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Theorem itgsin0pilem1 27744
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 2419 . . . . . . . . . . . . . 14  |-  F/_ x -u ( cos `  t
)
3 nfcv 2419 . . . . . . . . . . . . . 14  |-  F/_ t -u ( cos `  x
)
4 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( t  =  x  ->  ( cos `  t )  =  ( cos `  x
) )
54negeqd 9046 . . . . . . . . . . . . . 14  |-  ( t  =  x  ->  -u ( cos `  t )  = 
-u ( cos `  x
) )
62, 3, 5cbvmpt 4110 . . . . . . . . . . . . 13  |-  ( t  e.  ( 0 [,] pi )  |->  -u ( cos `  t ) )  =  ( x  e.  ( 0 [,] pi )  |->  -u ( cos `  x
) )
71, 6eqtri 2303 . . . . . . . . . . . 12  |-  C  =  ( x  e.  ( 0 [,] pi ) 
|->  -u ( cos `  x
) )
87oveq2i 5869 . . . . . . . . . . 11  |-  ( RR 
_D  C )  =  ( RR  _D  (
x  e.  ( 0 [,] pi )  |->  -u ( cos `  x ) ) )
9 recn 8827 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR  ->  x  e.  CC )
109ssriv 3184 . . . . . . . . . . . . . 14  |-  RR  C_  CC
1110a1i 10 . . . . . . . . . . . . 13  |-  (  T. 
->  RR  C_  CC )
12 0re 8838 . . . . . . . . . . . . . . . 16  |-  0  e.  RR
13 pire 19832 . . . . . . . . . . . . . . . 16  |-  pi  e.  RR
1412, 13pm3.2i 441 . . . . . . . . . . . . . . 15  |-  ( 0  e.  RR  /\  pi  e.  RR )
15 iccssre 10731 . . . . . . . . . . . . . . 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 3188 . . . . . . . . . . . . . . . . 17  |-  ( 0 [,] pi )  C_  CC
1918sseli 3176 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( 0 [,] pi )  ->  x  e.  CC )
2019coscld 12411 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( 0 [,] pi )  ->  ( cos `  x )  e.  CC )
2120adantl 452 . . . . . . . . . . . . . 14  |-  ( (  T.  /\  x  e.  ( 0 [,] pi ) )  ->  ( cos `  x )  e.  CC )
2221negcld 9144 . . . . . . . . . . . . 13  |-  ( (  T.  /\  x  e.  ( 0 [,] pi ) )  ->  -u ( cos `  x )  e.  CC )
23 eqid 2283 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2423tgioo2 18309 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
25 iccntr 18326 . . . . . . . . . . . . . . 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 19320 . . . . . . . . . . . 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 2303 . . . . . . . . . 10  |-  ( RR 
_D  C )  =  ( RR  _D  (
x  e.  ( 0 (,) pi )  |->  -u ( cos `  x ) ) )
31 reex 8828 . . . . . . . . . . . . . 14  |-  RR  e.  _V
3231prid1 3734 . . . . . . . . . . . . 13  |-  RR  e.  { RR ,  CC }
3332a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  RR  e.  { RR ,  CC } )
349coscld 12411 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  ( cos `  x )  e.  CC )
3534adantl 452 . . . . . . . . . . . . 13  |-  ( (  T.  /\  x  e.  RR )  ->  ( cos `  x )  e.  CC )
3635negcld 9144 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  RR )  ->  -u ( cos `  x )  e.  CC )
379sincld 12410 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  ( sin `  x )  e.  CC )
3837adantl 452 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  RR )  ->  ( sin `  x )  e.  CC )
3937negcld 9144 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR  ->  -u ( sin `  x )  e.  CC )
4039adantl 452 . . . . . . . . . . . . . . . 16  |-  ( (  T.  /\  x  e.  RR )  ->  -u ( sin `  x )  e.  CC )
41 dvcosre 27741 . . . . . . . . . . . . . . . . 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 19315 . . . . . . . . . . . . . . 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 9148 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR  ->  -u -u ( sin `  x )  =  ( sin `  x
) )
4645mpteq2ia 4102 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  |->  -u -u ( sin `  x ) )  =  ( x  e.  RR  |->  ( sin `  x
