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Theorem ftc2nc 29663
Description: Choice-free proof of ftc2 22173. (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  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ftc2nc.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
21rexrd 9632 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
3 ftc2nc.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
43rexrd 9632 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
5 ftc2nc.le . . . . . 6  |-  ( ph  ->  A  <_  B )
6 ubicc2 11626 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
72, 4, 5, 6syl3anc 1223 . . . . 5  |-  ( ph  ->  B  e.  ( A [,] B ) )
8 fvex 5867 . . . . . 6  |-  ( ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) `  A )  e.  _V
98fvconst2 6107 . . . . 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 2460 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1211subcn 21098 . . . . . . . . 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 2460 . . . . . . . . 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 3516 . . . . . . . . . 10  |-  ( A (,) B )  C_  ( A (,) B )
1615a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( A (,) B
)  C_  ( A (,) B ) )
17 ioossre 11575 . . . . . . . . . 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 21125 . . . . . . . . . 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 11611 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
24 ffun 5724 . . . . . . . . . . . . 13  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
2523, 24ax-mp 5 . . . . . . . . . . . 12  |-  Fun  (,)
26 fvelima 5910 . . . . . . . . . . . 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 6811 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2928fveq2d 5861 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( (,) `  x )  =  ( (,) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
30 df-ov 6278 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  =  ( (,) `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
3129, 30syl6eqr 2519 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( (,) `  x )  =  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )
3231eqeq1d 2462 . . . . . . . . . . . . . 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 6804 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( 1st `  x )  e.  ( A [,] B
) )
37 elicc1 11562 . . . . . . . . . . . . . . . . . . . . . . 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 1004 . . . . . . . . . . . . . . . . . . . 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 6805 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( ( A [,] B )  X.  ( A [,] B
) )  ->  ( 2nd `  x )  e.  ( A [,] B
) )
43 iccleub 11569 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( 2nd `  x )  e.  ( A [,] B ) )  ->  ( 2nd `  x )  <_  B
)
44433expa 1191 . . . . . . . . . . . . . . . . . . . 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 df-ioo 11522 . . . . . . . . . . . . . . . . . . . 20  |-  (,)  =  ( s  e.  RR* ,  t  e.  RR*  |->  { y  e.  RR*  |  (
s  <  y  /\  y  <  t ) } )
47 xrlelttr 11348 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  RR*  /\  ( 1st `  x )  e. 
RR*  /\  z  e.  RR* )  ->  ( ( A  <_  ( 1st `  x
)  /\  ( 1st `  x )  <  z
)  ->  A  <  z ) )
48 xrltletr 11349 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  e.  RR*  /\  ( 2nd `  x )  e. 
RR*  /\  B  e.  RR* )  ->  ( (
z  <  ( 2nd `  x )  /\  ( 2nd `  x )  <_  B )  ->  z  <  B ) )
4946, 46, 47, 48ixxss12 11538 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  ( 1st `  x )  /\  ( 2nd `  x )  <_  B ) )  ->  ( ( 1st `  x ) (,) ( 2nd `  x ) ) 
C_  ( A (,) B ) )
5035, 41, 45, 49syl12anc 1221 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  ( A (,) B ) )
5150sselda 3497 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  t  e.  ( A (,) B ) )
5222ffvelrnda 6012 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  CC )
5352adantlr 714 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  F
) `  t )  e.  CC )
5451, 53syldan 470 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  ( ( RR 
_D  F ) `  t )  e.  CC )
55 ioombl 21703 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  e.  dom  vol
5655a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( 1st `  x ) (,) ( 2nd `  x
) )  e.  dom  vol )
57 fvex 5867 . . . . . . . . . . . . . . . . . 18  |-  ( ( RR  _D  F ) `
 t )  e. 
