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Theorem ftc2nc 32090
Description: Choice-free proof of ftc2 23075. (Contributed by Brendan Leahy, 19-Jun-2018.)
Hypotheses
Ref Expression
ftc2nc.a  |-  ( ph  ->  A  e.  RR )
ftc2nc.b  |-  ( ph  ->  B  e.  RR )
ftc2nc.le  |-  ( ph  ->  A  <_  B )
ftc2nc.c  |-  ( ph  ->  ( RR  _D  F
)  e.  ( ( A (,) B )
-cn-> CC ) )
ftc2nc.i  |-  ( ph  ->  ( RR  _D  F
)  e.  L^1 )
ftc2nc.f  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
Assertion
Ref Expression
ftc2nc  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
Distinct variable groups:    t, A    t, B    t, F    ph, t

Proof of Theorem ftc2nc
Dummy variables  s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ftc2nc.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
21rexrd 9708 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
3 ftc2nc.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
43rexrd 9708 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
5 ftc2nc.le . . . . . 6  |-  ( ph  ->  A  <_  B )
6 ubicc2 11775 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
72, 4, 5, 6syl3anc 1292 . . . . 5  |-  ( ph  ->  B  e.  ( A [,] B ) )
8 fvex 5889 . . . . . 6  |-  ( ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) `  A )  e.  _V
98fvconst2 6136 . . . . 5  |-  ( B  e.  ( A [,] B )  ->  (
( ( A [,] B )  X.  {
( ( x  e.  ( A [,] B
)  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) ) ) `  A
) } ) `  B )  =  ( ( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) ) `  A
) )
107, 9syl 17 . . . 4  |-  ( ph  ->  ( ( ( A [,] B )  X. 
{ ( ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) ) ) `
 A ) } ) `  B )  =  ( ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) ) ) `
 A ) )
11 eqid 2471 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1211subcn 21976 . . . . . . . . 9  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
1312a1i 11 . . . . . . . 8  |-  ( ph  ->  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
14 eqid 2471 . . . . . . . . 9  |-  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t )  =  ( x  e.  ( A [,] B
)  |->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t )
15 ssid 3437 . . . . . . . . . 10  |-  ( A (,) B )  C_  ( A (,) B )
1615a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( A (,) B
)  C_  ( A (,) B ) )
17 ioossre 11721 . . . . . . . . . 10  |-  ( A (,) B )  C_  RR
1817a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( A (,) B
)  C_  RR )
19 ftc2nc.i . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  e.  L^1 )
20 ftc2nc.c . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  F
)  e.  ( ( A (,) B )
-cn-> CC ) )
21 cncff 22003 . . . . . . . . . 10  |-  ( ( RR  _D  F )  e.  ( ( A (,) B ) -cn-> CC )  ->  ( RR  _D  F ) : ( A (,) B ) --> CC )
2220, 21syl 17 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> CC )
23 ioof 11757 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
24 ffun 5742 . . . . . . . . . . . . 13  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
2523, 24ax-mp 5 . . . . . . . . . . . 12  |-  Fun  (,)
26 fvelima 5931 . . . . . . . . . . . 12  |-  ( ( Fun  (,)  /\  s  e.  ( (,) " (
( A [,] B
)  X.  ( A [,] B ) ) ) )  ->  E. x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) ( (,) `  x
)  =  s )
2725, 26mpan 684 . . . . . . . . . . 11  |-  ( s  e.  ( (,) " (
( A [,] B
)  X.  ( A [,] B ) ) )  ->  E. x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) ( (,) `  x
)  =  s )
28 1st2nd2 6849 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2928fveq2d 5883 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( (,) `  x )  =  ( (,) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
30 df-ov 6311 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  =  ( (,) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
3129, 30syl6eqr 2523 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( (,) `  x )  =  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )
3231eqeq1d 2473 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  (
( (,) `  x
)  =  s  <->  ( ( 1st `  x ) (,) ( 2nd `  x
) )  =  s ) )
3332adantl 473 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( (,) `  x )  =  s  <->  ( ( 1st `  x ) (,) ( 2nd `  x ) )  =  s ) )
342, 4jca 541 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( A  e.  RR*  /\  B  e.  RR* )
)
3534adantr 472 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( A  e.  RR*  /\  B  e. 
