Step |
Hyp |
Ref |
Expression |
1 |
|
sinval |
|- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) ) |
2 |
1
|
eqeq1d |
|- ( A e. CC -> ( ( sin ` A ) = 0 <-> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = 0 ) ) |
3 |
|
ax-icn |
|- _i e. CC |
4 |
|
mulcl |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC ) |
5 |
3 4
|
mpan |
|- ( A e. CC -> ( _i x. A ) e. CC ) |
6 |
|
efcl |
|- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) e. CC ) |
7 |
5 6
|
syl |
|- ( A e. CC -> ( exp ` ( _i x. A ) ) e. CC ) |
8 |
|
negicn |
|- -u _i e. CC |
9 |
|
mulcl |
|- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC ) |
10 |
8 9
|
mpan |
|- ( A e. CC -> ( -u _i x. A ) e. CC ) |
11 |
|
efcl |
|- ( ( -u _i x. A ) e. CC -> ( exp ` ( -u _i x. A ) ) e. CC ) |
12 |
10 11
|
syl |
|- ( A e. CC -> ( exp ` ( -u _i x. A ) ) e. CC ) |
13 |
7 12
|
subcld |
|- ( A e. CC -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC ) |
14 |
|
2mulicn |
|- ( 2 x. _i ) e. CC |
15 |
|
2muline0 |
|- ( 2 x. _i ) =/= 0 |
16 |
|
diveq0 |
|- ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC /\ ( 2 x. _i ) e. CC /\ ( 2 x. _i ) =/= 0 ) -> ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = 0 <-> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) = 0 ) ) |
17 |
14 15 16
|
mp3an23 |
|- ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC -> ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = 0 <-> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) = 0 ) ) |
18 |
13 17
|
syl |
|- ( A e. CC -> ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = 0 <-> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) = 0 ) ) |
19 |
7 12
|
subeq0ad |
|- ( A e. CC -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) = 0 <-> ( exp ` ( _i x. A ) ) = ( exp ` ( -u _i x. A ) ) ) ) |
20 |
2 18 19
|
3bitrd |
|- ( A e. CC -> ( ( sin ` A ) = 0 <-> ( exp ` ( _i x. A ) ) = ( exp ` ( -u _i x. A ) ) ) ) |
21 |
|
oveq2 |
|- ( ( exp ` ( _i x. A ) ) = ( exp ` ( -u _i x. A ) ) -> ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) ) |
22 |
|
2cn |
|- 2 e. CC |
23 |
|
mul12 |
|- ( ( _i e. CC /\ 2 e. CC /\ A e. CC ) -> ( _i x. ( 2 x. A ) ) = ( 2 x. ( _i x. A ) ) ) |
24 |
3 22 23
|
mp3an12 |
|- ( A e. CC -> ( _i x. ( 2 x. A ) ) = ( 2 x. ( _i x. A ) ) ) |
25 |
5
|
2timesd |
|- ( A e. CC -> ( 2 x. ( _i x. A ) ) = ( ( _i x. A ) + ( _i x. A ) ) ) |
26 |
24 25
|
eqtrd |
|- ( A e. CC -> ( _i x. ( 2 x. A ) ) = ( ( _i x. A ) + ( _i x. A ) ) ) |
27 |
26
|
fveq2d |
|- ( A e. CC -> ( exp ` ( _i x. ( 2 x. A ) ) ) = ( exp ` ( ( _i x. A ) + ( _i x. A ) ) ) ) |
28 |
|
efadd |
|- ( ( ( _i x. A ) e. CC /\ ( _i x. A ) e. CC ) -> ( exp ` ( ( _i x. A ) + ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) ) |
29 |
5 5 28
|
syl2anc |
|- ( A e. CC -> ( exp ` ( ( _i x. A ) + ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) ) |
30 |
27 29
|
eqtr2d |
|- ( A e. CC -> ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) = ( exp ` ( _i x. ( 2 x. A ) ) ) ) |
