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Theorem ftc2nc 31933
Description: Choice-free proof of ftc2 22938. (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 9641 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
3 ftc2nc.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
43rexrd 9641 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
5 ftc2nc.le . . . . . 6  |-  ( ph  ->  A  <_  B )
6 ubicc2 11700 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
72, 4, 5, 6syl3anc 1264 . . . . 5  |-  ( ph  ->  B  e.  ( A [,] B ) )
8 fvex 5835 . . . . . 6  |-  ( ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) `  A )  e.  _V
98fvconst2 6079 . . . . 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 2428 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1211subcn 21840 . . . . . . . . 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 2428 . . . . . . . . 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 3426 . . . . . . . . . 10  |-  ( A (,) B )  C_  ( A (,) B )
1615a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( A (,) B
)  C_  ( A (,) B ) )
17 ioossre 11647 . . . . . . . . . 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 21867 . . . . . . . . . 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 11683 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
24 ffun 5691 . . . . . . . . . . . . 13  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
2523, 24ax-mp 5 . . . . . . . . . . . 12  |-  Fun  (,)
26 fvelima 5877 . . . . . . . . . . . 12  |-  ( ( Fun  (,)  /\  s  e.  ( (,) " (
( A [,] B
)  X.  ( A [,] B ) ) ) )  ->  E. x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) ( (,) `  x
)  =  s )
2725, 26mpan 674 . . . . . . . . . . 11  |-  ( s  e.  ( (,) " (
( A [,] B
)  X.  ( A [,] B ) ) )  ->  E. x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) ( (,) `  x
)  =  s )
28 1st2nd2 6788 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2928fveq2d 5829 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( (,) `  x )  =  ( (,) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
30 df-ov 6252 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  =  ( (,) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
3129, 30syl6eqr 2480 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( (,) `  x )  =  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )
3231eqeq1d 2430 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  (
( (,) `  x
)  =  s  <->  ( ( 1st `  x ) (,) ( 2nd `  x
) )  =  s ) )
3332adantl 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( (,) `  x )  =  s  <->  ( ( 1st `  x ) (,) ( 2nd `  x ) )  =  s ) )
342, 4jca 534 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( A  e.  RR*  /\  B  e.  RR* )
)
3534adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( A  e.  RR*  /\  B  e. 
RR* ) )
36 xp1st 6781 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( 1st `  x )  e.  ( A [,] B
) )
37 elicc1 11631 . . . . . . . . . . . . . . . . . . . . . . 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 665 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( ( 1st `  x
)  e.  ( A [,] B )  <->  ( ( 1st `  x )  e. 
RR*  /\  A  <_  ( 1st `  x )  /\  ( 1st `  x
)  <_  B )
) )
3938biimpa 486 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  ( 1st `  x )  e.  ( A [,] B ) )  ->  ( ( 1st `  x )  e. 
RR*  /\  A  <_  ( 1st `  x )  /\  ( 1st `  x
)  <_  B )
)
4039simp2d 1018 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( 1st `  x )  e.  ( A [,] B ) )  ->  A  <_  ( 1st `  x ) )
4136, 40sylan2 476 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  A  <_  ( 1st `  x ) )
42 xp2nd 6782 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( 2nd `  x )  e.  ( A [,] B
) )
43 iccleub 11641 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( 2nd `  x )  e.  ( A [,] B ) )  ->  ( 2nd `  x )  <_  B
)
44433expa 1205 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 2nd `  x
)  e.  ( A [,] B ) )  ->  ( 2nd `  x
)  <_  B )
4534, 42, 44syl2an 479 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( 2nd `  x )  <_  B
)
46 ioossioo 11677 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  ( 1st `  x )  /\  ( 2nd `  x )  <_  B ) )  ->  ( ( 1st `  x ) (,) ( 2nd `  x ) ) 
C_  ( A (,) B ) )
4735, 41, 45, 46syl12anc 1262 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  ( A (,) B ) )
4847sselda 3407 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  t  e.  ( A (,) B ) )
4922ffvelrnda 5981 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  CC )
5049adantlr 719 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  F
) `  t )  e.  CC )
5148, 50syldan 472 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  ( ( RR 
_D  F ) `  t )  e.  CC )
52 ioombl 22460 . . . . . . . . . . . . . . . . . 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 5835 . . . . . . . . . . . . . . . . . 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 5878 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