RR* ) )
36 xp1st 6842 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( 1st `  x )  e.  ( A [,] B
) )
37 elicc1 11705 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 1st `  x
)  e.  ( A [,] B )  <->  ( ( 1st `  x )  e. 
RR*  /\  A  <_  ( 1st `  x )  /\  ( 1st `  x
)  <_  B )
) )
382, 4, 37syl2anc 673 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( ( 1st `  x
)  e.  ( A [,] B )  <->  ( ( 1st `  x )  e. 
RR*  /\  A  <_  ( 1st `  x )  /\  ( 1st `  x
)  <_  B )
) )
3938biimpa 492 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  ( 1st `  x )  e.  ( A [,] B ) )  ->  ( ( 1st `  x )  e. 
RR*  /\  A  <_  ( 1st `  x )  /\  ( 1st `  x
)  <_  B )
)
4039simp2d 1043 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( 1st `  x )  e.  ( A [,] B ) )  ->  A  <_  ( 1st `  x ) )
4136, 40sylan2 482 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  A  <_  ( 1st `  x ) )
42 xp2nd 6843 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( 2nd `  x )  e.  ( A [,] B
) )
43 iccleub 11715 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( 2nd `  x )  e.  ( A [,] B ) )  ->  ( 2nd `  x )  <_  B
)
44433expa 1231 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 2nd `  x
)  e.  ( A [,] B ) )  ->  ( 2nd `  x
)  <_  B )
4534, 42, 44syl2an 485 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( 2nd `  x )  <_  B
)
46 ioossioo 11751 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  ( 1st `  x )  /\  ( 2nd `  x )  <_  B ) )  ->  ( ( 1st `  x ) (,) ( 2nd `  x ) ) 
C_  ( A (,) B ) )
4735, 41, 45, 46syl12anc 1290 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  ( A (,) B ) )
4847sselda 3418 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  t  e.  ( A (,) B ) )
4922ffvelrnda 6037 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  CC )
5049adantlr 729 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  F
) `  t )  e.  CC )
5148, 50syldan 478 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  ( ( RR 
_D  F ) `  t )  e.  CC )
52 ioombl 22597 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  e.  dom  vol
5352a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( 1st `  x ) (,) ( 2nd `  x
) )  e.  dom  vol )
54 fvex 5889 . . . . . . . . . . . . . . . . . 18  |-  ( ( RR  _D  F ) `
 t )  e. 
_V
5554a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
5622feqmptd 5932 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
5756, 19eqeltrrd 2550 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( t  e.  ( A (,) B ) 
|->  ( ( RR  _D  F ) `  t
) )  e.  L^1 )
5857adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
5947, 53, 55, 58iblss 22841 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( ( RR  _D  F
) `  t )
)  e.  L^1 )
60 ax-resscn 9614 . . . . . . . . . . . . . . . . . . . . 21  |-  RR  C_  CC
61 ssid 3437 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  C_  CC
62 cncfss 22009 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( CC -cn-> RR )  C_  ( CC -cn-> CC ) )
6360, 61, 62mp2an 686 . . . . . . . . . . . . . . . . . . . 20  |-  ( CC
-cn-> RR )  C_  ( CC -cn-> CC )
64 abscncf 22011 . . . . . . . . . . . . . . . . . . . 20  |-  abs  e.  ( CC -cn-> RR )
6563, 64sselii 3415 . . . . . . . . . . . . . . . . . . 19  |-  abs  e.  ( CC -cn-> CC )
6665a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  abs  e.  ( CC -cn-> CC ) )
6756reseq1d 5110 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( 1st `  x
) (,) ( 2nd `  x ) ) )  =  ( ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) )  |`  ( ( 1st `  x
) (,) ( 2nd `  x ) ) ) )
6867adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( RR  _D  F )  |`  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  =  ( ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) )  |`  ( ( 1st `  x
) (,) ( 2nd `  x ) ) ) )
6947resmptd 5162 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( (
t  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  t )
)  |`  ( ( 1st `  x ) (,) ( 2nd `  x ) ) )  =  ( t  e.  ( ( 1st `  x ) (,) ( 2nd `  x ) ) 
|->  ( ( RR  _D  F ) `  t
) ) )