) )
4744, 46eqtri 2303 . . . . . . . . . . . . 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 10712 . . . . . . . . . . . . 13  |-  ( 0 (,) pi )  C_  RR
5049a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  ( 0 (,) pi )  C_  RR )
51 iooretop 18275 . . . . . . . . . . . . 13  |-  ( 0 (,) pi )  e.  ( topGen `  ran  (,) )
5251a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  ( 0 (,) pi )  e.  ( topGen ` 
ran  (,) ) )
5333, 36, 38, 48, 50, 24, 23, 52dvmptres 19312 . . . . . . . . . . 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 2303 . . . . . . . . 9  |-  ( RR 
_D  C )  =  ( x  e.  ( 0 (,) pi ) 
|->  ( sin `  x
) )
5655fveq1i 5526 . . . . . . . 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 3188 . . . . . . . . . . 11  |-  ( 0 (,) pi )  C_  CC
6059sseli 3176 . . . . . . . . . 10  |-  ( x  e.  ( 0 (,) pi )  ->  x  e.  CC )
6160sincld 12410 . . . . . . . . 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 2283 . . . . . . . . 9  |-  ( x  e.  ( 0 (,) pi )  |->  ( sin `  x ) )  =  ( x  e.  ( 0 (,) pi ) 
|->  ( sin `  x
) )
6463fvmpt2 5608 . . . . . . . 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 2315 . . . . . 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 19136 . . . 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 19833 . . . . . . 7  |-  0  <  pi
73 ltle 8910 . . . . . . 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 2419 . . . . . . . . 9  |-  F/_ x sin
77 sincn 19820 . . . . . . . . . 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 27717 . . . . . . . 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 2353 . . . . . 6  |-  ( RR 
_D  C )  e.  ( ( 0 (,) pi ) -cn-> CC )
8382a1i 10 . . . . 5  |-  (  T. 
->  ( RR  _D  C
)  e.  ( ( 0 (,) pi )
-cn-> CC ) )
84 ioossicc 10735 . . . . . . . . . 10  |-  ( 0 (,) pi )  C_  ( 0 [,] pi )
8584a1i 10 . . . . . . . . 9  |-  (  T. 
->  ( 0 (,) pi )  C_  ( 0 [,] pi ) )
86 ioombl 18922 . . . . . . . . . 10  |-  ( 0 (,) pi )  e. 
dom  vol
8786a1i 10 . . . . . . . . 9  |-  (  T. 
->  ( 0 (,) pi )  e.  dom  vol )
8819sincld 12410 . . . . . . . . . 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 27717 . . . . . . . . . . . . 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 19195 . . . . . . . . . . 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 19159 . . . . . . . 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 2353 . . . . . 6  |-  ( RR 
_D  C )  e.  L ^1
10099a1i 10 . . . . 5  |-  (  T. 
->  ( RR  _D  C
)  e.  L ^1 )
10120negcld 9144 . . . . . . . . . . . 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 2283 . . . . . . . . . . . 12  |-  ( x  e.  CC  |->  -u ( cos `  x ) )  =  ( x  e.  CC  |->  -u ( cos `  x
) )
104103fvmpt2 5608 . . . . . . . . . . 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 2288 . . . . . . . . 9  |-  ( x  e.  ( 0 [,] pi )  ->  -u ( cos `  x )  =  ( ( x  e.  CC  |->  -u ( cos `  x
) ) `  x
) )
107106mpteq2ia 4102 . . . . . . . 8  |-  ( x  e.  ( 0 [,] pi )  |->  -u ( cos `  x ) )  =  ( x  e.  ( 0 [,] pi )  |->  ( ( x  e.  CC  |->  -u ( cos `  x ) ) `
 x ) )
108 nfmpt1 4109 . . . . . . . . . 10  |-  F/_ x
( x  e.  CC  |->  -u ( cos `  x
) )
109 coscn 19821 . . . . . . . . . . . 12  |-  cos  e.  ( CC -cn-> CC )
110103negfcncf 18422 . . . . . . . . . . . 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 27717 . . . . . . . . 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 2353 . . . . . . 7  |-  ( x  e.  ( 0 [,] pi )  |->  -u ( cos `  x ) )  e.  ( ( 0 [,] pi ) -cn-> CC )
1167, 115eqeltri 2353 . . . . . 6  |-  C  e.  ( ( 0 [,] pi ) -cn-> CC )
117116a1i 10 . . . . 5  |-  (  T. 
