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Theorem ftc2 19795
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 9067 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
3 ftc2.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
43rexrd 9067 . . . . . 6  |-  ( ph  ->  B  e.  RR* )
5 ftc2.le . . . . . 6  |-  ( ph  ->  A  <_  B )
6 ubicc2 10946 . . . . . 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 5682 . . . . . 6  |-  ( ( x  e.  ( A [,] B )  |->  ( S. ( A (,) x ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 x ) ) ) `  A )  e.  _V
98fvconst2 5886 . . . . 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 2387 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1211subcn 18767 . . . . . . . . 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 2387 . . . . . . . . 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 3310 . . . . . . . . . 10  |-  ( A (,) B )  C_  ( A (,) B )
1615a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( A (,) B
)  C_  ( A (,) B ) )
17 ioossre 10904 . . . . . . . . . 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 18794 . . . . . . . . . 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 19788 . . . . . . . 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 18794 . . . . . . . . . . 11  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
2624, 25syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : ( A [,] B ) --> CC )
2726feqmptd 5718 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  ( A [,] B )  |->  ( F `
 x ) ) )
2827, 24eqeltrrd 2462 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( A [,] B ) 
|->  ( F `  x
) )  e.  ( ( A [,] B
) -cn-> CC ) )
2911, 13, 23, 28cncfmpt2f 18815 . . . . . . 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 8980 . . . . . . . . . . 11  |-  RR  C_  CC
3130a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
32 iccssre 10924 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
331, 3, 32syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( A [,] B
)  C_  RR )
34 fvex 5682 . . . . . . . . . . . . 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 9067 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  B  e.  RR* )
38 elicc2 10907 . . . . . . . . . . . . . . . . 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 10884 . . . . . . . . . . . . . 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 19326 . . . . . . . . . . . . . 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 5718 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
4847, 19eqeltrrd 2462 . . . . . . . . . . . . . 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 19563 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( t  e.  ( A (,) x
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
5135, 50itgcl 19542 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  S. ( A (,) x ) ( ( RR  _D  F
) `  t )  _d t  e.  CC )
5226ffvelrnda 5809 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  CC )
5351, 52subcld 9343 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  -  ( F `  x ) )  e.  CC )
5411tgioo2 18705 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
55 iccntr 18723 . . . . . . . . . . 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 19724 . . . . . . . . 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 9014 . . . . . . . . . . . 12  |-  RR  e.  _V
5958prid1 3855 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
6059a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
61 ioossicc 10928 . . . . . . . . . . . 12  |-  ( A (,) B )  C_  ( A [,] B )
6261sseli 3287 . . . . . . . . . . 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 5809 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
6514, 1, 3, 5, 20, 19ftc1cn 19794 . . . . . . . . . . 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 19724 . . . . . . . . . . 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 5718 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  F
)  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 x ) ) )
6865, 66, 673eqtr3d 2427 . . . . . . . . . 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 19724 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( RR  _D  ( x  e.  ( A (,) B )  |->  ( F `
 x ) ) ) )
7127oveq2d 6036 . . . . . . . . . . . 12  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( x  e.  ( A [,] B
)  |->  ( F `  x ) ) ) )
7271, 67eqtr3d 2421 . . . . . . . . . . 11  |-  ( ph  ->  ( RR  _D  (
x  e.  ( A [,] B )  |->  ( F `  x ) ) )  =  ( x  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  x )
) )
7370, 72eqtr3d 2421 . . . . . . . . . 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 19720 . . . . . . . . 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 9331 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( RR  _D  F
) `  x )  -  ( ( RR 
_D  F ) `  x ) )  =  0 )
7675mpteq2dva 4236 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A (,) B ) 
|->  ( ( ( RR 
_D  F ) `  x )  -  (
( RR  _D  F
) `  x )
) )  =  ( x  e.  ( A (,) B )  |->  0 ) )
7757, 74, 763eqtrd 2423 . . . . . . . 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 4861 . . . . . . . 8  |-  ( ( A (,) B )  X.  { 0 } )  =  ( x  e.  ( A (,) B )  |->  0 )
7977, 78syl6eqr 2437 . . . . . . 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 19751 . . . . . 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 5670 . . . . 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 6028 . . . . . . . . 9  |-  ( x  =  B  ->  ( A (,) x )  =  ( A (,) B
) )
83 itgeq1 19531 . . . . . . . . 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 5668 . . . . . . . 8  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
8684, 85oveq12d 6038 . . . . . . 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 2387 . . . . . . 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 6045 . . . . . . 7  |-  ( S. ( A (,) B
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  B ) )  e. 
