| Step |
Hyp |
Ref |
Expression |
| 1 |
|
0re |
⊢ 0 ∈ ℝ |
| 2 |
|
pire |
⊢ π ∈ ℝ |
| 3 |
2
|
rexri |
⊢ π ∈ ℝ* |
| 4 |
|
icossre |
⊢ ( ( 0 ∈ ℝ ∧ π ∈ ℝ* ) → ( 0 [,) π ) ⊆ ℝ ) |
| 5 |
1 3 4
|
mp2an |
⊢ ( 0 [,) π ) ⊆ ℝ |
| 6 |
5
|
sseli |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → 𝐴 ∈ ℝ ) |
| 7 |
6
|
recnd |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → 𝐴 ∈ ℂ ) |
| 8 |
7
|
halfcld |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( 𝐴 / 2 ) ∈ ℂ ) |
| 9 |
6
|
rehalfcld |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( 𝐴 / 2 ) ∈ ℝ ) |
| 10 |
9
|
rered |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( ℜ ‘ ( 𝐴 / 2 ) ) = ( 𝐴 / 2 ) ) |
| 11 |
|
elico2 |
⊢ ( ( 0 ∈ ℝ ∧ π ∈ ℝ* ) → ( 𝐴 ∈ ( 0 [,) π ) ↔ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < π ) ) ) |
| 12 |
1 3 11
|
mp2an |
⊢ ( 𝐴 ∈ ( 0 [,) π ) ↔ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < π ) ) |
| 13 |
|
pipos |
⊢ 0 < π |
| 14 |
|
lt0neg2 |
⊢ ( π ∈ ℝ → ( 0 < π ↔ - π < 0 ) ) |
| 15 |
2 14
|
ax-mp |
⊢ ( 0 < π ↔ - π < 0 ) |
| 16 |
13 15
|
mpbi |
⊢ - π < 0 |
| 17 |
2
|
renegcli |
⊢ - π ∈ ℝ |
| 18 |
|
ltletr |
⊢ ( ( - π ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( - π < 0 ∧ 0 ≤ 𝐴 ) → - π < 𝐴 ) ) |
| 19 |
17 1 18
|
mp3an12 |
⊢ ( 𝐴 ∈ ℝ → ( ( - π < 0 ∧ 0 ≤ 𝐴 ) → - π < 𝐴 ) ) |
| 20 |
16 19
|
mpani |
⊢ ( 𝐴 ∈ ℝ → ( 0 ≤ 𝐴 → - π < 𝐴 ) ) |
| 21 |
|
2re |
⊢ 2 ∈ ℝ |
| 22 |
|
2pos |
⊢ 0 < 2 |
| 23 |
21 22
|
pm3.2i |
⊢ ( 2 ∈ ℝ ∧ 0 < 2 ) |
| 24 |
|
ltdiv1 |
⊢ ( ( - π ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ( 2 ∈ ℝ ∧ 0 < 2 ) ) → ( - π < 𝐴 ↔ ( - π / 2 ) < ( 𝐴 / 2 ) ) ) |
| 25 |
17 23 24
|
mp3an13 |
⊢ ( 𝐴 ∈ ℝ → ( - π < 𝐴 ↔ ( - π / 2 ) < ( 𝐴 / 2 ) ) ) |
| 26 |
|
picn |
⊢ π ∈ ℂ |
| 27 |
|
2cn |
⊢ 2 ∈ ℂ |
| 28 |
|
2ne0 |
⊢ 2 ≠ 0 |
| 29 |
|
divneg |
⊢ ( ( π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) → - ( π / 2 ) = ( - π / 2 ) ) |
| 30 |
26 27 28 29
|
mp3an |
⊢ - ( π / 2 ) = ( - π / 2 ) |
| 31 |
30
|
breq1i |
⊢ ( - ( π / 2 ) < ( 𝐴 / 2 ) ↔ ( - π / 2 ) < ( 𝐴 / 2 ) ) |
| 32 |
25 31
|
bitr4di |
⊢ ( 𝐴 ∈ ℝ → ( - π < 𝐴 ↔ - ( π / 2 ) < ( 𝐴 / 2 ) ) ) |
| 33 |
20 32
|
sylibd |
⊢ ( 𝐴 ∈ ℝ → ( 0 ≤ 𝐴 → - ( π / 2 ) < ( 𝐴 / 2 ) ) ) |
| 34 |
|
ltdiv1 |
⊢ ( ( 𝐴 ∈ ℝ ∧ π ∈ ℝ ∧ ( 2 ∈ ℝ ∧ 0 < 2 ) ) → ( 𝐴 < π ↔ ( 𝐴 / 2 ) < ( π / 2 ) ) ) |
| 35 |
2 23 34
|
mp3an23 |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < π ↔ ( 𝐴 / 2 ) < ( π / 2 ) ) ) |
