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Theorem fourierdlem82 38046
Description: Integral by substitution, adding a constant to the function's argument, for a function on an open interval with finite limits ad boundary points. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem82.1  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) ) ) )
fourierdlem82.2  |-  ( ph  ->  A  e.  RR )
fourierdlem82.3  |-  ( ph  ->  B  e.  RR )
fourierdlem82.4  |-  ( ph  ->  A  <  B )
fourierdlem82.5  |-  ( ph  ->  F : ( A [,] B ) --> CC )
fourierdlem82.6  |-  ( ph  ->  ( F  |`  ( A (,) B ) )  e.  ( ( A (,) B ) -cn-> CC ) )
fourierdlem82.7  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
fourierdlem82.8  |-  ( ph  ->  R  e.  ( F lim
CC  A ) )
fourierdlem82.9  |-  ( ph  ->  X  e.  RR )
Assertion
Ref Expression
fourierdlem82  |-  ( ph  ->  S. ( A [,] B ) ( F `
 t )  _d t  =  S. ( ( A  -  X
) [,] ( B  -  X ) ) ( F `  ( X  +  t )
)  _d t )
Distinct variable groups:    t, A, x    t, B, x    x, F    t, G    x, L    x, R    t, X, x    ph, t, x
Allowed substitution hints:    R( t)    F( t)    G( x)    L( t)

Proof of Theorem fourierdlem82
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 fourierdlem82.2 . . . . 5  |-  ( ph  ->  A  e.  RR )
2 fourierdlem82.3 . . . . 5  |-  ( ph  ->  B  e.  RR )
3 fourierdlem82.9 . . . . 5  |-  ( ph  ->  X  e.  RR )
4 fourierdlem82.4 . . . . . 6  |-  ( ph  ->  A  <  B )
51, 2, 4ltled 9780 . . . . 5  |-  ( ph  ->  A  <_  B )
61, 2, 3, 5lesub1dd 10226 . . . 4  |-  ( ph  ->  ( A  -  X
)  <_  ( B  -  X ) )
76ditgpos 22804 . . 3  |-  ( ph  ->  S__ [ ( A  -  X )  -> 
( B  -  X
) ] ( G `
 ( X  +  t ) )  _d t  =  S. ( ( A  -  X
) (,) ( B  -  X ) ) ( G `  ( X  +  t )
)  _d t )
8 fourierdlem82.1 . . . . . . 7  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) ) ) )
9 iftrue 3886 . . . . . . . . . . . 12  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  R )
109adantl 468 . . . . . . . . . . 11  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  R )
11 iftrue 3886 . . . . . . . . . . . 12  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
1211adantl 468 . . . . . . . . . . 11  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
1310, 12eqtr4d 2487 . . . . . . . . . 10  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( F `  x
) ) ) )
1413adantlr 720 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( F `  x
) ) ) )
15 iffalse 3889 . . . . . . . . . . . . 13  |-  ( -.  x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B
) ) `  x
) ) )
16 iftrue 3886 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) )  =  L )
1715, 16sylan9eq 2504 . . . . . . . . . . . 12  |-  ( ( -.  x  =  A  /\  x  =  B )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  L )
1817adantll 719 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  L )
19 iffalse 3889 . . . . . . . . . . . . 13  |-  ( -.  x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
20 iftrue 3886 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  L )
2119, 20sylan9eq 2504 . . . . . . . . . . . 12  |-  ( ( -.  x  =  A  /\  x  =  B )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  L )
2221adantll 719 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  L )
2318, 22eqtr4d 2487 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( F `  x
) ) ) )
24 iffalse 3889 . . . . . . . . . . . 12  |-  ( -.  x  =  B  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) )  =  ( ( F  |`  ( A (,) B
) ) `  x
) )
2524adantl 468 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) )  =  ( ( F  |`  ( A (,) B ) ) `  x ) )
2615ad2antlr 732 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B
) ) `  x
) ) )
27 iffalse 3889 . . . . . . . . . . . . 13  |-  ( -.  x  =  B  ->  if ( x  =  B ,  L ,  ( F `  x ) )  =  ( F `
 x ) )
2827adantl 468 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  ( F `  x ) )
