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Theorem fourierdlem82 38164
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 9800 . . . . 5  |-  ( ph  ->  A  <_  B )
61, 2, 3, 5lesub1dd 10250 . . . 4  |-  ( ph  ->  ( A  -  X
)  <_  ( B  -  X ) )
76ditgpos 22890 . . 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 3878 . . . . . . . . . . . 12  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  R )
109adantl 473 . . . . . . . . . . 11  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  R )
11 iftrue 3878 . . . . . . . . . . . 12  |-  ( x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
1211adantl 473 . . . . . . . . . . 11  |-  ( (
ph  /\  x  =  A )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  R )
1310, 12eqtr4d 2508 . . . . . . . . . 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 729 . . . . . . . . 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 3881 . . . . . . . . . . . . 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 3878 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) )  =  L )
1715, 16sylan9eq 2525 . . . . . . . . . . . 12  |-  ( ( -.  x  =  A  /\  x  =  B )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  L )
1817adantll 728 . . . . . . . . . . 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 3881 . . . . . . . . . . . . 13  |-  ( -.  x  =  A  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
20 iftrue 3878 . . . . . . . . . . . . 13  |-  ( x  =  B  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  L )
2119, 20sylan9eq 2525 . . . . . . . . . . . 12  |-  ( ( -.  x  =  A  /\  x  =  B )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  L )
2221adantll 728 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  x  =  B )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) )  =  L )
2318, 22eqtr4d 2508 . . . . . . . . . 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 3881 . . . . . . . . . . . 12  |-  ( -.  x  =  B  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) )  =  ( ( F  |`  ( A (,) B
) ) `  x
) )
2524adantl 473 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) )  =  ( ( F  |`  ( A (,) B ) ) `  x ) )
2615ad2antlr 741 . . . . . . . . . . 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 3881 . . . . . . . . . . . . 13  |-  ( -.  x  =  B  ->  if ( x  =  B ,  L ,  ( F `  x ) )  =  ( F `
 x ) )
2827adantl 473 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  ( F `  x ) )
2919ad2antlr 741 . . . . . . . . . . . 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 9708 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  RR* )
3130ad3antrrr 744 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  e.  RR* )
322rexrd 9708 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  RR* )
3332ad3antrrr 744 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  B  e.  RR* )
341adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  e.  RR )
352adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
36 simpr 468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  ( A [,] B ) )
37 eliccre 37699 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  x  e.  ( A [,] B
) )  ->  x  e.  RR )
3834, 35, 36, 37syl3anc 1292 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
3938ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  RR )
401ad2antrr 740 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  e.  RR )
4138adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  e.  RR )
42 elicc2 11724 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4334, 35, 42syl2anc 673 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  ( A [,] B
)  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) ) )
4436, 43mpbid 215 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
4544simp2d 1043 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
4645adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <_  x )
47 neqne 2651 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  =  A  ->  x  =/=  A )
4847adantl 473 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  x  =/=  A )
4940, 41, 46, 48leneltd 9806 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  A )  ->  A  <  x )
5049adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  A  <  x )
5138adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  e.  RR )
522ad2antrr 740 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  e.  RR )
5344simp3d 1044 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
5453adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <_  B )
55 nesym 2699 . . . . . . . . . . . . . . . . . 18  |-  ( B  =/=  x  <->  -.  x  =  B )
5655biimpri 211 . . . . . . . . . . . . . . . . 17  |-  ( -.  x  =  B  ->  B  =/=  x )
5756adantl 473 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  B  =/=  x )
