Step |
Hyp |
Ref |
Expression |
1 |
|
itgsin0pilem1.1 |
⊢ 𝐶 = ( 𝑡 ∈ ( 0 [,] π ) ↦ - ( cos ‘ 𝑡 ) ) |
2 |
|
fveq2 |
⊢ ( 𝑡 = 𝑥 → ( cos ‘ 𝑡 ) = ( cos ‘ 𝑥 ) ) |
3 |
2
|
negeqd |
⊢ ( 𝑡 = 𝑥 → - ( cos ‘ 𝑡 ) = - ( cos ‘ 𝑥 ) ) |
4 |
3
|
cbvmptv |
⊢ ( 𝑡 ∈ ( 0 [,] π ) ↦ - ( cos ‘ 𝑡 ) ) = ( 𝑥 ∈ ( 0 [,] π ) ↦ - ( cos ‘ 𝑥 ) ) |
5 |
1 4
|
eqtri |
⊢ 𝐶 = ( 𝑥 ∈ ( 0 [,] π ) ↦ - ( cos ‘ 𝑥 ) ) |
6 |
5
|
oveq2i |
⊢ ( ℝ D 𝐶 ) = ( ℝ D ( 𝑥 ∈ ( 0 [,] π ) ↦ - ( cos ‘ 𝑥 ) ) ) |
7 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
8 |
7
|
a1i |
⊢ ( ⊤ → ℝ ⊆ ℂ ) |
9 |
|
0re |
⊢ 0 ∈ ℝ |
10 |
|
pire |
⊢ π ∈ ℝ |
11 |
|
iccssre |
⊢ ( ( 0 ∈ ℝ ∧ π ∈ ℝ ) → ( 0 [,] π ) ⊆ ℝ ) |
12 |
9 10 11
|
mp2an |
⊢ ( 0 [,] π ) ⊆ ℝ |
13 |
12
|
a1i |
⊢ ( ⊤ → ( 0 [,] π ) ⊆ ℝ ) |
14 |
12 7
|
sstri |
⊢ ( 0 [,] π ) ⊆ ℂ |
15 |
14
|
sseli |
⊢ ( 𝑥 ∈ ( 0 [,] π ) → 𝑥 ∈ ℂ ) |
16 |
15
|
coscld |
⊢ ( 𝑥 ∈ ( 0 [,] π ) → ( cos ‘ 𝑥 ) ∈ ℂ ) |
17 |
16
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( 0 [,] π ) ) → ( cos ‘ 𝑥 ) ∈ ℂ ) |
18 |
17
|
negcld |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( 0 [,] π ) ) → - ( cos ‘ 𝑥 ) ∈ ℂ ) |
19 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
20 |
19
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
21 |
|
iccntr |
⊢ ( ( 0 ∈ ℝ ∧ π ∈ ℝ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 0 [,] π ) ) = ( 0 (,) π ) ) |
22 |
9 10 21
|
mp2an |
⊢ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 0 [,] π ) ) = ( 0 (,) π ) |
23 |
22
|
a1i |
⊢ ( ⊤ → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 0 [,] π ) ) = ( 0 (,) π ) ) |
24 |
8 13 18 20 19 23
|
dvmptntr |
⊢ ( ⊤ → ( ℝ D ( 𝑥 ∈ ( 0 [,] π ) ↦ - ( cos ‘ 𝑥 ) ) ) = ( ℝ D ( 𝑥 ∈ ( 0 (,) π ) ↦ - ( cos ‘ 𝑥 ) ) ) ) |
25 |
24
|
mptru |
⊢ ( ℝ D ( 𝑥 ∈ ( 0 [,] π ) ↦ - ( cos ‘ 𝑥 ) ) ) = ( ℝ D ( 𝑥 ∈ ( 0 (,) π ) ↦ - ( cos ‘ 𝑥 ) ) ) |
26 |
|
reelprrecn |
⊢ ℝ ∈ { ℝ , ℂ } |
27 |
26
|
a1i |
⊢ ( ⊤ → ℝ ∈ { ℝ , ℂ } ) |
28 |
|
recn |
⊢ ( 𝑥 ∈ ℝ → 𝑥 ∈ ℂ ) |
