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Theorem ftc2 19354
Description: The Fundamental Theorem of Calculus, part two. If  F is a function continuous on  [ A ,  B ] and continuously differentiable on  ( A ,  B ), then the integral of the derivative of  F is equal to  F ( B )  -  F ( A ). (Contributed by Mario Carneiro, 2-Sep-2014.)
Hypotheses
Ref Expression
ftc2.a  |-  ( ph  ->  A  e.  RR )
ftc2.b  |-  ( ph  ->  B  e.  RR )
ftc2.le  |-  ( ph  ->  A  <_  B )
ftc2.c  |-  ( ph  ->  ( RR  _D  F
)  e.  ( ( A (,) B )
-cn-> CC ) )
ftc2.i  |-  ( ph  ->  ( RR  _D  F
)  e.  L ^1 )
ftc2.f  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
Assertion
Ref Expression
ftc2  |-  ( 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 ftc2
StepHypRef Expression
1 ftc2.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
21rexrd 8849 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
3 ftc2.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
43rexrd 8849 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
5 ftc2.le . . . . . 6  |-  ( ph  ->  A  <_  B )
6 ubicc2 10720 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
72, 4, 5, 6syl3anc 1187 . . . . 5  |-  ( ph  ->  B  e.  ( A [,] B ) )
8 fvex 5472 . . . . . 6  |-  ( ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) `  A )  e.  _V
98fvconst2 5663 . . . . 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 2258 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1211subcn 18333 . . . . . . . . 9  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
1312a1i 12 . . . . . . . 8  |-  ( ph  ->  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
14 eqid 2258 . . . . . . . . 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 3172 . . . . . . . . . 10  |-  ( A (,) B )  C_  ( A (,) B )
1615a1i 12 . . . . . . . . 9  |-  ( ph  ->  ( A (,) B
)  C_  ( A (,) B ) )
17 ioossre 10679 . . . . . . . . . 10  |-  ( A (,) B )  C_  RR
1817a1i 12 . . . . . . . . 9  |-  ( ph  ->  ( A (,) B
)  C_  RR )
19 ftc2.i . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  e.  L ^1 )
20 ftc2.c . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  F
)  e.  ( ( A (,) B )
-cn-> CC ) )
21 cncff 18360 . . . . . . . . . 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 )
2314, 1, 3, 5, 16, 18, 19, 22ftc1a 19347 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t )  e.  ( ( A [,] B
) -cn-> CC ) )
24 ftc2.f . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
25 cncff 18360 . . . . . . . . . . 11  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
2624, 25syl 17 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> CC )
2726feqmptd 5509 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  ( A [,] B )  |->  ( F `
 x ) ) )
2827, 24eqeltrrd 2333 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( F `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
2911, 13, 23, 28cncfmpt2f 18381 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) )  e.  ( ( A [,] B
) -cn-> CC ) )
30 ax-resscn 8762 . . . . . . . . . . 11  |-  RR  C_  CC
3130a1i 12 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
32 iccssre 10698 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
331, 3, 32syl2anc 645 . . . . . . . . . 10  |-  ( ph  ->  ( A [,] B
)  C_  RR )
34 fvex 5472 . . . . . . . . . . . . 13  |-  ( ( RR  _D  F ) `
 t )  e. 
_V
3534a1i 12 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  t  e.  ( A (,) x
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
363adantr 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
3736rexrd 8849 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
38 elicc2 10682 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
391, 3, 38syl2anc 645 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4039biimpa 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
4140simp3d 974 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
42 iooss2 10659 . . . . . . . . . . . . . 14  |-  ( ( B  e.  RR*  /\  x  <_  B )  ->  ( A (,) x )  C_  ( A (,) B ) )
4337, 41, 42syl2anc 645 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  C_  ( A (,) B ) )
44 ioombl 18885 . . . . . . . . . . . . . 14  |-  ( A (,) x )  e. 
