| Step |
Hyp |
Ref |
Expression |
| 1 |
|
sinval |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ 𝐴 ) = ( ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) / ( 2 · i ) ) ) |
| 2 |
1
|
eqeq1d |
⊢ ( 𝐴 ∈ ℂ → ( ( sin ‘ 𝐴 ) = 0 ↔ ( ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) / ( 2 · i ) ) = 0 ) ) |
| 3 |
|
ax-icn |
⊢ i ∈ ℂ |
| 4 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( i · 𝐴 ) ∈ ℂ ) |
| 5 |
3 4
|
mpan |
⊢ ( 𝐴 ∈ ℂ → ( i · 𝐴 ) ∈ ℂ ) |
| 6 |
|
efcl |
⊢ ( ( i · 𝐴 ) ∈ ℂ → ( exp ‘ ( i · 𝐴 ) ) ∈ ℂ ) |
| 7 |
5 6
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( exp ‘ ( i · 𝐴 ) ) ∈ ℂ ) |
| 8 |
|
negicn |
⊢ - i ∈ ℂ |
| 9 |
|
mulcl |
⊢ ( ( - i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( - i · 𝐴 ) ∈ ℂ ) |
| 10 |
8 9
|
mpan |
⊢ ( 𝐴 ∈ ℂ → ( - i · 𝐴 ) ∈ ℂ ) |
| 11 |
|
efcl |
⊢ ( ( - i · 𝐴 ) ∈ ℂ → ( exp ‘ ( - i · 𝐴 ) ) ∈ ℂ ) |
| 12 |
10 11
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( exp ‘ ( - i · 𝐴 ) ) ∈ ℂ ) |
| 13 |
7 12
|
subcld |
⊢ ( 𝐴 ∈ ℂ → ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) ∈ ℂ ) |
| 14 |
|
2mulicn |
⊢ ( 2 · i ) ∈ ℂ |
| 15 |
|
2muline0 |
⊢ ( 2 · i ) ≠ 0 |
| 16 |
|
diveq0 |
⊢ ( ( ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) ∈ ℂ ∧ ( 2 · i ) ∈ ℂ ∧ ( 2 · i ) ≠ 0 ) → ( ( ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) / ( 2 · i ) ) = 0 ↔ ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) = 0 ) ) |
| 17 |
14 15 16
|
mp3an23 |
⊢ ( ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) ∈ ℂ → ( ( ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) / ( 2 · i ) ) = 0 ↔ ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) = 0 ) ) |
| 18 |
13 17
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) / ( 2 · i ) ) = 0 ↔ ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) = 0 ) ) |
| 19 |
7 12
|
subeq0ad |
⊢ ( 𝐴 ∈ ℂ → ( ( ( exp ‘ ( i · 𝐴 ) ) − ( exp ‘ ( - i · 𝐴 ) ) ) = 0 ↔ ( exp ‘ ( i · 𝐴 ) ) = ( exp ‘ ( - i · 𝐴 ) ) ) ) |
| 20 |
2 18 19
|
3bitrd |
⊢ ( 𝐴 ∈ ℂ → ( ( sin ‘ 𝐴 ) = 0 ↔ ( exp ‘ ( i · 𝐴 ) ) = ( exp ‘ ( - i · 𝐴 ) ) ) ) |
| 21 |
|
oveq2 |
⊢ ( ( exp ‘ ( i · 𝐴 ) ) = ( exp ‘ ( - i · 𝐴 ) ) → ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( i · 𝐴 ) ) ) = ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( - i · 𝐴 ) ) ) ) |
| 22 |
|
2cn |
⊢ 2 ∈ ℂ |
| 23 |
|
mul12 |
⊢ ( ( i ∈ ℂ ∧ 2 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( i · ( 2 · 𝐴 ) ) = ( 2 · ( i · 𝐴 ) ) ) |
| 24 |
3 22 23
|
mp3an12 |
⊢ ( 𝐴 ∈ ℂ → ( i · ( 2 · 𝐴 ) ) = ( 2 · ( i · 𝐴 ) ) ) |
| 25 |
5
|
2timesd |
⊢ ( 𝐴 ∈ ℂ → ( 2 · ( i · 𝐴 ) ) = ( ( i · 𝐴 ) + ( i · 𝐴 ) ) ) |
| 26 |
24 25
|
eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( i · ( 2 · 𝐴 ) ) = ( ( i · 𝐴 ) + ( i · 𝐴 ) ) ) |
| 27 |
26
|
fveq2d |
⊢ ( 𝐴 ∈ ℂ → ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) = ( exp ‘ ( ( i · 𝐴 ) + ( i · 𝐴 ) ) ) ) |
| 28 |
|
efadd |
⊢ ( ( ( i · 𝐴 ) ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( exp ‘ ( ( i · 𝐴 ) + ( i · 𝐴 ) ) ) = ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( i · 𝐴 ) ) ) ) |
| 29 |
5 5 28
|
syl2anc |
⊢ ( 𝐴 ∈ ℂ → ( exp ‘ ( ( i · 𝐴 ) + ( i · 𝐴 ) ) ) = ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( i · 𝐴 ) ) ) ) |
| 30 |
27 29
|
eqtr2d |
⊢ ( 𝐴 ∈ ℂ → ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( i · 𝐴 ) ) ) = ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) ) |
| 31 |
|
efadd |
⊢ ( ( ( i · 𝐴 ) ∈ ℂ ∧ ( - i · 𝐴 ) ∈ ℂ ) → ( exp ‘ ( ( i · 𝐴 ) + ( - i · 𝐴 ) ) ) = ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( - i · 𝐴 ) ) ) ) |
| 32 |
5 10 31
|
syl2anc |
⊢ ( 𝐴 ∈ ℂ → ( exp ‘ ( ( i · 𝐴 ) + ( - i · 𝐴 ) ) ) = ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( - i · 𝐴 ) ) ) ) |
| 33 |
3
|
negidi |
⊢ ( i + - i ) = 0 |
| 34 |
33
|
oveq1i |
⊢ ( ( i + - i ) · 𝐴 ) = ( 0 · 𝐴 ) |
| 35 |
|
adddir |
⊢ ( ( i ∈ ℂ ∧ - i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( i + - i ) · 𝐴 ) = ( ( i · 𝐴 ) + ( - i · 𝐴 ) ) ) |
| 36 |
3 8 35
|
mp3an12 |
⊢ ( 𝐴 ∈ ℂ → ( ( i + - i ) · 𝐴 ) = ( ( i · 𝐴 ) + ( - i · 𝐴 ) ) ) |
| 37 |
|
mul02 |
⊢ ( 𝐴 ∈ ℂ → ( 0 · 𝐴 ) = 0 ) |
| 38 |
34 36 37
|
3eqtr3a |
⊢ ( 𝐴 ∈ ℂ → ( ( i · 𝐴 ) + ( - i · 𝐴 ) ) = 0 ) |
| 39 |
38
|
fveq2d |
⊢ ( 𝐴 ∈ ℂ → ( exp ‘ ( ( i · 𝐴 ) + ( - i · 𝐴 ) ) ) = ( exp ‘ 0 ) ) |
| 40 |
|
ef0 |
⊢ ( exp ‘ 0 ) = 1 |
| 41 |
39 40
|
eqtrdi |
⊢ ( 𝐴 ∈ ℂ → ( exp ‘ ( ( i · 𝐴 ) + ( - i · 𝐴 ) ) ) = 1 ) |
| 42 |
32 41
|
eqtr3d |
⊢ ( 𝐴 ∈ ℂ → ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( - i · 𝐴 ) ) ) = 1 ) |
| 43 |
30 42
|
eqeq12d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( i · 𝐴 ) ) ) = ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( - i · 𝐴 ) ) ) ↔ ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) = 1 ) ) |
| 44 |
|
fveq2 |
⊢ ( ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) = 1 → ( abs ‘ ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) ) = ( abs ‘ 1 ) ) |
| 45 |
43 44
|
biimtrdi |
⊢ ( 𝐴 ∈ ℂ → ( ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( i · 𝐴 ) ) ) = ( ( exp ‘ ( i · 𝐴 ) ) · ( exp ‘ ( - i · 𝐴 ) ) ) → ( abs ‘ ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) ) = ( abs ‘ 1 ) ) ) |
| 46 |
21 45
|
syl5 |
⊢ ( 𝐴 ∈ ℂ → ( ( exp ‘ ( i · 𝐴 ) ) = ( exp ‘ ( - i · 𝐴 ) ) → ( abs ‘ ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) ) = ( abs ‘ 1 ) ) ) |