5851, 52, 54, 57leneltd 9806 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  -.  x  =  B )  ->  x  <  B )
5958adantlr 729 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  <  B )
6031, 33, 39, 50, 59eliood 37691 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  x  e.  ( A [,] B ) )  /\  -.  x  =  A
)  /\  -.  x  =  B )  ->  x  e.  ( A (,) B
) )
61 fvres 5893 . . . . . . . . . . . . 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 2515 . . . . . . . . . . 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 2515 . . . . . . . . . 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 808 . . . . . . . . 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 808 . . . . . . . 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 4482 . . . . . . 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 2517 . . . . . 6  |-  ( ph  ->  G  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) ) ) )
6968adantr 472 . . . . 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 2475 . . . . . . 7  |-  ( x  =  ( X  +  t )  ->  (
x  =  A  <->  ( X  +  t )  =  A ) )
71 eqeq1 2475 . . . . . . . 8  |-  ( x  =  ( X  +  t )  ->  (
x  =  B  <->  ( X  +  t )  =  B ) )
72 fveq2 5879 . . . . . . . 8  |-  ( x  =  ( X  +  t )  ->  ( F `  x )  =  ( F `  ( X  +  t
) ) )
7371, 72ifbieq2d 3897 . . . . . . 7  |-  ( x  =  ( X  +  t )  ->  if ( x  =  B ,  L ,  ( F `
 x ) )  =  if ( ( X  +  t )  =  B ,  L ,  ( F `  ( X  +  t
) ) ) )
7470, 73ifbieq2d 3897 . . . . . 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 472 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  A  e.  RR )
76 simpr 468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )
771, 3resubcld 10068 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  -  X
)  e.  RR )
7877rexrd 9708 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  -  X
)  e.  RR* )
7978adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( A  -  X )  e.  RR* )
802, 3resubcld 10068 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( B  -  X
)  e.  RR )
8180rexrd 9708 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( B  -  X
)  e.  RR* )
8281adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( B  -  X )  e.  RR* )
83 elioo2 11702 . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . 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 215 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  (
t  e.  RR  /\  ( A  -  X
)  <  t  /\  t  <  ( B  -  X ) ) )
8685simp2d 1043 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( A  -  X )  <  t )
873adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  X  e.  RR )
8885simp1d 1042 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  t  e.  RR )
8975, 87, 88ltsubadd2d 10232 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  (
( A  -  X
)  <  t  <->  A  <  ( X  +  t ) ) )
9086, 89mpbid 215 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  A  <  ( X  +  t ) )
9175, 90gtned 9787 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  =/=  A )
9291neneqd 2648 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  -.  ( X  +  t
)  =  A )
9392iffalsed 3883 . . . . . . 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 9688 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  e.  RR )
9585simp3d 1044 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  t  <  ( B  -  X
) )
962adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  B  e.  RR )
9787, 88, 96ltaddsub2d 10235 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  (
( X  +  t )  <  B  <->  t  <  ( B  -  X ) ) )
9895, 97mpbird 240 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  <  B )
9994, 98ltned 9788 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  =/=  B )
10099neneqd 2648 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  -.  ( X  +  t
)  =  B )
101100iffalsed 3883 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  if ( ( X  +  t )  =  B ,  L ,  ( F `  ( X  +  t ) ) )  =  ( F `
 ( X  +  t ) ) )
10293, 101eqtrd 2505 . . . . . 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 2527 . . . . 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 9800 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  A  <_  ( X  +  t ) )
10594, 96, 98ltled 9800 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  <_  B )
10675, 96, 94, 104, 105eliccd 37697 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  e.  ( A [,] B
) )
107 fourierdlem82.5 . . . . . . . 8  |-  ( ph  ->  F : ( A [,] B ) --> CC )
108 ffun 5742 . . . . . . . 8  |-  ( F : ( A [,] B ) --> CC  ->  Fun 
F )
109107, 108syl 17 . . . . . . 7  |-  ( ph  ->  Fun  F )
110109adantr 472 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  Fun  F )
111 fdm 5745 . . . . . . . . . 10  |-  ( F : ( A [,] B ) --> CC  ->  dom 
F  =  ( A [,] B ) )