2919ad2antlr 732 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
301rexrd 9687 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  RR* )
3130ad3antrrr 735 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  e.  RR* )
322rexrd 9687 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  RR* )
3332ad3antrrr 735 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  B  e.  RR* )
341adantr 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
352adantr 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
36 simpr 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
37 eliccre 37597 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR )
3834, 35, 36, 37syl3anc 1267 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
3938ad2antrr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  RR )
401ad2antrr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  e.  RR )
4138adantr 467 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  e.  RR )
42 elicc2 11696 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4334, 35, 42syl2anc 666 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  ( A [,] B
)  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) ) )
4436, 43mpbid 214 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
4544simp2d 1020 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
4645adantr 467 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <_  x )
47 neqne 37368 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  =  A  ->  x  =/=  A )
4847adantl 468 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  =/=  A )
4940, 41, 46, 48leneltd 9786 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <  x )
5049adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  <  x )
5138adantr 467 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  e.  RR )
522ad2antrr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  e.  RR )
5344simp3d 1021 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
5453adantr 467 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <_  B )
55 nesym 2679 . . . . . . . . . . . . . . . . . 18  |-  ( B  =/=  x  <->  -.  x  =  B )
5655biimpri 210 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  =  B  ->  B  =/=  x )
5756adantl 468 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  =/=  x )
5851, 52, 54, 57leneltd 9786 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <  B )
5958adantlr 720 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  <  B )
6031, 33, 39, 50, 59eliood 37589 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  ( A (,) B
) )
61 fvres 5877 . . . . . . . . . . . . 13  |-  ( x  e.  ( A (,) B )  ->  (
( F  |`  ( A (,) B ) ) `
 x )  =  ( F `  x
) )
6260, 61syl 17 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  (
( F  |`  ( A (,) B ) ) `
 x )  =  ( F `  x
) )
6328, 29, 623eqtr4d 2494 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  ( ( F  |`  ( A (,) B
) ) `  x
) )
6425, 26, 633eqtr4d 2494 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( F `  x
) ) ) )
6523, 64pm2.61dan 799 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) ) )  =  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) ) )
6614, 65pm2.61dan 799 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( F `  x
) ) ) )
6766mpteq2dva 4488 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) ) ) )  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) ) )
688, 67syl5eq 2496 . . . . . 6  |-  ( ph  ->  G  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) ) ) )
6968adantr 467 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  G  =  ( x  e.  ( A [,] B
)  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L , 
( F `  x
) ) ) ) )
70 eqeq1 2454 . . . . . . 7  |-  ( x  =  ( X  +  t )  ->  (
x  =  A  <->  ( X  +  t )  =  A ) )
71 eqeq1 2454 . . . . . . . 8  |-  ( x  =  ( X  +  t )  ->  (