31 |
|
efadd |
|- ( ( ( _i x. A ) e. CC /\ ( -u _i x. A ) e. CC ) -> ( exp ` ( ( _i x. A ) + ( -u _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) ) |
32 |
5 10 31
|
syl2anc |
|- ( A e. CC -> ( exp ` ( ( _i x. A ) + ( -u _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) ) |
33 |
3
|
negidi |
|- ( _i + -u _i ) = 0 |
34 |
33
|
oveq1i |
|- ( ( _i + -u _i ) x. A ) = ( 0 x. A ) |
35 |
|
adddir |
|- ( ( _i e. CC /\ -u _i e. CC /\ A e. CC ) -> ( ( _i + -u _i ) x. A ) = ( ( _i x. A ) + ( -u _i x. A ) ) ) |
36 |
3 8 35
|
mp3an12 |
|- ( A e. CC -> ( ( _i + -u _i ) x. A ) = ( ( _i x. A ) + ( -u _i x. A ) ) ) |
37 |
|
mul02 |
|- ( A e. CC -> ( 0 x. A ) = 0 ) |
38 |
34 36 37
|
3eqtr3a |
|- ( A e. CC -> ( ( _i x. A ) + ( -u _i x. A ) ) = 0 ) |
39 |
38
|
fveq2d |
|- ( A e. CC -> ( exp ` ( ( _i x. A ) + ( -u _i x. A ) ) ) = ( exp ` 0 ) ) |
40 |
|
ef0 |
|- ( exp ` 0 ) = 1 |
41 |
39 40
|
eqtrdi |
|- ( A e. CC -> ( exp ` ( ( _i x. A ) + ( -u _i x. A ) ) ) = 1 ) |
42 |
32 41
|
eqtr3d |
|- ( A e. CC -> ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) = 1 ) |
43 |
30 42
|
eqeq12d |
|- ( A e. CC -> ( ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) <-> ( exp ` ( _i x. ( 2 x. A ) ) ) = 1 ) ) |
44 |
|
fveq2 |
|- ( ( exp ` ( _i x. ( 2 x. A ) ) ) = 1 -> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) ) |
45 |
43 44
|
syl6bi |
|- ( A e. CC -> ( ( ( exp ` ( _i x. A ) ) x. ( exp ` ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) x. ( exp ` ( -u _i x. A ) ) ) -> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) ) ) |
46 |
21 45
|
syl5 |
|- ( A e. CC -> ( ( exp ` ( _i x. A ) ) = ( exp ` ( -u _i x. A ) ) -> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) ) ) |
47 |
20 46
|
sylbid |
|- ( A e. CC -> ( ( sin ` A ) = 0 -> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) ) ) |
48 |
|
abs1 |
|- ( abs ` 1 ) = 1 |
49 |
48
|
eqeq2i |
|- ( ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) <-> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = 1 ) |
50 |
|
2re |
|- 2 e. RR |
51 |
|
2ne0 |
|- 2 =/= 0 |
52 |
|
mulre |
|- ( ( A e. CC /\ 2 e. RR /\ 2 =/= 0 ) -> ( A e. RR <-> ( 2 x. A ) e. RR ) ) |
53 |
50 51 52
|
mp3an23 |
|- ( A e. CC -> ( A e. RR <-> ( 2 x. A ) e. RR ) ) |
54 |
|
mulcl |
|- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC ) |
55 |
22 54
|
mpan |
|- ( A e. CC -> ( 2 x. A ) e. CC ) |
56 |
|
absefib |
|- ( ( 2 x. A ) e. CC -> ( ( 2 x. A ) e. RR <-> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = 1 ) ) |
57 |
55 56
|
syl |
|- ( A e. CC -> ( ( 2 x. A ) e. RR <-> ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = 1 ) ) |
58 |
53 57
|
bitr2d |
|- ( A e. CC -> ( ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = 1 <-> A e. RR ) ) |
59 |
49 58
|
syl5bb |
|- ( A e. CC -> ( ( abs ` ( exp ` ( _i x. ( 2 x. A ) ) ) ) = ( abs ` 1 ) <-> A e. RR ) ) |
60 |
47 59
|
sylibd |
|- ( A e. CC -> ( ( sin ` A ) = 0 -> A e. RR ) ) |
