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Theorem fourierdlem109 38079
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 38062 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 467 . . 3  |-  ( (
ph  /\  0  <  X )  ->  A  e.  RR )
3 fourierdlem109.b . . . 4  |-  ( ph  ->  B  e.  RR )
43adantr 467 . . 3  |-  ( (
ph  /\  0  <  X )  ->  B  e.  RR )
5 fourierdlem109.t . . 3  |-  T  =  ( B  -  A
)
6 fourierdlem109.x . . . . 5  |-  ( ph  ->  X  e.  RR )
76adantr 467 . . . 4  |-  ( (
ph  /\  0  <  X )  ->  X  e.  RR )
8 simpr 463 . . . 4  |-  ( (
ph  /\  0  <  X )  ->  0  <  X )
97, 8elrpd 11338 . . 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 467 . . 3  |-  ( (
ph  /\  0  <  X )  ->  M  e.  NN )
13 fourierdlem109.q . . . 4  |-  ( ph  ->  Q  e.  ( P `
 M ) )
1413adantr 467 . . 3  |-  ( (
ph  /\  0  <  X )  ->  Q  e.  ( P `  M ) )
15 fourierdlem109.f . . . 4  |-  ( ph  ->  F : RR --> CC )
1615adantr 467 . . 3  |-  ( (
ph  /\  0  <  X )  ->  F : RR
--> CC )
17 fourierdlem109.fper . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
1817adantlr 721 . . 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 721 . . 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 721 . . 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 721 . . 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 38078 . 2  |-  ( (
ph  /\  0  <  X )  ->  S. (
( A  -  X
) [,] ( B  -  X ) ) ( F `  x
)  _d x  =  S. ( A [,] B ) ( F `
 x )  _d x )
26 oveq2 6298 . . . . . . 7  |-  ( X  =  0  ->  ( A  -  X )  =  ( A  - 
0 ) )
271recnd 9669 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
2827subid1d 9975 . . . . . . 7  |-  ( ph  ->  ( A  -  0 )  =  A )
2926, 28sylan9eqr 2507 . . . . . 6  |-  ( (
ph  /\  X  = 
0 )  ->  ( A  -  X )  =  A )
30 oveq2 6298 . . . . . . 7  |-  ( X  =  0  ->  ( B  -  X )  =  ( B  - 
0 ) )
313recnd 9669 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
3231subid1d 9975 . . . . . . 7  |-  ( ph  ->  ( B  -  0 )  =  B )
3330, 32sylan9eqr 2507 . . . . . 6  |-  ( (
ph  /\  X  = 
0 )  ->  ( B  -  X )  =  B )
3429, 33oveq12d 6308 . . . . 5  |-  ( (
ph  /\  X  = 
0 )  ->  (
( A  -  X
) [,] ( B  -  X ) )  =  ( A [,] B ) )
3534itgeq1d 37833 . . . 4  |-  ( (
ph  /\  X  = 
0 )  ->  S. ( ( A  -  X ) [,] ( B  -  X )
) ( F `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x )
3635adantlr 721 . . 3  |-  ( ( ( ph  /\  -.  0  <  X )  /\  X  =  0 )  ->  S. ( ( A  -  X ) [,] ( B  -  X ) ) ( F `  x )  _d x  =  S. ( A [,] B
) ( F `  x )  _d x )
37 simpll 760 . . . 4  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  ph )
3837, 6syl 17 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  X  e.  RR )
39 0red 9644 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  0  e.  RR )
40 simpr 463 . . . . . 6  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  -.  X  =  0 )
4140neqned 2631 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  X  =/=  0 )
42 simplr 762 . . . . 5  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  -.  0  <  X )
4338, 39, 41, 42lttri5d 37517 . . . 4  |-  ( ( ( ph  /\  -.  0  <  X )  /\  -.  X  =  0
)  ->  X  <  0 )
446recnd 9669 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  CC )
4527, 44subcld 9986 . . . . . . . . . . 11  |-  ( ph  ->  ( A  -  X
)  e.  CC )
4645, 44subnegd 9993 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  -  X )  -  -u X