5756, 19eqeltrrd 2507 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( t  e.  ( A (,) B ) 
|->  ( ( RR  _D  F ) `  t
) )  e.  L^1 )
5857adantr 466 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
5947, 53, 55, 58iblss 22704 . . . . . . . . . . . . . . . 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 9547 . . . . . . . . . . . . . . . . . . . . 21  |-  RR  C_  CC
61 ssid 3426 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  C_  CC
62 cncfss 21873 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( CC -cn-> RR )  C_  ( CC -cn-> CC ) )
6360, 61, 62mp2an 676 . . . . . . . . . . . . . . . . . . . 20  |-  ( CC
-cn-> RR )  C_  ( CC -cn-> CC )
64 abscncf 21875 . . . . . . . . . . . . . . . . . . . 20  |-  abs  e.  ( CC -cn-> RR )
6563, 64sselii 3404 . . . . . . . . . . . . . . . . . . 19  |-  abs  e.  ( CC -cn-> CC )
6665a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  abs  e.  ( CC -cn-> CC ) )
6756reseq1d 5066 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( 1st `  x
) (,) ( 2nd `  x ) ) )  =  ( ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) )  |`  ( ( 1st `  x
) (,) ( 2nd `  x ) ) ) )
6867adantr 466 . . . . . . . . . . . . . . . . . . . 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 5118 . . . . . . . . . . . . . . . . . . . 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 2462 . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( RR  _D  F )  e.  ( ( A (,) B
) -cn-> CC ) )
72 rescncf 21871 . . . . . . . . . . . . . . . . . . . 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 62 . . . . . . . . . . . . . . . . . . 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 2507 . . . . . . . . . . . . . . . . . 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 21887 . . . . . . . . . . . . . . . . 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 22557 . . . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( abs `  ( ( RR  _D  F ) `
 t ) ) )  e. MblFn )
7851, 59itgcl 22683 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  S. (
( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
7978cjcld 13203 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( * `  S. ( ( 1st `  x ) (,) ( 2nd `  x ) ) ( ( RR  _D  F ) `  t
)  _d t )  e.  CC )
80 ioossre 11647 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  RR
8180, 60sstri 3416 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  CC
82 cncfmptc 21885 . . . . . . . . . . . . . . . . . . . 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 1352 . . . . . . . . . . . . . . . . . . 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 2569 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ s
( ( RR  _D  F ) `  t
)
86 nfcsb1v 3354 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ t [_ s  /  t ]_ ( ( RR  _D  F ) `  t
)
87 csbeq1a 3347 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  s  ->  (
( RR  _D  F
) `  t )  =  [_ s  /  t ]_ ( ( RR  _D  F ) `  t
) )
8885, 86, 87cbvmpt 4458 . . . . . . . . . . . . . . . . . . 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 2511 . . . . . . . . . . . . . . . . . 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 22340 . . . . . . . . . . . . . . . . 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 22557 . . . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . . . 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 31918 . . . . . . . . . . . . . . 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 13441 . . . . . . . . . . . . . . . 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 31913 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( abs `  ( ( RR  _D  F ) `
 t ) ) )  e.  L^1 )
9751absge0d 13449 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  0  <_  ( abs `  ( ( RR 
_D  F ) `  t ) ) )
9894, 96, 97itgposval 22695 . . . . . . . . . . . . . . 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 4391 . . . . . . . . . . . . . 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 22672 . . . . . . . . . . . . . . . 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 5829 . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  =  s  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  <->  t  e.  s ) )
103102ifbid 3876 . . . . . . . . . . . . . . . . 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 4454 . . . . . . . . . . . . . . . 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 5829 . . . . . . . . . . . . . . 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 4379 . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 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 218 . . . . . . . . . . . 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 2856 . . . . . . . . . . 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 33 . . . . . . . . . 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 2777 . . . . . . . . 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 31932 . . . . . . . 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 21867 . . . . . . . . . . 11  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