61 |
60
|
imp |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> A e. RR ) |
62 |
|
pirp |
|- _pi e. RR+ |
63 |
|
modval |
|- ( ( A e. RR /\ _pi e. RR+ ) -> ( A mod _pi ) = ( A - ( _pi x. ( |_ ` ( A / _pi ) ) ) ) ) |
64 |
61 62 63
|
sylancl |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( A mod _pi ) = ( A - ( _pi x. ( |_ ` ( A / _pi ) ) ) ) ) |
65 |
|
picn |
|- _pi e. CC |
66 |
|
pire |
|- _pi e. RR |
67 |
|
pipos |
|- 0 < _pi |
68 |
66 67
|
gt0ne0ii |
|- _pi =/= 0 |
69 |
|
redivcl |
|- ( ( A e. RR /\ _pi e. RR /\ _pi =/= 0 ) -> ( A / _pi ) e. RR ) |
70 |
66 68 69
|
mp3an23 |
|- ( A e. RR -> ( A / _pi ) e. RR ) |
71 |
61 70
|
syl |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( A / _pi ) e. RR ) |
72 |
71
|
flcld |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( |_ ` ( A / _pi ) ) e. ZZ ) |
73 |
72
|
zcnd |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( |_ ` ( A / _pi ) ) e. CC ) |
74 |
|
mulcl |
|- ( ( _pi e. CC /\ ( |_ ` ( A / _pi ) ) e. CC ) -> ( _pi x. ( |_ ` ( A / _pi ) ) ) e. CC ) |
75 |
65 73 74
|
sylancr |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( _pi x. ( |_ ` ( A / _pi ) ) ) e. CC ) |
76 |
|
negsub |
|- ( ( A e. CC /\ ( _pi x. ( |_ ` ( A / _pi ) ) ) e. CC ) -> ( A + -u ( _pi x. ( |_ ` ( A / _pi ) ) ) ) = ( A - ( _pi x. ( |_ ` ( A / _pi ) ) ) ) ) |
77 |
75 76
|
syldan |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( A + -u ( _pi x. ( |_ ` ( A / _pi ) ) ) ) = ( A - ( _pi x. ( |_ ` ( A / _pi ) ) ) ) ) |
78 |
|
mulcom |
|- ( ( _pi e. CC /\ ( |_ ` ( A / _pi ) ) e. CC ) -> ( _pi x. ( |_ ` ( A / _pi ) ) ) = ( ( |_ ` ( A / _pi ) ) x. _pi ) ) |
79 |
65 73 78
|
sylancr |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( _pi x. ( |_ ` ( A / _pi ) ) ) = ( ( |_ ` ( A / _pi ) ) x. _pi ) ) |
80 |
79
|
negeqd |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> -u ( _pi x. ( |_ ` ( A / _pi ) ) ) = -u ( ( |_ ` ( A / _pi ) ) x. _pi ) ) |
81 |
|
mulneg1 |
|- ( ( ( |_ ` ( A / _pi ) ) e. CC /\ _pi e. CC ) -> ( -u ( |_ ` ( A / _pi ) ) x. _pi ) = -u ( ( |_ ` ( A / _pi ) ) x. _pi ) ) |
82 |
73 65 81
|
sylancl |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( -u ( |_ ` ( A / _pi ) ) x. _pi ) = -u ( ( |_ ` ( A / _pi ) ) x. _pi ) ) |
83 |
80 82
|
eqtr4d |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> -u ( _pi x. ( |_ ` ( A / _pi ) ) ) = ( -u ( |_ ` ( A / _pi ) ) x. _pi ) ) |
84 |
83
|
oveq2d |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( A + -u ( _pi x. ( |_ ` ( A / _pi ) ) ) ) = ( A + ( -u ( |_ ` ( A / _pi ) ) x. _pi ) ) ) |
85 |
64 77 84
|
3eqtr2d |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( A mod _pi ) = ( A + ( -u ( |_ ` ( A / _pi ) ) x. _pi ) ) ) |
86 |
85
|
fveq2d |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( sin ` ( A mod _pi ) ) = ( sin ` ( A + ( -u ( |_ ` ( A / _pi ) ) x. _pi ) ) ) ) |
87 |
86
|
fveq2d |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( abs ` ( sin ` ( A mod _pi ) ) ) = ( abs ` ( sin ` ( A + ( -u ( |_ ` ( A / _pi ) ) x. _pi ) ) ) ) ) |
