Step |
Hyp |
Ref |
Expression |
1 |
|
itgsin0pilem1.1 |
|- C = ( t e. ( 0 [,] _pi ) |-> -u ( cos ` t ) ) |
2 |
|
fveq2 |
|- ( t = x -> ( cos ` t ) = ( cos ` x ) ) |
3 |
2
|
negeqd |
|- ( t = x -> -u ( cos ` t ) = -u ( cos ` x ) ) |
4 |
3
|
cbvmptv |
|- ( t e. ( 0 [,] _pi ) |-> -u ( cos ` t ) ) = ( x e. ( 0 [,] _pi ) |-> -u ( cos ` x ) ) |
5 |
1 4
|
eqtri |
|- C = ( x e. ( 0 [,] _pi ) |-> -u ( cos ` x ) ) |
6 |
5
|
oveq2i |
|- ( RR _D C ) = ( RR _D ( x e. ( 0 [,] _pi ) |-> -u ( cos ` x ) ) ) |
7 |
|
ax-resscn |
|- RR C_ CC |
8 |
7
|
a1i |
|- ( T. -> RR C_ CC ) |
9 |
|
0re |
|- 0 e. RR |
10 |
|
pire |
|- _pi e. RR |
11 |
|
iccssre |
|- ( ( 0 e. RR /\ _pi e. RR ) -> ( 0 [,] _pi ) C_ RR ) |
12 |
9 10 11
|
mp2an |
|- ( 0 [,] _pi ) C_ RR |
13 |
12
|
a1i |
|- ( T. -> ( 0 [,] _pi ) C_ RR ) |
14 |
12 7
|
sstri |
|- ( 0 [,] _pi ) C_ CC |
15 |
14
|
sseli |
|- ( x e. ( 0 [,] _pi ) -> x e. CC ) |
16 |
15
|
coscld |
|- ( x e. ( 0 [,] _pi ) -> ( cos ` x ) e. CC ) |
17 |
16
|
adantl |
|- ( ( T. /\ x e. ( 0 [,] _pi ) ) -> ( cos ` x ) e. CC ) |
18 |
17
|
negcld |
|- ( ( T. /\ x e. ( 0 [,] _pi ) ) -> -u ( cos ` x ) e. CC ) |
19 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
20 |
19
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
21 |
|
iccntr |
|- ( ( 0 e. RR /\ _pi e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( 0 [,] _pi ) ) = ( 0 (,) _pi ) ) |
22 |
9 10 21
|
mp2an |
|- ( ( int ` ( topGen ` ran (,) ) ) ` ( 0 [,] _pi ) ) = ( 0 (,) _pi ) |
23 |
22
|
a1i |
|- ( T. -> ( ( int ` ( topGen ` ran (,) ) ) ` ( 0 [,] _pi ) ) = ( 0 (,) _pi ) ) |
24 |
8 13 18 20 19 23
|
dvmptntr |
|- ( T. -> ( RR _D ( x e. ( 0 [,] _pi ) |-> -u ( cos ` x ) ) ) = ( RR _D ( x e. ( 0 (,) _pi ) |-> -u ( cos ` x ) ) ) ) |
25 |
24
|
mptru |
|- ( RR _D ( x e. ( 0 [,] _pi ) |-> -u ( cos ` x ) ) ) = ( RR _D ( x e. ( 0 (,) _pi ) |-> -u ( cos ` x ) ) ) |
26 |
|
reelprrecn |
|- RR e. { RR , CC } |
27 |
26
|
a1i |
|- ( T. -> RR e. { RR , CC } ) |
28 |
|
recn |
|- ( x e. RR -> x e. CC ) |
29 |
28
|
coscld |
|- ( x e. RR -> ( cos ` x ) e. CC ) |
30 |
29
|
adantl |
|- ( ( T. /\ x e. RR ) -> ( cos ` x ) e. CC ) |
31 |
30
|
negcld |
|- ( ( T. /\ x e. RR ) -> -u ( cos ` x ) e. CC ) |
32 |
28
|
sincld |
|- ( x e. RR -> ( sin ` x ) e. CC ) |
33 |
32
|
adantl |
|- ( ( T. /\ x e. RR ) -> ( sin ` x ) e. CC ) |
34 |
32
|
negcld |
|- ( x e. RR -> -u ( sin ` x ) e. CC ) |
35 |
34
|
adantl |