| 47 |
20 46
|
sylbid |
⊢ ( 𝐴 ∈ ℂ → ( ( sin ‘ 𝐴 ) = 0 → ( abs ‘ ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) ) = ( abs ‘ 1 ) ) ) |
| 48 |
|
abs1 |
⊢ ( abs ‘ 1 ) = 1 |
| 49 |
48
|
eqeq2i |
⊢ ( ( abs ‘ ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) ) = ( abs ‘ 1 ) ↔ ( abs ‘ ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) ) = 1 ) |
| 50 |
|
2re |
⊢ 2 ∈ ℝ |
| 51 |
|
2ne0 |
⊢ 2 ≠ 0 |
| 52 |
|
mulre |
⊢ ( ( 𝐴 ∈ ℂ ∧ 2 ∈ ℝ ∧ 2 ≠ 0 ) → ( 𝐴 ∈ ℝ ↔ ( 2 · 𝐴 ) ∈ ℝ ) ) |
| 53 |
50 51 52
|
mp3an23 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℝ ↔ ( 2 · 𝐴 ) ∈ ℝ ) ) |
| 54 |
|
mulcl |
⊢ ( ( 2 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 2 · 𝐴 ) ∈ ℂ ) |
| 55 |
22 54
|
mpan |
⊢ ( 𝐴 ∈ ℂ → ( 2 · 𝐴 ) ∈ ℂ ) |
| 56 |
|
absefib |
⊢ ( ( 2 · 𝐴 ) ∈ ℂ → ( ( 2 · 𝐴 ) ∈ ℝ ↔ ( abs ‘ ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) ) = 1 ) ) |
| 57 |
55 56
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ( 2 · 𝐴 ) ∈ ℝ ↔ ( abs ‘ ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) ) = 1 ) ) |
| 58 |
53 57
|
bitr2d |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) ) = 1 ↔ 𝐴 ∈ ℝ ) ) |
| 59 |
49 58
|
bitrid |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ ( exp ‘ ( i · ( 2 · 𝐴 ) ) ) ) = ( abs ‘ 1 ) ↔ 𝐴 ∈ ℝ ) ) |
| 60 |
47 59
|
sylibd |
⊢ ( 𝐴 ∈ ℂ → ( ( sin ‘ 𝐴 ) = 0 → 𝐴 ∈ ℝ ) ) |
| 61 |
60
|
imp |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → 𝐴 ∈ ℝ ) |
| 62 |
|
pirp |
⊢ π ∈ ℝ+ |
| 63 |
|
modval |
⊢ ( ( 𝐴 ∈ ℝ ∧ π ∈ ℝ+ ) → ( 𝐴 mod π ) = ( 𝐴 − ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) ) ) |
| 64 |
61 62 63
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 𝐴 mod π ) = ( 𝐴 − ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) ) ) |
| 65 |
|
picn |
⊢ π ∈ ℂ |
| 66 |
|
pire |
⊢ π ∈ ℝ |
| 67 |
|
pipos |
⊢ 0 < π |
| 68 |
66 67
|
gt0ne0ii |
⊢ π ≠ 0 |
| 69 |
|
redivcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ π ∈ ℝ ∧ π ≠ 0 ) → ( 𝐴 / π ) ∈ ℝ ) |
| 70 |
66 68 69
|
mp3an23 |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 / π ) ∈ ℝ ) |
| 71 |
61 70
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 𝐴 / π ) ∈ ℝ ) |
| 72 |
71
|
flcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( ⌊ ‘ ( 𝐴 / π ) ) ∈ ℤ ) |
| 73 |
72
|
zcnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( ⌊ ‘ ( 𝐴 / π ) ) ∈ ℂ ) |
| 74 |
|
mulcl |
⊢ ( ( π ∈ ℂ ∧ ( ⌊ ‘ ( 𝐴 / π ) ) ∈ ℂ ) → ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) ∈ ℂ ) |
| 75 |
65 73 74
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) ∈ ℂ ) |
| 76 |
|