| 36 |
35
|
biimpd |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < π → ( 𝐴 / 2 ) < ( π / 2 ) ) ) |
| 37 |
33 36
|
anim12d |
⊢ ( 𝐴 ∈ ℝ → ( ( 0 ≤ 𝐴 ∧ 𝐴 < π ) → ( - ( π / 2 ) < ( 𝐴 / 2 ) ∧ ( 𝐴 / 2 ) < ( π / 2 ) ) ) ) |
| 38 |
|
rehalfcl |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 / 2 ) ∈ ℝ ) |
| 39 |
38
|
rexrd |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 / 2 ) ∈ ℝ* ) |
| 40 |
|
halfpire |
⊢ ( π / 2 ) ∈ ℝ |
| 41 |
40
|
renegcli |
⊢ - ( π / 2 ) ∈ ℝ |
| 42 |
41
|
rexri |
⊢ - ( π / 2 ) ∈ ℝ* |
| 43 |
40
|
rexri |
⊢ ( π / 2 ) ∈ ℝ* |
| 44 |
|
elioo5 |
⊢ ( ( - ( π / 2 ) ∈ ℝ* ∧ ( π / 2 ) ∈ ℝ* ∧ ( 𝐴 / 2 ) ∈ ℝ* ) → ( ( 𝐴 / 2 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ↔ ( - ( π / 2 ) < ( 𝐴 / 2 ) ∧ ( 𝐴 / 2 ) < ( π / 2 ) ) ) ) |
| 45 |
42 43 44
|
mp3an12 |
⊢ ( ( 𝐴 / 2 ) ∈ ℝ* → ( ( 𝐴 / 2 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ↔ ( - ( π / 2 ) < ( 𝐴 / 2 ) ∧ ( 𝐴 / 2 ) < ( π / 2 ) ) ) ) |
| 46 |
39 45
|
syl |
⊢ ( 𝐴 ∈ ℝ → ( ( 𝐴 / 2 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ↔ ( - ( π / 2 ) < ( 𝐴 / 2 ) ∧ ( 𝐴 / 2 ) < ( π / 2 ) ) ) ) |
| 47 |
37 46
|
sylibrd |
⊢ ( 𝐴 ∈ ℝ → ( ( 0 ≤ 𝐴 ∧ 𝐴 < π ) → ( 𝐴 / 2 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) ) |
| 48 |
47
|
3impib |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 < π ) → ( 𝐴 / 2 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) |
| 49 |
12 48
|
sylbi |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( 𝐴 / 2 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) |
| 50 |
10 49
|
eqeltrd |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( ℜ ‘ ( 𝐴 / 2 ) ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) |
| 51 |
|
cosne0 |
⊢ ( ( ( 𝐴 / 2 ) ∈ ℂ ∧ ( ℜ ‘ ( 𝐴 / 2 ) ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) → ( cos ‘ ( 𝐴 / 2 ) ) ≠ 0 ) |
| 52 |
8 50 51
|
syl2anc |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( cos ‘ ( 𝐴 / 2 ) ) ≠ 0 ) |
| 53 |
|
tanval |
⊢ ( ( ( 𝐴 / 2 ) ∈ ℂ ∧ ( cos ‘ ( 𝐴 / 2 ) ) ≠ 0 ) → ( tan ‘ ( 𝐴 / 2 ) ) = ( ( sin ‘ ( 𝐴 / 2 ) ) / ( cos ‘ ( 𝐴 / 2 ) ) ) ) |
| 54 |
8 52 53
|
syl2anc |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( tan ‘ ( 𝐴 / 2 ) ) = ( ( sin ‘ ( 𝐴 / 2 ) ) / ( cos ‘ ( 𝐴 / 2 ) ) ) ) |
| 55 |
|
0xr |
⊢ 0 ∈ ℝ* |
| 56 |
|
elico1 |
