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Theorem fourierdlem105 38187
Description: A piecewise continuous function is integrable on any closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem105.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 ) ) ) } )
fourierdlem105.t  |-  T  =  ( B  -  A
)
fourierdlem105.m  |-  ( ph  ->  M  e.  NN )
fourierdlem105.q  |-  ( ph  ->  Q  e.  ( P `
 M ) )
fourierdlem105.f  |-  ( ph  ->  F : RR --> CC )
fourierdlem105.6  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
fourierdlem105.fcn  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
fourierdlem105.r  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 i ) ) )
fourierdlem105.l  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 ( i  +  1 ) ) ) )
fourierdlem105.c  |-  ( ph  ->  C  e.  RR )
fourierdlem105.d  |-  ( ph  ->  D  e.  ( C (,) +oo ) )
Assertion
Ref Expression
fourierdlem105  |-  ( ph  ->  ( x  e.  ( C [,] D ) 
|->  ( F `  x
) )  e.  L^1 )
Distinct variable groups:    A, i, x    A, m, p, i    B, i, x    B, m, p    C, i, x    C, m, p    D, i, x    D, 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    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 fourierdlem105
Dummy variables  f 
j  k  w  y  z  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem105.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 ) ) ) } )
2 fourierdlem105.t . 2  |-  T  =  ( B  -  A
)
3 fourierdlem105.m . 2  |-  ( ph  ->  M  e.  NN )
4 fourierdlem105.q . 2  |-  ( ph  ->  Q  e.  ( P `
 M ) )
5 fourierdlem105.f . 2  |-  ( ph  ->  F : RR --> CC )
6 fourierdlem105.6 . 2  |-  ( (
ph  /\  x  e.  RR )  ->  ( F `
 ( x  +  T ) )  =  ( F `  x
) )
7 fourierdlem105.fcn . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
8 fourierdlem105.r . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 i ) ) )
9 fourierdlem105.l . 2  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 ( i  +  1 ) ) ) )
10 fourierdlem105.c . 2  |-  ( ph  ->  C  e.  RR )
11 fourierdlem105.d . 2  |-  ( ph  ->  D  e.  ( C (,) +oo ) )
12 eqid 2471 . 2  |-  ( m  e.  NN  |->  { p  e.  ( RR  ^m  (
0 ... m ) )  |  ( ( ( p `  0 )  =  C  /\  (
p `  m )  =  D )  /\  A. i  e.  ( 0..^ m ) ( p `
 i )  < 
( p `  (
i  +  1 ) ) ) } )  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  (
0 ... m ) )  |  ( ( ( p `  0 )  =  C  /\  (
p `  m )  =  D )  /\  A. i  e.  ( 0..^ m ) ( p `
 i )  < 
( p `  (
i  +  1 ) ) ) } )
13 eqid 2471 . 2  |-  ( (
# `  ( { C ,  D }  u.  { w  e.  ( C [,] D )  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q } ) )  - 
1 )  =  ( ( # `  ( { C ,  D }  u.  { w  e.  ( C [,] D )  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q } ) )  - 
1 )
14 oveq1 6315 . . . . . . 7  |-  ( w  =  y  ->  (
w  +  ( j  x.  T ) )  =  ( y  +  ( j  x.  T
) ) )
1514eleq1d 2533 . . . . . 6  |-  ( w  =  y  ->  (
( w  +  ( j  x.  T ) )  e.  ran  Q  <->  ( y  +  ( j  x.  T ) )  e.  ran  Q ) )
1615rexbidv 2892 . . . . 5  |-  ( w  =  y  ->  ( E. j  e.  ZZ  ( w  +  (
j  x.  T ) )  e.  ran  Q  <->  E. j  e.  ZZ  (
y  +  ( j  x.  T ) )  e.  ran  Q ) )
17 oveq1 6315 . . . . . . . 8  |-  ( j  =  k  ->  (
j  x.  T )  =  ( k  x.  T ) )
1817oveq2d 6324 . . . . . . 7  |-  ( j  =  k  ->  (
y  +  ( j  x.  T ) )  =  ( y  +  ( k  x.  T
) ) )
1918eleq1d 2533 . . . . . 6  |-  ( j  =  k  ->  (