negsub |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) ∈ ℂ ) → ( 𝐴 + - ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) ) = ( 𝐴 − ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) ) ) |
| 77 |
75 76
|
syldan |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 𝐴 + - ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) ) = ( 𝐴 − ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) ) ) |
| 78 |
|
mulcom |
⊢ ( ( π ∈ ℂ ∧ ( ⌊ ‘ ( 𝐴 / π ) ) ∈ ℂ ) → ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) = ( ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) |
| 79 |
65 73 78
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) = ( ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) |
| 80 |
79
|
negeqd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → - ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) = - ( ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) |
| 81 |
|
mulneg1 |
⊢ ( ( ( ⌊ ‘ ( 𝐴 / π ) ) ∈ ℂ ∧ π ∈ ℂ ) → ( - ( ⌊ ‘ ( 𝐴 / π ) ) · π ) = - ( ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) |
| 82 |
73 65 81
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( - ( ⌊ ‘ ( 𝐴 / π ) ) · π ) = - ( ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) |
| 83 |
80 82
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → - ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) = ( - ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) |
| 84 |
83
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 𝐴 + - ( π · ( ⌊ ‘ ( 𝐴 / π ) ) ) ) = ( 𝐴 + ( - ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) ) |
| 85 |
64 77 84
|
3eqtr2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 𝐴 mod π ) = ( 𝐴 + ( - ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) ) |
| 86 |
85
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( sin ‘ ( 𝐴 mod π ) ) = ( sin ‘ ( 𝐴 + ( - ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) ) ) |
| 87 |
86
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) = ( abs ‘ ( sin ‘ ( 𝐴 + ( - ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) ) ) ) |
| 88 |
72
|
znegcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → - ( ⌊ ‘ ( 𝐴 / π ) ) ∈ ℤ ) |
| 89 |
|
abssinper |
⊢ ( ( 𝐴 ∈ ℂ ∧ - ( ⌊ ‘ ( 𝐴 / π ) ) ∈ ℤ ) → ( abs ‘ ( sin ‘ ( 𝐴 + ( - ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) ) ) = ( abs ‘ ( sin ‘ 𝐴 ) ) ) |
| 90 |
88 89
|
syldan |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( abs ‘ ( sin ‘ ( 𝐴 + ( - ( ⌊ ‘ ( 𝐴 / π ) ) · π ) ) ) ) = ( abs ‘ ( sin ‘ 𝐴 ) ) ) |
| 91 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( sin ‘ 𝐴 ) = 0 ) |
| 92 |
91
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( abs ‘ ( sin ‘ 𝐴 ) ) = ( abs ‘ 0 ) ) |
