Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itgsin0pilem1 Unicode version

Theorem itgsin0pilem1 27847
Description: Calculation of the integral for sine on the (0,π) interval (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypothesis
Ref Expression
itgsin0pilem1.1  |-  C  =  ( t  e.  ( 0 [,] pi ) 
|->  -u ( cos `  t
) )
Assertion
Ref Expression
itgsin0pilem1  |-  S. ( 0 (,) pi ) ( sin `  x
)  _d x  =  2
Distinct variable groups:    x, t    x, C
Allowed substitution hint:    C( t)

Proof of Theorem itgsin0pilem1
StepHypRef Expression
1 itgsin0pilem1.1 . . . . . . . . . . . . 13  |-  C  =  ( t  e.  ( 0 [,] pi ) 
|->  -u ( cos `  t
) )
2 nfcv 2432 . . . . . . . . . . . . . 14  |-  F/_ x -u ( cos `  t
)
3 nfcv 2432 . . . . . . . . . . . . . 14  |-  F/_ t -u ( cos `  x
)
4 fveq2 5541 . . . . . . . . . . . . . . 15  |-  ( t  =  x  ->  ( cos `  t )  =  ( cos `  x
) )
54negeqd 9062 . . . . . . . . . . . . . 14  |-  ( t  =  x  ->  -u ( cos `  t )  = 
-u ( cos `  x
) )
62, 3, 5cbvmpt 4126 . . . . . . . . . . . . 13  |-  ( t  e.  ( 0 [,] pi )  |->  -u ( cos `  t ) )  =  ( x  e.  ( 0 [,] pi )  |->  -u ( cos `  x
) )
71, 6eqtri 2316 . . . . . . . . . . . 12  |-  C  =  ( x  e.  ( 0 [,] pi ) 
|->  -u ( cos `  x
) )
87oveq2i 5885 . . . . . . . . . . 11  |-  ( RR 
_D  C )  =  ( RR  _D  (
x  e.  ( 0 [,] pi )  |->  -u ( cos `  x ) ) )
9 recn 8843 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR  ->  x  e.  CC )
109ssriv 3197 . . . . . . . . . . . . . 14  |-  RR  C_  CC
1110a1i 10 . . . . . . . . . . . . 13  |-  (  T. 
->  RR  C_  CC )
12 0re 8854 . . . . . . . . . . . . . . . 16  |-  0  e.  RR
13 pire 19848 . . . . . . . . . . . . . . . 16  |-  pi  e.  RR
1412, 13pm3.2i 441 . . . . . . . . . . . . . . 15  |-  ( 0  e.  RR  /\  pi  e.  RR )
15 iccssre 10747 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  pi  e.  RR )  -> 
( 0 [,] pi )  C_  RR )
1614, 15ax-mp 8 . . . . . . . . . . . . . 14  |-  ( 0 [,] pi )  C_  RR
1716a1i 10 . . . . . . . . . . . . 13  |-  (  T. 
->  ( 0 [,] pi )  C_  RR )
1816, 10sstri 3201 . . . . . . . . . . . . . . . . 17  |-  ( 0 [,] pi )  C_  CC
1918sseli 3189 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( 0 [,] pi )  ->  x  e.  CC )
2019coscld 12427 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( 0 [,] pi )  ->  ( cos `  x )  e.  CC )
2120adantl 452 . . . . . . . . . . . . . 14  |-  ( (  T.  /\  x  e.  ( 0 [,] pi ) )  ->  ( cos `  x )  e.  CC )
2221negcld 9160 . . . . . . . . . . . . 13  |-  ( (  T.  /\  x  e.  ( 0 [,] pi ) )  ->  -u ( cos `  x )  e.  CC )
23 eqid 2296 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2423tgioo2 18325 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
25 iccntr 18342 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  pi  e.  RR )  -> 
( ( int `  ( topGen `
 ran  (,) )
) `  ( 0 [,] pi ) )  =  ( 0 (,) pi ) )
2614, 25ax-mp 8 . . . . . . . . . . . . . 14  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( 0 [,] pi ) )  =  ( 0 (,) pi )
2726a1i 10 . . . . . . . . . . . . 13  |-  (  T. 