⊢ ( ( 0 ∈ ℝ* ∧ π ∈ ℝ* ) → ( 𝐴 ∈ ( 0 [,) π ) ↔ ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < π ) ) ) |
| 57 |
55 3 56
|
mp2an |
⊢ ( 𝐴 ∈ ( 0 [,) π ) ↔ ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < π ) ) |
| 58 |
21 2
|
remulcli |
⊢ ( 2 · π ) ∈ ℝ |
| 59 |
58
|
rexri |
⊢ ( 2 · π ) ∈ ℝ* |
| 60 |
|
1lt2 |
⊢ 1 < 2 |
| 61 |
|
ltmulgt12 |
⊢ ( ( π ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < π ) → ( 1 < 2 ↔ π < ( 2 · π ) ) ) |
| 62 |
2 21 13 61
|
mp3an |
⊢ ( 1 < 2 ↔ π < ( 2 · π ) ) |
| 63 |
60 62
|
mpbi |
⊢ π < ( 2 · π ) |
| 64 |
|
xrlttr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ π ∈ ℝ* ∧ ( 2 · π ) ∈ ℝ* ) → ( ( 𝐴 < π ∧ π < ( 2 · π ) ) → 𝐴 < ( 2 · π ) ) ) |
| 65 |
3 64
|
mp3an2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ ( 2 · π ) ∈ ℝ* ) → ( ( 𝐴 < π ∧ π < ( 2 · π ) ) → 𝐴 < ( 2 · π ) ) ) |
| 66 |
63 65
|
mpan2i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ ( 2 · π ) ∈ ℝ* ) → ( 𝐴 < π → 𝐴 < ( 2 · π ) ) ) |
| 67 |
|
xrltle |
⊢ ( ( 𝐴 ∈ ℝ* ∧ ( 2 · π ) ∈ ℝ* ) → ( 𝐴 < ( 2 · π ) → 𝐴 ≤ ( 2 · π ) ) ) |
| 68 |
66 67
|
syld |
⊢ ( ( 𝐴 ∈ ℝ* ∧ ( 2 · π ) ∈ ℝ* ) → ( 𝐴 < π → 𝐴 ≤ ( 2 · π ) ) ) |
| 69 |
59 68
|
mpan2 |
⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 < π → 𝐴 ≤ ( 2 · π ) ) ) |
| 70 |
69
|
anim2d |
⊢ ( 𝐴 ∈ ℝ* → ( ( 0 ≤ 𝐴 ∧ 𝐴 < π ) → ( 0 ≤ 𝐴 ∧ 𝐴 ≤ ( 2 · π ) ) ) ) |
| 71 |
|
elicc4 |
⊢ ( ( 0 ∈ ℝ* ∧ ( 2 · π ) ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐴 ∈ ( 0 [,] ( 2 · π ) ) ↔ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ ( 2 · π ) ) ) ) |
| 72 |
55 59 71
|
mp3an12 |
⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 ∈ ( 0 [,] ( 2 · π ) ) ↔ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ ( 2 · π ) ) ) ) |
| 73 |
70 72
|
sylibrd |
⊢ ( 𝐴 ∈ ℝ* → ( ( 0 ≤ 𝐴 ∧ 𝐴 < π ) → 𝐴 ∈ ( 0 [,] ( 2 · π ) ) ) ) |
| 74 |
73
|
3impib |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < π ) → 𝐴 ∈ ( 0 [,] ( 2 · π ) ) ) |
| 75 |
57 74
|
sylbi |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → 𝐴 ∈ ( 0 [,] ( 2 · π ) ) ) |
| 76 |
|
sin2h |
⊢ ( 𝐴 ∈ ( 0 [,] ( 2 · π ) ) → ( sin ‘ ( 𝐴 / 2 ) ) = ( √ ‘ ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ) ) |
| 77 |
75 76
|
syl |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( sin ‘ ( 𝐴 / 2 ) ) = ( √ ‘ ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ) ) |
| 78 |
1 2 13
|
ltleii |
⊢ 0 ≤ π |
| 79 |
|
le0neg2 |
⊢ ( π ∈ ℝ → ( 0 ≤ π ↔ - π ≤ 0 ) ) |