| 93 |
87 90 92
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) = ( abs ‘ 0 ) ) |
| 94 |
|
abs0 |
⊢ ( abs ‘ 0 ) = 0 |
| 95 |
93 94
|
eqtrdi |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) = 0 ) |
| 96 |
|
modcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ π ∈ ℝ+ ) → ( 𝐴 mod π ) ∈ ℝ ) |
| 97 |
61 62 96
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 𝐴 mod π ) ∈ ℝ ) |
| 98 |
|
modlt |
⊢ ( ( 𝐴 ∈ ℝ ∧ π ∈ ℝ+ ) → ( 𝐴 mod π ) < π ) |
| 99 |
61 62 98
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 𝐴 mod π ) < π ) |
| 100 |
97 99
|
jca |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( ( 𝐴 mod π ) ∈ ℝ ∧ ( 𝐴 mod π ) < π ) ) |
| 101 |
100
|
biantrurd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 0 < ( 𝐴 mod π ) ↔ ( ( ( 𝐴 mod π ) ∈ ℝ ∧ ( 𝐴 mod π ) < π ) ∧ 0 < ( 𝐴 mod π ) ) ) ) |
| 102 |
|
0re |
⊢ 0 ∈ ℝ |
| 103 |
|
rexr |
⊢ ( 0 ∈ ℝ → 0 ∈ ℝ* ) |
| 104 |
|
rexr |
⊢ ( π ∈ ℝ → π ∈ ℝ* ) |
| 105 |
|
elioo2 |
⊢ ( ( 0 ∈ ℝ* ∧ π ∈ ℝ* ) → ( ( 𝐴 mod π ) ∈ ( 0 (,) π ) ↔ ( ( 𝐴 mod π ) ∈ ℝ ∧ 0 < ( 𝐴 mod π ) ∧ ( 𝐴 mod π ) < π ) ) ) |
| 106 |
103 104 105
|
syl2an |
⊢ ( ( 0 ∈ ℝ ∧ π ∈ ℝ ) → ( ( 𝐴 mod π ) ∈ ( 0 (,) π ) ↔ ( ( 𝐴 mod π ) ∈ ℝ ∧ 0 < ( 𝐴 mod π ) ∧ ( 𝐴 mod π ) < π ) ) ) |
| 107 |
102 66 106
|
mp2an |
⊢ ( ( 𝐴 mod π ) ∈ ( 0 (,) π ) ↔ ( ( 𝐴 mod π ) ∈ ℝ ∧ 0 < ( 𝐴 mod π ) ∧ ( 𝐴 mod π ) < π ) ) |
| 108 |
|
3anan32 |
⊢ ( ( ( 𝐴 mod π ) ∈ ℝ ∧ 0 < ( 𝐴 mod π ) ∧ ( 𝐴 mod π ) < π ) ↔ ( ( ( 𝐴 mod π ) ∈ ℝ ∧ ( 𝐴 mod π ) < π ) ∧ 0 < ( 𝐴 mod π ) ) ) |
| 109 |
107 108
|
bitri |
⊢ ( ( 𝐴 mod π ) ∈ ( 0 (,) π ) ↔ ( ( ( 𝐴 mod π ) ∈ ℝ ∧ ( 𝐴 mod π ) < π ) ∧ 0 < ( 𝐴 mod π ) ) ) |
| 110 |
101 109
|
bitr4di |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 0 < ( 𝐴 mod π ) ↔ ( 𝐴 mod π ) ∈ ( 0 (,) π ) ) ) |
| 111 |
|
sinq12gt0 |
⊢ ( ( 𝐴 mod π ) ∈ ( 0 (,) π ) → 0 < ( sin ‘ ( 𝐴 mod π ) ) ) |
| 112 |
|
elioore |
⊢ ( ( 𝐴 mod π ) ∈ ( 0 (,) π ) → ( 𝐴 mod π ) ∈ ℝ ) |
| 113 |
112
|
resincld |
⊢ ( ( 𝐴 mod π ) ∈ ( 0 (,) π ) → ( sin ‘ ( 𝐴 mod π ) ) ∈ ℝ ) |
| 114 |
|
ltle |
⊢ ( ( 0 ∈ ℝ ∧ ( sin ‘ ( 𝐴 mod π ) ) ∈ ℝ ) → ( 0 < ( sin ‘ ( 𝐴 mod π ) ) → 0 ≤ ( sin ‘ ( 𝐴 mod π ) ) ) ) |
| 115 |
102 113 114
|
sylancr |
⊢ ( ( 𝐴 mod π ) ∈ ( 0 (,) π ) → ( 0 < ( sin ‘ ( 𝐴 mod π ) ) → 0 ≤ ( sin ‘ ( 𝐴 mod π ) ) ) ) |
| 116 |
111 115
|
mpd |
⊢ ( ( 𝐴 mod π ) ∈ ( 0 (,) π ) → 0 ≤ ( sin ‘ ( 𝐴 mod π ) ) ) |
| 117 |
113 116
|
absidd |
⊢ ( ( 𝐴 mod π ) ∈ ( 0 (,) π ) → ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) = ( sin ‘ ( 𝐴 mod π ) ) ) |
| 118 |
111 117
|
breqtrrd |
⊢ ( ( 𝐴 mod π ) ∈ ( 0 (,) π ) → 0 < ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) ) |
| 119 |
110 118
