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Theorem ftc2nc 30304
Description: Choice-free proof of ftc2 22571. (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 9660 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
3 ftc2nc.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
43rexrd 9660 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
5 ftc2nc.le . . . . . 6  |-  ( ph  ->  A  <_  B )
6 ubicc2 11662 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
72, 4, 5, 6syl3anc 1228 . . . . 5  |-  ( ph  ->  B  e.  ( A [,] B ) )
8 fvex 5882 . . . . . 6  |-  ( ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) `  A )  e.  _V
98fvconst2 6128 . . . . 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 16 . . . 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 2457 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1211subcn 21496 . . . . . . . . 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 2457 . . . . . . . . 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 3518 . . . . . . . . . 10  |-  ( A (,) B )  C_  ( A (,) B )
1615a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( A (,) B
)  C_  ( A (,) B ) )
17 ioossre 11611 . . . . . . . . . 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 21523 . . . . . . . . . 10  |-  ( ( RR  _D  F )  e.  ( ( A (,) B ) -cn-> CC )  ->  ( RR  _D  F ) : ( A (,) B ) --> CC )
2220, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> CC )
23 ioof 11647 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
24 ffun 5739 . . . . . . . . . . . . 13  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
2523, 24ax-mp 5 . . . . . . . . . . . 12  |-  Fun  (,)
26 fvelima 5925 . . . . . . . . . . . 12  |-  ( ( Fun  (,)  /\  s  e.  ( (,) " (
( A [,] B
)  X.  ( A [,] B ) ) ) )  ->  E. x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) ( (,) `  x
)  =  s )
2725, 26mpan 670 . . . . . . . . . . 11  |-  ( s  e.  ( (,) " (
( A [,] B
)  X.  ( A [,] B ) ) )  ->  E. x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) ( (,) `  x
)  =  s )
28 1st2nd2 6836 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2928fveq2d 5876 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( (,) `  x )  =  ( (,) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
30 df-ov 6299 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  =  ( (,) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
3129, 30syl6eqr 2516 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( (,) `  x )  =  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )
3231eqeq1d 2459 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  (
( (,) `  x
)  =  s  <->  ( ( 1st `  x ) (,) ( 2nd `  x
) )  =  s ) )
3332adantl 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( (,) `  x )  =  s  <->  ( ( 1st `  x ) (,) ( 2nd `  x ) )  =  s ) )
342, 4jca 532 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( A  e.  RR*  /\  B  e.  RR* )
)
3534adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( A  e.  RR*  /\  B  e. 
RR* ) )
36 xp1st 6829 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( 1st `  x )  e.  ( A [,] B
) )
37 elicc1 11598 . . . . . . . . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( ( 1st `  x
)  e.  ( A [,] B )  <->  ( ( 1st `  x )  e. 
RR*  /\  A  <_  ( 1st `  x )  /\  ( 1st `  x
)  <_  B )
) )
3938biimpa 484 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  ( 1st `  x )  e.  ( A [,] B ) )  ->  ( ( 1st `  x )  e. 
RR*  /\  A  <_  ( 1st `  x )  /\  ( 1st `  x
)  <_  B )
)
4039simp2d 1009 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( 1st `  x )  e.  ( A [,] B ) )  ->  A  <_  ( 1st `  x ) )
4136, 40sylan2 474 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  A  <_  ( 1st `  x ) )
42 xp2nd 6830 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( 2nd `  x )  e.  ( A [,] B
) )
43 iccleub 11605 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( 2nd `  x )  e.  ( A [,] B ) )  ->  ( 2nd `  x )  <_  B
)
44433expa 1196 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 2nd `  x
)  e.  ( A [,] B ) )  ->  ( 2nd `  x
)  <_  B )
4534, 42, 44syl2an 477 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( 2nd `  x )  <_  B
)
46 ioossioo 11641 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  ( 1st `  x )  /\  ( 2nd `  x )  <_  B ) )  ->  ( ( 1st `  x ) (,) ( 2nd `  x ) ) 
C_  ( A (,) B ) )
4735, 41, 45, 46syl12anc 1226 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  ( A (,) B ) )
4847sselda 3499 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  t  e.  ( A (,) B ) )
4922ffvelrnda 6032 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  CC )
5049adantlr 714 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  F
) `  t )  e.  CC )
5148, 50syldan 470 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  ( ( RR 
_D  F ) `  t )  e.  CC )
52 ioombl 22101 . . . . . . . . . . . . . . . . . 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 5882 . . . . . . . . . . . . . . . . . 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 5926 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
5756, 19eqeltrrd 2546 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( t  e.  ( A (,) B ) 
|->  ( ( RR  _D  F ) `  t
) )  e.  L^1 )
5857adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
5947, 53, 55, 58iblss 22337 . . . . . . . . . . . . . . . 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 9566 . . . . . . . . . . . . . . . . . . . . 21  |-  RR  C_  CC
61 ssid 3518 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  C_  CC
62 cncfss 21529 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( CC -cn-> RR )  C_  ( CC -cn-> CC ) )
6360, 61, 62mp2an 672 . . . . . . . . . . . . . . . . . . . 20  |-  ( CC
-cn-> RR )  C_  ( CC -cn-> CC )
64 abscncf 21531 . . . . . . . . . . . . . . . . . . . 20  |-  abs  e.  ( CC -cn-> RR )
6563, 64sselii 3496 . . . . . . . . . . . . . . . . . . 19  |-  abs  e.  ( CC -cn-> CC )
6665a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  abs  e.  ( CC -cn-> CC ) )
6756reseq1d 5282 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( 1st `  x