7068, 69eqtrd 2505 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( RR  _D  F )  |`  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  =  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( ( RR  _D  F
) `  t )
) )
7120adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( RR  _D  F )  e.  ( ( A (,) B
) -cn-> CC ) )
72 rescncf 22007 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  C_  ( A (,) B )  ->  ( ( RR 
_D  F )  e.  ( ( A (,) B ) -cn-> CC )  ->  ( ( RR 
_D  F )  |`  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  e.  ( ( ( 1st `  x ) (,) ( 2nd `  x
) ) -cn-> CC ) ) )
7347, 71, 72sylc 61 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( RR  _D  F )  |`  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  e.  ( ( ( 1st `  x ) (,) ( 2nd `  x
) ) -cn-> CC ) )
7470, 73eqeltrrd 2550 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( ( RR  _D  F
) `  t )
)  e.  ( ( ( 1st `  x
) (,) ( 2nd `  x ) ) -cn-> CC ) )
7566, 74cncfmpt1f 22023 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( abs `  ( ( RR  _D  F ) `
 t ) ) )  e.  ( ( ( 1st `  x
) (,) ( 2nd `  x ) ) -cn-> CC ) )
76 cnmbf 22694 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  e. 
dom  vol  /\  ( t  e.  ( ( 1st `  x ) (,) ( 2nd `  x ) ) 
|->  ( abs `  (
( RR  _D  F
) `  t )
) )  e.  ( ( ( 1st `  x
) (,) ( 2nd `  x ) ) -cn-> CC ) )  ->  (
t  e.  ( ( 1st `  x ) (,) ( 2nd `  x
) )  |->  ( abs `  ( ( RR  _D  F ) `  t
) ) )  e. MblFn
)
7752, 75, 76sylancr 676 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( abs `  ( ( RR  _D  F ) `
 t ) ) )  e. MblFn )
7851, 59itgcl 22820 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  S. (
( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
7978cjcld 13336 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( * `  S. ( ( 1st `  x ) (,) ( 2nd `  x ) ) ( ( RR  _D  F ) `  t
)  _d t )  e.  CC )
80 ioossre 11721 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  RR
8180, 60sstri 3427 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  CC
82 cncfmptc 22021 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( * `  S. ( ( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t )  e.  CC  /\  ( ( 1st `  x
) (,) ( 2nd `  x ) )  C_  CC  /\  CC  C_  CC )  ->  ( s  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( * `  S. ( ( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t ) )  e.  ( ( ( 1st `  x ) (,) ( 2nd `  x ) )
-cn-> CC ) )
8381, 61, 82mp3an23 1382 . . . . . . . . . . . . . . . . . . 19  |-  ( ( * `  S. ( ( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t )  e.  CC  ->  ( s  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( * `  S. ( ( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t ) )  e.  ( ( ( 1st `  x ) (,) ( 2nd `  x ) )
-cn-> CC ) )
8479, 83syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( s  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( * `  S. ( ( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t ) )  e.  ( ( ( 1st `  x ) (,) ( 2nd `  x ) )
-cn-> CC ) )
85 nfcv 2612 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ s
( ( RR  _D  F ) `  t
)
86 nfcsb1v 3365 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ t [_ s  /  t ]_ ( ( RR  _D  F ) `  t
)
87 csbeq1a 3358 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  s  ->  (
( RR  _D  F
) `  t )  =  [_ s  /  t ]_ ( ( RR  _D  F ) `  t
) )
8885, 86, 87cbvmpt 4487 . . . . . . . . . . . . . . . . . . 19  |-  ( t  e.  ( ( 1st `  x ) (,) ( 2nd `  x ) ) 
|->  ( ( RR  _D  F ) `  t
) )  =  ( s  e.  ( ( 1st `  x ) (,) ( 2nd `  x
) )  |->  [_ s  /  t ]_ (
( RR  _D  F
) `  t )
)
8988, 74syl5eqelr 2554 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( s  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  [_ s  /  t ]_ (
( RR  _D  F
) `  t )
)  e.  ( ( ( 1st `  x
) (,) ( 2nd `  x ) ) -cn-> CC ) )
9084, 89mulcncf 22476 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( s  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( ( * `  S. ( ( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t )  x.  [_ s  /  t ]_ (
( RR  _D  F
) `  t )
) )  e.  ( ( ( 1st `  x
) (,) ( 2nd `  x ) ) -cn-> CC ) )
91 cnmbf 22694 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  e. 