|- ( ( T. /\ x e. RR ) -> -u ( sin ` x ) e. CC ) |
36 |
|
dvcosre |
|- ( RR _D ( x e. RR |-> ( cos ` x ) ) ) = ( x e. RR |-> -u ( sin ` x ) ) |
37 |
36
|
a1i |
|- ( T. -> ( RR _D ( x e. RR |-> ( cos ` x ) ) ) = ( x e. RR |-> -u ( sin ` x ) ) ) |
38 |
27 30 35 37
|
dvmptneg |
|- ( T. -> ( RR _D ( x e. RR |-> -u ( cos ` x ) ) ) = ( x e. RR |-> -u -u ( sin ` x ) ) ) |
39 |
32
|
negnegd |
|- ( x e. RR -> -u -u ( sin ` x ) = ( sin ` x ) ) |
40 |
39
|
mpteq2ia |
|- ( x e. RR |-> -u -u ( sin ` x ) ) = ( x e. RR |-> ( sin ` x ) ) |
41 |
38 40
|
eqtrdi |
|- ( T. -> ( RR _D ( x e. RR |-> -u ( cos ` x ) ) ) = ( x e. RR |-> ( sin ` x ) ) ) |
42 |
|
ioossre |
|- ( 0 (,) _pi ) C_ RR |
43 |
42
|
a1i |
|- ( T. -> ( 0 (,) _pi ) C_ RR ) |
44 |
|
iooretop |
|- ( 0 (,) _pi ) e. ( topGen ` ran (,) ) |
45 |
44
|
a1i |
|- ( T. -> ( 0 (,) _pi ) e. ( topGen ` ran (,) ) ) |
46 |
27 31 33 41 43 20 19 45
|
dvmptres |
|- ( T. -> ( RR _D ( x e. ( 0 (,) _pi ) |-> -u ( cos ` x ) ) ) = ( x e. ( 0 (,) _pi ) |-> ( sin ` x ) ) ) |
47 |
46
|
mptru |
|- ( RR _D ( x e. ( 0 (,) _pi ) |-> -u ( cos ` x ) ) ) = ( x e. ( 0 (,) _pi ) |-> ( sin ` x ) ) |
48 |
6 25 47
|
3eqtri |
|- ( RR _D C ) = ( x e. ( 0 (,) _pi ) |-> ( sin ` x ) ) |
49 |
48
|
fveq1i |
|- ( ( RR _D C ) ` x ) = ( ( x e. ( 0 (,) _pi ) |-> ( sin ` x ) ) ` x ) |
50 |
42 7
|
sstri |
|- ( 0 (,) _pi ) C_ CC |
51 |
50
|
sseli |
|- ( x e. ( 0 (,) _pi ) -> x e. CC ) |
52 |
51
|
sincld |
|- ( x e. ( 0 (,) _pi ) -> ( sin ` x ) e. CC ) |
53 |
|
eqid |
|- ( x e. ( 0 (,) _pi ) |-> ( sin ` x ) ) = ( x e. ( 0 (,) _pi ) |-> ( sin ` x ) ) |
54 |
53
|
fvmpt2 |
|- ( ( x e. ( 0 (,) _pi ) /\ ( sin ` x ) e. CC ) -> ( ( x e. ( 0 (,) _pi ) |-> ( sin ` x ) ) ` x ) = ( sin ` x ) ) |
55 |
52 54
|
mpdan |
|- ( x e. ( 0 (,) _pi ) -> ( ( x e. ( 0 (,) _pi ) |-> ( sin ` x ) ) ` x ) = ( sin ` x ) ) |
56 |
49 55
|
syl5eq |
|- ( x e. ( 0 (,) _pi ) -> ( ( RR _D C ) ` x ) = ( sin ` x ) ) |
57 |
56
|
adantl |
|- ( ( T. /\ x e. ( 0 (,) _pi ) ) -> ( ( RR _D C ) ` x ) = ( sin ` x ) ) |
58 |
57
|
itgeq2dv |
|- ( T. -> S. ( 0 (,) _pi ) ( ( RR _D C ) ` x ) _d x = S. ( 0 (,) _pi ) ( sin ` x ) _d x ) |
59 |
58
|
mptru |
|- S. ( 0 (,) _pi ) ( ( RR _D C ) ` x ) _d x = S. ( 0 (,) _pi ) ( sin ` x ) _d x |
60 |
9
|
a1i |
|- ( T. -> 0 e. RR ) |
61 |
10
|
a1i |
|- ( T. -> _pi e. RR ) |
62 |
|
pipos |
|- 0 < _pi |
63 |
9 10 62
|
ltleii |
|- 0 <_ _pi |
64 |
63
|
a1i |
|- ( T. -> 0 <_ _pi ) |
65 |
|
nfcv |
|- F/_ x sin |