( y  +  ( j  x.  T ) )  e.  ran  Q  <->  ( y  +  ( k  x.  T ) )  e.  ran  Q ) )
2019cbvrexv 3006 . . . . 5  |-  ( E. j  e.  ZZ  (
y  +  ( j  x.  T ) )  e.  ran  Q  <->  E. k  e.  ZZ  ( y  +  ( k  x.  T
) )  e.  ran  Q )
2116, 20syl6bb 269 . . . 4  |-  ( w  =  y  ->  ( E. j  e.  ZZ  ( w  +  (
j  x.  T ) )  e.  ran  Q  <->  E. k  e.  ZZ  (
y  +  ( k  x.  T ) )  e.  ran  Q ) )
2221cbvrabv 3030 . . 3  |-  { w  e.  ( C [,] D
)  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q }  =  { y  e.  ( C [,] D )  |  E. k  e.  ZZ  (
y  +  ( k  x.  T ) )  e.  ran  Q }
2322uneq2i 3576 . 2  |-  ( { C ,  D }  u.  { w  e.  ( C [,] D )  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q } )  =  ( { C ,  D }  u.  { y  e.  ( C [,] D
)  |  E. k  e.  ZZ  ( y  +  ( k  x.  T
) )  e.  ran  Q } )
24 isoeq1 6228 . . 3  |-  ( g  =  f  ->  (
g  Isom  <  ,  <  ( ( 0 ... (
( # `  ( { C ,  D }  u.  { w  e.  ( C [,] D )  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { C ,  D }  u.  { w  e.  ( C [,] D
)  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q } ) )  <->  f  Isom  <  ,  <  ( ( 0 ... ( ( # `  ( { C ,  D }  u.  { w  e.  ( C [,] D
)  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { C ,  D }  u.  { w  e.  ( C [,] D
)  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q } ) ) ) )
2524cbviotav 5559 . 2  |-  ( iota g g  Isom  <  ,  <  ( ( 0 ... ( ( # `  ( { C ,  D }  u.  { w  e.  ( C [,] D
)  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { C ,  D }  u.  { w  e.  ( C [,] D
)  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q } ) ) )  =  ( iota f
f  Isom  <  ,  <  ( ( 0 ... (
( # `  ( { C ,  D }  u.  { w  e.  ( C [,] D )  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q } ) )  - 
1 ) ) ,  ( { C ,  D }  u.  { w  e.  ( C [,] D
)  |  E. j  e.  ZZ  ( w  +  ( j  x.  T
) )  e.  ran  Q } ) ) )
26 id 22 . . . 4  |-  ( w  =  x  ->  w  =  x )
27 oveq2 6316 . . . . . . 7  |-  ( w  =  x  ->  ( B  -  w )  =  ( B  -  x ) )
2827oveq1d 6323 . . . . . 6  |-  ( w  =  x  ->  (
( B  -  w
)  /  T )  =  ( ( B  -  x )  /  T ) )
2928fveq2d 5883 . . . . 5  |-  ( w  =  x  ->  ( |_ `  ( ( B  -  w )  /  T ) )  =  ( |_ `  (
( B  -  x
)  /  T ) ) )
3029oveq1d 6323 . . . 4  |-  ( w  =  x  ->  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T )  =  ( ( |_
`  ( ( B  -  x )  /  T ) )  x.  T ) )
3126, 30oveq12d 6326 . . 3  |-  ( w  =  x  ->  (
w  +  ( ( |_ `  ( ( B  -  w )  /  T ) )  x.  T ) )  =  ( x  +  ( ( |_ `  ( ( B  -  x )  /  T
) )  x.  T
) ) )
3231cbvmptv 4488 . 2  |-  ( w  e.  RR  |->  ( w  +  ( ( |_
`  ( ( B  -  w )  /  T ) )  x.  T ) ) )  =  ( x  e.  RR  |->  ( x  +  ( ( |_ `  ( ( B  -  x )  /  T
) )  x.  T
) ) )
33 eqeq1 2475 . . . 4  |-  ( w  =  y  ->  (
w  =  B  <->  y  =  B ) )
34 id 22 . . . 4  |-  ( w  =  y  ->  w  =  y )
3533, 34ifbieq2d 3897 . . 3  |-  ( w  =  y  ->  if ( w  =  B ,  A ,  w )  =  if ( y  =  B ,  A ,  y ) )
3635cbvmptv 4488 . 2  |-  ( w  e.  ( A (,] B )  |->  if ( w  =  B ,  A ,  w )
)  =  ( y  e.  ( A (,] B )  |->  if ( y  =  B ,  A ,  y )
)
37 fveq2 5879 . . . . . . . 8  |-  ( z  =  x  ->  (
( w  e.  RR  |->  ( w  +  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T ) ) ) `  z
)  =  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T ) )  x.  T ) ) ) `  x ) )
