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Theorem ftc2 19881
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 9090 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
3 ftc2.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
43rexrd 9090 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
5 ftc2.le . . . . . 6  |-  ( ph  ->  A  <_  B )
6 ubicc2 10970 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
72, 4, 5, 6syl3anc 1184 . . . . 5  |-  ( ph  ->  B  e.  ( A [,] B ) )
8 fvex 5701 . . . . . 6  |-  ( ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) `  A )  e.  _V
98fvconst2 5906 . . . . 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 2404 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1211subcn 18849 . . . . . . . . 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 2404 . . . . . . . . 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 3327 . . . . . . . . . 10  |-  ( A (,) B )  C_  ( A (,) B )
1615a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( A (,) B
)  C_  ( A (,) B ) )
17 ioossre 10928 . . . . . . . . . 10  |-  ( A (,) B )  C_  RR
1817a1i 11 . . . . . . . . 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 18876 . . . . . . . . . 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 )
2314, 1, 3, 5, 16, 18, 19, 22ftc1a 19874 . . . . . . . 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 18876 . . . . . . . . . . 11  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
2624, 25syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> CC )
2726feqmptd 5738 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  ( A [,] B )  |->  ( F `
 x ) ) )
2827, 24eqeltrrd 2479 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( F `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
2911, 13, 23, 28cncfmpt2f 18897 . . . . . . 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 9003 . . . . . . . . . . 11  |-  RR  C_  CC
3130a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
32 iccssre 10948 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
331, 3, 32syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( A [,] B
)  C_  RR )
34 fvex 5701 . . . . . . . . . . . . 13  |-  ( ( RR  _D  F ) `
 t )  e. 
_V
3534a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  t  e.  ( A (,) x
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
363adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR )
3736rexrd 9090 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
38 elicc2 10931 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
391, 3, 38syl2anc 643 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4039biimpa 471 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
4140simp3d 971 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
42 iooss2 10908 . . . . . . . . . . . . . 14  |-  ( ( B  e.  RR*  /\  x  <_  B )  ->  ( A (,) x )  C_  ( A (,) B ) )
4337, 41, 42syl2anc 643 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  C_  ( A (,) B ) )
44 ioombl 19412 . . . . . . . . . . . . . 14  |-  ( A (,) x )  e. 
dom  vol
4544a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( A (,) x )  e.  dom  vol )
4634a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
4722feqmptd 5738 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
4847, 19eqeltrrd 2479 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( t  e.  ( A (,) B ) 
|->  ( ( RR  _D  F ) `  t
) )  e.  L ^1 )
4948adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
5043, 45, 46, 49iblss 19649 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) x
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
5135, 50itgcl 19628 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
5226ffvelrnda 5829 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
5351, 52subcld 9367 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) )  e.  CC )
5411tgioo2 18787 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
55 iccntr 18805 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
561, 3, 55syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
5731, 33, 53, 54, 11, 56dvmptntr 19810 . . . . . . . . 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 ) ) ) ) )
58 reex 9037 . . . . . . . . . . . 12  |-  RR  e.  _V
5958prid1 3872 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
6059a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
61 ioossicc 10952 . . . . . . . . . . . 12  |-  ( A (,) B )  C_  ( A [,] B )
6261sseli 3304 . . . . . . . . . . 11  |-  ( x  e.  ( A (,) B )  ->  x  e.  ( A [,] B
) )
6362, 51sylan2 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
6422ffvelrnda 5829 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
6514, 1, 3, 5, 20, 19ftc1cn 19880 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t ) )  =  ( RR  _D  F ) )
6631, 33, 51, 54, 11, 56dvmptntr 19810 . . . . . . . . . . 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 ) ) )
6722feqmptd 5738 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 x ) ) )
6865, 66, 673eqtr3d 2444 . . . . . . . . . 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 )
) )
6962, 52sylan2 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  e.  CC )
7031, 33, 52, 54, 11, 56dvmptntr 19810 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( RR  _D  ( x  e.  ( A (,) B )  |->  ( F `
 x ) ) ) )
7127oveq2d 6056 . . . . . . . . . . . 12  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( x  e.  ( A [,] B
)  |->  ( F `  x ) ) ) )
7271, 67eqtr3d 2438 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  x )
) )
7370, 72eqtr3d 2438 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A (,) B )  |->  ( F `  x ) ) )  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  x )
) )
7460, 63, 64, 68, 69, 64, 73dvmptsub 19806 . . . . . . . . 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 ) ) ) )
7564subidd 9355 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( RR  _D  F
) `  x )  -  ( ( RR 
_D  F ) `  x ) )  =  0 )
7675mpteq2dva 4255 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( ( ( RR 
_D  F ) `  x )  -  (
( RR  _D  F
) `  x )
) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
7757, 74, 763eqtrd 2440 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
78 fconstmpt 4880 . . . . . . . 8  |-  ( ( A (,) B )  X.  { 0 } )  =  ( x  e.  ( A (,) B )  |->  0 )
7977, 78syl6eqr 2454 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) )  =  ( ( A (,) B
)  X.  { 0 } ) )
801, 3, 29, 79dveq0 19837 . . . . . 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
) } ) )
8180fveq1d 5689 . . . . 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 ) )
82 oveq2 6048 . . . . . . . . 9  |-  ( x  =  B  ->  ( A (,) x )  =  ( A (,) B
) )
83 itgeq1 19617 . . . . . . . . 9  |-  ( ( A (,) x )  =  ( A (,) B )  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t )
8482, 83syl 16 . . . . . . . 8  |-  ( x  =  B  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t )
85 fveq2 5687 . . . . . . . 8  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
8684, 85oveq12d 6058 . . . . . . 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 )
) )
87 eqid 2404 . . . . . . 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 ) ) )
88 ovex 6065 . . . . . . 7  |-  ( S. ( A (,) B
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  B ) )  e. 