| 80 |
2 79
|
ax-mp |
⊢ ( 0 ≤ π ↔ - π ≤ 0 ) |
| 81 |
78 80
|
mpbi |
⊢ - π ≤ 0 |
| 82 |
17
|
rexri |
⊢ - π ∈ ℝ* |
| 83 |
|
xrletr |
⊢ ( ( - π ∈ ℝ* ∧ 0 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( ( - π ≤ 0 ∧ 0 ≤ 𝐴 ) → - π ≤ 𝐴 ) ) |
| 84 |
82 55 83
|
mp3an12 |
⊢ ( 𝐴 ∈ ℝ* → ( ( - π ≤ 0 ∧ 0 ≤ 𝐴 ) → - π ≤ 𝐴 ) ) |
| 85 |
81 84
|
mpani |
⊢ ( 𝐴 ∈ ℝ* → ( 0 ≤ 𝐴 → - π ≤ 𝐴 ) ) |
| 86 |
|
xrltle |
⊢ ( ( 𝐴 ∈ ℝ* ∧ π ∈ ℝ* ) → ( 𝐴 < π → 𝐴 ≤ π ) ) |
| 87 |
3 86
|
mpan2 |
⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 < π → 𝐴 ≤ π ) ) |
| 88 |
85 87
|
anim12d |
⊢ ( 𝐴 ∈ ℝ* → ( ( 0 ≤ 𝐴 ∧ 𝐴 < π ) → ( - π ≤ 𝐴 ∧ 𝐴 ≤ π ) ) ) |
| 89 |
|
elicc4 |
⊢ ( ( - π ∈ ℝ* ∧ π ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐴 ∈ ( - π [,] π ) ↔ ( - π ≤ 𝐴 ∧ 𝐴 ≤ π ) ) ) |
| 90 |
82 3 89
|
mp3an12 |
⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 ∈ ( - π [,] π ) ↔ ( - π ≤ 𝐴 ∧ 𝐴 ≤ π ) ) ) |
| 91 |
88 90
|
sylibrd |
⊢ ( 𝐴 ∈ ℝ* → ( ( 0 ≤ 𝐴 ∧ 𝐴 < π ) → 𝐴 ∈ ( - π [,] π ) ) ) |
| 92 |
91
|
3impib |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < π ) → 𝐴 ∈ ( - π [,] π ) ) |
| 93 |
57 92
|
sylbi |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → 𝐴 ∈ ( - π [,] π ) ) |
| 94 |
|
cos2h |
⊢ ( 𝐴 ∈ ( - π [,] π ) → ( cos ‘ ( 𝐴 / 2 ) ) = ( √ ‘ ( ( 1 + ( cos ‘ 𝐴 ) ) / 2 ) ) ) |
| 95 |
93 94
|
syl |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( cos ‘ ( 𝐴 / 2 ) ) = ( √ ‘ ( ( 1 + ( cos ‘ 𝐴 ) ) / 2 ) ) ) |
| 96 |
77 95
|
oveq12d |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( ( sin ‘ ( 𝐴 / 2 ) ) / ( cos ‘ ( 𝐴 / 2 ) ) ) = ( ( √ ‘ ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ) / ( √ ‘ ( ( 1 + ( cos ‘ 𝐴 ) ) / 2 ) ) ) ) |
| 97 |
54 96
|
eqtrd |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( tan ‘ ( 𝐴 / 2 ) ) = ( ( √ ‘ ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ) / ( √ ‘ ( ( 1 + ( cos ‘ 𝐴 ) ) / 2 ) ) ) ) |
| 98 |
|
1re |
⊢ 1 ∈ ℝ |
| 99 |
6
|
recoscld |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( cos ‘ 𝐴 ) ∈ ℝ ) |
| 100 |
|
resubcl |
⊢ ( ( 1 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ∈ ℝ ) → ( 1 − ( cos ‘ 𝐴 ) ) ∈ ℝ ) |
| 101 |
98 99 100
|
sylancr |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( 1 − ( cos ‘ 𝐴 ) ) ∈ ℝ ) |
| 102 |
101
|
rehalfcld |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ∈ ℝ ) |
| 103 |
|
cosbnd |