3837fveq2d 5883 . . . . . . 7  |-  ( z  =  x  ->  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  z ) )  =  ( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) )
3938breq2d 4407 . . . . . 6  |-  ( z  =  x  ->  (
( Q `  j
)  <_  ( (
w  e.  ( A (,] B )  |->  if ( w  =  B ,  A ,  w
) ) `  (
( w  e.  RR  |->  ( w  +  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T ) ) ) `  z
) )  <->  ( Q `  j )  <_  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) ) )
4039rabbidv 3022 . . . . 5  |-  ( z  =  x  ->  { j  e.  ( 0..^ M )  |  ( Q `
 j )  <_ 
( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  z ) ) }  =  { j  e.  ( 0..^ M )  |  ( Q `  j )  <_  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) } )
41 fveq2 5879 . . . . . . 7  |-  ( j  =  i  ->  ( Q `  j )  =  ( Q `  i ) )
4241breq1d 4405 . . . . . 6  |-  ( j  =  i  ->  (
( Q `  j
)  <_  ( (
w  e.  ( A (,] B )  |->  if ( w  =  B ,  A ,  w
) ) `  (
( w  e.  RR  |->  ( w  +  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T ) ) ) `  x
) )  <->  ( Q `  i )  <_  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) ) )
4342cbvrabv 3030 . . . . 5  |-  { j  e.  ( 0..^ M )  |  ( Q `
 j )  <_ 
( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) }  =  { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) }
4440, 43syl6eq 2521 . . . 4  |-  ( z  =  x  ->  { j  e.  ( 0..^ M )  |  ( Q `
 j )  <_ 
( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  z ) ) }  =  { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  (
( w  e.  ( A (,] B ) 
|->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) } )
4544supeq1d 7978 . . 3  |-  ( z  =  x  ->  sup ( { j  e.  ( 0..^ M )  |  ( Q `  j
)  <_  ( (
w  e.  ( A (,] B )  |->  if ( w  =  B ,  A ,  w
) ) `  (
( w  e.  RR  |->  ( w  +  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T ) ) ) `  z
) ) } ,  RR ,  <  )  =  sup ( { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) } ,  RR ,  <  ) )
4645cbvmptv 4488 . 2  |-  ( z  e.  RR  |->  sup ( { j  e.  ( 0..^ M )  |  ( Q `  j
)  <_  ( (
w  e.  ( A (,] B )  |->  if ( w  =  B ,  A ,  w
) ) `  (
( w  e.  RR  |->  ( w  +  (
( |_ `  (
( B  -  w
)  /  T ) )  x.  T ) ) ) `  z
) ) } ,  RR ,  <  ) )  =  ( x  e.  RR  |->  sup ( { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( ( w  e.  ( A (,] B
)  |->  if ( w  =  B ,  A ,  w ) ) `  ( ( w  e.  RR  |->  ( w  +  ( ( |_ `  ( ( B  -  w )  /  T
) )  x.  T
) ) ) `  x ) ) } ,  RR ,  <  ) )
471, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 23, 25, 32, 36, 46fourierdlem100 38182 1  |-  ( ph  ->  ( x  e.  ( C [,] D ) 
|->  ( F `  x
) )  e.  L^1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ 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    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   ZZcz 10961   (,)cioo 11660   (,]cioc 11661   [,]cicc 11663   ...cfz 11810  ..^cfzo 11942   |_cfl 12059   #chash 12553   -cn->ccncf 21986   L^1cibl 22654   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-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-cn 20320  df-cnp 20321  df-cmp 20479  df-tx 20654  df-hmeo 20847  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-limc 22900
This theorem is referenced by:  fourierdlem107  38189  fourierdlem111  38193
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