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Theorem fourierdlem37 38119
Description:  I is a function that maps any real point to the point that in the partition that immediately precedes the corresponding periodic point in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem37.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 ) ) ) } )
fourierdlem37.m  |-  ( ph  ->  M  e.  NN )
fourierdlem37.q  |-  ( ph  ->  Q  e.  ( P `
 M ) )
fourierdlem37.t  |-  T  =  ( B  -  A
)
fourierdlem37.e  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( B  -  x
)  /  T ) )  x.  T ) ) )
fourierdlem37.l  |-  L  =  ( y  e.  ( A (,] B ) 
|->  if ( y  =  B ,  A , 
y ) )
fourierdlem37.i  |-  I  =  ( x  e.  RR  |->  sup ( { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `  ( E `  x ) ) } ,  RR ,  <  ) )
Assertion
Ref Expression
fourierdlem37  |-  ( ph  ->  ( I : RR --> ( 0..^ M )  /\  ( x  e.  RR  ->  sup ( { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( L `  ( E `  x )
) } ,  RR ,  <  )  e.  {
i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `  ( E `  x ) ) } ) ) )
Distinct variable groups:    A, m, p    x, A, y    B, m, p    x, B, y   
i, E    y, E    i, L    i, M, m, p    x, M, i    Q, i, p    x, T    ph, i, x    ph, y
Allowed substitution hints:    ph( m, p)    A( i)    B( i)    P( x, y, i, m, p)    Q( x, y, m)    T( y, i, m, p)    E( x, m, p)    I( x, y, i, m, p)    L( x, y, m, p)    M( y)

Proof of Theorem fourierdlem37
StepHypRef Expression
1 ssrab2 3500 . . . 4  |-  { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( L `  ( E `  x )
) }  C_  (
0..^ M )
2 ltso 9732 . . . . . 6  |-  <  Or  RR
32a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  RR )  ->  <  Or  RR )
4 fzfi 12223 . . . . . . 7  |-  ( 0 ... M )  e. 
Fin
5 fzossfz 11965 . . . . . . . 8  |-  ( 0..^ M )  C_  (
0 ... M )
61, 5sstri 3427 . . . . . . 7  |-  { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( L `  ( E `  x )
) }  C_  (
0 ... M )
7 ssfi 7810 . . . . . . 7  |-  ( ( ( 0 ... M
)  e.  Fin  /\  { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `  ( E `  x ) ) }  C_  (
0 ... M ) )  ->  { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `  ( E `  x ) ) }  e.  Fin )
84, 6, 7mp2an 686 . . . . . 6  |-  { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( L `  ( E `  x )
) }  e.  Fin
98a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  RR )  ->  { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( L `  ( E `  x )
) }  e.  Fin )
10 0zd 10973 . . . . . . . . 9  |-  ( ph  ->  0  e.  ZZ )
11 fourierdlem37.m . . . . . . . . . 10  |-  ( ph  ->  M  e.  NN )
1211nnzd 11062 . . . . . . . . 9  |-  ( ph  ->  M  e.  ZZ )
1311nngt0d 10675 . . . . . . . . 9  |-  ( ph  ->  0  <  M )
14 fzolb 11953 . . . . . . . . 9  |-  ( 0  e.  ( 0..^ M )  <->  ( 0  e.  ZZ  /\  M  e.  ZZ  /\  0  < 
M ) )
1510, 12, 13, 14syl3anbrc 1214 . . . . . . . 8  |-  ( ph  ->  0  e.  ( 0..^ M ) )
1615adantr 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  0  e.  ( 0..^ M ) )
17 fourierdlem37.q . . . . . . . . . . . . . . . 16  |-  ( ph  ->  Q  e.  ( P `
 M ) )
18 fourierdlem37.p . . . . . . . . . . . . . . . . . 18  |-  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 ) ) ) } )