->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( 0 [,] pi ) )  =  ( 0 (,) pi ) )
2811, 17, 22, 24, 23, 27dvmptntr 19336 . . . . . . . . . . . 12  |-  (  T. 
->  ( RR  _D  (
x  e.  ( 0 [,] pi )  |->  -u ( cos `  x ) ) )  =  ( RR  _D  ( x  e.  ( 0 (,) pi )  |->  -u ( cos `  x ) ) ) )
2928trud 1314 . . . . . . . . . . 11  |-  ( RR 
_D  ( x  e.  ( 0 [,] pi )  |->  -u ( cos `  x
) ) )  =  ( RR  _D  (
x  e.  ( 0 (,) pi )  |->  -u ( cos `  x ) ) )
308, 29eqtri 2316 . . . . . . . . . 10  |-  ( RR 
_D  C )  =  ( RR  _D  (
x  e.  ( 0 (,) pi )  |->  -u ( cos `  x ) ) )
31 reex 8844 . . . . . . . . . . . . . 14  |-  RR  e.  _V
3231prid1 3747 . . . . . . . . . . . . 13  |-  RR  e.  { RR ,  CC }
3332a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  RR  e.  { RR ,  CC } )
349coscld 12427 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  ( cos `  x )  e.  CC )
3534adantl 452 . . . . . . . . . . . . 13  |-  ( (  T.  /\  x  e.  RR )  ->  ( cos `  x )  e.  CC )
3635negcld 9160 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  RR )  ->  -u ( cos `  x )  e.  CC )
379sincld 12426 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  ( sin `  x )  e.  CC )
3837adantl 452 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  RR )  ->  ( sin `  x )  e.  CC )
3937negcld 9160 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR  ->  -u ( sin `  x )  e.  CC )
4039adantl 452 . . . . . . . . . . . . . . . 16  |-  ( (  T.  /\  x  e.  RR )  ->  -u ( sin `  x )  e.  CC )
41 dvcosre 27844 . . . . . . . . . . . . . . . . 17  |-  ( RR 
_D  ( x  e.  RR  |->  ( cos `  x
) ) )  =  ( x  e.  RR  |->  -u ( sin `  x
) )
4241a1i 10 . . . . . . . . . . . . . . . 16  |-  (  T. 
->  ( RR  _D  (
x  e.  RR  |->  ( cos `  x ) ) )  =  ( x  e.  RR  |->  -u ( sin `  x ) ) )
4333, 35, 40, 42dvmptneg 19331 . . . . . . . . . . . . . . 15  |-  (  T. 
->  ( RR  _D  (
x  e.  RR  |->  -u ( cos `  x ) ) )  =  ( x  e.  RR  |->  -u -u ( sin `  x
) ) )
4443trud 1314 . . . . . . . . . . . . . 14  |-  ( RR 
_D  ( x  e.  RR  |->  -u ( cos `  x
) ) )  =  ( x  e.  RR  |->  -u -u ( sin `  x
) )
4537negnegd 9164 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR  ->  -u -u ( sin `  x )  =  ( sin `  x
) )
4645mpteq2ia 4118 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  |->  -u -u ( sin `  x ) )  =  ( x  e.  RR  |->  ( sin `  x
) )
4744, 46eqtri 2316 . . . . . . . . . . . . 13  |-  ( RR 
_D  ( x  e.  RR  |->  -u ( cos `  x
) ) )  =  ( x  e.  RR  |->  ( sin `  x ) )
4847a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  ( RR  _D  (
x  e.  RR  |->  -u ( cos `  x ) ) )  =  ( x  e.  RR  |->  ( sin `  x ) ) )
49 ioossre 10728 . . . . . . . . . . . . 13  |-  ( 0 (,) pi )  C_  RR
5049a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  ( 0 (,) pi )  C_  RR )
51 iooretop 18291 . . . . . . . . . . . . 13  |-  ( 0 (,) pi )  e.  ( topGen `  ran  (,) )
5251a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  ( 0 (,) pi )  e.  ( topGen ` 
ran  (,) ) )
5333, 36, 38, 48, 50, 24, 23, 52dvmptres 19328 . . . . . . . . . . 11  |-  (  T. 