_V
5857a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
5922feqmptd 5911 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
6059, 19eqeltrrd 2549 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( t  e.  ( A (,) B ) 
|->  ( ( RR  _D  F ) `  t
) )  e.  L^1 )
6160adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
6250, 56, 58, 61iblss 21939 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( ( RR  _D  F
) `  t )
)  e.  L^1 )
63 ax-resscn 9538 . . . . . . . . . . . . . . . . . . . . 21  |-  RR  C_  CC
64 ssid 3516 . . . . . . . . . . . . . . . . . . . . 21  |-  CC  C_  CC
65 cncfss 21131 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  ( CC -cn-> RR )  C_  ( CC -cn-> CC ) )
6663, 64, 65mp2an 672 . . . . . . . . . . . . . . . . . . . 20  |-  ( CC
-cn-> RR )  C_  ( CC -cn-> CC )
67 abscncf 21133 . . . . . . . . . . . . . . . . . . . 20  |-  abs  e.  ( CC -cn-> RR )
6866, 67sselii 3494 . . . . . . . . . . . . . . . . . . 19  |-  abs  e.  ( CC -cn-> CC )
6968a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  abs  e.  ( CC -cn-> CC ) )
7059reseq1d 5263 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( 1st `  x
) (,) ( 2nd `  x ) ) )  =  ( ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) )  |`  ( ( 1st `  x
) (,) ( 2nd `  x ) ) ) )
7170adantr 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 ) ) ) )
72 resmpt 5314 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  C_  ( A (,) B )  ->  ( ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) )  |`  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  =  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( ( RR  _D  F
) `  t )
) )
7350, 72syl 16 . . . . . . . . . . . . . . . . . . . 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
) ) )
7471, 73eqtrd 2501 . . . . . . . . . . . . . . . . . . 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 )
) )
7520adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( RR  _D  F )  e.  ( ( A (,) B
) -cn-> CC ) )
76 rescncf 21129 . . . . . . . . . . . . . . . . . . . 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 ) ) )
7750, 75, 76sylc 60 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( ( RR  _D  F )  |`  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  e.  ( ( ( 1st `  x ) (,) ( 2nd `  x
) ) -cn-> CC ) )
7874, 77eqeltrrd 2549 . . . . . . . . . . . . . . . . . 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 ) )
7969, 78cncfmpt1f 21145 . . . . . . . . . . . . . . . . 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 ) )
80 cnmbf 21794 . . . . . . . . . . . . . . . . 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
)
8155, 79, 80sylancr 663 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( abs `  ( ( RR  _D  F ) `
 t ) ) )  e. MblFn )
8211mulcn 21099 . . . . . . . . . . . . . . . . . . 19  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
8382a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) ) )
8454, 62itgcl 21918 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  S. (
( 1st `  x
) (,) ( 2nd `  x ) ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
8584cjcld 12979 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( * `  S. ( ( 1st `  x ) (,) ( 2nd `  x ) ) ( ( RR  _D  F ) `  t
)  _d t )  e.  CC )
86 ioossre 11575 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  RR
8786, 63sstri 3506 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  x ) (,) ( 2nd `  x
) )  C_  CC
88 cncfmptc 21143 . . . . . . . . . . . . . . . . . . . 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 ) )
8987, 64, 88mp3an23 1311 . . . . . . . . . . . . . . . . . . 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 ) )
9085, 89syl 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 ) )
91 nfcv 2622 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ s
( ( RR  _D  F ) `  t
)
92 nfcsb1v 3444 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ t [_ s  /  t ]_ ( ( RR  _D  F ) `  t
)
93 csbeq1a 3437 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  s  ->  (
( RR  _D  F
) `  t )  =  [_ s  /  t ]_ ( ( RR  _D  F ) `  t
) )
9491, 92, 93cbvmpt 4530 . . . . . . . . . . . . . . . . . . 19  |-  ( t  e.  ( ( 1st `  x ) (,) ( 2nd `  x ) ) 
|->  ( ( RR  _D  F ) `  t
) )  =  ( s  e.  ( ( 1st `  x ) (,) ( 2nd `  x
) )  |->  [_ s  /  t ]_ (
( RR  _D  F
) `  t )
)
9594, 78syl5eqelr 2553 . . . . . . . . . . . . . . . . . 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 ) )
9611, 83, 90, 95cncfmpt2f 21146 . . . . . . . . . . . . . . . . 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 ) )
97 cnmbf 21794 . . . . . . . . . . . . . . . . 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 )
9855, 96, 97sylancr 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 )
9954, 62, 81, 98itgabsnc 29648 . . . . . . . . . . . . . . 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 )
10054abscld 13216 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  ( abs `  (
( RR  _D  F
) `  t )
)  e.  RR )
10157a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  ( ( RR 
_D  F ) `  t )  e.  _V )
102101, 62, 81iblabsnc 29643 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  |->  ( abs `  ( ( RR  _D  F ) `
 t ) ) )  e.  L^1 )
10354absge0d 13224 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  ( ( A [,] B )  X.  ( A [,] B ) ) )  /\  t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) ) )  ->  0  <_  ( abs `  ( ( RR 
_D  F ) `  t ) ) )
104100, 102, 103itgposval 21930 . . . . . . . . . . . . . . 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 ) ) ) )
10599, 104breqtrd 4464 . . . . . . . . . . . . . 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 ) ) ) )
106 itgeq1 21907 . . . . . . . . . . . . . . . 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 )
107106fveq2d 5861 . . . . . . . . . . . . . . 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 ) )
108 eleq2 2533 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  x
) (,) ( 2nd `  x ) )  =  s  ->  ( t  e.  ( ( 1st `  x
) (,) ( 2nd `  x ) )  <->  t  e.  s ) )
109108ifbid 3954 . . . . . . . . . . . . . . . . 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 ) )
110109mpteq2dv 4527 . . . . . . . . . . . . . . . 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 ) ) )
111110fveq2d 5861 . . . . . . . . . . . . . . 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 ) ) ) )