1918fourierdlem2 38083 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  NN  ->  ( Q  e.  ( P `  M )  <->  ( Q  e.  ( RR  ^m  (
0 ... M ) )  /\  ( ( ( Q `  0 )  =  A  /\  ( Q `  M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) ) )
2011, 19syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( Q  e.  ( P `  M )  <-> 
( Q  e.  ( RR  ^m  ( 0 ... M ) )  /\  ( ( ( Q `  0 )  =  A  /\  ( Q `  M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) ) )
2117, 20mpbid 215 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Q  e.  ( RR  ^m  ( 0 ... M ) )  /\  ( ( ( Q `  0 )  =  A  /\  ( Q `  M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) )
2221simprd 470 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( Q `
 0 )  =  A  /\  ( Q `
 M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) )
2322simplld 769 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q `  0
)  =  A )
2418, 11, 17fourierdlem11 38092 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) )
2524simp1d 1042 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  RR )
2623, 25eqeltrd 2549 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q `  0
)  e.  RR )
2726, 23eqled 9755 . . . . . . . . . . 11  |-  ( ph  ->  ( Q `  0
)  <_  A )
2827ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  ( E `  x )  =  B )  ->  ( Q `  0 )  <_  A )
29 iftrue 3878 . . . . . . . . . . . 12  |-  ( ( E `  x )  =  B  ->  if ( ( E `  x )  =  B ,  A ,  ( E `  x ) )  =  A )
3029eqcomd 2477 . . . . . . . . . . 11  |-  ( ( E `  x )  =  B  ->  A  =  if ( ( E `
 x )  =  B ,  A , 
( E `  x
) ) )
3130adantl 473 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  ( E `  x )  =  B )  ->  A  =  if ( ( E `
 x )  =  B ,  A , 
( E `  x
) ) )
3228, 31breqtrd 4420 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR )  /\  ( E `  x )  =  B )  ->  ( Q `  0 )  <_  if ( ( E `
 x )  =  B ,  A , 
( E `  x
) ) )
3326adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  RR )  ->  ( Q `
 0 )  e.  RR )
3425adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  RR )  ->  A  e.  RR )
3534rexrd 9708 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  RR )  ->  A  e. 
RR* )
3624simp2d 1043 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  RR )
3736adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  RR )  ->  B  e.  RR )
38 iocssre 11739 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  ( A (,] B )  C_  RR )
3935, 37, 38syl2anc 673 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  RR )  ->  ( A (,] B )  C_  RR )
4024simp3d 1044 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  <  B )
41 fourierdlem37.t . . . . . . . . . . . . . . 15  |-  T  =  ( B  -  A
)
42 fourierdlem37.e . . . . . . . . . . . . . . 15  |-  E  =  ( x  e.  RR  |->  ( x  +  (
( |_ `  (
( B  -  x
)  /  T ) )  x.  T ) ) )
4325, 36, 40, 41, 42fourierdlem4 38085 . . . . . . . . . . . . . 14  |-  ( ph  ->  E : RR --> ( A (,] B ) )
4443fnvinran 37398 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  RR )  ->  ( E `
 x )  e.  ( A (,] B
) )
4539, 44sseldd 3419 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  RR )  ->  ( E `
 x )  e.  RR )
4623adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  RR )  ->  ( Q `
 0 )  =  A )
47 elioc2 11722 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  B  e.  RR )  ->  (