x  =  B  <->  ( X  +  t )  =  B ) )
72 fveq2 5863 . . . . . . . 8  |-  ( x  =  ( X  +  t )  ->  ( F `  x )  =  ( F `  ( X  +  t
) ) )
7371, 72ifbieq2d 3905 . . . . . . 7  |-  ( x  =  ( X  +  t )  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  if ( ( X  +  t )  =  B ,  L ,  ( F `  ( X  +  t
) ) ) )
7470, 73ifbieq2d 3905 . . . . . 6  |-  ( x  =  ( X  +  t )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  if ( ( X  +  t )  =  A ,  R ,  if ( ( X  +  t )  =  B ,  L , 
( F `  ( X  +  t )
) ) ) )
751adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  A  e.  RR )
76 simpr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )
771, 3resubcld 10044 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  -  X
)  e.  RR )
7877rexrd 9687 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  -  X
)  e.  RR* )
7978adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( A  -  X )  e.  RR* )
802, 3resubcld 10044 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( B  -  X
)  e.  RR )
8180rexrd 9687 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( B  -  X
)  e.  RR* )
8281adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( B  -  X )  e.  RR* )
83 elioo2 11674 . . . . . . . . . . . . . 14  |-  ( ( ( A  -  X
)  e.  RR*  /\  ( B  -  X )  e.  RR* )  ->  (
t  e.  ( ( A  -  X ) (,) ( B  -  X ) )  <->  ( t  e.  RR  /\  ( A  -  X )  < 
t  /\  t  <  ( B  -  X ) ) ) )
8479, 82, 83syl2anc 666 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  (
t  e.  ( ( A  -  X ) (,) ( B  -  X ) )  <->  ( t  e.  RR  /\  ( A  -  X )  < 
t  /\  t  <  ( B  -  X ) ) ) )
8576, 84mpbid 214 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  (
t  e.  RR  /\  ( A  -  X
)  <  t  /\  t  <  ( B  -  X ) ) )
8685simp2d 1020 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( A  -  X )  <  t )
873adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  X  e.  RR )
8885simp1d 1019 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  t  e.  RR )
8975, 87, 88ltsubadd2d 10208 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  (
( A  -  X
)  <  t  <->  A  <  ( X  +  t ) ) )
9086, 89mpbid 214 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  A  <  ( X  +  t ) )
9175, 90gtned 9767 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  =/=  A )
9291neneqd 2628 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  -.  ( X  +  t
)  =  A )
9392iffalsed 3891 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  if ( ( X  +  t )  =  A ,  R ,  if ( ( X  +  t )  =  B ,  L ,  ( F `  ( X  +  t ) ) ) )  =  if ( ( X  +  t )  =  B ,  L ,  ( F `  ( X  +  t ) ) ) )
9487, 88readdcld 9667 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  e.  RR )
9585simp3d 1021 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  t  <  ( B  -  X
) )
962adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  B  e.  RR )
9787, 88, 96ltaddsub2d 10211 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  (
( X  +  t )  <  B  <->  t  <  ( B  -  X ) ) )
9895, 97mpbird 236 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  <  B )
9994, 98ltned 9768 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  =/=  B )
10099neneqd 2628 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  -.  ( X  +  t
)  =  B )
101100iffalsed 3891 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  if ( ( X  +  t )  =  B ,  L ,  ( F `  ( X  +  t ) ) )  =  ( F `
 ( X  +  t ) ) )
10293, 101eqtrd 2484 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  if ( ( X  +  t )  =  A ,  R ,  if ( ( X  +  t )  =  B ,  L ,  ( F `  ( X  +  t ) ) ) )  =  ( F `  ( X  +  t ) ) )
10374, 102sylan9eqr 2506 . . . . 5  |-  ( ( ( ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  /\  x  =  ( X  +  t ) )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) )  =  ( F `
 ( X  +  t ) ) )
10475, 94, 90ltled 9780 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  A  <_  ( X  +  t ) )