⊢ ( 𝐴 ∈ ℝ → ( - 1 ≤ ( cos ‘ 𝐴 ) ∧ ( cos ‘ 𝐴 ) ≤ 1 ) ) |
| 104 |
103
|
simprd |
⊢ ( 𝐴 ∈ ℝ → ( cos ‘ 𝐴 ) ≤ 1 ) |
| 105 |
|
recoscl |
⊢ ( 𝐴 ∈ ℝ → ( cos ‘ 𝐴 ) ∈ ℝ ) |
| 106 |
|
subge0 |
⊢ ( ( 1 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ∈ ℝ ) → ( 0 ≤ ( 1 − ( cos ‘ 𝐴 ) ) ↔ ( cos ‘ 𝐴 ) ≤ 1 ) ) |
| 107 |
|
halfnneg2 |
⊢ ( ( 1 − ( cos ‘ 𝐴 ) ) ∈ ℝ → ( 0 ≤ ( 1 − ( cos ‘ 𝐴 ) ) ↔ 0 ≤ ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ) ) |
| 108 |
100 107
|
syl |
⊢ ( ( 1 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ∈ ℝ ) → ( 0 ≤ ( 1 − ( cos ‘ 𝐴 ) ) ↔ 0 ≤ ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ) ) |
| 109 |
106 108
|
bitr3d |
⊢ ( ( 1 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ∈ ℝ ) → ( ( cos ‘ 𝐴 ) ≤ 1 ↔ 0 ≤ ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ) ) |
| 110 |
98 105 109
|
sylancr |
⊢ ( 𝐴 ∈ ℝ → ( ( cos ‘ 𝐴 ) ≤ 1 ↔ 0 ≤ ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ) ) |
| 111 |
104 110
|
mpbid |
⊢ ( 𝐴 ∈ ℝ → 0 ≤ ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ) |
| 112 |
6 111
|
syl |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → 0 ≤ ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ) |
| 113 |
|
readdcl |
⊢ ( ( 1 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ∈ ℝ ) → ( 1 + ( cos ‘ 𝐴 ) ) ∈ ℝ ) |
| 114 |
98 99 113
|
sylancr |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( 1 + ( cos ‘ 𝐴 ) ) ∈ ℝ ) |
| 115 |
103
|
simpld |
⊢ ( 𝐴 ∈ ℝ → - 1 ≤ ( cos ‘ 𝐴 ) ) |
| 116 |
98
|
renegcli |
⊢ - 1 ∈ ℝ |
| 117 |
|
subge0 |
⊢ ( ( ( cos ‘ 𝐴 ) ∈ ℝ ∧ - 1 ∈ ℝ ) → ( 0 ≤ ( ( cos ‘ 𝐴 ) − - 1 ) ↔ - 1 ≤ ( cos ‘ 𝐴 ) ) ) |
| 118 |
105 116 117
|
sylancl |
⊢ ( 𝐴 ∈ ℝ → ( 0 ≤ ( ( cos ‘ 𝐴 ) − - 1 ) ↔ - 1 ≤ ( cos ‘ 𝐴 ) ) ) |
| 119 |
|
recn |
⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ ) |
| 120 |
119
|
coscld |
⊢ ( 𝐴 ∈ ℝ → ( cos ‘ 𝐴 ) ∈ ℂ ) |
| 121 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 122 |
|
subneg |
⊢ ( ( ( cos ‘ 𝐴 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( cos ‘ 𝐴 ) − - 1 ) = ( ( cos ‘ 𝐴 ) + 1 ) ) |
| 123 |
|
addcom |
⊢ ( ( ( cos ‘ 𝐴 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( cos ‘ 𝐴 ) + 1 ) = ( 1 + ( cos ‘ 𝐴 ) ) ) |
| 124 |
122 123
|
eqtrd |
⊢ ( ( ( cos ‘ 𝐴 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( cos ‘ 𝐴 ) − - 1 ) = ( 1 + ( cos ‘ 𝐴 ) ) ) |
| 125 |
120 121 124
|
sylancl |