112107, 111syl 17 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  ( A [,] B ) )
113112eqcomd 2477 . . . . . . . 8  |-  ( ph  ->  ( A [,] B
)  =  dom  F
)
114113adantr 472 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( A [,] B )  =  dom  F )
115106, 114eleqtrd 2551 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( X  +  t )  e.  dom  F )
116 fvelrn 6030 . . . . . 6  |-  ( ( Fun  F  /\  ( X  +  t )  e.  dom  F )  -> 
( F `  ( X  +  t )
)  e.  ran  F
)
117110, 115, 116syl2anc 673 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( F `  ( X  +  t ) )  e.  ran  F )
11869, 103, 106, 117fvmptd 5969 . . . 4  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) (,) ( B  -  X )
) )  ->  ( G `  ( X  +  t ) )  =  ( F `  ( X  +  t
) ) )
119118itgeq2dv 22818 . . 3  |-  ( ph  ->  S. ( ( A  -  X ) (,) ( B  -  X
) ) ( G `
 ( X  +  t ) )  _d t  =  S. ( ( A  -  X
) (,) ( B  -  X ) ) ( F `  ( X  +  t )
)  _d t )
120 frn 5747 . . . . . . 7  |-  ( F : ( A [,] B ) --> CC  ->  ran 
F  C_  CC )
121107, 120syl 17 . . . . . 6  |-  ( ph  ->  ran  F  C_  CC )
122121adantr 472 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ran  F 
C_  CC )
123109adantr 472 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  Fun  F )
1241adantr 472 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  A  e.  RR )
1252adantr 472 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  B  e.  RR )
1263adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  X  e.  RR )
12777adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( A  -  X )  e.  RR )
12880adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( B  -  X )  e.  RR )
129 simpr 468 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )
130 eliccre 37699 . . . . . . . . . 10  |-  ( ( ( A  -  X
)  e.  RR  /\  ( B  -  X
)  e.  RR  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X
) ) )  -> 
t  e.  RR )
131127, 128, 129, 130syl3anc 1292 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  t  e.  RR )
132126, 131readdcld 9688 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( X  +  t )  e.  RR )
133 elicc2 11724 . . . . . . . . . . . 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 673 . . . . . . . . . . 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 215 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  (
t  e.  RR  /\  ( A  -  X
)  <_  t  /\  t  <_  ( B  -  X ) ) )
136135simp2d 1043 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( A  -  X )  <_  t )
137124, 126, 131lesubadd2d 10233 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  (
( A  -  X
)  <_  t  <->  A  <_  ( X  +  t ) ) )
138136, 137mpbid 215 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  A  <_  ( X  +  t ) )
139135simp3d 1044 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  t  <_  ( B  -  X
) )
140126, 131, 125leaddsub2d 10236 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  (
( X  +  t )  <_  B  <->  t  <_  ( B  -  X ) ) )
141139, 140mpbird 240 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( X  +  t )  <_  B )
142124, 125, 132, 138, 141eliccd 37697 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( X  +  t )  e.  ( A [,] B
) )
143113adantr 472 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( A [,] B )  =  dom  F )
144142, 143eleqtrd 2551 . . . . . 6  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( X  +  t )  e.  dom  F )
145123, 144, 116syl2anc 673 . . . . 5  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( F `  ( X  +  t ) )  e.  ran  F )
146122, 145sseldd 3419 . . . 4  |-  ( (
ph  /\  t  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( F `  ( X  +  t ) )  e.  CC )
14777, 80, 146itgioo 22852 . . 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 2510 . 2  |-  ( ph  ->  S. ( ( A  -  X ) [,] ( B  -  X
) ) ( F `
 ( X  +  t ) )  _d t  =  S__ [
( A  -  X
)  ->  ( B  -  X ) ] ( G `  ( X  +  t ) )  _d t )
149 nfv 1769 . . . 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 37814 . . . . 5  |-  ( ph  ->  ( ( F  |`  ( A (,) B ) ) lim CC  B )  =  ( F lim CC  B ) )
153151, 152eleqtrrd 2552 . . . 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 37798 . . . . 5  |-  ( ph  ->  ( ( F  |`  ( A (,) B ) ) lim CC  A )  =  ( F lim CC  A ) )
156154, 155eleqtrrd 2552 . . . 4  |-  ( ph  ->  R  e.  ( ( F  |`  ( A (,) B ) ) lim CC  A ) )
157149, 8, 1, 2, 150, 153, 156cncfiooicc 37869 . . 3  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> CC ) )
1581, 2, 5, 3, 157itgsbtaddcnst 37956 . 2  |-  ( ph  ->  S__ [ ( A  -  X )  -> 
( B  -  X
) ] ( G `
 ( X  +  t ) )  _d t  =  S__ [ A  ->  B ] ( G `  s )  _d s )
1595ditgpos 22890 . . 3  |-  ( ph  ->  S__ [ A  ->  B ] ( G `  s )  _d s  =  S. ( A (,) B ) ( G `  s )  _d s )
160 fveq2 5879 . . . . 5  |-  ( s  =  t  ->  ( G `  s )  =  ( G `  t ) )