112107, 111breq12d 4453 . . . . . . . . . . . . . 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 ) ) ) ) )
113105, 112syl5ibcom 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 ) ) ) ) )
11433, 113sylbid 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 ) ) ) ) )
115114rexlimdva 2948 . . . . . . . . . . 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 ) ) ) ) )
11627, 115syl5 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 ) ) ) ) )
117116ralrimiv 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 ) ) ) )
11814, 1, 3, 5, 16, 18, 19, 22, 117ftc1anc 29662 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t )  e.  ( ( A [,] B
) -cn-> CC ) )
119 ftc2nc.f . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
120 cncff 21125 . . . . . . . . . . 11  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
121119, 120syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> CC )
122121feqmptd 5911 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  ( A [,] B )  |->  ( F `
 x ) ) )
123122, 119eqeltrrd 2549 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( F `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
12411, 13, 118, 123cncfmpt2f 21146 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) )  e.  ( ( A [,] B
) -cn-> CC ) )
12563a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
126 iccssre 11595 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
1271, 3, 126syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( A [,] B
)  C_  RR )
12857a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  t  e.  ( A (,) x
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
1293adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
130129rexrd 9632 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
131 elicc2 11578 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
1321, 3, 131syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
133132biimpa 484 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
134133simp3d 1005 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
135 iooss2 11554 . . . . . . . . . . . . . 14  |-  ( ( B  e.  RR*  /\  x  <_  B )  ->  ( A (,) x )  C_  ( A (,) B ) )
136130, 134, 135syl2anc 661 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  C_  ( A (,) B ) )
137 ioombl 21703 . . . . . . . . . . . . . 14  |-  ( A (,) x )  e. 
dom  vol
138137a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  e.  dom  vol )
13957a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
14060adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
141136, 138, 139, 140iblss 21939 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) x
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
142128, 141itgcl 21918 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
143121ffvelrnda 6012 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
144142, 143subcld 9919 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) )  e.  CC )
14511tgioo2 21036 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
146 iccntr 21054 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
1471, 3, 146syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
148125, 127, 144, 145, 11, 147dvmptntr 22102 . . . . . . . . 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 ) ) ) ) )
149 reelprrecn 9573 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
150149a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
151 ioossicc 11599 . . . . . . . . . . . 12  |-  ( A (,) B )  C_  ( A [,] B )
152151sseli 3493 . . . . . . . . . . 11  |-  ( x  e.  ( A (,) B )  ->  x  e.  ( A [,] B
) )
153152, 142sylan2 474 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
15422ffvelrnda 6012 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
15514, 1, 3, 5, 20, 19ftc1cnnc 29653 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t ) )  =  ( RR  _D  F ) )
156125, 127, 142, 145, 11, 147dvmptntr 22102 . . . . . . . . . . 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 ) ) )
15722feqmptd 5911 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 x ) ) )
158155, 156, 1573eqtr3d 2509 . . . . . . . . . 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 )
) )
159152, 143sylan2 474 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  e.  CC )
160122oveq2d 6291 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( x  e.  ( A [,] B
)  |->  ( F `  x ) ) ) )
161125, 127, 143, 145, 11, 147dvmptntr 22102 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( RR  _D  ( x  e.  ( A (,) B )  |->  ( F `
 x ) ) ) )
162160, 157, 1613eqtr3rd 2510 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A (,) B )  |->  ( F `  x ) ) )  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  x )
) )
163150, 153, 154, 158, 159, 154, 162dvmptsub 22098 . . . . . . . . 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 ) ) ) )
164154subidd 9907 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( RR  _D  F
) `  x )  -  ( ( RR 
_D  F ) `  x ) )  =  0 )
165164mpteq2dva 4526 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( ( ( RR 
_D  F ) `  x )  -  (
( RR  _D  F
) `  x )
) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
166148, 163, 1653eqtrd 2505 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
167 fconstmpt 5035 . . . . . . . 8  |-  ( ( A (,) B )  X.  { 0 } )  =  ( x  e.  ( A (,) B )  |->  0 )
168166, 167syl6eqr 2519 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) )  =  ( ( A (,) B
)  X.  { 0 } ) )
1691, 3, 124, 168dveq0 22129 . . . . . 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
) } ) )
170169fveq1d 5859 . . . . 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 ) )
171 oveq2 6283 . . . . . . . . 9  |-  ( x  =  B  ->  ( A (,) x )  =  ( A (,) B
) )
172 itgeq1 21907 . . . . . . . . 9  |-  ( ( A (,) x )  =  ( A (,) B )  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t )
173171, 172syl 16 . . . . . . . 8  |-  ( x  =  B  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t )
174 fveq2 5857 . . . . . . . 8  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
175173, 174oveq12d 6293 . . . . . . 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 )
) )
176 eqid 2460 . . . . . . 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 ) ) )
177 ovex 6300 . . . . . . 7  |-  ( S. ( A (,) B
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  B ) )  e. 