29 |
28
|
coscld |
⊢ ( 𝑥 ∈ ℝ → ( cos ‘ 𝑥 ) ∈ ℂ ) |
30 |
29
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ ) → ( cos ‘ 𝑥 ) ∈ ℂ ) |
31 |
30
|
negcld |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ ) → - ( cos ‘ 𝑥 ) ∈ ℂ ) |
32 |
28
|
sincld |
⊢ ( 𝑥 ∈ ℝ → ( sin ‘ 𝑥 ) ∈ ℂ ) |
33 |
32
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
34 |
32
|
negcld |
⊢ ( 𝑥 ∈ ℝ → - ( sin ‘ 𝑥 ) ∈ ℂ ) |
35 |
34
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ ) → - ( sin ‘ 𝑥 ) ∈ ℂ ) |
36 |
|
dvcosre |
⊢ ( ℝ D ( 𝑥 ∈ ℝ ↦ ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℝ ↦ - ( sin ‘ 𝑥 ) ) |
37 |
36
|
a1i |
⊢ ( ⊤ → ( ℝ D ( 𝑥 ∈ ℝ ↦ ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℝ ↦ - ( sin ‘ 𝑥 ) ) ) |
38 |
27 30 35 37
|
dvmptneg |
⊢ ( ⊤ → ( ℝ D ( 𝑥 ∈ ℝ ↦ - ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℝ ↦ - - ( sin ‘ 𝑥 ) ) ) |
39 |
32
|
negnegd |
⊢ ( 𝑥 ∈ ℝ → - - ( sin ‘ 𝑥 ) = ( sin ‘ 𝑥 ) ) |
40 |
39
|
mpteq2ia |
⊢ ( 𝑥 ∈ ℝ ↦ - - ( sin ‘ 𝑥 ) ) = ( 𝑥 ∈ ℝ ↦ ( sin ‘ 𝑥 ) ) |
41 |
38 40
|
eqtrdi |
⊢ ( ⊤ → ( ℝ D ( 𝑥 ∈ ℝ ↦ - ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( sin ‘ 𝑥 ) ) ) |
42 |
|
ioossre |
⊢ ( 0 (,) π ) ⊆ ℝ |
43 |
42
|
a1i |
⊢ ( ⊤ → ( 0 (,) π ) ⊆ ℝ ) |
44 |
|
iooretop |
⊢ ( 0 (,) π ) ∈ ( topGen ‘ ran (,) ) |
45 |
44
|
a1i |
⊢ ( ⊤ → ( 0 (,) π ) ∈ ( topGen ‘ ran (,) ) ) |
46 |
27 31 33 41 43 20 19 45
|
dvmptres |
⊢ ( ⊤ → ( ℝ D ( 𝑥 ∈ ( 0 (,) π ) ↦ - ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( 0 (,) π ) ↦ ( sin ‘ 𝑥 ) ) ) |
47 |
46
|
mptru |
⊢ ( ℝ D ( 𝑥 ∈ ( 0 (,) π ) ↦ - ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( 0 (,) π ) ↦ ( sin ‘ 𝑥 ) ) |
48 |
6 25 47
|
3eqtri |
⊢ ( ℝ D 𝐶 ) = ( 𝑥 ∈ ( 0 (,) π ) ↦ ( sin ‘ 𝑥 ) ) |
49 |
48
|
fveq1i |
⊢ ( ( ℝ D 𝐶 ) ‘ 𝑥 ) = ( ( 𝑥 ∈ ( 0 (,) π ) ↦ ( sin ‘ 𝑥 ) ) ‘ 𝑥 ) |
50 |
42 7
|
sstri |
⊢ ( 0 (,) π ) ⊆ ℂ |
51 |
50
|
sseli |
⊢ ( 𝑥 ∈ ( 0 (,) π ) → 𝑥 ∈ ℂ ) |
52 |
51
|
sincld |
