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Theorem fourierdlem109 31952
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 31935 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 465 . . 3  |-  ( (
ph  /\  0  <  X )  ->  A  e.  RR )
3 fourierdlem109.b . . . 4  |-  ( ph  ->  B  e.  RR )
43adantr 465 . . 3  |-  ( (
ph  /\  0  <  X )  ->  B  e.  RR )
5 fourierdlem109.t . . 3  |-  T  =  ( B  -  A
)
6 fourierdlem109.x . . . . 5  |-  ( ph  ->  X  e.  RR )
76adantr 465 . . . 4  |-  ( (
ph  /\  0  <  X )  ->  X  e.  RR )
8 simpr 461 . . . 4  |-  ( (
ph  /\  0  <  X )  ->  0  <  X )
97, 8elrpd 11265 . . 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 465 . . 3  |-  ( (
ph  /\  0  <  X )  ->  M  e.  NN )
13 fourierdlem109.q . . . 4  |-  ( ph  ->  Q  e.  ( P `
 M ) )
1413adantr 465 . . 3  |-  ( (
ph  /\  0  <  X )  ->  Q  e.  ( P `  M ) )
15 fourierdlem109.f . . . 4  |-  ( ph  ->  F : RR --> CC )
1615adantr 465 . . 3  |-  ( (
ph  /\  0  <  X )  ->  F : RR
--> CC )
17 fourierdlem109.fper . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
1817adantlr 714 . . 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 714 . . 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 714 . . 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 714 . . 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 31951 . 2  |-  ( (
ph  /\  0  <  X )  ->  S. (
( A  -  X
) [,] ( B  -  X ) ) ( F `  x
)  _d x  =  S. ( A [,] B ) ( F `
 x )  _d x )
26 oveq2 6289 . . . . . . 7  |-  ( X  =  0  ->  ( A  -  X )  =  ( A  - 
0 ) )
271recnd 9625 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
2827subid1d 9925 . . . . . . 7  |-  ( ph  ->  ( A  -  0 )  =  A )
2926, 28sylan9eqr 2506 . . . . . 6  |-  ( (
ph  /\  X  = 
0 )  ->  ( A  -  X )  =  A )
30 oveq2 6289 . . . . . . 7  |-  ( X  =  0  ->  ( B  -  X )  =  ( B  - 
0 ) )
313recnd 9625 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
3231subid1d 9925 . . . . . . 7  |-  ( ph  ->  ( B  -  0 )  =  B )
3330, 32sylan9eqr 2506 . . . . . 6  |-  ( (
ph  /\  X  = 
0 )  ->  ( B  -  X )  =  B )
3429, 33oveq12d 6299 . . . . 5  |-  ( (
ph  /\  X  = 
0 )  ->  (
( A  -  X
) [,] ( B  -  X ) )  =  ( A [,] B ) )
3534itgeq1d 31709 . . . 4  |-  ( (
ph  /\  X  = 
0 )  ->  S. ( ( A  -  X ) [,] ( B  -  X )
) ( F `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x )
3635adantlr 714 . . 3  |-  ( ( ( ph  /\  -.  0  <  X )  /\  X  =  0 )  ->  S. ( ( A  -  X ) [,] ( B  -  X ) ) ( F `  x )  _d x  =  S. ( A [,] B
) ( F `  x )  _d x )
37 simpll 753 . . . 4  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  ph )
3837, 6syl 16 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  X  e.  RR )
39 0red 9600 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  0  e.  RR )
40 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  -.  X  =  0 )
4140neqned 2646 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  X  =/=  0 )
42 simplr 755 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  -.  0  <  X )
4338, 39, 41, 42lttri5d 31453 . . . 4  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  X  <  0 )
446recnd 9625 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  CC )
4527, 44subcld 9936 . . . . . . . . . . 11  |-  ( ph  ->  ( A  -  X
)  e.  CC )
4645, 44subnegd 9943 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  -  X )  -  -u X
)  =  ( ( A  -  X )  +  X ) )
4727, 44npcand 9940 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  -  X )  +  X
)  =  A )
4846, 47eqtrd 2484 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  X )  -  -u X
)  =  A )