dom  vol  /\  ( s  e.  ( ( 1st `  x ) (,) ( 2nd `  x ) ) 
|->  ( ( * `  S. ( ( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t )  x.  [_ s  /  t ]_ (
( RR  _D  F
) `  t )
) )  e.  ( ( ( 1st `  x
) (,) ( 2nd `  x ) ) -cn-> CC ) )  ->  (
s  e.  ( ( 1st `  x ) (,) ( 2nd `  x
) )  |->  ( ( * `  S. ( ( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t )  x.  [_ s  /  t ]_ (
( RR  _D  F
) `  t )
) )  e. MblFn )
9252, 90, 91sylancr 676 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( s  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( ( * `  S. ( ( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t )  x.  [_ s  /  t ]_ (
( RR  _D  F
) `  t )
) )  e. MblFn )
9351, 59, 77, 92itgabsnc 32075 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( abs `  S. ( ( 1st `  x ) (,) ( 2nd `  x ) ) ( ( RR  _D  F ) `  t
)  _d t )  <_  S. ( ( 1st `  x ) (,) ( 2nd `  x
) ) ( abs `  ( ( RR  _D  F ) `  t
) )  _d t )
9451abscld 13575 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  ( abs `  (
( RR  _D  F
) `  t )
)  e.  RR )
9554a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  ( ( RR 
_D  F ) `  t )  e.  _V )
9695, 59, 77iblabsnc 32070 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( abs `  ( ( RR  _D  F ) `
 t ) ) )  e.  L^1 )
9751absge0d 13583 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  0  <_  ( abs `  ( ( RR 
_D  F ) `  t ) ) )
9894, 96, 97itgposval 22832 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  S. (
( 1st `  x
) (,) ( 2nd `  x ) ) ( abs `  ( ( RR  _D  F ) `
 t ) )  _d t  =  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  ( ( 1st `  x ) (,) ( 2nd `  x
) ) ,  ( abs `  ( ( RR  _D  F ) `
 t ) ) ,  0 ) ) ) )
9993, 98breqtrd 4420 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( abs `  S. ( ( 1st `  x ) (,) ( 2nd `  x ) ) ( ( RR  _D  F ) `  t
)  _d t )  <_  ( S.2 `  (
t  e.  RR  |->  if ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) ,  ( abs `  (
( RR  _D  F
) `  t )
) ,  0 ) ) ) )
100 itgeq1 22809 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  =  s  ->  S. (
( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t  =  S. s ( ( RR 
_D  F ) `  t )  _d t )
101100fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  =  s  ->  ( abs `  S. ( ( 1st `  x ) (,) ( 2nd `  x ) ) ( ( RR  _D  F ) `  t
)  _d t )  =  ( abs `  S. s ( ( RR 
_D  F ) `  t )  _d t ) )
102 eleq2 2538 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  =  s  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  <->  t  e.  s ) )
103102ifbid 3894 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  =  s  ->  if (
t  e.  ( ( 1st `  x ) (,) ( 2nd `  x
) ) ,  ( abs `  ( ( RR  _D  F ) `
 t ) ) ,  0 )  =  if ( t  e.  s ,  ( abs `  ( ( RR  _D  F ) `  t
) ) ,  0 ) )
104103mpteq2dv 4483 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  =  s  ->  ( t  e.  RR  |->  if ( t  e.  ( ( 1st `  x ) (,) ( 2nd `  x ) ) ,  ( abs `  (
( RR  _D  F
) `  t )
) ,  0 ) )  =  ( t  e.  RR  |->  if ( t  e.  s ,  ( abs `  (
( RR  _D  F
) `  t )
) ,  0 ) ) )
105104fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  =  s  ->  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) ,  ( abs `  (
( RR  _D  F
) `  t )
) ,  0 ) ) )  =  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  s ,  ( abs `  (
( RR  _D  F
) `  t )
) ,  0 ) ) ) )
106101, 105breq12d 4408 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  =  s  ->  ( ( abs `  S. ( ( 1st `  x ) (,) ( 2nd `  x
) ) ( ( RR  _D  F ) `