)  =  ( ( A  -  X )  +  X ) )
4727, 44npcand 9990 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  -  X )  +  X
)  =  A )
4846, 47eqtrd 2485 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  X )  -  -u X
)  =  A )
4931, 44subcld 9986 . . . . . . . . . . 11  |-  ( ph  ->  ( B  -  X
)  e.  CC )
5049, 44subnegd 9993 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  -  X )  -  -u X
)  =  ( ( B  -  X )  +  X ) )
5131, 44npcand 9990 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  -  X )  +  X
)  =  B )
5250, 51eqtrd 2485 . . . . . . . . 9  |-  ( ph  ->  ( ( B  -  X )  -  -u X
)  =  B )
5348, 52oveq12d 6308 . . . . . . . 8  |-  ( ph  ->  ( ( ( A  -  X )  -  -u X ) [,] (
( B  -  X
)  -  -u X
) )  =  ( A [,] B ) )
5453eqcomd 2457 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  =  ( ( ( A  -  X
)  -  -u X
) [,] ( ( B  -  X )  -  -u X ) ) )
5554itgeq1d 37833 . . . . . 6  |-  ( ph  ->  S. ( A [,] B ) ( F `
 x )  _d x  =  S. ( ( ( A  -  X )  -  -u X
) [,] ( ( B  -  X )  -  -u X ) ) ( F `  x
)  _d x )
5655adantr 467 . . . . 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 10047 . . . . . . 7  |-  ( ph  ->  ( A  -  X
)  e.  RR )
5857adantr 467 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  ( A  -  X )  e.  RR )
593, 6resubcld 10047 . . . . . . 7  |-  ( ph  ->  ( B  -  X
)  e.  RR )
6059adantr 467 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  ( B  -  X )  e.  RR )
61 eqid 2451 . . . . . 6  |-  ( ( B  -  X )  -  ( A  -  X ) )  =  ( ( B  -  X )  -  ( A  -  X )
)
626renegcld 10046 . . . . . . . 8  |-  ( ph  -> 
-u X  e.  RR )
6362adantr 467 . . . . . . 7  |-  ( (
ph  /\  X  <  0 )  ->  -u X  e.  RR )
646lt0neg1d 10183 . . . . . . . 8  |-  ( ph  ->  ( X  <  0  <->  0  <  -u X ) )
6564biimpa 487 . . . . . . 7  |-  ( (
ph  /\  X  <  0 )  ->  0  <  -u X )
6663, 65elrpd 11338 . . . . . 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 5865 . . . . . . . . . . . . 13  |-  ( i  =  j  ->  (
p `  i )  =  ( p `  j ) )
69 oveq1 6297 . . . . . . . . . . . . . 14  |-  ( i  =  j  ->  (
i  +  1 )  =  ( j  +  1 ) )
7069fveq2d 5869 . . . . . . . . . . . . 13  |-  ( i  =  j  ->  (
p `  ( i  +  1 ) )  =  ( p `  ( j  +  1 ) ) )
7168, 70breq12d 4415 . . . . . . . . . . . 12  |-  ( i  =  j  ->  (
( p `  i
)  <  ( p `  ( i  +  1 ) )  <->  ( p `  j )  <  (
p `  ( j  +  1 ) ) ) )
7271cbvralv 3019 . . . . . . . . . . 11  |-  ( A. i  e.  ( 0..^ m ) ( p `
 i )  < 
( p `  (
i  +  1 ) )  <->  A. j  e.  ( 0..^ m ) ( p `  j )  <  ( p `  ( j  +  1 ) ) )
7372anbi2i 700 . . . . . . . . . 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 3033 . . . . . . . 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 4486 . . . . . . 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 2473 . . . . . 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 37980 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) )
7978simp3d 1022 . . . . . . . . . . 11  |-  ( ph  ->  A  <  B )
801, 3, 6, 79ltsub1dd 10225 . . . . . . . . . 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 38024 . . . . . . . . 9  |-  ( ph  ->  ( ( N  e.  NN  /\  S  e.  ( O `  N
) )  /\  S  Isom  <  ,  <  (
( 0 ... N
) ,  H ) ) )
8584simpld 461 . . . . . . . 8  |-  ( ph  ->  ( N  e.  NN  /\  S  e.  ( O `
 N ) ) )
8685simpld 461 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
8786adantr 467 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  N  e.  NN )