) (,) ( 2nd `  x ) ) )  =  ( ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) )  |`  ( ( 1st `  x
) (,) ( 2nd `  x ) ) ) )
6867adantr 465 . . . . . . . . . . . . . . . . . . . 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 5335 . . . . . . . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( RR  _D  F )  e.  ( ( A (,) B
) -cn-> CC ) )
72 rescncf 21527 . . . . . . . . . . . . . . . . . . . 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 60 . . . . . . . . . . . . . . . . . . 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 2546 . . . . . . . . . . . . . . . . . 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 21543 . . . . . . . . . . . . . . . . 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 22192 . . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( abs `  ( ( RR  _D  F ) `
 t ) ) )  e. MblFn )
7851, 59itgcl 22316 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  S. (
( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
7978cjcld 13041 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( * `  S. ( ( 1st `  x ) (,) ( 2nd `  x ) ) ( ( RR  _D  F ) `  t
)  _d t )  e.  CC )
80 ioossre 11611 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  RR
8180, 60sstri 3508 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  CC
82 cncfmptc 21541 . . . . . . . . . . . . . . . . . . . 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 1316 . . . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . . . 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 2619 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ s
( ( RR  _D  F ) `  t
)
86 nfcsb1v 3446 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ t [_ s  /  t ]_ ( ( RR  _D  F ) `  t
)
87 csbeq1a 3439 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  s  ->  (
( RR  _D  F
) `  t )  =  [_ s  /  t ]_ ( ( RR  _D  F ) `  t
) )
8885, 86, 87cbvmpt 4547 . . . . . . . . . . . . . . . . . . 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 2550 . . . . . . . . . . . . . . . . . 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 21985 . . . . . . . . . . . . . . . . 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 22192 . . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . . 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 30289 . . . . . . . . . . . . . . 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 13279 . . . . . . . . . . . . . . . 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 30284 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( abs `  ( ( RR  _D  F ) `
 t ) ) )  e.  L^1 )
9751absge0d 13287 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  0  <_  ( abs `  ( ( RR 
_D  F ) `  t ) ) )
9894, 96, 97itgposval 22328 . . . . . . . . . . . . . . 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 4480 . . . . . . . . . . . . . 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 22305 . . . . . . . . . . . . . . . 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 5876 . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  =  s  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  <->  t  e.  s ) )
103102ifbid 3966 . . . . . . . . . . . . . . . . 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 4544 . . . . . . . . . . . . . . . 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 5876 . . . . . . . . . . . . . . 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 4469 . . . . . . . . . . . . . 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 220 . . . . . . . . . . . . 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 215 . . . . . . . . . . . 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 2949 . . . . . . . . . . 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 2869 . . . . . . . . 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 30303 . . . . . . . 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 21523 . . . . . . . . . . 11  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
115113, 114syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> CC )
116115feqmptd 5926 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  ( A [,] B )  |->  ( F `
 x ) ) )
117116, 113eqeltrrd 2546 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( F `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
11811, 13, 112, 117cncfmpt2f 21544 . . . . . . 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 11631 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
1211, 3, 120syl2anc 661 . . . . . . . . . 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 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
124123rexrd 9660 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
125 elicc2 11614 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
1261, 3, 125syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
127126biimpa 484 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
128127simp3d 1010 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
129 iooss2 11590 . . . . . . . . . . . . . 14  |-  ( ( B  e.  RR*  /\  x  <_  B )  ->  ( A (,) x )  C_  ( A (,) B ) )
130124, 128, 129syl2anc 661 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  C_  ( A (,) B ) )
131 ioombl 22101 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
135130, 132, 133, 134iblss 22337 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) x
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
136122, 135itgcl 22316 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
137115ffvelrnda 6032 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
138136, 137subcld 9950 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) )  e.  CC )
13911tgioo2 21434 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
140 iccntr 21452 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
1411, 3, 140syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
142119, 121, 138, 139, 11, 141dvmptntr 22500 . . . . . . . . 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 9601 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