10594, 96, 98ltled 9780 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  <_  B )
10675, 96, 94, 104, 105eliccd 37595 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  e.  ( A [,] B
) )
107 fourierdlem82.5 . . . . . . . 8  |-  ( ph  ->  F : ( A [,] B ) --> CC )
108 ffun 5729 . . . . . . . 8  |-  ( F : ( A [,] B ) --> CC  ->  Fun 
F )
109107, 108syl 17 . . . . . . 7  |-  ( ph  ->  Fun  F )
110109adantr 467 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  Fun  F )
111 fdm 5731 . . . . . . . . . 10  |-  ( F : ( A [,] B ) --> CC  ->  dom 
F  =  ( A [,] B ) )
112107, 111syl 17 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  ( A [,] B ) )
113112eqcomd 2456 . . . . . . . 8  |-  ( ph  ->  ( A [,] B
)  =  dom  F
)
114113adantr 467 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( A [,] B )  =  dom  F )
115106, 114eleqtrd 2530 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  e.  dom  F )
116 fvelrn 6013 . . . . . 6  |-  ( ( Fun  F  /\  ( X  +  t )  e.  dom  F )  -> 
( F `  ( X  +  t )
)  e.  ran  F
)
117110, 115, 116syl2anc 666 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( F `  ( X  +  t ) )  e.  ran  F )
11869, 103, 106, 117fvmptd 5952 . . . 4  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( G `  ( X  +  t ) )  =  ( F `  ( X  +  t
) ) )
119118itgeq2dv 22732 . . 3  |-  ( ph  ->  S. ( ( A  -  X ) (,) ( B  -  X
) ) ( G `
 ( X  +  t ) )  _d t  =  S. ( ( A  -  X
) (,) ( B  -  X ) ) ( F `  ( X  +  t )
)  _d t )
120 frn 5733 . . . . . . 7  |-  ( F : ( A [,] B ) --> CC  ->  ran 
F  C_  CC )
121107, 120syl 17 . . . . . 6  |-  ( ph  ->  ran  F  C_  CC )
122121adantr 467 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ran  F 
C_  CC )
123109adantr 467 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  Fun  F )
1241adantr 467 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  A  e.  RR )
1252adantr 467 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  B  e.  RR )
1263adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  X  e.  RR )
12777adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( A  -  X )  e.  RR )
12880adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( B  -  X )  e.  RR )
129 simpr 463 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )
130 eliccre 37597 . . . . . . . . . 10  |-  ( ( ( A  -  X
)  e.  RR  /\  ( B  -  X
)  e.  RR  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X
) ) )  -> 
t  e.  RR )
131127, 128, 129, 130syl3anc 1267 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  t  e.  RR )
132126, 131readdcld 9667 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( X  +  t )  e.  RR )
133 elicc2 11696 . . . . . . . . . . . 12  |-  ( ( ( A  -  X
)  e.  RR  /\  ( B  -  X
)  e.  RR )  ->  ( t  e.  ( ( A  -  X ) [,] ( B  -  X )
)  <->  ( t  e.  RR  /\  ( A  -  X )  <_ 
t  /\  t  <_  ( B  -  X ) ) ) )
134127, 128, 133syl2anc 666 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  (
t  e.  ( ( A  -  X ) [,] ( B  -  X ) )  <->  ( t  e.  RR  /\  ( A  -  X )  <_ 
t  /\  t  <_  ( B  -  X ) ) ) )
135129, 134mpbid 214 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  (
t  e.  RR  /\  ( A  -  X
)  <_  t  /\  t  <_  ( B  -  X ) ) )
136135simp2d 1020 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( A  -  X )  <_  t )
137124, 126, 131lesubadd2d 10209 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  (
( A  -  X
)  <_  t  <->  A  <_  ( X  +  t ) ) )
138136, 137mpbid 214 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  A  <_  ( X  +  t ) )
139135simp3d 1021 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  t  <_  ( B  -  X
) )
140126, 131, 125leaddsub2d 10212 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  (
( X  +  t )  <_  B  <->  t  <_  ( B  -  X ) ) )
141139, 140mpbird 236 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( X  +  t )  <_  B )