8885simprd 465 . . . . . . 7  |-  ( ph  ->  S  e.  ( O `
 N ) )
8988adantr 467 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  S  e.  ( O `  N
) )
9015adantr 467 . . . . . 6  |-  ( (
ph  /\  X  <  0 )  ->  F : RR --> CC )
9131, 27, 44nnncan2d 10021 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( B  -  X )  -  ( A  -  X )
)  =  ( B  -  A ) )
9291, 5syl6eqr 2503 . . . . . . . . . . 11  |-  ( ph  ->  ( ( B  -  X )  -  ( A  -  X )
)  =  T )
9392oveq2d 6306 . . . . . . . . . 10  |-  ( ph  ->  ( x  +  ( ( B  -  X
)  -  ( A  -  X ) ) )  =  ( x  +  T ) )
9493adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR )  ->  ( x  +  ( ( B  -  X )  -  ( A  -  X
) ) )  =  ( x  +  T
) )
9594fveq2d 5869 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  ( ( B  -  X )  -  ( A  -  X )
) ) )  =  ( F `  (
x  +  T ) ) )
9695, 17eqtrd 2485 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  ( ( B  -  X )  -  ( A  -  X )
) ) )  =  ( F `  x
) )
9796adantlr 721 . . . . . 6  |-  ( ( ( ph  /\  X  <  0 )  /\  x  e.  RR )  ->  ( F `  ( x  +  ( ( B  -  X )  -  ( A  -  X
) ) ) )  =  ( F `  x ) )
9811adantr 467 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  M  e.  NN )
9913adantr 467 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  Q  e.  ( P `  M ) )
10015adantr 467 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  F : RR --> CC )
10117adantlr 721 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 0..^ N ) )  /\  x  e.  RR )  ->  ( F `  ( x  +  T ) )  =  ( F `  x
) )
10219adantlr 721 . . . . . . . 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 467 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  ( A  -  X )  e.  RR )
10457rexrd 9690 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  X
)  e.  RR* )
105 pnfxr 11412 . . . . . . . . . . 11  |- +oo  e.  RR*
106105a1i 11 . . . . . . . . . 10  |-  ( ph  -> +oo  e.  RR* )
10759ltpnfd 11423 . . . . . . . . . 10  |-  ( ph  ->  ( B  -  X
)  < +oo )
108104, 106, 59, 80, 107eliood 37595 . . . . . . . . 9  |-  ( ph  ->  ( B  -  X
)  e.  ( ( A  -  X ) (,) +oo ) )
109108adantr 467 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  ( B  -  X )  e.  ( ( A  -  X
) (,) +oo )
)
110 oveq1 6297 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
x  +  ( k  x.  T ) )  =  ( y  +  ( k  x.  T
) ) )
111110eleq1d 2513 . . . . . . . . . . . 12  |-  ( x  =  y  ->  (
( x  +  ( k  x.  T ) )  e.  ran  Q  <->  ( y  +  ( k  x.  T ) )  e.  ran  Q ) )
112111rexbidv 2901 . . . . . . . . . . 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 3044 . . . . . . . . . 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 3585 . . . . . . . . 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 2473 . . . . . . . 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 463 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  j  e.  ( 0..^ N ) )
119 eqid 2451 . . . . . . . 8  |-  ( ( S `  ( j  +  1 ) )  -  ( E `  ( S `  ( j  +  1 ) ) ) )  =  ( ( S `  (
j  +  1 ) )  -  ( E `
 ( S `  ( j  +  1 ) ) ) )
120 eqid 2451 . . . . . . . 8  |-  ( F  |`  ( ( J `  ( E `  ( S `
 j ) ) ) (,) ( E `
 ( S `  ( j  +  1 ) ) ) ) )  =  ( F  |`  ( ( J `  ( E `  ( S `
 j ) ) ) (,) ( E `
 ( S `  ( j  +  1 ) ) ) ) )
121 eqid 2451 . . . . . . . 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 5865 . . . . . . . . . . . . 13  |-  ( j  =  i  ->  ( Q `  j )  =  ( Q `  i ) )