⊢ ( 𝑥 ∈ ( 0 (,) π ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
53 |
|
eqid |
⊢ ( 𝑥 ∈ ( 0 (,) π ) ↦ ( sin ‘ 𝑥 ) ) = ( 𝑥 ∈ ( 0 (,) π ) ↦ ( sin ‘ 𝑥 ) ) |
54 |
53
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ ( 0 (,) π ) ∧ ( sin ‘ 𝑥 ) ∈ ℂ ) → ( ( 𝑥 ∈ ( 0 (,) π ) ↦ ( sin ‘ 𝑥 ) ) ‘ 𝑥 ) = ( sin ‘ 𝑥 ) ) |
55 |
52 54
|
mpdan |
⊢ ( 𝑥 ∈ ( 0 (,) π ) → ( ( 𝑥 ∈ ( 0 (,) π ) ↦ ( sin ‘ 𝑥 ) ) ‘ 𝑥 ) = ( sin ‘ 𝑥 ) ) |
56 |
49 55
|
syl5eq |
⊢ ( 𝑥 ∈ ( 0 (,) π ) → ( ( ℝ D 𝐶 ) ‘ 𝑥 ) = ( sin ‘ 𝑥 ) ) |
57 |
56
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( 0 (,) π ) ) → ( ( ℝ D 𝐶 ) ‘ 𝑥 ) = ( sin ‘ 𝑥 ) ) |
58 |
57
|
itgeq2dv |
⊢ ( ⊤ → ∫ ( 0 (,) π ) ( ( ℝ D 𝐶 ) ‘ 𝑥 ) d 𝑥 = ∫ ( 0 (,) π ) ( sin ‘ 𝑥 ) d 𝑥 ) |
59 |
58
|
mptru |
⊢ ∫ ( 0 (,) π ) ( ( ℝ D 𝐶 ) ‘ 𝑥 ) d 𝑥 = ∫ ( 0 (,) π ) ( sin ‘ 𝑥 ) d 𝑥 |
60 |
9
|
a1i |
⊢ ( ⊤ → 0 ∈ ℝ ) |
61 |
10
|
a1i |
⊢ ( ⊤ → π ∈ ℝ ) |
62 |
|
pipos |
⊢ 0 < π |
63 |
9 10 62
|
ltleii |
⊢ 0 ≤ π |
64 |
63
|
a1i |
⊢ ( ⊤ → 0 ≤ π ) |
65 |
|
nfcv |
⊢ Ⅎ 𝑥 sin |
66 |
|
sincn |
⊢ sin ∈ ( ℂ –cn→ ℂ ) |
67 |
66
|
a1i |
⊢ ( ⊤ → sin ∈ ( ℂ –cn→ ℂ ) ) |
68 |
50
|
a1i |
⊢ ( ⊤ → ( 0 (,) π ) ⊆ ℂ ) |
69 |
65 67 68
|
cncfmptss |
⊢ ( ⊤ → ( 𝑥 ∈ ( 0 (,) π ) ↦ ( sin ‘ 𝑥 ) ) ∈ ( ( 0 (,) π ) –cn→ ℂ ) ) |
70 |
48 69
|
eqeltrid |
⊢ ( ⊤ → ( ℝ D 𝐶 ) ∈ ( ( 0 (,) π ) –cn→ ℂ ) ) |
71 |
|
ioossicc |
⊢ ( 0 (,) π ) ⊆ ( 0 [,] π ) |
72 |
71
|
a1i |
⊢ ( ⊤ → ( 0 (,) π ) ⊆ ( 0 [,] π ) ) |
73 |
|
ioombl |
⊢ ( 0 (,) π ) ∈ dom vol |
74 |
73
|
a1i |
⊢ ( ⊤ → ( 0 (,) π ) ∈ dom vol ) |
75 |
15
|
sincld |
⊢ ( 𝑥 ∈ ( 0 [,] π ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
76 |
75
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( 0 [,] π ) ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
77 |
14
|
a1i |
⊢ ( ⊤ → ( 0 [,] π ) ⊆ ℂ ) |
78 |
65 67 77
|
cncfmptss |
⊢ ( ⊤ → ( 𝑥 ∈ ( 0 [,] π ) ↦ ( sin ‘ 𝑥 ) ) ∈ ( ( 0 [,] π ) –cn→ ℂ ) ) |
79 |
78
|
mptru |
⊢ ( 𝑥 ∈ ( 0 [,] π ) ↦ ( sin ‘ 𝑥 ) ) ∈ ( ( 0 [,] π ) –cn→ ℂ ) |