->  ( RR  _D  (
x  e.  ( 0 (,) pi )  |->  -u ( cos `  x ) ) )  =  ( x  e.  ( 0 (,) pi )  |->  ( sin `  x ) ) )
5453trud 1314 . . . . . . . . . 10  |-  ( RR 
_D  ( x  e.  ( 0 (,) pi )  |->  -u ( cos `  x
) ) )  =  ( x  e.  ( 0 (,) pi ) 
|->  ( sin `  x
) )
5530, 54eqtri 2316 . . . . . . . . 9  |-  ( RR 
_D  C )  =  ( x  e.  ( 0 (,) pi ) 
|->  ( sin `  x
) )
5655fveq1i 5542 . . . . . . . 8  |-  ( ( RR  _D  C ) `
 x )  =  ( ( x  e.  ( 0 (,) pi )  |->  ( sin `  x
) ) `  x
)
5756a1i 10 . . . . . . 7  |-  ( x  e.  ( 0 (,) pi )  ->  (
( RR  _D  C
) `  x )  =  ( ( x  e.  ( 0 (,) pi )  |->  ( sin `  x ) ) `  x ) )
58 id 19 . . . . . . . . 9  |-  ( x  e.  ( 0 (,) pi )  ->  x  e.  ( 0 (,) pi ) )
5949, 10sstri 3201 . . . . . . . . . . 11  |-  ( 0 (,) pi )  C_  CC
6059sseli 3189 . . . . . . . . . 10  |-  ( x  e.  ( 0 (,) pi )  ->  x  e.  CC )
6160sincld 12426 . . . . . . . . 9  |-  ( x  e.  ( 0 (,) pi )  ->  ( sin `  x )  e.  CC )
6258, 61jca 518 . . . . . . . 8  |-  ( x  e.  ( 0 (,) pi )  ->  (
x  e.  ( 0 (,) pi )  /\  ( sin `  x )  e.  CC ) )
63 eqid 2296 . . . . . . . . 9  |-  ( x  e.  ( 0 (,) pi )  |->  ( sin `  x ) )  =  ( x  e.  ( 0 (,) pi ) 
|->  ( sin `  x
) )
6463fvmpt2 5624 . . . . . . . 8  |-  ( ( x  e.  ( 0 (,) pi )  /\  ( sin `  x )  e.  CC )  -> 
( ( x  e.  ( 0 (,) pi )  |->  ( sin `  x
) ) `  x
)  =  ( sin `  x ) )
6562, 64syl 15 . . . . . . 7  |-  ( x  e.  ( 0 (,) pi )  ->  (
( x  e.  ( 0 (,) pi ) 
|->  ( sin `  x
) ) `  x
)  =  ( sin `  x ) )
6657, 65eqtrd 2328 . . . . . 6  |-  ( x  e.  ( 0 (,) pi )  ->  (
( RR  _D  C
) `  x )  =  ( sin `  x
) )
6766adantl 452 . . . . 5  |-  ( (  T.  /\  x  e.  ( 0 (,) pi ) )  ->  (
( RR  _D  C
) `  x )  =  ( sin `  x
) )
6867itgeq2dv 19152 . . . 4  |-  (  T. 
->  S. ( 0 (,) pi ) ( ( RR  _D  C ) `
 x )  _d x  =  S. ( 0 (,) pi ) ( sin `  x
)  _d x )
6968trud 1314 . . 3  |-  S. ( 0 (,) pi ) ( ( RR  _D  C ) `  x
)  _d x  =  S. ( 0 (,) pi ) ( sin `  x )  _d x
7012a1i 10 . . . . 5  |-  (  T. 
->  0  e.  RR )
7113a1i 10 . . . . 5  |-  (  T. 
->  pi  e.  RR )
72 pipos 19849 . . . . . . 7  |-  0  <  pi
73 ltle 8926 . . . . . . 7  |-  ( ( 0  e.  RR  /\  pi  e.  RR )  -> 
( 0  <  pi  ->  0  <_  pi )
)
7414, 72, 73mp2 17 . . . . . 6  |-  0  <_  pi
7574a1i 10 . . . . 5  |-  (  T. 
->  0  <_  pi )
76 nfcv 2432 . . . . . . . . 9  |-  F/_ x sin
77 sincn 19836 . . . . . . . . . 10  |-  sin  e.  ( CC -cn-> CC )
7877a1i 10 . . . . . . . . 9  |-  (  T. 