4931, 44subcld 9936 . . . . . . . . . . 11  |-  ( ph  ->  ( B  -  X
)  e.  CC )
5049, 44subnegd 9943 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  -  X )  -  -u X
)  =  ( ( B  -  X )  +  X ) )
5131, 44npcand 9940 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  -  X )  +  X
)  =  B )
5250, 51eqtrd 2484 . . . . . . . . 9  |-  ( ph  ->  ( ( B  -  X )  -  -u X
)  =  B )
5348, 52oveq12d 6299 . . . . . . . 8  |-  ( ph  ->  ( ( ( A  -  X )  -  -u X ) [,] (
( B  -  X
)  -  -u X
) )  =  ( A [,] B ) )
5453eqcomd 2451 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  =  ( ( ( A  -  X
)  -  -u X
) [,] ( ( B  -  X )  -  -u X ) ) )
5554itgeq1d 31709 . . . . . 6  |-  ( ph  ->  S. ( A [,] B ) ( F `
 x )  _d x  =  S. ( ( ( A  -  X )  -  -u X
) [,] ( ( B  -  X )  -  -u X ) ) ( F `  x
)  _d x )
5655adantr 465 . . . . 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 9994 . . . . . . 7  |-  ( ph  ->  ( A  -  X
)  e.  RR )
5857adantr 465 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  ( A  -  X )  e.  RR )
593, 6resubcld 9994 . . . . . . 7  |-  ( ph  ->  ( B  -  X
)  e.  RR )
6059adantr 465 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  ( B  -  X )  e.  RR )
61 eqid 2443 . . . . . 6  |-  ( ( B  -  X )  -  ( A  -  X ) )  =  ( ( B  -  X )  -  ( A  -  X )
)
626renegcld 9993 . . . . . . . 8  |-  ( ph  -> 
-u X  e.  RR )
6362adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  <  0 )  ->  -u X  e.  RR )
646lt0neg1d 10129 . . . . . . . 8  |-  ( ph  ->  ( X  <  0  <->  0  <  -u X ) )
6564biimpa 484 . . . . . . 7  |-  ( (
ph  /\  X  <  0 )  ->  0  <  -u X )
6663, 65elrpd 11265 . . . . . 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 5856 . . . . . . . . . . . . 13  |-  ( i  =  j  ->  (
p `  i )  =  ( p `  j ) )
69 oveq1 6288 . . . . . . . . . . . . . 14  |-  ( i  =  j  ->  (
i  +  1 )  =  ( j  +  1 ) )
7069fveq2d 5860 . . . . . . . . . . . . 13  |-  ( i  =  j  ->  (
p `  ( i  +  1 ) )  =  ( p `  ( j  +  1 ) ) )
7168, 70breq12d 4450 . . . . . . . . . . . 12  |-  ( i  =  j  ->  (
( p `  i
)  <  ( p `  ( i  +  1 ) )  <->  ( p `  j )  <  (
p `  ( j  +  1 ) ) ) )
7271cbvralv 3070 . . . . . . . . . . 11  |-  ( A. i  e.  ( 0..^ m ) ( p `
 i )  < 
( p `  (
i  +  1 ) )  <->  A. j  e.  ( 0..^ m ) ( p `  j )  <  ( p `  ( j  +  1 ) ) )
7372anbi2i 694 . . . . . . . . . 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 3084 . . . . . . . 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 4520 . . . . . . 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 2472 . . . . . 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 31854 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) )
7978simp3d 1011 . . . . . . . . . . 11  |-  ( ph  ->  A  <  B )
801, 3, 6, 79ltsub1dd 10171 . . . . . . . . . 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 31897 . . . . . . . . 9  |-  ( ph  ->  ( ( N  e.  NN  /\  S  e.  ( O `  N
) )  /\  S  Isom  <  ,  <  (
( 0 ... N
) ,  H ) ) )
8584simpld 459 . . . . . . . 8  |-  ( ph  ->  ( N  e.  NN  /\  S  e.  ( O `
 N ) ) )
8685simpld 459 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
8786adantr 465 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  N  e.  NN )
8885simprd 463 . . . . . . 7  |-  ( ph  ->  S  e.  ( O `
 N ) )
8988adantr 465 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  S  e.  ( O `  N
) )
9015adantr 465 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  F : RR --> CC )
9131, 27, 44nnncan2d 9971 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( B  -  X )  -  ( A  -  X )
)  =  ( B  -  A ) )