80 |
|
cniccibl |
⊢ ( ( 0 ∈ ℝ ∧ π ∈ ℝ ∧ ( 𝑥 ∈ ( 0 [,] π ) ↦ ( sin ‘ 𝑥 ) ) ∈ ( ( 0 [,] π ) –cn→ ℂ ) ) → ( 𝑥 ∈ ( 0 [,] π ) ↦ ( sin ‘ 𝑥 ) ) ∈ 𝐿1 ) |
81 |
9 10 79 80
|
mp3an |
⊢ ( 𝑥 ∈ ( 0 [,] π ) ↦ ( sin ‘ 𝑥 ) ) ∈ 𝐿1 |
82 |
81
|
a1i |
⊢ ( ⊤ → ( 𝑥 ∈ ( 0 [,] π ) ↦ ( sin ‘ 𝑥 ) ) ∈ 𝐿1 ) |
83 |
72 74 76 82
|
iblss |
⊢ ( ⊤ → ( 𝑥 ∈ ( 0 (,) π ) ↦ ( sin ‘ 𝑥 ) ) ∈ 𝐿1 ) |
84 |
48 83
|
eqeltrid |
⊢ ( ⊤ → ( ℝ D 𝐶 ) ∈ 𝐿1 ) |
85 |
16
|
negcld |
⊢ ( 𝑥 ∈ ( 0 [,] π ) → - ( cos ‘ 𝑥 ) ∈ ℂ ) |
86 |
|
eqid |
⊢ ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) = ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) |
87 |
86
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ ℂ ∧ - ( cos ‘ 𝑥 ) ∈ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) ‘ 𝑥 ) = - ( cos ‘ 𝑥 ) ) |
88 |
15 85 87
|
syl2anc |
⊢ ( 𝑥 ∈ ( 0 [,] π ) → ( ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) ‘ 𝑥 ) = - ( cos ‘ 𝑥 ) ) |
89 |
88
|
eqcomd |
⊢ ( 𝑥 ∈ ( 0 [,] π ) → - ( cos ‘ 𝑥 ) = ( ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) ‘ 𝑥 ) ) |
90 |
89
|
mpteq2ia |
⊢ ( 𝑥 ∈ ( 0 [,] π ) ↦ - ( cos ‘ 𝑥 ) ) = ( 𝑥 ∈ ( 0 [,] π ) ↦ ( ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) ‘ 𝑥 ) ) |
91 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) |
92 |
|
coscn |
⊢ cos ∈ ( ℂ –cn→ ℂ ) |
93 |
86
|
negfcncf |
⊢ ( cos ∈ ( ℂ –cn→ ℂ ) → ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
94 |
92 93
|
ax-mp |
⊢ ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) ∈ ( ℂ –cn→ ℂ ) |
95 |
94
|
a1i |
⊢ ( ⊤ → ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
96 |
91 95 77
|
cncfmptss |
⊢ ( ⊤ → ( 𝑥 ∈ ( 0 [,] π ) ↦ ( ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) ‘ 𝑥 ) ) ∈ ( ( 0 [,] π ) –cn→ ℂ ) ) |
97 |
96
|
mptru |
⊢ ( 𝑥 ∈ ( 0 [,] π ) ↦ ( ( 𝑥 ∈ ℂ ↦ - ( cos ‘ 𝑥 ) ) ‘ 𝑥 ) ) ∈ ( ( 0 [,] π ) –cn→ ℂ ) |
98 |
90 97
|
eqeltri |
⊢ ( 𝑥 ∈ ( 0 [,] π ) ↦ - ( cos ‘ 𝑥 ) ) ∈ ( ( 0 [,] π ) –cn→ ℂ ) |
99 |
5 98
|
eqeltri |