_V
178175, 176, 177fvmpt 5941 . . . . . 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 ) ) )
1797, 178syl 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 ) ) )
180170, 179eqtr3d 2503 . . . 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 ) ) )
181 lbicc2 11625 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
1822, 4, 5, 181syl3anc 1223 . . . . 5  |-  ( ph  ->  A  e.  ( A [,] B ) )
183 oveq2 6283 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( A (,) x )  =  ( A (,) A
) )
184 iooid 11546 . . . . . . . . . . 11  |-  ( A (,) A )  =  (/)
185183, 184syl6eq 2517 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A (,) x )  =  (/) )
186 itgeq1 21907 . . . . . . . . . 10  |-  ( ( A (,) x )  =  (/)  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
187185, 186syl 16 . . . . . . . . 9  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
188 itg0 21914 . . . . . . . . 9  |-  S. (/) ( ( RR  _D  F ) `  t
)  _d t  =  0
189187, 188syl6eq 2517 . . . . . . . 8  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  0 )
190 fveq2 5857 . . . . . . . 8  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
191189, 190oveq12d 6293 . . . . . . 7  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  =  ( 0  -  ( F `  A )
) )
192 df-neg 9797 . . . . . . 7  |-  -u ( F `  A )  =  ( 0  -  ( F `  A
) )
193191, 192syl6eqr 2519 . . . . . 6  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  = 
-u ( F `  A ) )
194 negex 9807 . . . . . 6  |-  -u ( F `  A )  e.  _V
195193, 176, 194fvmpt 5941 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) ) `  A
)  =  -u ( F `  A )
)
196182, 195syl 16 . . . 4  |-  ( ph  ->  ( ( x  e.  ( A [,] B
)  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) ) ) `  A
)  =  -u ( F `  A )
)
19710, 180, 1963eqtr3d 2509 . . 3  |-  ( ph  ->  ( S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  B )
)  =  -u ( F `  A )
)
198197oveq2d 6291 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  ( ( F `  B )  +  -u ( F `  A ) ) )
199121, 7ffvelrnd 6013 . . 3  |-  ( ph  ->  ( F `  B
)  e.  CC )
20057a1i 11 . . . 4  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  _V )
201200, 60itgcl 21918 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  e.  CC )
202199, 201pncan3d 9922 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  S. ( A (,) B ) ( ( RR  _D  F ) `  t
)  _d t )
203121, 182ffvelrnd 6013 . . 3  |-  ( ph  ->  ( F `  A
)  e.  CC )
204199, 203negsubd 9925 . 2  |-  ( ph  ->  ( ( F `  B )  +  -u ( F `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
205198, 202, 2043eqtr3d 2509 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 968    = wceq 1374    e. wcel 1762   E.wrex 2808   _Vcvv 3106   [_csb 3428    C_ wss 3469   (/)c0 3778   ifcif 3932   ~Pcpw 4003   {csn 4020   {cpr 4022   <.cop 4026   class class class wbr 4440    |-> cmpt 4498    X. cxp 4990   dom cdm 4992   ran crn 4993    |` cres 4994   "cima 4995   Fun wfun 5573   -->wf 5575   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484    x. cmul 9486   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9794   -ucneg 9795   (,)cioo 11518   [,]cicc 11521   *ccj 12879   abscabs 13017   TopOpenctopn 14666   topGenctg 14682  ℂfldccnfld 18184   intcnt 19277    Cn ccn 19484    tX ctx 19789   -cn->ccncf 21108   volcvol 21603  MblFncmbf 21751   S.2citg2 21753   L^1cibl 21754   S.citg 21755    _D cdv 21995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-disj 4411  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-ofr 6516  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-omul 7125  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-acn 8312  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-rlim 13261  df-sum 13458  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-cmp 19646  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-ovol 21604  df-vol 21605  df-mbf 21756  df-itg1 21757  df-itg2 21758  df-ibl 21759  df-itg 21760  df-0p 21805  df-limc 21998  df-dv 21999
This theorem is referenced by:  areacirc  29676
  Copyright terms: Public domain W3C validator