144143a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
145 ioossicc 11635 . . . . . . . . . . . 12  |-  ( A (,) B )  C_  ( A [,] B )
146145sseli 3495 . . . . . . . . . . 11  |-  ( x  e.  ( A (,) B )  ->  x  e.  ( A [,] B
) )
147146, 136sylan2 474 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
14822ffvelrnda 6032 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
14914, 1, 3, 5, 20, 19ftc1cnnc 30294 . . . . . . . . . . 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 22500 . . . . . . . . . . 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 5926 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 x ) ) )
152149, 150, 1513eqtr3d 2506 . . . . . . . . . 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 474 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  e.  CC )
154116oveq2d 6312 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( x  e.  ( A [,] B
)  |->  ( F `  x ) ) ) )
155119, 121, 137, 139, 11, 141dvmptntr 22500 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( RR  _D  ( x  e.  ( A (,) B )  |->  ( F `
 x ) ) ) )
156154, 151, 1553eqtr3rd 2507 . . . . . . . . . 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 22496 . . . . . . . . 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 9938 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( RR  _D  F
) `  x )  -  ( ( RR 
_D  F ) `  x ) )  =  0 )
159158mpteq2dva 4543 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( ( ( RR 
_D  F ) `  x )  -  (
( RR  _D  F
) `  x )
) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
160142, 157, 1593eqtrd 2502 . . . . . . . 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 5052 . . . . . . . 8  |-  ( ( A (,) B )  X.  { 0 } )  =  ( x  e.  ( A (,) B )  |->  0 )
162160, 161syl6eqr 2516 . . . . . . 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 22527 . . . . . 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 5874 . . . . 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 6304 . . . . . . . . 9  |-  ( x  =  B  ->  ( A (,) x )  =  ( A (,) B
) )
166 itgeq1 22305 . . . . . . . . 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 16 . . . . . . . 8  |-  ( x  =  B  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t )
168 fveq2 5872 . . . . . . . 8  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
169167, 168oveq12d 6314 . . . . . . 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 2457 . . . . . . 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 6324 . . . . . . 7  |-  ( S. ( A (,) B
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  B ) )  e. 
_V
172169, 170, 171fvmpt 5956 . . . . . 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 16 . . . . 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 2500 . . . 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 11661 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
1762, 4, 5, 175syl3anc 1228 . . . . 5  |-  ( ph  ->  A  e.  ( A [,] B ) )
177 oveq2 6304 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( A (,) x )  =  ( A (,) A
) )
178 iooid 11582 . . . . . . . . . . 11  |-  ( A (,) A )  =  (/)
179177, 178syl6eq 2514 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A (,) x )  =  (/) )
180 itgeq1 22305 . . . . . . . . . 10  |-  ( ( A (,) x )  =  (/)  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
181179, 180syl 16 . . . . . . . . 9  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
182 itg0 22312 . . . . . . . . 9  |-  S. (/) ( ( RR  _D  F ) `  t
)  _d t  =  0
183181, 182syl6eq 2514 . . . . . . . 8  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  0 )
184 fveq2 5872 . . . . . . . 8  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
185183, 184oveq12d 6314 . . . . . . 7  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  =  ( 0  -  ( F `  A )
) )
186 df-neg 9827 . . . . . . 7  |-  -u ( F `  A )  =  ( 0  -  ( F `  A
) )
187185, 186syl6eqr 2516 . . . . . 6  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  = 
-u ( F `  A ) )
188 negex 9837 . . . . . 6  |-  -u ( F `  A )  e.  _V
189187, 170, 188fvmpt 5956 . . . . 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 16 . . . 4  |-  ( ph  ->  ( ( x  e.  ( A [,] B
)  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) ) ) `  A
)  =  -u ( F `  A )
)
19110, 174, 1903eqtr3d 2506 . . 3  |-  ( ph  ->  ( S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  B )
)  =  -u ( F `  A )
)
192191oveq2d 6312 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  ( ( F `  B )  +  -u ( F `  A ) ) )
193115, 7ffvelrnd 6033 . . 3  |-  ( ph  ->  ( F `  B
)  e.  CC )
19454a1i 11 . . . 4  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  _V )
195194, 57itgcl 22316 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  e.  CC )
196193, 195pncan3d 9953 . 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 6033 . . 3  |-  ( ph  ->  ( F `  A
)  e.  CC )
198193, 197negsubd 9956 . 2  |-  ( ph  ->  ( ( F `  B )  +  -u ( F `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
199192, 196, 1983eqtr3d 2506 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 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109   [_csb 3430    C_ wss 3471   (/)c0 3793   ifcif 3944   ~Pcpw 4015   {csn 4032   {cpr 4034   <.cop 4038   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   CCcc 9507   RRcr 9508   0cc0 9509    + caddc 9512    x. cmul 9514   RR*cxr 9644    <_ cle 9646    - cmin 9824   -ucneg 9825   (,)cioo 11554   [,]cicc 11557   *ccj 12941   abscabs 13079   TopOpenctopn 14839   topGenctg 14855  ℂfldccnfld 18547   intcnt 19645    Cn ccn 19852    tX ctx 20187   -cn->ccncf 21506   volcvol 22001  MblFncmbf 22149   S.2citg2 22151   L^1cibl 22152   S.citg 22153    _D cdv 22393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-cmp 20014  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-ovol 22002  df-vol 22003  df-mbf 22154  df-itg1 22155  df-itg2 22156  df-ibl 22157  df-itg 22158  df-0p 22203  df-limc 22396  df-dv 22397
This theorem is referenced by:  areacirc  30317
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