 t )  _d t )  <_  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  ( ( 1st `  x ) (,) ( 2nd `  x ) ) ,  ( abs `  (
( RR  _D  F
) `  t )
) ,  0 ) ) )  <->  ( abs `  S. s ( ( RR  _D  F ) `
 t )  _d t )  <_  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  s ,  ( abs `  ( ( RR  _D  F ) `
 t ) ) ,  0 ) ) ) ) )
10799, 106syl5ibcom 228 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( (
( 1st `  x
) (,) ( 2nd `  x ) )  =  s  ->  ( abs `  S. s ( ( RR  _D  F ) `
 t )  _d t )  <_  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  s ,  ( abs `  ( ( RR  _D  F ) `
 t ) ) ,  0 ) ) ) ) )
10833, 107sylbid 223 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( (,) `  x )  =  s  ->  ( abs `  S. s ( ( RR  _D  F ) `
 t )  _d t )  <_  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  s ,  ( abs `  ( ( RR  _D  F ) `
 t ) ) ,  0 ) ) ) ) )
109108rexlimdva 2871 . . . . . . . . . . 11  |-  ( ph  ->  ( E. x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) ( (,) `  x
)  =  s  -> 
( abs `  S. s ( ( RR 
_D  F ) `  t )  _d t )  <_  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  s ,  ( abs `  (
( RR  _D  F
) `  t )
) ,  0 ) ) ) ) )
11027, 109syl5 32 . . . . . . . . . 10  |-  ( ph  ->  ( s  e.  ( (,) " ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( abs `  S. s ( ( RR 
_D  F ) `  t )  _d t )  <_  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  s ,  ( abs `  (
( RR  _D  F
) `  t )
) ,  0 ) ) ) ) )
111110ralrimiv 2808 . . . . . . . . 9  |-  ( ph  ->  A. s  e.  ( (,) " ( ( A [,] B )  X.  ( A [,] B ) ) ) ( abs `  S. s ( ( RR 
_D  F ) `  t )  _d t )  <_  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  s ,  ( abs `  (
( RR  _D  F
) `  t )
) ,  0 ) ) ) )
11214, 1, 3, 5, 16, 18, 19, 22, 111ftc1anc 32089 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t )  e.  ( ( A [,] B
) -cn-> CC ) )
113 ftc2nc.f . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
114 cncff 22003 . . . . . . . . . . 11  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
115113, 114syl 17 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> CC )
116115feqmptd 5932 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  ( A [,] B )  |->  ( F `
 x ) ) )
117116, 113eqeltrrd 2550 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( F `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
11811, 13, 112, 117cncfmpt2f 22024 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) )  e.  ( ( A [,] B
) -cn-> CC ) )
11960a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
120 iccssre 11741 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
1211, 3, 120syl2anc 673 . . . . . . . . . 10  |-  ( ph  ->  ( A [,] B
)  C_  RR )
12254a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  t  e.  ( A (,) x
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
1233adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
124123rexrd 9708 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
125 elicc2 11724 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
1261, 3, 125syl2anc 673 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
127126biimpa 492 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
128127simp3d 1044 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
129 iooss2 11697 . . . . . . . . . . . . . 14  |-  ( ( B  e.  RR*  /\  x  <_  B )  ->  ( A (,) x )  C_  ( A (,) B ) )
130124, 128, 129syl2anc 673 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  C_  ( A (,) B ) )
131 ioombl 22597 . . . . . . . . . . . . . 14  |-  ( A (,) x )  e. 