( E `  x
)  e.  ( A (,] B )  <->  ( ( E `  x )  e.  RR  /\  A  < 
( E `  x
)  /\  ( E `  x )  <_  B
) ) )
4835, 37, 47syl2anc 673 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( E `  x )  e.  ( A (,] B )  <->  ( ( E `  x )  e.  RR  /\  A  < 
( E `  x
)  /\  ( E `  x )  <_  B
) ) )
4944, 48mpbid 215 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( E `  x )  e.  RR  /\  A  <  ( E `  x
)  /\  ( E `  x )  <_  B
) )
5049simp2d 1043 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  RR )  ->  A  < 
( E `  x
) )
5146, 50eqbrtrd 4416 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  RR )  ->  ( Q `
 0 )  < 
( E `  x
) )
5233, 45, 51ltled 9800 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  RR )  ->  ( Q `
 0 )  <_ 
( E `  x
) )
5352adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  -.  ( E `  x )  =  B )  -> 
( Q `  0
)  <_  ( E `  x ) )
54 iffalse 3881 . . . . . . . . . . . 12  |-  ( -.  ( E `  x
)  =  B  ->  if ( ( E `  x )  =  B ,  A ,  ( E `  x ) )  =  ( E `
 x ) )
5554eqcomd 2477 . . . . . . . . . . 11  |-  ( -.  ( E `  x
)  =  B  -> 
( E `  x
)  =  if ( ( E `  x
)  =  B ,  A ,  ( E `  x ) ) )
5655adantl 473 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  -.  ( E `  x )  =  B )  -> 
( E `  x
)  =  if ( ( E `  x
)  =  B ,  A ,  ( E `  x ) ) )
5753, 56breqtrd 4420 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR )  /\  -.  ( E `  x )  =  B )  -> 
( Q `  0
)  <_  if (
( E `  x
)  =  B ,  A ,  ( E `  x ) ) )
5832, 57pm2.61dan 808 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR )  ->  ( Q `
 0 )  <_  if ( ( E `  x )  =  B ,  A ,  ( E `  x ) ) )
59 fourierdlem37.l . . . . . . . . . 10  |-  L  =  ( y  e.  ( A (,] B ) 
|->  if ( y  =  B ,  A , 
y ) )
6059a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR )  ->  L  =  ( y  e.  ( A (,] B ) 
|->  if ( y  =  B ,  A , 
y ) ) )
61 eqeq1 2475 . . . . . . . . . . 11  |-  ( y  =  ( E `  x )  ->  (
y  =  B  <->  ( E `  x )  =  B ) )
62 id 22 . . . . . . . . . . 11  |-  ( y  =  ( E `  x )  ->  y  =  ( E `  x ) )
6361, 62ifbieq2d 3897 . . . . . . . . . 10  |-  ( y  =  ( E `  x )  ->  if ( y  =  B ,  A ,  y )  =  if ( ( E `  x
)  =  B ,  A ,  ( E `  x ) ) )
6463adantl 473 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR )  /\  y  =  ( E `  x ) )  ->  if ( y  =  B ,  A ,  y )  =  if ( ( E `  x
)  =  B ,  A ,  ( E `  x ) ) )
6534, 45ifcld 3915 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR )  ->  if ( ( E `  x
)  =  B ,  A ,  ( E `  x ) )  e.  RR )
6660, 64, 44, 65fvmptd 5969 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR )  ->  ( L `
 ( E `  x ) )  =  if ( ( E `
 x )  =  B ,  A , 
( E `  x
) ) )
6758, 66breqtrrd 4422 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  ( Q `
 0 )  <_ 
( L `  ( E `  x )
) )
68 fveq2 5879 . . . . . . . . 9  |-  ( i  =  0  ->  ( Q `  i )  =  ( Q ` 
0 ) )
6968breq1d 4405 . . . . . . . 8  |-  ( i  =  0  ->  (
( Q `  i
)  <_  ( L `  ( E `  x
) )  <->  ( Q `  0 )  <_ 
( L `  ( E `  x )
) ) )
7069elrab 3184 . . . . . . 7  |-  ( 0  e.  { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `  ( E `  x ) ) }  <-> 
( 0  e.  ( 0..^ M )  /\  ( Q `  0 )  <_  ( L `  ( E `  x ) ) ) )
7116, 67, 70sylanbrc 677 . . . . . 6  |-  ( (
ph  /\  x  e.  RR )  ->  0  e. 