142124, 125, 132, 138, 141eliccd 37595 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( X  +  t )  e.  ( A [,] B
) )
143113adantr 467 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( A [,] B )  =  dom  F )
144142, 143eleqtrd 2530 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( X  +  t )  e.  dom  F )
145123, 144, 116syl2anc 666 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( F `  ( X  +  t ) )  e.  ran  F )
146122, 145sseldd 3432 . . . 4  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( F `  ( X  +  t ) )  e.  CC )
14777, 80, 146itgioo 22766 . . 3  |-  ( ph  ->  S. ( ( A  -  X ) (,) ( B  -  X
) ) ( F `
 ( X  +  t ) )  _d t  =  S. ( ( A  -  X
) [,] ( B  -  X ) ) ( F `  ( X  +  t )
)  _d t )
1487, 119, 1473eqtrrd 2489 . 2  |-  ( ph  ->  S. ( ( A  -  X ) [,] ( B  -  X
) ) ( F `
 ( X  +  t ) )  _d t  =  S__ [
( A  -  X
)  ->  ( B  -  X ) ] ( G `  ( X  +  t ) )  _d t )
149 nfv 1760 . . . 4  |-  F/ x ph
150 fourierdlem82.6 . . . 4  |-  ( ph  ->  ( F  |`  ( A (,) B ) )  e.  ( ( A (,) B ) -cn-> CC ) )
151 fourierdlem82.7 . . . . 5  |-  ( ph  ->  L  e.  ( F lim
CC  B ) )
1521, 2, 4, 107limcicciooub 37711 . . . . 5  |-  ( ph  ->  ( ( F  |`  ( A (,) B ) ) lim CC  B )  =  ( F lim CC  B ) )
153151, 152eleqtrrd 2531 . . . 4  |-  ( ph  ->  L  e.  ( ( F  |`  ( A (,) B ) ) lim CC  B ) )
154 fourierdlem82.8 . . . . 5  |-  ( ph  ->  R  e.  ( F lim
CC  A ) )
1551, 2, 4, 107limciccioolb 37695 . . . . 5  |-  ( ph  ->  ( ( F  |`  ( A (,) B ) ) lim CC  A )  =  ( F lim CC  A ) )
156154, 155eleqtrrd 2531 . . . 4  |-  ( ph  ->  R  e.  ( ( F  |`  ( A (,) B ) ) lim CC  A ) )
157149, 8, 1, 2, 150, 153, 156cncfiooicc 37766 . . 3  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> CC ) )
1581, 2, 5, 3, 157itgsbtaddcnst 37853 . 2  |-  ( ph  ->  S__ [ ( A  -  X )  -> 
( B  -  X
) ] ( G `
 ( X  +  t ) )  _d t  =  S__ [ A  ->  B ] ( G `  s )  _d s )
1595ditgpos 22804 . . 3  |-  ( ph  ->  S__ [ A  ->  B ] ( G `  s )  _d s  =  S. ( A (,) B ) ( G `  s )  _d s )
160 fveq2 5863 . . . . 5  |-  ( s  =  t  ->  ( G `  s )  =  ( G `  t ) )
161160cbvitgv 22727 . . . 4  |-  S. ( A (,) B ) ( G `  s
)  _d s  =  S. ( A (,) B ) ( G `
 t )  _d t
1628a1i 11 . . . . . 6  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  G  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) ) ) )
1631ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  A  e.  RR )
164 simplr 761 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
t  e.  ( A (,) B ) )
16530ad2antrr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  A  e.  RR* )
16632ad2antrr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  B  e.  RR* )
167 elioo2 11674 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
t  e.  ( A (,) B )  <->  ( t  e.  RR  /\  A  < 
t  /\  t  <  B ) ) )
168165, 166, 167syl2anc 666 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
( t  e.  ( A (,) B )  <-> 
( t  e.  RR  /\  A  <  t  /\  t  <  B ) ) )
169164, 168mpbid 214 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
( t  e.  RR  /\  A  <  t  /\  t  <  B ) )
170169simp2d 1020 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  A  <  t )
171 simpr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  x  =  t )
172170, 171breqtrrd 4428 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  A  <  x )
173163, 172gtned 9767 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  x  =/=  A )
174173neneqd 2628 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  -.  x  =  A
)
175174iffalsed 3891 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B
) ) `  x
) ) )
176169simp1d 1019 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
t  e.  RR )
177171, 176eqeltrd 2528 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  x  e.  RR )
178169simp3d 1021 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
t  <  B )