dom  vol
132131a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  e.  dom  vol )
13354a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
13457adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
135130, 132, 133, 134iblss 22841 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) x
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
136122, 135itgcl 22820 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
137115ffvelrnda 6037 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
138136, 137subcld 10005 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) )  e.  CC )
13911tgioo2 21899 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
140 iccntr 21917 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
1411, 3, 140syl2anc 673 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
142119, 121, 138, 139, 11, 141dvmptntr 23004 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) )  =  ( RR  _D  ( x  e.  ( A (,) B )  |->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) ) ) ) )
143 reelprrecn 9649 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
144143a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
145 ioossicc 11745 . . . . . . . . . . . 12  |-  ( A (,) B )  C_  ( A [,] B )
146145sseli 3414 . . . . . . . . . . 11  |-  ( x  e.  ( A (,) B )  ->  x  e.  ( A [,] B
) )
147146, 136sylan2 482 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
14822ffvelrnda 6037 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
14914, 1, 3, 5, 20, 19ftc1cnnc 32080 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t ) )  =  ( RR  _D  F ) )
150119, 121, 136, 139, 11, 141dvmptntr 23004 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t ) )  =  ( RR  _D  ( x  e.  ( A (,) B )  |->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t ) ) )
15122feqmptd 5932 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 x ) ) )
152149, 150, 1513eqtr3d 2513 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A (,) B )  |->  S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t ) )  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  x )
) )
153146, 137sylan2 482 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  e.  CC )
154116oveq2d 6324 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( x  e.  ( A [,] B
)  |->  ( F `  x ) ) ) )
155119, 121, 137, 139, 11, 141dvmptntr 23004 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( RR  _D  ( x  e.  ( A (,) B )  |->  ( F `
 x ) ) ) )
156154, 151, 1553eqtr3rd 2514 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A (,) B )  |->  ( F `  x ) ) )  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  x )
) )
157144, 147, 148, 152, 153, 148, 156dvmptsub 23000 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A (,) B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) )  =  ( x  e.  ( A (,) B )  |->  ( ( ( RR  _D  F ) `  x
)  -  ( ( RR  _D  F ) `
 x ) ) ) )
158148subidd 9993 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( RR  _D  F
) `  x )  -  ( ( RR 
_D  F ) `  x ) )  =  0 )
159158mpteq2dva 4482 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( ( ( RR 
_D  F ) `  x )  -  (
( RR  _D  F
) `  x )
) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
160142, 157, 1593eqtrd 2509 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
161 fconstmpt 4883 . . . . . . . 8  |-  ( ( A (,) B )  X.  { 0 } )  =  ( x  e.  ( A (,) B )  |->  0 )
162160, 161syl6eqr 2523 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) )  =  ( ( A (,) B
)  X.  { 0 } ) )
1631, 3, 118, 162dveq0 23031 . . . . . 6  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) )  =  ( ( A [,] B
)  X.  { ( ( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) ) `  A
) } ) )
164163fveq1d 5881 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( A [,] B
)  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) ) ) `  B
)  =  ( ( ( A [,] B
)  X.  { ( ( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) ) `  A
) } ) `  B ) )
165 oveq2 6316 . . . . . . . . 9  |-  ( x  =  B  ->  ( A (,) x )  =  ( A (,) B
) )
166 itgeq1 22809 . . . . . . . . 9  |-  ( ( A (,) x )  =  ( A (,) B )  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t )
167165, 166syl 17 . . . . . . . 8  |-  ( x  =  B  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t )
168 fveq2 5879 . . . . . . . 8  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
169167, 168oveq12d 6326 . . . . . . 7  |-  ( x  =  B  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  =  ( S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  B )
) )
170 eqid 2471 . . . . . . 7  |-  ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) ) )  =  ( x  e.  ( A [,] B
)  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) ) )
171 ovex 6336 . . . . . . 7  |-  ( S. ( A (,) B
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  B ) )  e. 