9291, 5syl6eqr 2502 . . . . . . . . . . 11  |-  ( ph  ->  ( ( B  -  X )  -  ( A  -  X )
)  =  T )
9392oveq2d 6297 . . . . . . . . . 10  |-  ( ph  ->  ( x  +  ( ( B  -  X
)  -  ( A  -  X ) ) )  =  ( x  +  T ) )
9493adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR )  ->  ( x  +  ( ( B  -  X )  -  ( A  -  X
) ) )  =  ( x  +  T
) )
9594fveq2d 5860 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  ( ( B  -  X )  -  ( A  -  X )
) ) )  =  ( F `  (
x  +  T ) ) )
9695, 17eqtrd 2484 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  ( ( B  -  X )  -  ( A  -  X )
) ) )  =  ( F `  x
) )
9796adantlr 714 . . . . . 6  |-  ( ( ( ph  /\  X  <  0 )  /\  x  e.  RR )  ->  ( F `  ( x  +  ( ( B  -  X )  -  ( A  -  X
) ) ) )  =  ( F `  x ) )
9811adantr 465 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  M  e.  NN )
9913adantr 465 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  Q  e.  ( P `  M ) )
10015adantr 465 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  F : RR --> CC )
10117adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 0..^ N ) )  /\  x  e.  RR )  ->  ( F `  ( x  +  T ) )  =  ( F `  x
) )
10219adantlr 714 . . . . . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  ( A  -  X )  e.  RR )
10457rexrd 9646 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  X
)  e.  RR* )
105 pnfxr 11332 . . . . . . . . . . 11  |- +oo  e.  RR*
106105a1i 11 . . . . . . . . . 10  |-  ( ph  -> +oo  e.  RR* )
10759ltpnfd 31434 . . . . . . . . . 10  |-  ( ph  ->  ( B  -  X
)  < +oo )
108104, 106, 59, 80, 107eliood 31485 . . . . . . . . 9  |-  ( ph  ->  ( B  -  X
)  e.  ( ( A  -  X ) (,) +oo ) )
109108adantr 465 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  ( B  -  X )  e.  ( ( A  -  X
) (,) +oo )
)
110 oveq1 6288 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
x  +  ( k  x.  T ) )  =  ( y  +  ( k  x.  T
) ) )
111110eleq1d 2512 . . . . . . . . . . . 12  |-  ( x  =  y  ->  (
( x  +  ( k  x.  T ) )  e.  ran  Q  <->  ( y  +  ( k  x.  T ) )  e.  ran  Q ) )
112111rexbidv 2954 . . . . . . . . . . 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 3094 . . . . . . . . . 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 3640 . . . . . . . . 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 2472 . . . . . . . 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 461 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  j  e.  ( 0..^ N ) )
119 eqid 2443 . . . . . . . 8  |-  ( ( S `  ( j  +  1 ) )  -  ( E `  ( S `  ( j  +  1 ) ) ) )  =  ( ( S `  (
j  +  1 ) )  -  ( E `
 ( S `  ( j  +  1 ) ) ) )
120 eqid 2443 . . . . . . . 8  |-  ( F  |`  ( ( J `  ( E `  ( S `
 j ) ) ) (,) ( E `
 ( S `  ( j  +  1 ) ) ) ) )  =  ( F  |`  ( ( J `  ( E `  ( S `
 j ) ) ) (,) ( E `
 ( S `  ( j  +  1 ) ) ) ) )
121 eqid 2443 . . . . . . . 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 5856 . . . . . . . . . . . . 13  |-  ( j  =  i  ->  ( Q `  j )  =  ( Q `  i ) )
124123breq1d 4447 . . . . . . . . . . . 12  |-  ( j  =  i  ->  (
( Q `  j
)  <_  ( J `  ( E `  x
) )  <->  ( Q `  i )  <_  ( J `  ( E `  x ) ) ) )
125124cbvrabv 3094 . . . . . . . . . . 11  |-  { j  e.  ( 0..^ M )  |  ( Q `
 j )  <_ 
( J `  ( E `  x )
) }  =  {
i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( J `  ( E `  x ) ) }
126125supeq1i 7909 . . . . . . . . . 10  |-  sup ( { j  e.  ( 0..^ M )  |  ( Q `  j
)  <_  ( J `  ( E `  x
) ) } ,  RR ,  <  )  =  sup ( { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( J `  ( E `  x )
) } ,  RR ,  <  )