115113, 114syl 17 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> CC )
116115feqmptd 5878 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  ( A [,] B )  |->  ( F `
 x ) ) )
117116, 113eqeltrrd 2507 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( F `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
11811, 13, 112, 117cncfmpt2f 21888 . . . . . . 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 11667 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
1211, 3, 120syl2anc 665 . . . . . . . . . 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 466 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
124123rexrd 9641 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
125 elicc2 11650 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
1261, 3, 125syl2anc 665 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
127126biimpa 486 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
128127simp3d 1019 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
129 iooss2 11623 . . . . . . . . . . . . . 14  |-  ( ( B  e.  RR*  /\  x  <_  B )  ->  ( A (,) x )  C_  ( A (,) B ) )
130124, 128, 129syl2anc 665 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  C_  ( A (,) B ) )
131 ioombl 22460 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
135130, 132, 133, 134iblss 22704 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) x
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
136122, 135itgcl 22683 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
137115ffvelrnda 5981 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
138136, 137subcld 9937 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) )  e.  CC )
13911tgioo2 21763 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
140 iccntr 21781 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
1411, 3, 140syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
142119, 121, 138, 139, 11, 141dvmptntr 22867 . . . . . . . . 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 9582 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
144143a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
145 ioossicc 11671 . . . . . . . . . . . 12  |-  ( A (,) B )  C_  ( A [,] B )
146145sseli 3403 . . . . . . . . . . 11  |-  ( x  e.  ( A (,) B )  ->  x  e.  ( A [,] B
) )
147146, 136sylan2 476 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
14822ffvelrnda 5981 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
14914, 1, 3, 5, 20, 19ftc1cnnc 31923 . . . . . . . . . . 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 22867 . . . . . . . . . . 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 5878 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 x ) ) )
152149, 150, 1513eqtr3d 2470 . . . . . . . . . 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 476 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  e.  CC )
154116oveq2d 6265 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( x  e.  ( A [,] B
)  |->  ( F `  x ) ) ) )
155119, 121, 137, 139, 11, 141dvmptntr 22867 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( RR  _D  ( x  e.  ( A (,) B )  |->  ( F `
 x ) ) ) )
156154, 151, 1553eqtr3rd 2471 . . . . . . . . . 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 22863 . . . . . . . . 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 9925 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( RR  _D  F
) `  x )  -  ( ( RR 
_D  F ) `  x ) )  =  0 )
159158mpteq2dva 4453 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( ( ( RR 
_D  F ) `  x )  -  (
( RR  _D  F
) `  x )
) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
160142, 157, 1593eqtrd 2466 . . . . . . . 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 4840 . . . . . . . 8  |-  ( ( A (,) B )  X.  { 0 } )  =  ( x  e.  ( A (,) B )  |->  0 )
162160, 161syl6eqr 2480 . . . . . . 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 22894 . . . . . 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 5827 . . . . 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 6257 . . . . . . . . 9  |-  ( x  =  B  ->  ( A (,) x )  =  ( A (,) B
) )
166 itgeq1 22672 . . . . . . . . 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 5825 . . . . . . . 8  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
169167, 168oveq12d 6267 . . . . . . 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 2428 . . . . . . 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 6277 . . . . . . 7  |-  ( S. ( A (,) B
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  B ) )  e. 
_V
172169, 170, 171fvmpt 5908 . . . . . 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 2464 . . . 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 11699 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