_V
172169, 170, 171fvmpt 5963 . . . . . 6  |-  ( B  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) ) `  B
)  =  ( S. ( A (,) B
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  B ) ) )
1737, 172syl 17 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( A [,] B
)  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) ) ) `  B
)  =  ( S. ( A (,) B
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  B ) ) )
174164, 173eqtr3d 2507 . . . 4  |-  ( ph  ->  ( ( ( A [,] B )  X. 
{ ( ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) ) ) `
 A ) } ) `  B )  =  ( S. ( A (,) B ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  B ) ) )
175 lbicc2 11774 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
1762, 4, 5, 175syl3anc 1292 . . . . 5  |-  ( ph  ->  A  e.  ( A [,] B ) )
177 oveq2 6316 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( A (,) x )  =  ( A (,) A
) )
178 iooid 11689 . . . . . . . . . . 11  |-  ( A (,) A )  =  (/)
179177, 178syl6eq 2521 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A (,) x )  =  (/) )
180 itgeq1 22809 . . . . . . . . . 10  |-  ( ( A (,) x )  =  (/)  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
181179, 180syl 17 . . . . . . . . 9  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
182 itg0 22816 . . . . . . . . 9  |-  S. (/) ( ( RR  _D  F ) `  t
)  _d t  =  0
183181, 182syl6eq 2521 . . . . . . . 8  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  0 )
184 fveq2 5879 . . . . . . . 8  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
185183, 184oveq12d 6326 . . . . . . 7  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  =  ( 0  -  ( F `  A )
) )
186 df-neg 9883 . . . . . . 7  |-  -u ( F `  A )  =  ( 0  -  ( F `  A
) )
187185, 186syl6eqr 2523 . . . . . 6  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  = 
-u ( F `  A ) )
188 negex 9893 . . . . . 6  |-  -u ( F `  A )  e.  _V
189187, 170, 188fvmpt 5963 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) ) `  A
)  =  -u ( F `  A )
)
190176, 189syl 17 . . . 4  |-  ( ph  ->  ( ( x  e.  ( A [,] B
)  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) ) ) `  A
)  =  -u ( F `  A )
)
19110, 174, 1903eqtr3d 2513 . . 3  |-  ( ph  ->  ( S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  B )
)  =  -u ( F `  A )
)
192191oveq2d 6324 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  ( ( F `  B )  +  -u ( F `  A ) ) )
193115, 7ffvelrnd 6038 . . 3  |-  ( ph  ->  ( F `  B
)  e.  CC )
19454a1i 11 . . . 4  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  _V )
195194, 57itgcl 22820 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  e.  CC )
196193, 195pncan3d 10008 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  S. ( A (,) B ) ( ( RR  _D  F ) `  t
)  _d t )
197115, 176ffvelrnd 6038 . . 3  |-  ( ph  ->  ( F `  A
)  e.  CC )
198193, 197negsubd 10011 . 2  |-  ( ph  ->  ( ( F `  B )  +  -u ( F `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
199192, 196, 1983eqtr3d 2513 1  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031   [_csb 3349    C_ wss 3390   (/)c0 3722   ifcif 3872   ~Pcpw 3942   {csn 3959   {cpr 3961   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   CCcc 9555   RRcr 9556   0cc0 9557    + caddc 9560    x. cmul 9562   RR*cxr 9692    <_ cle 9694    - cmin 9880   -ucneg 9881   (,)cioo 11660   [,]cicc 11663   *ccj 13236   abscabs 13374   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047   intcnt 20109    Cn ccn 20317    tX ctx 20652   -cn->ccncf 21986   volcvol 22493  MblFncmbf 22651   S.2citg2 22653   L^1cibl 22654   S.citg 22655    _D cdv 22897
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-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-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-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-limc 22900  df-dv 22901
This theorem is referenced by:  areacirc  32101
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