_V
8986, 87, 88fvmpt 5765 . . . . . 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 ) ) )
907, 89syl 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 ) ) )
9181, 90eqtr3d 2438 . . . 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 ) ) )
92 lbicc2 10969 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
932, 4, 5, 92syl3anc 1184 . . . . 5  |-  ( ph  ->  A  e.  ( A [,] B ) )
94 oveq2 6048 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( A (,) x )  =  ( A (,) A
) )
95 iooid 10900 . . . . . . . . . . 11  |-  ( A (,) A )  =  (/)
9694, 95syl6eq 2452 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A (,) x )  =  (/) )
97 itgeq1 19617 . . . . . . . . . 10  |-  ( ( A (,) x )  =  (/)  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
9896, 97syl 16 . . . . . . . . 9  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  S. (/) ( ( RR 
_D  F ) `  t )  _d t )
99 itg0 19624 . . . . . . . . 9  |-  S. (/) ( ( RR  _D  F ) `  t
)  _d t  =  0
10098, 99syl6eq 2452 . . . . . . . 8  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  0 )
101 fveq2 5687 . . . . . . . 8  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
102100, 101oveq12d 6058 . . . . . . 7  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  =  ( 0  -  ( F `  A )
) )
103 df-neg 9250 . . . . . . 7  |-  -u ( F `  A )  =  ( 0  -  ( F `  A
) )
104102, 103syl6eqr 2454 . . . . . 6  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  = 
-u ( F `  A ) )
105 negex 9260 . . . . . 6  |-  -u ( F `  A )  e.  _V
106104, 87, 105fvmpt 5765 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  (
( x  e.  ( A [,] B ) 
|->  ( S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  x )
) ) `  A
)  =  -u ( F `  A )
)
10793, 106syl 16 . . . 4  |-  ( ph  ->  ( ( x  e.  ( A [,] B
)  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) ) ) `  A
)  =  -u ( F `  A )
)
10810, 91, 1073eqtr3d 2444 . . 3  |-  ( ph  ->  ( S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  B )
)  =  -u ( F `  A )
)
109108oveq2d 6056 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  ( ( F `  B )  +  -u ( F `  A ) ) )
11026, 7ffvelrnd 5830 . . 3  |-  ( ph  ->  ( F `  B
)  e.  CC )
11134a1i 11 . . . 4  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  _V )
112111, 48itgcl 19628 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  e.  CC )
113110, 112pncan3d 9370 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  S. ( A (,) B ) ( ( RR  _D  F ) `  t
)  _d t )
11426, 93ffvelrnd 5830 . . 3  |-  ( ph  ->  ( F `  A
)  e.  CC )
115110, 114negsubd 9373 . 2  |-  ( ph  ->  ( ( F `  B )  +  -u ( F `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
116109, 113, 1153eqtr3d 2444 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   (/)c0 3588   {csn 3774   {cpr 3775   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   dom cdm 4837   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949   RR*cxr 9075    <_ cle 9077    - cmin 9247   -ucneg 9248   (,)cioo 10872   [,]cicc 10875   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658   intcnt 17036    Cn ccn 17242    tX ctx 17545   -cn->ccncf 18859   volcvol 19313   L ^1cibl 19462   S.citg 19463    _D cdv 19703
This theorem is referenced by:  ftc2ditglem  19882  itgparts  19884  itgsubstlem  19885  areacirc  26187  lhe4.4ex1a  27414  itgsin0pilem1  27611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-ofr 6265  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-ovol 19314  df-vol 19315  df-mbf 19465  df-itg1 19466  df-itg2 19467  df-ibl 19468  df-itg 19469  df-0p 19515  df-limc 19706  df-dv 19707
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