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Theorem ftc2 19393
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ftc2.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
21rexrd 8883 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
3 ftc2.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
43rexrd 8883 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
5 ftc2.le . . . . . 6  |-  ( ph  ->  A  <_  B )
6 ubicc2 10755 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
72, 4, 5, 6syl3anc 1182 . . . . 5  |-  ( ph  ->  B  e.  ( A [,] B ) )
8 fvex 5541 . . . . . 6  |-  ( ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) `  A )  e.  _V
98fvconst2 5731 . . . . 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 15 . . . 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 2285 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1211subcn 18372 . . . . . . . . 9  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
1312a1i 10 . . . . . . . 8  |-  ( ph  ->  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
14 eqid 2285 . . . . . . . . 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 3199 . . . . . . . . . 10  |-  ( A (,) B )  C_  ( A (,) B )
1615a1i 10 . . . . . . . . 9  |-  ( ph  ->  ( A (,) B
)  C_  ( A (,) B ) )
17 ioossre 10714 . . . . . . . . . 10  |-  ( A (,) B )  C_  RR
1817a1i 10 . . . . . . . . 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 18399 . . . . . . . . . 10  |-  ( ( RR  _D  F )  e.  ( ( A (,) B ) -cn-> CC )  ->  ( RR  _D  F ) : ( A (,) B ) --> CC )
2220, 21syl 15 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> CC )
2314, 1, 3, 5, 16, 18, 19, 22ftc1a 19386 . . . . . . . 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 18399 . . . . . . . . . . 11  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
2624, 25syl 15 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> CC )
2726feqmptd 5577 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  ( A [,] B )  |->  ( F `
 x ) ) )
2827, 24eqeltrrd 2360 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( F `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
2911, 13, 23, 28cncfmpt2f 18420 . . . . . . 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 8796 . . . . . . . . . . 11  |-  RR  C_  CC
3130a1i 10 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
32 iccssre 10733 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
331, 3, 32syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( A [,] B
)  C_  RR )
34 fvex 5541 . . . . . . . . . . . . 13  |-  ( ( RR  _D  F ) `
 t )  e. 
_V
3534a1i 10 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  t  e.  ( A (,) x
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
363adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
3736rexrd 8883 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
38 elicc2 10717 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
391, 3, 38syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4039biimpa 470 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
4140simp3d 969 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
42 iooss2 10694 . . . . . . . . . . . . . 14  |-  ( ( B  e.  RR*  /\  x  <_  B )  ->  ( A (,) x )  C_  ( A (,) B ) )
4337, 41, 42syl2anc 642 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  C_  ( A (,) B ) )
44 ioombl 18924 . . . . . . . . . . . . . 14  |-  ( A (,) x )  e. 
dom  vol
4544a1i 10 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  e.  dom  vol )
4634a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
4722feqmptd 5577 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
4847, 19eqeltrrd 2360 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( t  e.  ( A (,) B ) 
|->  ( ( RR  _D  F ) `  t
) )  e.  L ^1 )
4948adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
5043, 45, 46, 49iblss 19161 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) x
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
5135, 50itgcl 19140 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
52 ffvelrn 5665 . . . . . . . . . . . 12  |-  ( ( F : ( A [,] B ) --> CC 
/\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
5326, 52sylan 457 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
5451, 53subcld 9159 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) )  e.  CC )
5511tgioo2 18311 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
56 iccntr 18328 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
571, 3, 56syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
5831, 33, 54, 55, 11, 57dvmptntr 19322 . . . . . . . . 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 8830 . . . . . . . . . . . 12  |-  RR  e.  _V
6059prid1 3736 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
6160a1i 10 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
62 ioossicc 10737 . . . . . . . . . . . 12  |-  ( A (,) B )  C_  ( A [,] B )
6362sseli 3178 . . . . . . . . . . 11  |-  ( x  e.  ( A (,) B )  ->  x  e.  ( A [,] B
) )
6463, 51sylan2 460 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
65 ffvelrn 5665 . . . . . . . . . . 11  |-  ( ( ( RR  _D  F
) : ( A (,) B ) --> CC 
/\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
6622, 65sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
6714, 1, 3, 5, 20, 19ftc1cn 19392 . . . . . . . . . . 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 19322 . . . . . . . . . . 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 5577 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 x ) ) )
7067, 68, 693eqtr3d 2325 . . . . . . . . . 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 460 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  e.  CC )
7231, 33, 53, 55, 11, 57dvmptntr 19322 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( RR  _D  ( x  e.  ( A (,) B )  |->  ( F `
 x ) ) ) )
7327oveq2d 5876 . . . . . . . . . . . 12  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( x  e.  ( A [,] B
)  |->  ( F `  x ) ) ) )
7473, 69eqtr3d 2319 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  x )
) )
7572, 74eqtr3d 2319 . . . . . . . . . 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 19318 . . . . . . . . 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 9147 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( RR  _D  F
) `  x )  -  ( ( RR 
_D  F ) `  x ) )  =  0 )
7877mpteq2dva 4108 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( ( ( RR 
_D  F ) `  x )  -  (
( RR  _D  F
) `  x )
) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
7958, 76, 783eqtrd 2321 . . . . . . . 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 4734 . . . . . . . 8  |-  ( ( A (,) B )  X.  { 0 } )  =  ( x  e.  ( A (,) B )  |->  0 )
8179, 80syl6eqr 2335 . . . . . . 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 19349 . . . . . 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 5529 . . . . 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 5868 . . . . . . . . 9  |-  ( x  =  B  ->  ( A (,) x )  =  ( A (,) B
) )
85 itgeq1 19129 . . . . . . . . 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 15 . . . . . . . 8  |-  ( x  =  B  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t )
87 fveq2 5527 . . . . . . . 8  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
8886, 87oveq12d 5878 . . . . . . 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 2285 . . . . . . 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 5885 . . . . . . 7  |-  ( S. ( A (,) B
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  B ) )  e. 