dom  vol
4544a1i 12 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  e.  dom  vol )
4634a1i 12 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
4722feqmptd 5509 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
4847, 19eqeltrrd 2333 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( t  e.  ( A (,) B ) 
|->  ( ( RR  _D  F ) `  t
) )  e.  L ^1 )
4948adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
5043, 45, 46, 49iblss 19122 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) x
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
5135, 50itgcl 19101 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
52 ffvelrn 5597 . . . . . . . . . . . 12  |-  ( ( F : ( A [,] B ) --> CC 
/\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
5326, 52sylan 459 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
5451, 53subcld 9125 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) )  e.  CC )
5511tgioo2 18272 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
56 iccntr 18289 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
571, 3, 56syl2anc 645 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
5831, 33, 54, 55, 11, 57dvmptntr 19283 . . . . . . . . 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 ) ) ) ) )
59 reex 8796 . . . . . . . . . . . 12  |-  RR  e.  _V
6059prid1 3708 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
6160a1i 12 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
62 ioossicc 10702 . . . . . . . . . . . 12  |-  ( A (,) B )  C_  ( A [,] B )
6362sseli 3151 . . . . . . . . . . 11  |-  ( x  e.  ( A (,) B )  ->  x  e.  ( A [,] B
) )
6463, 51sylan2 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
65 ffvelrn 5597 . . . . . . . . . . 11  |-  ( ( ( RR  _D  F
) : ( A (,) B ) --> CC 
/\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
6622, 65sylan 459 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
6714, 1, 3, 5, 20, 19ftc1cn 19353 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t ) )  =  ( RR  _D  F ) )
6831, 33, 51, 55, 11, 57dvmptntr 19283 . . . . . . . . . . 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 ) ) )
6922feqmptd 5509 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 x ) ) )
7067, 68, 693eqtr3d 2298 . . . . . . . . . 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 )
) )
7163, 53sylan2 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  e.  CC )
7231, 33, 53, 55, 11, 57dvmptntr 19283 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( RR  _D  ( x  e.  ( A (,) B )  |->  ( F `
 x ) ) ) )
7327oveq2d 5808 . . . . . . . . . . . 12  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( x  e.  ( A [,] B
)  |->  ( F `  x ) ) ) )
7473, 69eqtr3d 2292 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  x )
) )
7572, 74eqtr3d 2292 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A (,) B )  |->  ( F `  x ) ) )  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  x )
) )
7661, 64, 66, 70, 71, 66, 75dvmptsub 19279 . . . . . . . . 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 ) ) ) )
7766subidd 9113 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( RR  _D  F
) `  x )  -  ( ( RR 
_D  F ) `  x ) )  =  0 )
7877mpteq2dva 4080 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( ( ( RR 
_D  F ) `  x )  -  (
( RR  _D  F
) `  x )
) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
7958, 76, 783eqtrd 2294 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
80 fconstmpt 4720 . . . . . . . 8  |-  ( ( A (,) B )  X.  { 0 } )  =  ( x  e.  ( A (,) B )  |->  0 )
8179, 80syl6eqr 2308 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) )  =  ( ( A (,) B
)  X.  { 0 } ) )
821, 3, 29, 81dveq0 19310 . . . . . 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
) } ) )
8382fveq1d 5460 . . . . 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 ) )
84 oveq2 5800 . . . . . . . . 9  |-  ( x  =  B  ->  ( A (,) x )  =  ( A (,) B
) )
85 itgeq1 19090 . . . . . . . . 9  |-  ( ( A (,) x )  =  ( A (,) B )  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t )
8684, 85syl 17 . . . . . . . 8  |-  ( x  =  B  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t )
87 fveq2 5458 . . . . . . . 8  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
8886, 87oveq12d 5810 . . . . . . 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 )
) )
89 eqid 2258 . . . . . . 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 ) ) )
90 ovex 5817 . . . . . . 7  |-  ( S. ( A (,) B
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  B ) )  e. 