66 |
|
sincn |
|- sin e. ( CC -cn-> CC ) |
67 |
66
|
a1i |
|- ( T. -> sin e. ( CC -cn-> CC ) ) |
68 |
50
|
a1i |
|- ( T. -> ( 0 (,) _pi ) C_ CC ) |
69 |
65 67 68
|
cncfmptss |
|- ( T. -> ( x e. ( 0 (,) _pi ) |-> ( sin ` x ) ) e. ( ( 0 (,) _pi ) -cn-> CC ) ) |
70 |
48 69
|
eqeltrid |
|- ( T. -> ( RR _D C ) e. ( ( 0 (,) _pi ) -cn-> CC ) ) |
71 |
|
ioossicc |
|- ( 0 (,) _pi ) C_ ( 0 [,] _pi ) |
72 |
71
|
a1i |
|- ( T. -> ( 0 (,) _pi ) C_ ( 0 [,] _pi ) ) |
73 |
|
ioombl |
|- ( 0 (,) _pi ) e. dom vol |
74 |
73
|
a1i |
|- ( T. -> ( 0 (,) _pi ) e. dom vol ) |
75 |
15
|
sincld |
|- ( x e. ( 0 [,] _pi ) -> ( sin ` x ) e. CC ) |
76 |
75
|
adantl |
|- ( ( T. /\ x e. ( 0 [,] _pi ) ) -> ( sin ` x ) e. CC ) |
77 |
14
|
a1i |
|- ( T. -> ( 0 [,] _pi ) C_ CC ) |
78 |
65 67 77
|
cncfmptss |
|- ( T. -> ( x e. ( 0 [,] _pi ) |-> ( sin ` x ) ) e. ( ( 0 [,] _pi ) -cn-> CC ) ) |
79 |
78
|
mptru |
|- ( x e. ( 0 [,] _pi ) |-> ( sin ` x ) ) e. ( ( 0 [,] _pi ) -cn-> CC ) |
80 |
|
cniccibl |
|- ( ( 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 ) |
81 |
9 10 79 80
|
mp3an |
|- ( x e. ( 0 [,] _pi ) |-> ( sin ` x ) ) e. L^1 |
82 |
81
|
a1i |
|- ( T. -> ( x e. ( 0 [,] _pi ) |-> ( sin ` x ) ) e. L^1 ) |
83 |
72 74 76 82
|
iblss |
|- ( T. -> ( x e. ( 0 (,) _pi ) |-> ( sin ` x ) ) e. L^1 ) |
84 |
48 83
|
eqeltrid |
|- ( T. -> ( RR _D C ) e. L^1 ) |
85 |
16
|
negcld |
|- ( x e. ( 0 [,] _pi ) -> -u ( cos ` x ) e. CC ) |
86 |
|
eqid |
|- ( x e. CC |-> -u ( cos ` x ) ) = ( x e. CC |-> -u ( cos ` x ) ) |
87 |
86
|
fvmpt2 |
|- ( ( x e. CC /\ -u ( cos ` x ) e. CC ) -> ( ( x e. CC |-> -u ( cos ` x ) ) ` x ) = -u ( cos ` x ) ) |
88 |
15 85 87
|
syl2anc |
|- ( x e. ( 0 [,] _pi ) -> ( ( x e. CC |-> -u ( cos ` x ) ) ` x ) = -u ( cos ` x ) ) |
89 |
88
|
eqcomd |
|- ( x e. ( 0 [,] _pi ) -> -u ( cos ` x ) = ( ( x e. CC |-> -u ( cos ` x ) ) ` x ) ) |
90 |
89
|
mpteq2ia |
|- ( x e. ( 0 [,] _pi ) |-> -u ( cos ` x ) ) = ( x e. ( 0 [,] _pi ) |-> ( ( x e. CC |-> -u ( cos ` x ) ) ` x ) ) |
91 |
|
nfmpt1 |
|- F/_ x ( x e. CC |-> -u ( cos ` x ) ) |
92 |
|
coscn |
|- cos e. ( CC -cn-> CC ) |
93 |
86
|
negfcncf |
|- ( cos e. ( CC -cn-> CC ) -> ( x e. CC |-> -u ( cos ` x ) ) e. ( CC -cn-> CC ) ) |
94 |
92 93
|
ax-mp |
|- ( x e. CC |-> -u ( cos ` x ) ) e. ( CC -cn-> CC ) |
95 |
94
|
a1i |
|- ( T. -> ( x e. CC |-> -u ( cos ` x ) ) e. ( CC -cn-> CC ) ) |
96 |
91 95 77
|
cncfmptss |