->  sin  e.  ( CC
-cn-> CC ) )
7959a1i 10 . . . . . . . . 9  |-  (  T. 
->  ( 0 (,) pi )  C_  CC )
8076, 78, 79cncfmptss 27820 . . . . . . . 8  |-  (  T. 
->  ( x  e.  ( 0 (,) pi ) 
|->  ( sin `  x
) )  e.  ( ( 0 (,) pi ) -cn-> CC ) )
8180trud 1314 . . . . . . 7  |-  ( x  e.  ( 0 (,) pi )  |->  ( sin `  x ) )  e.  ( ( 0 (,) pi ) -cn-> CC )
8255, 81eqeltri 2366 . . . . . 6  |-  ( RR 
_D  C )  e.  ( ( 0 (,) pi ) -cn-> CC )
8382a1i 10 . . . . 5  |-  (  T. 
->  ( RR  _D  C
)  e.  ( ( 0 (,) pi )
-cn-> CC ) )
84 ioossicc 10751 . . . . . . . . . 10  |-  ( 0 (,) pi )  C_  ( 0 [,] pi )
8584a1i 10 . . . . . . . . 9  |-  (  T. 
->  ( 0 (,) pi )  C_  ( 0 [,] pi ) )
86 ioombl 18938 . . . . . . . . . 10  |-  ( 0 (,) pi )  e. 
dom  vol
8786a1i 10 . . . . . . . . 9  |-  (  T. 
->  ( 0 (,) pi )  e.  dom  vol )
8819sincld 12426 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] pi )  ->  ( sin `  x )  e.  CC )
8988adantl 452 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 0 [,] pi ) )  ->  ( sin `  x )  e.  CC )
9018a1i 10 . . . . . . . . . . . . . 14  |-  (  T. 
->  ( 0 [,] pi )  C_  CC )
9176, 78, 90cncfmptss 27820 . . . . . . . . . . . . 13  |-  (  T. 
->  ( x  e.  ( 0 [,] pi ) 
|->  ( sin `  x
) )  e.  ( ( 0 [,] pi ) -cn-> CC ) )
9291trud 1314 . . . . . . . . . . . 12  |-  ( x  e.  ( 0 [,] pi )  |->  ( sin `  x ) )  e.  ( ( 0 [,] pi ) -cn-> CC )
9312, 13, 923pm3.2i 1130 . . . . . . . . . . 11  |-  ( 0  e.  RR  /\  pi  e.  RR  /\  ( x  e.  ( 0 [,] pi )  |->  ( sin `  x ) )  e.  ( ( 0 [,] pi ) -cn-> CC ) )
94 cniccibl 19211 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  pi  e.  RR  /\  (
x  e.  ( 0 [,] pi )  |->  ( sin `  x ) )  e.  ( ( 0 [,] pi )
-cn-> CC ) )  -> 
( x  e.  ( 0 [,] pi ) 
|->  ( sin `  x
) )  e.  L ^1 )
9593, 94ax-mp 8 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] pi )  |->  ( sin `  x ) )  e.  L ^1
9695a1i 10 . . . . . . . . 9  |-  (  T. 
->  ( x  e.  ( 0 [,] pi ) 
|->  ( sin `  x
) )  e.  L ^1 )
9785, 87, 89, 96iblss 19175 . . . . . . . 8  |-  (  T. 
->  ( x  e.  ( 0 (,) pi ) 
|->  ( sin `  x
) )  e.  L ^1 )
9897trud 1314 . . . . . . 7  |-  ( x  e.  ( 0 (,) pi )  |->  ( sin `  x ) )  e.  L ^1
9955, 98eqeltri 2366 . . . . . 6  |-  ( RR 
_D  C )  e.  L ^1
10099a1i 10 . . . . 5  |-  (  T. 