_V
9188, 89, 90fvmpt 5536 . . . . . 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 ) ) )
927, 91syl 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 ) ) )
9383, 92eqtr3d 2292 . . . 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 ) ) )
94 lbicc2 10719 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
952, 4, 5, 94syl3anc 1187 . . . . 5  |-  ( ph  ->  A  e.  ( A [,] B ) )
96 oveq2 5800 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( A (,) x )  =  ( A (,) A
) )
97 iooid 10651 . . . . . . . . . . 11  |-  ( A (,) A )  =  (/)
9896, 97syl6eq 2306 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A (,) x )  =  (/) )
99 itgeq1 19090 . . . . . . . . . 10  |-  ( ( A (,) x )  =  (/)  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
10098, 99syl 17 . . . . . . . . 9  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
101 itg0 19097 . . . . . . . . 9  |-  S. (/) ( ( RR  _D  F ) `  t
)  _d t  =  0
102100, 101syl6eq 2306 . . . . . . . 8  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  0 )
103 fveq2 5458 . . . . . . . 8  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
104102, 103oveq12d 5810 . . . . . . 7  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  =  ( 0  -  ( F `  A )
) )
105 df-neg 9008 . . . . . . 7  |-  -u ( F `  A )  =  ( 0  -  ( F `  A
) )
106104, 105syl6eqr 2308 . . . . . 6  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  = 
-u ( F `  A ) )
107 negex 9018 . . . . . 6  |-  -u ( F `  A )  e.  _V
108106, 89, 107fvmpt 5536 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) ) `  A
)  =  -u ( F `  A )
)
10995, 108syl 17 . . . 4  |-  ( ph  ->  ( ( x  e.  ( A [,] B
)  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) ) ) `  A
)  =  -u ( F `  A )
)
11010, 93, 1093eqtr3d 2298 . . 3  |-  ( ph  ->  ( S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  B )
)  =  -u ( F `  A )
)
111110oveq2d 5808 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  ( ( F `  B )  +  -u ( F `  A ) ) )
112 ffvelrn 5597 . . . 4  |-  ( ( F : ( A [,] B ) --> CC 
/\  B  e.  ( A [,] B ) )  ->  ( F `  B )  e.  CC )
11326, 7, 112syl2anc 645 . . 3  |-  ( ph  ->  ( F `  B
)  e.  CC )
11434a1i 12 . . . 4  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  _V )
115114, 48itgcl 19101 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  e.  CC )
116113, 115pncan3d 9128 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  S. ( A (,) B ) ( ( RR  _D  F ) `  t
)  _d t )
117 ffvelrn 5597 . . . 4  |-  ( ( F : ( A [,] B ) --> CC 
/\  A  e.  ( A [,] B ) )  ->  ( F `  A )  e.  CC )
11826, 95, 117syl2anc 645 . . 3  |-  ( ph  ->  ( F `  A
)  e.  CC )
119113, 118negsubd 9131 . 2  |-  ( ph  ->  ( ( F `  B )  +  -u ( F `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
120111, 116, 1193eqtr3d 2298 1  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   _Vcvv 2763    C_ wss 3127   (/)c0 3430   {csn 3614   {cpr 3615   class class class wbr 3997    e. cmpt 4051    X. cxp 4659   dom cdm 4661   ran crn 4662   -->wf 4669   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704   0cc0 8705    + caddc 8708   RR*cxr 8834    <_ cle 8836    - cmin 9005   -ucneg 9006   (,)cioo 10623   [,]cicc 10626   TopOpenctopn 13289   topGenctg 13305  ℂfldccnfld 16340   intcnt 16717    Cn ccn 16917    tX ctx 17218   -cn->ccncf 18343   volcvol 18786   L ^1cibl 18935   S.citg 18936    _D cdv 19176
This theorem is referenced by:  ftc2ditglem  19355  itgparts  19357  itgsubstlem  19358  lhe4.4ex1a  26914  itgsin0pilem1  27113
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cc 8029  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-disj 3968  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-ofr 6013  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-omul 6452  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-acn 7543  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9934  df-z 9993  df-dec 10093  df-uz 10199  df-q 10285  df-rp 10323  df-xneg 10420  df-xadd 10421  df-xmul 10422  df-ioo 10627  df-ioc 10628  df-ico 10629  df-icc 10630  df-fz 10750  df-fzo 10838  df-fl 10892  df-mod 10941  df-seq 11014  df-exp 11072  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-rlim 11929  df-sum 12125  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-starv 13186  df-sca 13187  df-vsca 13188  df-tset 13190  df-ple 13191  df-ds 13193  df-hom 13195  df-cco 13196  df-rest 13290  df-topn 13291  df-topgen 13307  df-pt 13308  df-prds 13311  df-xrs 13366  df-0g 13367  df-gsum 13368  df-qtop 13373  df-imas 13374  df-xps 13376  df-mre 13451  df-mrc 13452  df-acs 13454  df-mnd 14330  df-submnd 14379  df-mulg 14455  df-cntz 14756  df-cmn 15054  df-xmet 16336  df-met 16337  df-bl 16338  df-mopn 16339  df-cnfld 16341  df-top 16599  df-bases 16601  df-topon 16602  df-topsp 16603  df-cld 16719  df-ntr 16720  df-cls 16721  df-nei 16798  df-lp 16831  df-perf 16832  df-cn 16920  df-cnp 16921  df-haus 17006  df-cmp 17077  df-tx 17220  df-hmeo 17409  df-fbas 17483  df-fg 17484  df-fil 17504  df-fm 17596  df-flim 17597  df-flf 17598  df-xms 17848  df-ms 17849  df-tms 17850  df-cncf 18345  df-ovol 18787  df-vol 18788  df-mbf 18938  df-itg1 18939  df-itg2 18940  df-ibl 18941  df-itg 18942  df-0p 18988  df-limc 19179  df-dv 19180
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