_V
9188, 89, 90fvmpt 5604 . . . . . 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 15 . . . . 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 2319 . . . 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 10754 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
952, 4, 5, 94syl3anc 1182 . . . . 5  |-  ( ph  ->  A  e.  ( A [,] B ) )
96 oveq2 5868 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( A (,) x )  =  ( A (,) A
) )
97 iooid 10686 . . . . . . . . . . 11  |-  ( A (,) A )  =  (/)
9896, 97syl6eq 2333 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A (,) x )  =  (/) )
99 itgeq1 19129 . . . . . . . . . 10  |-  ( ( A (,) x )  =  (/)  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
10098, 99syl 15 . . . . . . . . 9  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
101 itg0 19136 . . . . . . . . 9  |-  S. (/) ( ( RR  _D  F ) `  t
)  _d t  =  0
102100, 101syl6eq 2333 . . . . . . . 8  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  0 )
103 fveq2 5527 . . . . . . . 8  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
104102, 103oveq12d 5878 . . . . . . 7  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  =  ( 0  -  ( F `  A )
) )
105 df-neg 9042 . . . . . . 7  |-  -u ( F `  A )  =  ( 0  -  ( F `  A
) )
106104, 105syl6eqr 2335 . . . . . 6  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  = 
-u ( F `  A ) )
107 negex 9052 . . . . . 6  |-  -u ( F `  A )  e.  _V
108106, 89, 107fvmpt 5604 . . . . 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 15 . . . 4  |-  ( ph  ->  ( ( x  e.  ( A [,] B
)  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) ) ) `  A
)  =  -u ( F `  A )
)
11010, 93, 1093eqtr3d 2325 . . 3  |-  ( ph  ->  ( S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  B )
)  =  -u ( F `  A )
)
111110oveq2d 5876 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  ( ( F `  B )  +  -u ( F `  A ) ) )
112 ffvelrn 5665 . . . 4  |-  ( ( F : ( A [,] B ) --> CC 
/\  B  e.  ( A [,] B ) )  ->  ( F `  B )  e.  CC )
11326, 7, 112syl2anc 642 . . 3  |-  ( ph  ->  ( F `  B
)  e.  CC )
11434a1i 10 . . . 4  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  _V )
115114, 48itgcl 19140 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  e.  CC )
116113, 115pncan3d 9162 . 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 5665 . . . 4  |-  ( ( F : ( A [,] B ) --> CC 
/\  A  e.  ( A [,] B ) )  ->  ( F `  A )  e.  CC )
11826, 95, 117syl2anc 642 . . 3  |-  ( ph  ->  ( F `  A
)  e.  CC )
119113, 118negsubd 9165 . 2  |-  ( ph  ->  ( ( F `  B )  +  -u ( F `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
120111, 116, 1193eqtr3d 2325 1  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   _Vcvv 2790    C_ wss 3154   (/)c0 3457   {csn 3642   {cpr 3643   class class class wbr 4025    e. cmpt 4079    X. cxp 4689   dom cdm 4691   ran crn 4692   -->wf 5253   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739    + caddc 8742   RR*cxr 8868    <_ cle 8870    - cmin 9039   -ucneg 9040   (,)cioo 10658   [,]cicc 10661   TopOpenctopn 13328   topGenctg 13344  ℂfldccnfld 16379   intcnt 16756    Cn ccn 16956    tX ctx 17257   -cn->ccncf 18382   volcvol 18825   L ^1cibl 18974   S.citg 18975    _D cdv 19215
This theorem is referenced by:  ftc2ditglem  19394  itgparts  19396  itgsubstlem  19397  areacirc  24942  lhe4.4ex1a  27557  itgsin0pilem1  27755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cc 8063  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-disj 3996  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-ofr 6081  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-omul 6486  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-acn 7577  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-rlim 11965  df-sum 12161  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-cmp 17116  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-ovol 18826  df-vol 18827  df-mbf 18977  df-itg1 18978  df-itg2 18979  df-ibl 18980  df-itg 18981  df-0p 19027  df-limc 19218  df-dv 19219
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