->  ( RR  _D  C
)  e.  L ^1 )
10120negcld 9160 . . . . . . . . . . . 12  |-  ( x  e.  ( 0 [,] pi )  ->  -u ( cos `  x )  e.  CC )
10219, 101jca 518 . . . . . . . . . . 11  |-  ( x  e.  ( 0 [,] pi )  ->  (
x  e.  CC  /\  -u ( cos `  x
)  e.  CC ) )
103 eqid 2296 . . . . . . . . . . . 12  |-  ( x  e.  CC  |->  -u ( cos `  x ) )  =  ( x  e.  CC  |->  -u ( cos `  x
) )
104103fvmpt2 5624 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  -u ( cos `  x
)  e.  CC )  ->  ( ( x  e.  CC  |->  -u ( cos `  x ) ) `
 x )  = 
-u ( cos `  x
) )
105102, 104syl 15 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] pi )  ->  (
( x  e.  CC  |->  -u ( cos `  x
) ) `  x
)  =  -u ( cos `  x ) )
106105eqcomd 2301 . . . . . . . . 9  |-  ( x  e.  ( 0 [,] pi )  ->  -u ( cos `  x )  =  ( ( x  e.  CC  |->  -u ( cos `  x
) ) `  x
) )
107106mpteq2ia 4118 . . . . . . . 8  |-  ( x  e.  ( 0 [,] pi )  |->  -u ( cos `  x ) )  =  ( x  e.  ( 0 [,] pi )  |->  ( ( x  e.  CC  |->  -u ( cos `  x ) ) `
 x ) )
108 nfmpt1 4125 . . . . . . . . . 10  |-  F/_ x
( x  e.  CC  |->  -u ( cos `  x
) )
109 coscn 19837 . . . . . . . . . . . 12  |-  cos  e.  ( CC -cn-> CC )
110103negfcncf 18438 . . . . . . . . . . . 12  |-  ( cos 
e.  ( CC -cn-> CC )  ->  ( x  e.  CC  |->  -u ( cos `  x
) )  e.  ( CC -cn-> CC ) )
111109, 110ax-mp 8 . . . . . . . . . . 11  |-  ( x  e.  CC  |->  -u ( cos `  x ) )  e.  ( CC -cn-> CC )
112111a1i 10 . . . . . . . . . 10  |-  (  T. 
->  ( x  e.  CC  |->  -u ( cos `  x
) )  e.  ( CC -cn-> CC ) )
113108, 112, 90cncfmptss 27820 . . . . . . . . 9  |-  (  T. 
->  ( x  e.  ( 0 [,] pi ) 
|->  ( ( x  e.  CC  |->  -u ( cos `  x
) ) `  x
) )  e.  ( ( 0 [,] pi ) -cn-> CC ) )
114113trud 1314 . . . . . . . 8  |-  ( x  e.  ( 0 [,] pi )  |->  ( ( x  e.  CC  |->  -u ( cos `  x ) ) `  x ) )  e.  ( ( 0 [,] pi )
-cn-> CC )
115107, 114eqeltri 2366 . . . . . . 7  |-  ( x  e.  ( 0 [,] pi )  |->  -u ( cos `  x ) )  e.  ( ( 0 [,] pi ) -cn-> CC )
1167, 115eqeltri 2366 . . . . . 6  |-  C  e.  ( ( 0 [,] pi ) -cn-> CC )
117116a1i 10 . . . . 5  |-  (  T. 
->  C  e.  (
( 0 [,] pi ) -cn-> CC ) )
11870, 71, 75, 83, 100, 117ftc2 19407 . . . 4  |-  (  T. 