⊢ ( 𝐴 ∈ ℝ → ( ( cos ‘ 𝐴 ) − - 1 ) = ( 1 + ( cos ‘ 𝐴 ) ) ) |
| 126 |
125
|
breq2d |
⊢ ( 𝐴 ∈ ℝ → ( 0 ≤ ( ( cos ‘ 𝐴 ) − - 1 ) ↔ 0 ≤ ( 1 + ( cos ‘ 𝐴 ) ) ) ) |
| 127 |
118 126
|
bitr3d |
⊢ ( 𝐴 ∈ ℝ → ( - 1 ≤ ( cos ‘ 𝐴 ) ↔ 0 ≤ ( 1 + ( cos ‘ 𝐴 ) ) ) ) |
| 128 |
115 127
|
mpbid |
⊢ ( 𝐴 ∈ ℝ → 0 ≤ ( 1 + ( cos ‘ 𝐴 ) ) ) |
| 129 |
6 128
|
syl |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → 0 ≤ ( 1 + ( cos ‘ 𝐴 ) ) ) |
| 130 |
|
snunioo |
⊢ ( ( 0 ∈ ℝ* ∧ π ∈ ℝ* ∧ 0 < π ) → ( { 0 } ∪ ( 0 (,) π ) ) = ( 0 [,) π ) ) |
| 131 |
55 3 13 130
|
mp3an |
⊢ ( { 0 } ∪ ( 0 (,) π ) ) = ( 0 [,) π ) |
| 132 |
131
|
eleq2i |
⊢ ( 𝐴 ∈ ( { 0 } ∪ ( 0 (,) π ) ) ↔ 𝐴 ∈ ( 0 [,) π ) ) |
| 133 |
|
elun |
⊢ ( 𝐴 ∈ ( { 0 } ∪ ( 0 (,) π ) ) ↔ ( 𝐴 ∈ { 0 } ∨ 𝐴 ∈ ( 0 (,) π ) ) ) |
| 134 |
132 133
|
bitr3i |
⊢ ( 𝐴 ∈ ( 0 [,) π ) ↔ ( 𝐴 ∈ { 0 } ∨ 𝐴 ∈ ( 0 (,) π ) ) ) |
| 135 |
|
elsni |
⊢ ( 𝐴 ∈ { 0 } → 𝐴 = 0 ) |
| 136 |
|
fveq2 |
⊢ ( 𝐴 = 0 → ( cos ‘ 𝐴 ) = ( cos ‘ 0 ) ) |
| 137 |
|
cos0 |
⊢ ( cos ‘ 0 ) = 1 |
| 138 |
136 137
|
eqtrdi |
⊢ ( 𝐴 = 0 → ( cos ‘ 𝐴 ) = 1 ) |
| 139 |
138
|
oveq2d |
⊢ ( 𝐴 = 0 → ( 1 + ( cos ‘ 𝐴 ) ) = ( 1 + 1 ) ) |
| 140 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
| 141 |
139 140
|
eqtr4di |
⊢ ( 𝐴 = 0 → ( 1 + ( cos ‘ 𝐴 ) ) = 2 ) |
| 142 |
28
|
a1i |
⊢ ( 𝐴 = 0 → 2 ≠ 0 ) |
| 143 |
141 142
|
eqnetrd |
⊢ ( 𝐴 = 0 → ( 1 + ( cos ‘ 𝐴 ) ) ≠ 0 ) |
| 144 |
135 143
|
syl |
⊢ ( 𝐴 ∈ { 0 } → ( 1 + ( cos ‘ 𝐴 ) ) ≠ 0 ) |
| 145 |
|
sinq12gt0 |
⊢ ( 𝐴 ∈ ( 0 (,) π ) → 0 < ( sin ‘ 𝐴 ) ) |
| 146 |
|
ltne |
⊢ ( ( 0 ∈ ℝ ∧ 0 < ( sin ‘ 𝐴 ) ) → ( sin ‘ 𝐴 ) ≠ 0 ) |
| 147 |
1 146
|
mpan |
⊢ ( 0 < ( sin ‘ 𝐴 ) → ( sin ‘ 𝐴 ) ≠ 0 ) |
| 148 |
|
elioore |
⊢ ( 𝐴 ∈ ( 0 (,) π ) → 𝐴 ∈ ℝ ) |
| 149 |
148
|
recnd |
⊢ ( 𝐴 ∈ ( 0 (,) π ) → 𝐴 ∈ ℂ ) |
| 150 |
|
oveq1 |
⊢ ( - 1 = ( cos ‘ 𝐴 ) → ( - 1 ↑ 2 ) = ( ( cos ‘ 𝐴 ) ↑ 2 ) ) |
| 151 |
150
|
a1i |
⊢ ( 𝐴 ∈ ℂ → ( - 1 = ( cos ‘ 𝐴 ) → ( - 1 ↑ 2 ) = ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ) |
| 152 |
|
df-neg |
⊢ - 1 = ( 0 − 1 ) |
| 153 |
152
|
eqeq1i |
⊢ ( - 1 = ( cos ‘ 𝐴 ) ↔ ( 0 − 1 ) = ( cos ‘ 𝐴 ) ) |
| 154 |
|
coscl |
⊢ ( 𝐴 ∈ ℂ → ( cos ‘ 𝐴 ) ∈ ℂ ) |
| 155 |
|
0cn |
⊢ 0 ∈ ℂ |
| 156 |
|
subadd |
⊢ ( ( 0 ∈ ℂ ∧ 1 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ∈ ℂ ) → ( ( 0 − 1 ) = ( cos ‘ 𝐴 ) ↔ ( 1 + ( cos ‘ 𝐴 ) ) = 0 ) ) |
| 157 |
155 121 156
|
mp3an12 |
⊢ ( ( cos ‘ 𝐴 ) ∈ ℂ → ( ( 0 − 1 ) = ( cos ‘ 𝐴 ) ↔ ( 1 + ( cos ‘ 𝐴 ) ) = 0 ) ) |