127126mpteq2i 4520 . . . . . . . . 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 2472 . . . . . . . 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 31933 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  ( F  |`  ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) )  e.  ( ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) -cn-> CC ) )
130129adantlr 714 . . . . . 6  |-  ( ( ( ph  /\  X  <  0 )  /\  j  e.  ( 0..^ N ) )  ->  ( F  |`  ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) )  e.  ( ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) -cn-> CC ) )
13121adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 0..^ N ) )  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
) )
132 eqid 2443 . . . . . . . 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 31932 . . . . . . 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 714 . . . . . 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 714 . . . . . . . 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 2443 . . . . . . . 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 31934 . . . . . . 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 714 . . . . . 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 31951 . . . . 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 2485 . . . 4  |-  ( (
ph  /\  X  <  0 )  ->  S. ( ( A  -  X ) [,] ( B  -  X )
) ( F `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x )
14137, 43, 140syl2anc 661 . . 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 791 . 2  |-  ( (
ph  /\  -.  0  <  X )  ->  S. ( ( A  -  X ) [,] ( B  -  X )
) ( F `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x )
14325, 142pm2.61dan 791 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 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   {crab 2797    u. cun 3459   ifcif 3926   {cpr 4016   class class class wbr 4437    |-> cmpt 4495   ran crn 4990    |` cres 4991   iotacio 5539   -->wf 5574   ` cfv 5578    Isom wiso 5579  (class class class)co 6281    ^m cmap 7422   supcsup 7902   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   +oocpnf 9628   RR*cxr 9630    < clt 9631    <_ cle 9632    - cmin 9810   -ucneg 9811    / cdiv 10213   NNcn 10543   ZZcz 10871   (,)cioo 11540   (,]cioc 11541   [,]cicc 11543   ...cfz 11683  ..^cfzo 11806   |_cfl 11909   #chash 12387   -cn->ccncf 21358   S.citg 22005   lim CC climc 22244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cc 8818  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ioo 11544  df-ioc 11545  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-fl 11911  df-mod 11979  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-limsup 13276  df-clim 13293  df-rlim 13294  df-sum 13491  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-pt 14824  df-prds 14827  df-xrs 14881  df-qtop 14886  df-imas 14887  df-xps 14889  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-mulg 16039  df-cntz 16334  df-cmn 16779  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-fbas 18395  df-fg 18396  df-cnfld 18400  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cld 19498  df-ntr 19499  df-cls 19500  df-nei 19577  df-lp 19615  df-perf 19616  df-cn 19706  df-cnp 19707  df-haus 19794  df-cmp 19865  df-tx 20041  df-hmeo 20234  df-fil 20325  df-fm 20417  df-flim 20418  df-flf 20419  df-xms 20801  df-ms 20802  df-tms 20803  df-cncf 21360  df-ovol 21854  df-vol 21855  df-mbf 22006  df-itg1 22007  df-itg2 22008  df-ibl 22009  df-itg 22010  df-0p 22055  df-ditg 22229  df-limc 22248  df-dv 22249
This theorem is referenced by:  fourierdlem110  31953
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