179171, 178eqbrtrd 4422 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  x  <  B )
180177, 179ltned 9768 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  x  =/=  B )
181180neneqd 2628 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  -.  x  =  B
)
182181iffalsed 3891 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) )  =  ( ( F  |`  ( A (,) B
) ) `  x
) )
183171, 164eqeltrd 2528 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  x  e.  ( A (,) B ) )
184183, 61syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
( ( F  |`  ( A (,) B ) ) `  x )  =  ( F `  x ) )
185 fveq2 5863 . . . . . . . . 9  |-  ( x  =  t  ->  ( F `  x )  =  ( F `  t ) )
186185adantl 468 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
( F `  x
)  =  ( F `
 t ) )
187184, 186eqtrd 2484 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
( ( F  |`  ( A (,) B ) ) `  x )  =  ( F `  t ) )
188175, 182, 1873eqtrd 2488 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  ( F `  t ) )
189 ioossicc 11717 . . . . . . 7  |-  ( A (,) B )  C_  ( A [,] B )
190 simpr 463 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  t  e.  ( A (,) B ) )
191189, 190sseldi 3429 . . . . . 6  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  t  e.  ( A [,] B ) )
192109adantr 467 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  Fun  F )
193113adantr 467 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( A [,] B )  =  dom  F )
194191, 193eleqtrd 2530 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  t  e.  dom  F )
195 fvelrn 6013 . . . . . . 7  |-  ( ( Fun  F  /\  t  e.  dom  F )  -> 
( F `  t
)  e.  ran  F
)
196192, 194, 195syl2anc 666 . . . . . 6  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( F `  t )  e.  ran  F )
197162, 188, 191, 196fvmptd 5952 . . . . 5  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( G `  t )  =  ( F `  t ) )
198197itgeq2dv 22732 . . . 4  |-  ( ph  ->  S. ( A (,) B ) ( G `
 t )  _d t  =  S. ( A (,) B ) ( F `  t
)  _d t )
199161, 198syl5eq 2496 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( G `
 s )  _d s  =  S. ( A (,) B ) ( F `  t
)  _d t )
200107ffvelrnda 6020 . . . 4  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( F `  t )  e.  CC )
2011, 2, 200itgioo 22766 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( F `
 t )  _d t  =  S. ( A [,] B ) ( F `  t
)  _d t )
202159, 199, 2013eqtrd 2488 . 2  |-  ( ph  ->  S__ [ A  ->  B ] ( G `  s )  _d s  =  S. ( A [,] B ) ( F `  t )  _d t )
203148, 158, 2023eqtrrd 2489 1  |-  ( ph  ->  S. ( A [,] B ) ( F `
 t )  _d t  =  S. ( ( A  -  X
) [,] ( B  -  X ) ) ( F `  ( X  +  t )
)  _d t )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621    C_ wss 3403   ifcif 3880   class class class wbr 4401    |-> cmpt 4460   dom cdm 4833   ran crn 4834    |` cres 4835   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535    + caddc 9539   RR*cxr 9671    < clt 9672    <_ cle 9673    - cmin 9857   (,)cioo 11632   [,]cicc 11635   -cn->ccncf 21901   S.citg 22569   S__cdit 22794   lim CC climc 22810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cc 8862  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-disj 4373  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-ofr 6529  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-omul 7184  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-acn 8373  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-lp 20145  df-perf 20146  df-cn 20236  df-cnp 20237  df-haus 20324  df-cmp 20395  df-tx 20570  df-hmeo 20763  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-xms 21328  df-ms 21329  df-tms 21330  df-cncf 21903  df-ovol 22409  df-vol 22411  df-mbf 22570  df-itg1 22571  df-itg2 22572  df-ibl 22573  df-itg 22574  df-0p 22621  df-ditg 22795  df-limc 22814  df-dv 22815
This theorem is referenced by:  fourierdlem93  38057
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