| 158 |
154 157
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ( 0 − 1 ) = ( cos ‘ 𝐴 ) ↔ ( 1 + ( cos ‘ 𝐴 ) ) = 0 ) ) |
| 159 |
153 158
|
syl5bb |
⊢ ( 𝐴 ∈ ℂ → ( - 1 = ( cos ‘ 𝐴 ) ↔ ( 1 + ( cos ‘ 𝐴 ) ) = 0 ) ) |
| 160 |
|
sincl |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ 𝐴 ) ∈ ℂ ) |
| 161 |
160
|
sqcld |
⊢ ( 𝐴 ∈ ℂ → ( ( sin ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) |
| 162 |
|
0cnd |
⊢ ( 𝐴 ∈ ℂ → 0 ∈ ℂ ) |
| 163 |
154
|
sqcld |
⊢ ( 𝐴 ∈ ℂ → ( ( cos ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) |
| 164 |
161 162 163
|
addcan2d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( sin ‘ 𝐴 ) ↑ 2 ) + ( ( cos ‘ 𝐴 ) ↑ 2 ) ) = ( 0 + ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ↔ ( ( sin ‘ 𝐴 ) ↑ 2 ) = 0 ) ) |
| 165 |
|
sincossq |
⊢ ( 𝐴 ∈ ℂ → ( ( ( sin ‘ 𝐴 ) ↑ 2 ) + ( ( cos ‘ 𝐴 ) ↑ 2 ) ) = 1 ) |
| 166 |
|
neg1sqe1 |
⊢ ( - 1 ↑ 2 ) = 1 |
| 167 |
165 166
|
eqtr4di |
⊢ ( 𝐴 ∈ ℂ → ( ( ( sin ‘ 𝐴 ) ↑ 2 ) + ( ( cos ‘ 𝐴 ) ↑ 2 ) ) = ( - 1 ↑ 2 ) ) |
| 168 |
163
|
addid2d |
⊢ ( 𝐴 ∈ ℂ → ( 0 + ( ( cos ‘ 𝐴 ) ↑ 2 ) ) = ( ( cos ‘ 𝐴 ) ↑ 2 ) ) |
| 169 |
167 168
|
eqeq12d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( sin ‘ 𝐴 ) ↑ 2 ) + ( ( cos ‘ 𝐴 ) ↑ 2 ) ) = ( 0 + ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ↔ ( - 1 ↑ 2 ) = ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ) |
| 170 |
|
sqeq0 |
⊢ ( ( sin ‘ 𝐴 ) ∈ ℂ → ( ( ( sin ‘ 𝐴 ) ↑ 2 ) = 0 ↔ ( sin ‘ 𝐴 ) = 0 ) ) |
| 171 |
160 170
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ( ( sin ‘ 𝐴 ) ↑ 2 ) = 0 ↔ ( sin ‘ 𝐴 ) = 0 ) ) |
| 172 |
164 169 171
|
3bitr3d |
⊢ ( 𝐴 ∈ ℂ → ( ( - 1 ↑ 2 ) = ( ( cos ‘ 𝐴 ) ↑ 2 ) ↔ ( sin ‘ 𝐴 ) = 0 ) ) |
| 173 |
151 159 172
|
3imtr3d |
⊢ ( 𝐴 ∈ ℂ → ( ( 1 + ( cos ‘ 𝐴 ) ) = 0 → ( sin ‘ 𝐴 ) = 0 ) ) |
| 174 |
149 173
|
syl |
⊢ ( 𝐴 ∈ ( 0 (,) π ) → ( ( 1 + ( cos ‘ 𝐴 ) ) = 0 → ( sin ‘ 𝐴 ) = 0 ) ) |
| 175 |
174
|
necon3d |
⊢ ( 𝐴 ∈ ( 0 (,) π ) → ( ( sin ‘ 𝐴 ) ≠ 0 → ( 1 + ( cos ‘ 𝐴 ) ) ≠ 0 ) ) |
| 176 |
147 175
|
syl5 |
⊢ ( 𝐴 ∈ ( 0 (,) π ) → ( 0 < ( sin ‘ 𝐴 ) → ( 1 + ( cos ‘ 𝐴 ) ) ≠ 0 ) ) |
| 177 |
145 176
|
mpd |
⊢ ( 𝐴 ∈ ( 0 (,) π ) → ( 1 + ( cos ‘ 𝐴 ) ) ≠ 0 ) |
| 178 |
144 177
|
jaoi |
⊢ ( ( 𝐴 ∈ { 0 } ∨ 𝐴 ∈ ( 0 (,) π ) ) → ( 1 + ( cos ‘ 𝐴 ) ) ≠ 0 ) |
| 179 |
134 178
|
sylbi |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( 1 + ( cos ‘ 𝐴 ) ) ≠ 0 ) |