|
biimtrdi |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 0 < ( 𝐴 mod π ) → 0 < ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) ) ) |
| 120 |
|
ltne |
⊢ ( ( 0 ∈ ℝ ∧ 0 < ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) ) → ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) ≠ 0 ) |
| 121 |
102 120
|
mpan |
⊢ ( 0 < ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) → ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) ≠ 0 ) |
| 122 |
119 121
|
syl6 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 0 < ( 𝐴 mod π ) → ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) ≠ 0 ) ) |
| 123 |
122
|
necon2bd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( ( abs ‘ ( sin ‘ ( 𝐴 mod π ) ) ) = 0 → ¬ 0 < ( 𝐴 mod π ) ) ) |
| 124 |
95 123
|
mpd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ¬ 0 < ( 𝐴 mod π ) ) |
| 125 |
|
modge0 |
⊢ ( ( 𝐴 ∈ ℝ ∧ π ∈ ℝ+ ) → 0 ≤ ( 𝐴 mod π ) ) |
| 126 |
61 62 125
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → 0 ≤ ( 𝐴 mod π ) ) |
| 127 |
|
leloe |
⊢ ( ( 0 ∈ ℝ ∧ ( 𝐴 mod π ) ∈ ℝ ) → ( 0 ≤ ( 𝐴 mod π ) ↔ ( 0 < ( 𝐴 mod π ) ∨ 0 = ( 𝐴 mod π ) ) ) ) |
| 128 |
102 97 127
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 0 ≤ ( 𝐴 mod π ) ↔ ( 0 < ( 𝐴 mod π ) ∨ 0 = ( 𝐴 mod π ) ) ) ) |
| 129 |
126 128
|
mpbid |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 0 < ( 𝐴 mod π ) ∨ 0 = ( 𝐴 mod π ) ) ) |
| 130 |
129
|
ord |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( ¬ 0 < ( 𝐴 mod π ) → 0 = ( 𝐴 mod π ) ) ) |
| 131 |
124 130
|
mpd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → 0 = ( 𝐴 mod π ) ) |
| 132 |
131
|
eqcomd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 𝐴 mod π ) = 0 ) |
| 133 |
|
mod0 |
⊢ ( ( 𝐴 ∈ ℝ ∧ π ∈ ℝ+ ) → ( ( 𝐴 mod π ) = 0 ↔ ( 𝐴 / π ) ∈ ℤ ) ) |
| 134 |
61 62 133
|
sylancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( ( 𝐴 mod π ) = 0 ↔ ( 𝐴 / π ) ∈ ℤ ) ) |
| 135 |
132 134
|
mpbid |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) = 0 ) → ( 𝐴 / π ) ∈ ℤ ) |
| 136 |
|
divcan1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ π ∈ ℂ ∧ π ≠ 0 ) → ( ( 𝐴 / π ) · π ) = 𝐴 ) |
| 137 |
65 68 136
|
mp3an23 |
⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 / π ) · π ) = 𝐴 ) |
| 138 |
137
|
fveq2d |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ ( ( 𝐴 / π ) · π ) ) = ( sin ‘ 𝐴 ) ) |
| 139 |
|
sinkpi |
⊢ ( ( 𝐴 / π ) ∈ ℤ → ( sin ‘ ( ( 𝐴 / π ) · π ) ) = 0 ) |
| 140 |
138 139
|
sylan9req |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐴 / π ) ∈ ℤ ) → ( sin ‘ 𝐴 ) = 0 ) |
| 141 |
135 140
|
impbida |
⊢ ( 𝐴 ∈ ℂ → ( ( sin ‘ 𝐴 ) = 0 ↔ ( 𝐴 / π ) ∈ ℤ ) ) |