161160cbvitgv 22813 . . . 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 740 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  A  e.  RR )
164 simplr 770 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
t  e.  ( A (,) B ) )
16530ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  A  e.  RR* )
16632ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  B  e.  RR* )
167 elioo2 11702 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
t  e.  ( A (,) B )  <->  ( t  e.  RR  /\  A  < 
t  /\  t  <  B ) ) )
168165, 166, 167syl2anc 673 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
( t  e.  ( A (,) B )  <-> 
( t  e.  RR  /\  A  <  t  /\  t  <  B ) ) )
169164, 168mpbid 215 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
( t  e.  RR  /\  A  <  t  /\  t  <  B ) )
170169simp2d 1043 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  A  <  t )
171 simpr 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  x  =  t )
172170, 171breqtrrd 4422 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  A  <  x )
173163, 172gtned 9787 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  x  =/=  A )
174173neneqd 2648 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  -.  x  =  A
)
175174iffalsed 3883 . . . . . . 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 1042 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
t  e.  RR )
177171, 176eqeltrd 2549 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  x  e.  RR )
178169simp3d 1044 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
t  <  B )
179171, 178eqbrtrd 4416 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  x  <  B )
180177, 179ltned 9788 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  x  =/=  B )
181180neneqd 2648 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  -.  x  =  B
)
182181iffalsed 3883 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `
 x ) )  =  ( ( F  |`  ( A (,) B
) ) `  x
) )
183171, 164eqeltrd 2549 . . . . . . . . 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 5879 . . . . . . . . 9  |-  ( x  =  t  ->  ( F `  x )  =  ( F `  t ) )
186185adantl 473 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
( F `  x
)  =  ( F `
 t ) )
187184, 186eqtrd 2505 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  -> 
( ( F  |`  ( A (,) B ) ) `  x )  =  ( F `  t ) )
188175, 182, 1873eqtrd 2509 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( A (,) B
) )  /\  x  =  t )  ->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( ( F  |`  ( A (,) B ) ) `  x ) ) )  =  ( F `  t ) )
189 ioossicc 11745 . . . . . . 7  |-  ( A (,) B )  C_  ( A [,] B )
190 simpr 468 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  t  e.  ( A (,) B ) )
191189, 190sseldi 3416 . . . . . 6  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  t  e.  ( A [,] B ) )
192109adantr 472 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  Fun  F )
193113adantr 472 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( A [,] B )  =  dom  F )
194191, 193eleqtrd 2551 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  t  e.  dom  F )
195 fvelrn 6030 . . . . . . 7  |-  ( ( Fun  F  /\  t  e.  dom  F )  -> 
( F `  t
)  e.  ran  F
)
196192, 194, 195syl2anc 673 . . . . . 6  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( F `  t )  e.  ran  F )
197162, 188, 191, 196fvmptd 5969 . . . . 5  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( G `  t )  =  ( F `  t ) )
198197itgeq2dv 22818 . . . 4  |-  ( ph  ->  S. ( A (,) B ) ( G `
 t )  _d t  =  S. ( A (,) B ) ( F `  t
)  _d t )
199161, 198syl5eq 2517 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( G `
 s )  _d s  =  S. ( A (,) B ) ( F `  t
)  _d t )
200107ffvelrnda 6037 . . . 4  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( F `  t )  e.  CC )
2011, 2, 200itgioo 22852 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( F `
 t )  _d t  =  S. ( A [,] B ) ( F `  t
)  _d t )
202159, 199, 2013eqtrd 2509 . 2  |-  ( ph  ->  S__ [ A  ->  B ] ( G `  s )  _d s  =  S. ( A [,] B ) ( F `  t )  _d t )
203148, 158, 2023eqtrrd 2510 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641    C_ wss 3390   ifcif 3872   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840    |` cres 4841   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556    + caddc 9560   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   (,)cioo 11660   [,]cicc 11663   -cn->ccncf 21986   S.citg 22655   S__cdit 22880   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-fbas 19044  df-fg 19045  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-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  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-ditg 22881  df-limc 22900  df-dv 22901
This theorem is referenced by:  fourierdlem93  38175
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