⊢ 𝐶 ∈ ( ( 0 [,] π ) –cn→ ℂ ) |
100 |
99
|
a1i |
⊢ ( ⊤ → 𝐶 ∈ ( ( 0 [,] π ) –cn→ ℂ ) ) |
101 |
60 61 64 70 84 100
|
ftc2 |
⊢ ( ⊤ → ∫ ( 0 (,) π ) ( ( ℝ D 𝐶 ) ‘ 𝑥 ) d 𝑥 = ( ( 𝐶 ‘ π ) − ( 𝐶 ‘ 0 ) ) ) |
102 |
101
|
mptru |
⊢ ∫ ( 0 (,) π ) ( ( ℝ D 𝐶 ) ‘ 𝑥 ) d 𝑥 = ( ( 𝐶 ‘ π ) − ( 𝐶 ‘ 0 ) ) |
103 |
59 102
|
eqtr3i |
⊢ ∫ ( 0 (,) π ) ( sin ‘ 𝑥 ) d 𝑥 = ( ( 𝐶 ‘ π ) − ( 𝐶 ‘ 0 ) ) |
104 |
|
0xr |
⊢ 0 ∈ ℝ* |
105 |
10
|
rexri |
⊢ π ∈ ℝ* |
106 |
|
ubicc2 |
⊢ ( ( 0 ∈ ℝ* ∧ π ∈ ℝ* ∧ 0 ≤ π ) → π ∈ ( 0 [,] π ) ) |
107 |
104 105 63 106
|
mp3an |
⊢ π ∈ ( 0 [,] π ) |
108 |
|
fveq2 |
⊢ ( 𝑡 = π → ( cos ‘ 𝑡 ) = ( cos ‘ π ) ) |
109 |
|
cospi |
⊢ ( cos ‘ π ) = - 1 |
110 |
108 109
|
eqtrdi |
⊢ ( 𝑡 = π → ( cos ‘ 𝑡 ) = - 1 ) |
111 |
110
|
negeqd |
⊢ ( 𝑡 = π → - ( cos ‘ 𝑡 ) = - - 1 ) |
112 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
113 |
112
|
a1i |
⊢ ( 𝑡 = π → 1 ∈ ℂ ) |
114 |
113
|
negnegd |
⊢ ( 𝑡 = π → - - 1 = 1 ) |
115 |
111 114
|
eqtrd |
⊢ ( 𝑡 = π → - ( cos ‘ 𝑡 ) = 1 ) |
116 |
|
1ex |
⊢ 1 ∈ V |
117 |
115 1 116
|
fvmpt |
⊢ ( π ∈ ( 0 [,] π ) → ( 𝐶 ‘ π ) = 1 ) |
118 |
107 117
|
ax-mp |
⊢ ( 𝐶 ‘ π ) = 1 |
119 |
|
lbicc2 |
⊢ ( ( 0 ∈ ℝ* ∧ π ∈ ℝ* ∧ 0 ≤ π ) → 0 ∈ ( 0 [,] π ) ) |
120 |
104 105 63 119
|
mp3an |
⊢ 0 ∈ ( 0 [,] π ) |
121 |
|
fveq2 |
⊢ ( 𝑡 = 0 → ( cos ‘ 𝑡 ) = ( cos ‘ 0 ) ) |
122 |
121
|
negeqd |
⊢ ( 𝑡 = 0 → - ( cos ‘ 𝑡 ) = - ( cos ‘ 0 ) ) |
123 |
|
negex |
⊢ - ( cos ‘ 0 ) ∈ V |
124 |
122 1 123
|
fvmpt |
⊢ ( 0 ∈ ( 0 [,] π ) → ( 𝐶 ‘ 0 ) = - ( cos ‘ 0 ) ) |
125 |
120 124
|
ax-mp |
⊢ ( 𝐶 ‘ 0 ) = - ( cos ‘ 0 ) |
126 |
|
cos0 |
⊢ ( cos ‘ 0 ) = 1 |
127 |
126
|
negeqi |
⊢ - ( cos ‘ 0 ) = - 1 |
128 |
125 127
|
eqtri |
⊢ ( 𝐶 ‘ 0 ) = - 1 |
129 |
118 128
|
oveq12i |
⊢ ( ( 𝐶 ‘ π ) − ( 𝐶 ‘ 0 ) ) = ( 1 − - 1 ) |
130 |
112 112
|
subnegi |
⊢ ( 1 − - 1 ) = ( 1 + 1 ) |
131 |
|
1p1e2 |
⊢ ( 1 + 1 ) = 2 |
132 |
130 131
|
eqtri |
⊢ ( 1 − - 1 ) = 2 |
133 |
103 129 132
|
3eqtri |
⊢ ∫ ( 0 (,) π ) ( sin ‘ 𝑥 ) d 𝑥 = 2 |