_V
8986, 87, 88fvmpt 5745 . . . . . 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 2421 . . . 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 10945 . . . . . 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 6028 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( A (,) x )  =  ( A (,) A
) )
95 iooid 10876 . . . . . . . . . . 11  |-  ( A (,) A )  =  (/)
9694, 95syl6eq 2435 . . . . . . . . . 10  |-  ( x  =  A  ->  ( A (,) x )  =  (/) )
97 itgeq1 19531 . . . . . . . . . 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 19538 . . . . . . . . 9  |-  S. (/) ( ( RR  _D  F ) `  t
)  _d t  =  0
10098, 99syl6eq 2435 . . . . . . . 8  |-  ( x  =  A  ->  S. ( A (,) x ) ( ( RR  _D  F ) `  t
)  _d t  =  0 )
101 fveq2 5668 . . . . . . . 8  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
102100, 101oveq12d 6038 . . . . . . 7  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  =  ( 0  -  ( F `  A )
) )
103 df-neg 9226 . . . . . . 7  |-  -u ( F `  A )  =  ( 0  -  ( F `  A
) )
104102, 103syl6eqr 2437 . . . . . 6  |-  ( x  =  A  ->  ( S. ( A (,) x
) ( ( RR 
_D  F ) `  t )  _d t  -  ( F `  x ) )  = 
-u ( F `  A ) )
105 negex 9236 . . . . . 6  |-  -u ( F `  A )  e.  _V
106104, 87, 105fvmpt 5745 . . . . 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 2427 . . 3  |-  ( ph  ->  ( S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  -  ( F `  B )
)  =  -u ( F `  A )
)
109108oveq2d 6036 . 2  |-  ( ph  ->  ( ( F `  B )  +  ( S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  -  ( F `
 B ) ) )  =  ( ( F `  B )  +  -u ( F `  A ) ) )
11026, 7ffvelrnd 5810 . . 3  |-  ( ph  ->  ( F `  B
)  e.  CC )
11134a1i 11 . . . 4  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  t )  e.  _V )
112111, 48itgcl 19542 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `
 t )  _d t  e.  CC )
113110, 112pncan3d 9346 . 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 5810 . . 3  |-  ( ph  ->  ( F `  A
)  e.  CC )
115110, 114negsubd 9349 . 2  |-  ( ph  ->  ( ( F `  B )  +  -u ( F `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
116109, 113, 1153eqtr3d 2427 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 1717   _Vcvv 2899    C_ wss 3263   (/)c0 3571   {csn 3757   {cpr 3758   class class class wbr 4153    e. cmpt 4207    X. cxp 4816   dom cdm 4818   ran crn 4819   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923    + caddc 8926   RR*cxr 9052    <_ cle 9054    - cmin 9223   -ucneg 9224   (,)cioo 10848   [,]cicc 10851   TopOpenctopn 13576   topGenctg 13592  ℂfldccnfld 16626   intcnt 17004    Cn ccn 17210    tX ctx 17513   -cn->ccncf 18777   volcvol 19227   L ^1cibl 19376   S.citg 19377    _D cdv 19617
This theorem is referenced by:  ftc2ditglem  19796  itgparts  19798  itgsubstlem  19799  areacirc  25988  lhe4.4ex1a  27215  itgsin0pilem1  27412
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cc 8248  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-disj 4124  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-ofr 6245  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-acn 7762  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ioc 10853  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-rlim 12210  df-sum 12407  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-fbas 16623  df-fg 16624  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123  df-perf 17124  df-cn 17213  df-cnp 17214  df-haus 17301  df-cmp 17372  df-tx 17515  df-hmeo 17708  df-fil 17799  df-fm 17891  df-flim 17892  df-flf 17893  df-xms 18259  df-ms 18260  df-tms 18261  df-cncf 18779  df-ovol 19228  df-vol 19229  df-mbf 19379  df-itg1 19380  df-itg2 19381  df-ibl 19382  df-itg 19383  df-0p 19429  df-limc 19620  df-dv 19621
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