  |-  I  =  ( x  e.  RR  |->  sup ( { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( J `  ( E `  x ) ) } ,  RR ,  <  ) )
12910, 5, 98, 99, 100, 101, 102, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 120, 121, 128fourierdlem90 38172 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  ( F  |`  ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) )  e.  ( ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) -cn-> CC ) )
130129adantlr 729 . . . . . 6  |-  ( ( ( ph  /\  X  <  0 )  /\  j  e.  ( 0..^ N ) )  ->  ( F  |`  ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) )  e.  ( ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) -cn-> CC ) )
13121adantlr 729 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 0..^ N ) )  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
) )
132 eqid 2471 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  |->  R )  =  ( i  e.  ( 0..^ M )  |->  R )
13310, 5, 98, 99, 100, 101, 102, 131, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 128, 132fourierdlem89 38171 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  if ( ( J `  ( E `
 ( S `  j ) ) )  =  ( Q `  ( I `  ( S `  j )
) ) ,  ( ( i  e.  ( 0..^ M )  |->  R ) `  ( I `
 ( S `  j ) ) ) ,  ( F `  ( J `  ( E `
 ( S `  j ) ) ) ) )  e.  ( ( F  |`  (
( S `  j
) (,) ( S `
 ( j  +  1 ) ) ) ) lim CC  ( S `
 j ) ) )
134133adantlr 729 . . . . . 6  |-  ( ( ( ph  /\  X  <  0 )  /\  j  e.  ( 0..^ N ) )  ->  if (
( J `  ( E `  ( S `  j ) ) )  =  ( Q `  ( I `  ( S `  j )
) ) ,  ( ( i  e.  ( 0..^ M )  |->  R ) `  ( I `
 ( S `  j ) ) ) ,  ( F `  ( J `  ( E `
 ( S `  j ) ) ) ) )  e.  ( ( F  |`  (
( S `  j
) (,) ( S `
 ( j  +  1 ) ) ) ) lim CC  ( S `
 j ) ) )
13523adantlr 729 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 0..^ N ) )  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) ) )
136 eqid 2471 . . . . . . . 8  |-  ( i  e.  ( 0..^ M )  |->  L )  =  ( i  e.  ( 0..^ M )  |->  L )
13710, 5, 98, 99, 100, 101, 102, 135, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 128, 136fourierdlem91 38173 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  if ( ( E `  ( S `
 ( j  +  1 ) ) )  =  ( Q `  ( ( I `  ( S `  j ) )  +  1 ) ) ,  ( ( i  e.  ( 0..^ M )  |->  L ) `
 ( I `  ( S `  j ) ) ) ,  ( F `  ( E `
 ( S `  ( j  +  1 ) ) ) ) )  e.  ( ( F  |`  ( ( S `  j ) (,) ( S `  (
j  +  1 ) ) ) ) lim CC  ( S `  ( j  +  1 ) ) ) )
138137adantlr 729 . . . . . 6  |-  ( ( ( ph  /\  X  <  0 )  /\  j  e.  ( 0..^ N ) )  ->  if (
( E `  ( S `  ( j  +  1 ) ) )  =  ( Q `
 ( ( I `
 ( S `  j ) )  +  1 ) ) ,  ( ( i  e.  ( 0..^ M ) 
|->  L ) `  (
I `  ( S `  j ) ) ) ,  ( F `  ( E `  ( S `
 ( j  +  1 ) ) ) ) )  e.  ( ( F  |`  (
( S `  j
) (,) ( S `
 ( j  +  1 ) ) ) ) lim CC  ( S `
 ( j  +  1 ) ) ) )
13958, 60, 61, 66, 77, 87, 89, 90, 97, 130, 134, 138fourierdlem108 38190 . . . . 5  |-  ( (
ph  /\  X  <  0 )  ->  S. ( ( ( A  -  X )  -  -u X ) [,] (
( B  -  X
)  -  -u X
) ) ( F `
 x )  _d x  =  S. ( ( A  -  X
) [,] ( B  -  X ) ) ( F `  x
)  _d x )
14056, 139eqtr2d 2506 . . . 4  |-  ( (
ph  /\  X  <  0 )  ->  S. ( ( A  -  X ) [,] ( B  -  X )
) ( F `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x )
14137, 43, 140syl2anc 673 . . 3  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  S. (
( A  -  X
) [,] ( B  -  X ) ) ( F `  x
)  _d x  =  S. ( A [,] B ) ( F `
 x )  _d x )
14236, 141pm2.61dan 808 . 2  |-  ( (
ph  /\  -.  0  <  X )  ->  S. ( ( A  -  X ) [,] ( B  -  X )
) ( F `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x )
14325, 142pm2.61dan 808 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:   -. wn 3    -> 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   +oocpnf 9690   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   -ucneg 9881    / cdiv 10291   NNcn 10631   ZZcz 10961   (,)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:  fourierdlem110  38192
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