1762, 4, 5, 175syl3anc 1264 . . . . 5  |-  ( ph  ->  A  e.  ( A [,] B ) )
177 oveq2 6257 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( A (,) x )  =  ( A (,) A
) )
178 iooid 11615 . . . . . . . . . . 11  |-  ( A (,) A )  =  (/)
179177, 178syl6eq 2478 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A (,) x )  =  (/) )
180 itgeq1 22672 . . . . . . . . . 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 22679 . . . . . . . . 9  |-  S. (/) ( ( RR  _D  F ) `  t
)  _d t  =  0
183181, 182syl6eq 2478 . . . . . . . 8  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  0 )
184 fveq2 5825 . . . . . . . 8  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
185183, 184oveq12d 6267 . . . . . . 7  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  =  ( 0  -  ( F `  A )
) )
186 df-neg 9814 . . . . . . 7  |-  -u ( F `  A )  =  ( 0  -  ( F `  A
) )
187185, 186syl6eqr 2480 . . . . . 6  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  = 
-u ( F `  A ) )
188 negex 9824 . . . . . 6  |-  -u ( F `  A )  e.  _V
189187, 170, 188fvmpt 5908 . . . . 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 2470 . . 3  |-  ( ph  ->  ( S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  B )
)  =  -u ( F `  A )
)
192191oveq2d 6265 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  ( ( F `  B )  +  -u ( F `  A ) ) )
193115, 7ffvelrnd 5982 . . 3  |-  ( ph  ->  ( F `  B
)  e.  CC )
19454a1i 11 . . . 4  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  _V )
195194, 57itgcl 22683 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  e.  CC )
196193, 195pncan3d 9940 . 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 5982 . . 3  |-  ( ph  ->  ( F `  A
)  e.  CC )
198193, 197negsubd 9943 . 2  |-  ( ph  ->  ( ( F `  B )  +  -u ( F `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
199192, 196, 1983eqtr3d 2470 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   E.wrex 2715   _Vcvv 3022   [_csb 3338    C_ wss 3379   (/)c0 3704   ifcif 3854   ~Pcpw 3924   {csn 3941   {cpr 3943   <.cop 3947   class class class wbr 4366    |-> cmpt 4425    X. cxp 4794   dom cdm 4796   ran crn 4797    |` cres 4798   "cima 4799   Fun wfun 5538   -->wf 5540   ` cfv 5544  (class class class)co 6249   1stc1st 6749   2ndc2nd 6750   CCcc 9488   RRcr 9489   0cc0 9490    + caddc 9493    x. cmul 9495   RR*cxr 9625    <_ cle 9627    - cmin 9811   -ucneg 9812   (,)cioo 11586   [,]cicc 11589   *ccj 13103   abscabs 13241   TopOpenctopn 15263   topGenctg 15279  ℂfldccnfld 18913   intcnt 19974    Cn ccn 20182    tX ctx 20517   -cn->ccncf 21850   volcvol 22357  MblFncmbf 22514   S.2citg2 22516   L^1cibl 22517   S.citg 22518    _D cdv 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-iin 4245  df-disj 4338  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-of 6489  df-ofr 6490  df-om 6651  df-1st 6751  df-2nd 6752  df-supp 6870  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-2o 7138  df-oadd 7141  df-omul 7142  df-er 7318  df-map 7429  df-pm 7430  df-ixp 7478  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-fsupp 7837  df-fi 7878  df-sup 7909  df-inf 7910  df-oi 7978  df-card 8325  df-acn 8328  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-7 10624  df-8 10625  df-9 10626  df-10 10627  df-n0 10821  df-z 10889  df-dec 11003  df-uz 11111  df-q 11216  df-rp 11254  df-xneg 11360  df-xadd 11361  df-xmul 11362  df-ioo 11590  df-ico 11592  df-icc 11593  df-fz 11736  df-fzo 11867  df-fl 11978  df-mod 12047  df-seq 12164  df-exp 12223  df-hash 12466  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-clim 13495  df-rlim 13496  df-sum 13696  df-struct 15066  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-ress 15071  df-plusg 15146  df-mulr 15147  df-starv 15148  df-sca 15149  df-vsca 15150  df-ip 15151  df-tset 15152  df-ple 15153  df-ds 15155  df-unif 15156  df-hom 15157  df-cco 15158  df-rest 15264  df-topn 15265  df-0g 15283  df-gsum 15284  df-topgen 15285  df-pt 15286  df-prds 15289  df-xrs 15343  df-qtop 15349  df-imas 15350  df-xps 15353  df-mre 15435  df-mrc 15436  df-acs 15438  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-submnd 16526  df-mulg 16619  df-cntz 16914  df-cmn 17375  df-psmet 18905  df-xmet 18906  df-met 18907  df-bl 18908  df-mopn 18909  df-fbas 18910  df-fg 18911  df-cnfld 18914  df-top 19863  df-bases 19864  df-topon 19865  df-topsp 19866  df-cld 19976  df-ntr 19977  df-cls 19978  df-nei 20056  df-lp 20094  df-perf 20095  df-cn 20185  df-cnp 20186  df-haus 20273  df-cmp 20344  df-tx 20519  df-hmeo 20712  df-fil 20803  df-fm 20895  df-flim 20896  df-flf 20897  df-xms 21277  df-ms 21278  df-tms 21279  df-cncf 21852  df-ovol 22358  df-vol 22360  df-mbf 22519  df-itg1 22520  df-itg2 22521  df-ibl 22522  df-itg 22523  df-0p 22570  df-limc 22763  df-dv 22764
This theorem is referenced by:  areacirc  31944
  Copyright terms: Public domain W3C validator