124123breq1d 4412 . . . . . . . . . . . 12  |-  ( j  =  i  ->  (
( Q `  j
)  <_  ( J `  ( E `  x
) )  <->  ( Q `  i )  <_  ( J `  ( E `  x ) ) ) )
125124cbvrabv 3044 . . . . . . . . . . 11  |-  { j  e.  ( 0..^ M )  |  ( Q `
 j )  <_ 
( J `  ( E `  x )
) }  =  {
i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( J `  ( E `  x ) ) }
126125supeq1i 7961 . . . . . . . . . 10  |-  sup ( { j  e.  ( 0..^ M )  |  ( Q `  j
)  <_  ( J `  ( E `  x
) ) } ,  RR ,  <  )  =  sup ( { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( J `  ( E `  x )
) } ,  RR ,  <  )
127126mpteq2i 4486 . . . . . . . . 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 2473 . . . . . . . 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 38060 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 0..^ N ) )  ->  ( F  |`  ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) )  e.  ( ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) -cn-> CC ) )
130129adantlr 721 . . . . . 6  |-  ( ( ( ph  /\  X  <  0 )  /\  j  e.  ( 0..^ N ) )  ->  ( F  |`  ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) )  e.  ( ( ( S `  j ) (,) ( S `  ( j  +  1 ) ) ) -cn-> CC ) )
13121adantlr 721 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  ( 0..^ N ) )  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
) )
132 eqid 2451 . . . . . . . 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 38059 . . . . . . 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 721 . . . . . 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 721 . . . . . . . 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 2451 . . . . . . . 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 38061 . . . . . . 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 721 . . . . . 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 38078 . . . . 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 2486 . . . 4  |-  ( (
ph  /\  X  <  0 )  ->  S. ( ( A  -  X ) [,] ( B  -  X )
) ( F `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x )
14137, 43, 140syl2anc 667 . . 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 800 . 2  |-  ( (
ph  /\  -.  0  <  X )  ->  S. ( ( A  -  X ) [,] ( B  -  X )
) ( F `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x )
14325, 142pm2.61dan 800 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738   {crab 2741    u. cun 3402   ifcif 3881   {cpr 3970   class class class wbr 4402    |-> cmpt 4461   ran crn 4835    |` cres 4836   iotacio 5544   -->wf 5578   ` cfv 5582    Isom wiso 5583  (class class class)co 6290    ^m cmap 7472   supcsup 7954   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676    - cmin 9860   -ucneg 9861    / cdiv 10269   NNcn 10609   ZZcz 10937   (,)cioo 11635   (,]cioc 11636   [,]cicc 11638   ...cfz 11784  ..^cfzo 11915   |_cfl 12026   #chash 12515   -cn->ccncf 21908   S.citg 22576   lim CC climc 22817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cc 8865  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-ofr 6532  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-acn 8376  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-ovol 22416  df-vol 22418  df-mbf 22577  df-itg1 22578  df-itg2 22579  df-ibl 22580  df-itg 22581  df-0p 22628  df-ditg 22802  df-limc 22821  df-dv 22822
This theorem is referenced by:  fourierdlem110  38080
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