| 180 |
114 129 179
|
ne0gt0d |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → 0 < ( 1 + ( cos ‘ 𝐴 ) ) ) |
| 181 |
114 180
|
elrpd |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( 1 + ( cos ‘ 𝐴 ) ) ∈ ℝ+ ) |
| 182 |
181
|
rphalfcld |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( ( 1 + ( cos ‘ 𝐴 ) ) / 2 ) ∈ ℝ+ ) |
| 183 |
102 112 182
|
sqrtdivd |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( √ ‘ ( ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) / ( ( 1 + ( cos ‘ 𝐴 ) ) / 2 ) ) ) = ( ( √ ‘ ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) ) / ( √ ‘ ( ( 1 + ( cos ‘ 𝐴 ) ) / 2 ) ) ) ) |
| 184 |
7
|
coscld |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( cos ‘ 𝐴 ) ∈ ℂ ) |
| 185 |
|
subcl |
⊢ ( ( 1 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ∈ ℂ ) → ( 1 − ( cos ‘ 𝐴 ) ) ∈ ℂ ) |
| 186 |
121 184 185
|
sylancr |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( 1 − ( cos ‘ 𝐴 ) ) ∈ ℂ ) |
| 187 |
|
addcl |
⊢ ( ( 1 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ∈ ℂ ) → ( 1 + ( cos ‘ 𝐴 ) ) ∈ ℂ ) |
| 188 |
121 184 187
|
sylancr |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( 1 + ( cos ‘ 𝐴 ) ) ∈ ℂ ) |
| 189 |
|
2cnne0 |
⊢ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) |
| 190 |
|
divcan7 |
⊢ ( ( ( 1 − ( cos ‘ 𝐴 ) ) ∈ ℂ ∧ ( ( 1 + ( cos ‘ 𝐴 ) ) ∈ ℂ ∧ ( 1 + ( cos ‘ 𝐴 ) ) ≠ 0 ) ∧ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) → ( ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) / ( ( 1 + ( cos ‘ 𝐴 ) ) / 2 ) ) = ( ( 1 − ( cos ‘ 𝐴 ) ) / ( 1 + ( cos ‘ 𝐴 ) ) ) ) |
| 191 |
189 190
|
mp3an3 |
⊢ ( ( ( 1 − ( cos ‘ 𝐴 ) ) ∈ ℂ ∧ ( ( 1 + ( cos ‘ 𝐴 ) ) ∈ ℂ ∧ ( 1 + ( cos ‘ 𝐴 ) ) ≠ 0 ) ) → ( ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) / ( ( 1 + ( cos ‘ 𝐴 ) ) / 2 ) ) = ( ( 1 − ( cos ‘ 𝐴 ) ) / ( 1 + ( cos ‘ 𝐴 ) ) ) ) |
| 192 |
186 188 179 191
|
syl12anc |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) / ( ( 1 + ( cos ‘ 𝐴 ) ) / 2 ) ) = ( ( 1 − ( cos ‘ 𝐴 ) ) / ( 1 + ( cos ‘ 𝐴 ) ) ) ) |
| 193 |
192
|
fveq2d |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( √ ‘ ( ( ( 1 − ( cos ‘ 𝐴 ) ) / 2 ) / ( ( 1 + ( cos ‘ 𝐴 ) ) / 2 ) ) ) = ( √ ‘ ( ( 1 − ( cos ‘ 𝐴 ) ) / ( 1 + ( cos ‘ 𝐴 ) ) ) ) ) |
| 194 |
97 183 193
|
3eqtr2d |
⊢ ( 𝐴 ∈ ( 0 [,) π ) → ( tan ‘ ( 𝐴 / 2 ) ) = ( √ ‘ ( ( 1 − ( cos ‘ 𝐴 ) ) / ( 1 + ( cos ‘ 𝐴 ) ) ) ) ) |