|- ( T. -> ( x e. ( 0 [,] _pi ) |-> ( ( x e. CC |-> -u ( cos ` x ) ) ` x ) ) e. ( ( 0 [,] _pi ) -cn-> CC ) ) |
97 |
96
|
mptru |
|- ( x e. ( 0 [,] _pi ) |-> ( ( x e. CC |-> -u ( cos ` x ) ) ` x ) ) e. ( ( 0 [,] _pi ) -cn-> CC ) |
98 |
90 97
|
eqeltri |
|- ( x e. ( 0 [,] _pi ) |-> -u ( cos ` x ) ) e. ( ( 0 [,] _pi ) -cn-> CC ) |
99 |
5 98
|
eqeltri |
|- C e. ( ( 0 [,] _pi ) -cn-> CC ) |
100 |
99
|
a1i |
|- ( T. -> C e. ( ( 0 [,] _pi ) -cn-> CC ) ) |
101 |
60 61 64 70 84 100
|
ftc2 |
|- ( T. -> S. ( 0 (,) _pi ) ( ( RR _D C ) ` x ) _d x = ( ( C ` _pi ) - ( C ` 0 ) ) ) |
102 |
101
|
mptru |
|- S. ( 0 (,) _pi ) ( ( RR _D C ) ` x ) _d x = ( ( C ` _pi ) - ( C ` 0 ) ) |
103 |
59 102
|
eqtr3i |
|- S. ( 0 (,) _pi ) ( sin ` x ) _d x = ( ( C ` _pi ) - ( C ` 0 ) ) |
104 |
|
0xr |
|- 0 e. RR* |
105 |
10
|
rexri |
|- _pi e. RR* |
106 |
|
ubicc2 |
|- ( ( 0 e. RR* /\ _pi e. RR* /\ 0 <_ _pi ) -> _pi e. ( 0 [,] _pi ) ) |
107 |
104 105 63 106
|
mp3an |
|- _pi e. ( 0 [,] _pi ) |
108 |
|
fveq2 |
|- ( t = _pi -> ( cos ` t ) = ( cos ` _pi ) ) |
109 |
|
cospi |
|- ( cos ` _pi ) = -u 1 |
110 |
108 109
|
eqtrdi |
|- ( t = _pi -> ( cos ` t ) = -u 1 ) |
111 |
110
|
negeqd |
|- ( t = _pi -> -u ( cos ` t ) = -u -u 1 ) |
112 |
|
ax-1cn |
|- 1 e. CC |
113 |
112
|
a1i |
|- ( t = _pi -> 1 e. CC ) |
114 |
113
|
negnegd |
|- ( t = _pi -> -u -u 1 = 1 ) |
115 |
111 114
|
eqtrd |
|- ( t = _pi -> -u ( cos ` t ) = 1 ) |
116 |
|
1ex |
|- 1 e. _V |
117 |
115 1 116
|
fvmpt |
|- ( _pi e. ( 0 [,] _pi ) -> ( C ` _pi ) = 1 ) |
118 |
107 117
|
ax-mp |
|- ( C ` _pi ) = 1 |
119 |
|
lbicc2 |
|- ( ( 0 e. RR* /\ _pi e. RR* /\ 0 <_ _pi ) -> 0 e. ( 0 [,] _pi ) ) |
120 |
104 105 63 119
|
mp3an |
|- 0 e. ( 0 [,] _pi ) |
121 |
|
fveq2 |
|- ( t = 0 -> ( cos ` t ) = ( cos ` 0 ) ) |
122 |
121
|
negeqd |
|- ( t = 0 -> -u ( cos ` t ) = -u ( cos ` 0 ) ) |
123 |
|
negex |
|- -u ( cos ` 0 ) e. _V |
124 |
122 1 123
|
fvmpt |
|- ( 0 e. ( 0 [,] _pi ) -> ( C ` 0 ) = -u ( cos ` 0 ) ) |
125 |
120 124
|
ax-mp |
|- ( C ` 0 ) = -u ( cos ` 0 ) |
126 |
|
cos0 |
|- ( cos ` 0 ) = 1 |
127 |
126
|
negeqi |
|- -u ( cos ` 0 ) = -u 1 |
128 |
125 127
|
eqtri |
|- ( C ` 0 ) = -u 1 |
129 |
118 128
|
oveq12i |
|- ( ( C ` _pi ) - ( C ` 0 ) ) = ( 1 - -u 1 ) |
130 |
112 112
|
subnegi |
|- ( 1 - -u 1 ) = ( 1 + 1 ) |
131 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
132 |
130 131
|
eqtri |
|- ( 1 - -u 1 ) = 2 |
133 |
103 129 132
|
3eqtri |
|- S. ( 0 (,) _pi ) ( sin ` x ) _d x = 2 |