->  S. ( 0 (,) pi ) ( ( RR  _D  C ) `
 x )  _d x  =  ( ( C `  pi )  -  ( C ` 
0 ) ) )
119118trud 1314 . . 3  |-  S. ( 0 (,) pi ) ( ( RR  _D  C ) `  x
)  _d x  =  ( ( C `  pi )  -  ( C `  0 )
)
12069, 119eqtr3i 2318 . 2  |-  S. ( 0 (,) pi ) ( sin `  x
)  _d x  =  ( ( C `  pi )  -  ( C `  0 )
)
121 0xr 8894 . . . . . . 7  |-  0  e.  RR*
12213rexri 8900 . . . . . . 7  |-  pi  e.  RR*
123121, 122, 743pm3.2i 1130 . . . . . 6  |-  ( 0  e.  RR*  /\  pi  e.  RR*  /\  0  <_  pi )
124 ubicc2 10769 . . . . . 6  |-  ( ( 0  e.  RR*  /\  pi  e.  RR*  /\  0  <_  pi )  ->  pi  e.  ( 0 [,] pi ) )
125123, 124ax-mp 8 . . . . 5  |-  pi  e.  ( 0 [,] pi )
126 fveq2 5541 . . . . . . . . 9  |-  ( t  =  pi  ->  ( cos `  t )  =  ( cos `  pi ) )
127 cospi 19856 . . . . . . . . . 10  |-  ( cos `  pi )  =  -u
1
128127a1i 10 . . . . . . . . 9  |-  ( t  =  pi  ->  ( cos `  pi )  = 
-u 1 )
129126, 128eqtrd 2328 . . . . . . . 8  |-  ( t  =  pi  ->  ( cos `  t )  = 
-u 1 )
130129negeqd 9062 . . . . . . 7  |-  ( t  =  pi  ->  -u ( cos `  t )  = 
-u -u 1 )
131 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
132131a1i 10 . . . . . . . 8  |-  ( t  =  pi  ->  1  e.  CC )
133132negnegd 9164 . . . . . . 7  |-  ( t  =  pi  ->  -u -u 1  =  1 )
134130, 133eqtrd 2328 . . . . . 6  |-  ( t  =  pi  ->  -u ( cos `  t )  =  1 )
135 1ex 8849 . . . . . 6  |-  1  e.  _V
136134, 1, 135fvmpt 5618 . . . . 5  |-  ( pi  e.  ( 0 [,] pi )  ->  ( C `  pi )  =  1 )
137125, 136ax-mp 8 . . . 4  |-  ( C `
 pi )  =  1
138 lbicc2 10768 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  pi  e.  RR*  /\  0  <_  pi )  ->  0  e.  ( 0 [,] pi ) )
139123, 138ax-mp 8 . . . . . 6  |-  0  e.  ( 0 [,] pi )
140 fveq2 5541 . . . . . . . 8  |-  ( t  =  0  ->  ( cos `  t )  =  ( cos `  0
) )
141140negeqd 9062 . . . . . . 7  |-  ( t  =  0  ->  -u ( cos `  t )  = 
-u ( cos `  0
) )
142 negex 9066 . . . . . . 7  |-  -u ( cos `  0 )  e. 
_V
143141, 1, 142fvmpt 5618 . . . . . 6  |-  ( 0  e.  ( 0 [,] pi )  ->  ( C `  0 )  =  -u ( cos `  0
) )
144139, 143ax-mp 8 . . . . 5  |-  ( C `
 0 )  = 
-u ( cos `  0
)
145 cos0 12446 . . . . . 6  |-  ( cos `  0 )  =  1
146145negeqi 9061 . . . . 5  |-  -u ( cos `  0 )  = 
-u 1
147144, 146eqtri 2316 . . . 4  |-  ( C `
 0 )  = 
-u 1
148137, 147oveq12i 5886 . . 3  |-  ( ( C `  pi )  -  ( C ` 
0 ) )  =  ( 1  -  -u 1
)
149131, 131pm3.2i 441 . . . . 5  |-  ( 1  e.  CC  /\  1  e.  CC )
150 subneg 9112 . . . . 5  |-  ( ( 1  e.  CC  /\  1  e.  CC )  ->  ( 1  -  -u 1
)  =  ( 1  +  1 ) )
151149, 150ax-mp 8 . . . 4  |-  ( 1  -  -u 1 )  =  ( 1  +  1 )
152 1p1e2 9856 . . . 4  |-  ( 1  +  1 )  =  2
153151, 152eqtri 2316 . . 3  |-  ( 1  -  -u 1 )  =  2
154148, 153eqtri 2316 . 2  |-  ( ( C `  pi )  -  ( C ` 
0 ) )  =  2
155120, 154eqtri 2316 1  |-  S. ( 0 (,) pi ) ( sin `  x
)  _d x  =  2
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1632    e. wcel 1696    C_ wss 3165   {cpr 3654   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054   2c2 9811   (,)cioo 10672   [,]cicc 10675   sincsin 12361   cosccos 12362   picpi 12364   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393   intcnt 16770   -cn->ccncf 18396   volcvol 18839   L ^1cibl 18988   S.citg 18989    _D cdv 19229
This theorem is referenced by:  itgsin0pi  27849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-ovol 18840  df-vol 18841  df-mbf 18991  df-itg1 18992  df-itg2 18993  df-ibl 18994  df-itg 18995  df-0p 19041  df-limc 19232  df-dv 19233
  Copyright terms: Public domain W3C validator