{ i  e.  ( 0..^ M )  |  ( Q `  i
)  <_  ( L `  ( E `  x
) ) } )
72 ne0i 3728 . . . . . 6  |-  ( 0  e.  { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `  ( E `  x ) ) }  ->  { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `  ( E `  x ) ) }  =/=  (/) )
7371, 72syl 17 . . . . 5  |-  ( (
ph  /\  x  e.  RR )  ->  { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( L `  ( E `  x )
) }  =/=  (/) )
74 fzssz 11827 . . . . . . . . 9  |-  ( 0 ... M )  C_  ZZ
755, 74sstri 3427 . . . . . . . 8  |-  ( 0..^ M )  C_  ZZ
76 zssre 10968 . . . . . . . 8  |-  ZZ  C_  RR
7775, 76sstri 3427 . . . . . . 7  |-  ( 0..^ M )  C_  RR
781, 77sstri 3427 . . . . . 6  |-  { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( L `  ( E `  x )
) }  C_  RR
7978a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  RR )  ->  { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( L `  ( E `  x )
) }  C_  RR )
80 fisupcl 8003 . . . . 5  |-  ( (  <  Or  RR  /\  ( { i  e.  ( 0..^ M )  |  ( Q `  i
)  <_  ( L `  ( E `  x
) ) }  e.  Fin  /\  { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `  ( E `  x ) ) }  =/=  (/)  /\  { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( L `  ( E `  x )
) }  C_  RR ) )  ->  sup ( { i  e.  ( 0..^ M )  |  ( Q `  i
)  <_  ( L `  ( E `  x
) ) } ,  RR ,  <  )  e. 
{ i  e.  ( 0..^ M )  |  ( Q `  i
)  <_  ( L `  ( E `  x
) ) } )
813, 9, 73, 79, 80syl13anc 1294 . . . 4  |-  ( (
ph  /\  x  e.  RR )  ->  sup ( { i  e.  ( 0..^ M )  |  ( Q `  i
)  <_  ( L `  ( E `  x
) ) } ,  RR ,  <  )  e. 
{ i  e.  ( 0..^ M )  |  ( Q `  i
)  <_  ( L `  ( E `  x
) ) } )
821, 81sseldi 3416 . . 3  |-  ( (
ph  /\  x  e.  RR )  ->  sup ( { i  e.  ( 0..^ M )  |  ( Q `  i
)  <_  ( L `  ( E `  x
) ) } ,  RR ,  <  )  e.  ( 0..^ M ) )
83 fourierdlem37.i . . 3  |-  I  =  ( x  e.  RR  |->  sup ( { i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `  ( E `  x ) ) } ,  RR ,  <  ) )
8482, 83fmptd 6061 . 2  |-  ( ph  ->  I : RR --> ( 0..^ M ) )
8581ex 441 . 2  |-  ( ph  ->  ( x  e.  RR  ->  sup ( { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( L `  ( E `  x )
) } ,  RR ,  <  )  e.  {
i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `  ( E `  x ) ) } ) )
8684, 85jca 541 1  |-  ( ph  ->  ( I : RR --> ( 0..^ M )  /\  ( x  e.  RR  ->  sup ( { i  e.  ( 0..^ M )  |  ( Q `
 i )  <_ 
( L `  ( E `  x )
) } ,  RR ,  <  )  e.  {
i  e.  ( 0..^ M )  |  ( Q `  i )  <_  ( L `  ( E `  x ) ) } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   {crab 2760    C_ wss 3390   (/)c0 3722   ifcif 3872   class class class wbr 4395    |-> cmpt 4454    Or wor 4759   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   Fincfn 7587   supcsup 7972   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   ZZcz 10961   (,]cioc 11661   ...cfz 11810  ..^cfzo 11942   |_cfl 12059
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-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  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
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  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-iun 4271  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-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-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  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-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-ioc 11665  df-fz 11811  df-fzo 11943  df-fl 12061
This theorem is referenced by:  fourierdlem79  38161  fourierdlem89  38171  fourierdlem90  38172  fourierdlem91  38173
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