->  C  e.  (
( 0 [,] pi ) -cn-> CC ) )
11870, 71, 75, 83, 100, 117ftc2 19391 . . . 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 2305 . 2  |-  S. ( 0 (,) pi ) ( sin `  x
)  _d x  =  ( ( C `  pi )  -  ( C `  0 )
)
121 0xr 8878 . . . . . . 7  |-  0  e.  RR*
12213rexri 8884 . . . . . . 7  |-  pi  e.  RR*
123121, 122, 743pm3.2i 1130 . . . . . 6  |-  ( 0  e.  RR*  /\  pi  e.  RR*  /\  0  <_  pi )
124 ubicc2 10753 . . . . . 6  |-  ( ( 0  e.  RR*  /\  pi  e.  RR*  /\  0  <_  pi )  ->  pi  e.  ( 0 [,] pi ) )
125123, 124ax-mp 8 . . . . 5  |-  pi  e.  ( 0 [,] pi )
126 fveq2 5525 . . . . . . . . 9  |-  ( t  =  pi  ->  ( cos `  t )  =  ( cos `  pi ) )
127 cospi 19840 . . . . . . . . . 10  |-  ( cos `  pi )  =  -u
1
128127a1i 10 . . . . . . . . 9  |-  ( t  =  pi  ->  ( cos `  pi )  = 
-u 1 )
129126, 128eqtrd 2315 . . . . . . . 8  |-  ( t  =  pi  ->  ( cos `  t )  = 
-u 1 )
130129negeqd 9046 . . . . . . 7  |-  ( t  =  pi  ->  -u ( cos `  t )  = 
-u -u 1 )
131 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
132131a1i 10 . . . . . . . 8  |-  ( t  =  pi  ->  1  e.  CC )
133132negnegd 9148 . . . . . . 7  |-  ( t  =  pi  ->  -u -u 1  =  1 )
134130, 133eqtrd 2315 . . . . . 6  |-  ( t  =  pi  ->  -u ( cos `  t )  =  1 )
135 1ex 8833 . . . . . 6  |-  1  e.  _V
136134, 1, 135fvmpt 5602 . . . . 5  |-  ( pi  e.  ( 0 [,] pi )  ->  ( C `  pi )  =  1 )
137125, 136ax-mp 8 . . . 4  |-  ( C `
 pi )  =  1
138 lbicc2 10752 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  pi  e.  RR*  /\  0  <_  pi )  ->  0  e.  ( 0 [,] pi ) )
139123, 138ax-mp 8 . . . . . 6  |-  0  e.  ( 0 [,] pi )
140 fveq2 5525 . . . . . . . 8  |-  ( t  =  0  ->  ( cos `  t )  =  ( cos `  0
) )
141140negeqd 9046 . . . . . . 7  |-  ( t  =  0  ->  -u ( cos `  t )  = 
-u ( cos `  0
) )
142 negex 9050 . . . . . . 7  |-  -u ( cos `  0 )  e. 
_V
143141, 1, 142fvmpt 5602 . . . . . 6  |-  ( 0  e.  ( 0 [,] pi )  ->  ( C `  0 )  =  -u ( cos `  0
) )
144139, 143ax-mp 8 . . . . 5  |-  ( C `
 0 )  = 
-u ( cos `  0
)
145 cos0 12430 . . . . . 6  |-  ( cos `  0 )  =  1
146145negeqi 9045 . . . . 5  |-  -u ( cos `  0 )  = 
-u 1
147144, 146eqtri 2303 . . . 4  |-  ( C `
 0 )  = 
-u 1
148137, 147oveq12i 5870 . . 3  |-  ( ( C `  pi )  -  ( C ` 
0 ) )  =  ( 1  -  -u 1
)
149131, 131pm3.2i 441 . . . . 5  |-  ( 1  e.  CC  /\  1  e.  CC )
150 subneg 9096 . . . . 5  |-  ( ( 1  e.  CC  /\  1  e.  CC )  ->  ( 1  -  -u 1
)  =  ( 1  +  1 ) )
151149, 150ax-mp 8 . . . 4  |-  ( 1  -  -u 1 )  =  ( 1  +  1 )
152 1p1e2 9840 . . . 4  |-  ( 1  +  1 )  =  2
153151, 152eqtri 2303 . . 3  |-  ( 1  -  -u 1 )  =  2
154148, 153eqtri 2303 . 2  |-  ( ( C `  pi )  -  ( C ` 
0 ) )  =  2
155120, 154eqtri 2303 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 1623    e. wcel 1684    C_ wss 3152   {cpr 3641   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ran crn 4690   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038   2c2 9795   (,)cioo 10656   [,]cicc 10659   sincsin 12345   cosccos 12346   picpi 12348   TopOpenctopn 13326   topGenctg 13342  ℂfldccnfld 16377   intcnt 16754   -cn->ccncf 18380   volcvol 18823   L ^1cibl 18972   S.citg 18973    _D cdv 19213
This theorem is referenced by:  itgsin0pi  27746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976  df-itg2 18977  df-ibl 18978  df-itg 18979  df-0p 19025  df-limc 19216  df-dv 19217
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