88 |
72
|
znegcld |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> -u ( |_ ` ( A / _pi ) ) e. ZZ ) |
89 |
|
abssinper |
|- ( ( A e. CC /\ -u ( |_ ` ( A / _pi ) ) e. ZZ ) -> ( abs ` ( sin ` ( A + ( -u ( |_ ` ( A / _pi ) ) x. _pi ) ) ) ) = ( abs ` ( sin ` A ) ) ) |
90 |
88 89
|
syldan |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( abs ` ( sin ` ( A + ( -u ( |_ ` ( A / _pi ) ) x. _pi ) ) ) ) = ( abs ` ( sin ` A ) ) ) |
91 |
|
simpr |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( sin ` A ) = 0 ) |
92 |
91
|
fveq2d |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( abs ` ( sin ` A ) ) = ( abs ` 0 ) ) |
93 |
87 90 92
|
3eqtrd |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( abs ` ( sin ` ( A mod _pi ) ) ) = ( abs ` 0 ) ) |
94 |
|
abs0 |
|- ( abs ` 0 ) = 0 |
95 |
93 94
|
eqtrdi |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( abs ` ( sin ` ( A mod _pi ) ) ) = 0 ) |
96 |
|
modcl |
|- ( ( A e. RR /\ _pi e. RR+ ) -> ( A mod _pi ) e. RR ) |
97 |
61 62 96
|
sylancl |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( A mod _pi ) e. RR ) |
98 |
|
modlt |
|- ( ( A e. RR /\ _pi e. RR+ ) -> ( A mod _pi ) < _pi ) |
99 |
61 62 98
|
sylancl |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( A mod _pi ) < _pi ) |
100 |
97 99
|
jca |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( ( A mod _pi ) e. RR /\ ( A mod _pi ) < _pi ) ) |
101 |
100
|
biantrurd |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( 0 < ( A mod _pi ) <-> ( ( ( A mod _pi ) e. RR /\ ( A mod _pi ) < _pi ) /\ 0 < ( A mod _pi ) ) ) ) |
102 |
|
0re |
|- 0 e. RR |
103 |
|
rexr |
|- ( 0 e. RR -> 0 e. RR* ) |
104 |
|
rexr |
|- ( _pi e. RR -> _pi e. RR* ) |
105 |
|
elioo2 |
|- ( ( 0 e. RR* /\ _pi e. RR* ) -> ( ( A mod _pi ) e. ( 0 (,) _pi ) <-> ( ( A mod _pi ) e. RR /\ 0 < ( A mod _pi ) /\ ( A mod _pi ) < _pi ) ) ) |
106 |
103 104 105
|
syl2an |
|- ( ( 0 e. RR /\ _pi e. RR ) -> ( ( A mod _pi ) e. ( 0 (,) _pi ) <-> ( ( A mod _pi ) e. RR /\ 0 < ( A mod _pi ) /\ ( A mod _pi ) < _pi ) ) ) |
107 |
102 66 106
|
mp2an |
|- ( ( A mod _pi ) e. ( 0 (,) _pi ) <-> ( ( A mod _pi ) e. RR /\ 0 < ( A mod _pi ) /\ ( A mod _pi ) < _pi ) ) |
108 |
|
3anan32 |
|- ( ( ( A mod _pi ) e. RR /\ 0 < ( A mod _pi ) /\ ( A mod _pi ) < _pi ) <-> ( ( ( A mod _pi ) e. RR /\ ( A mod _pi ) < _pi ) /\ 0 < ( A mod _pi ) ) ) |
109 |
107 108
|
bitri |
|- ( ( A mod _pi ) e. ( 0 (,) _pi ) <-> ( ( ( A mod _pi ) e. RR /\ ( A mod _pi ) < _pi ) /\ 0 < ( A mod _pi ) ) ) |
110 |
101 109
|
bitr4di |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( 0 < ( A mod _pi ) <-> ( A mod _pi ) e. ( 0 (,) _pi ) ) ) |
111 |
|
sinq12gt0 |
|- ( ( A mod _pi ) e. ( 0 (,) _pi ) -> 0 < ( sin ` ( A mod _pi ) ) ) |
112 |
|
elioore |
|- ( ( A mod _pi ) e. ( 0 (,) _pi ) -> ( A mod _pi ) e. RR ) |
113 |
112
|
resincld |
|- ( ( A mod _pi ) e. ( 0 (,) _pi ) -> ( sin ` ( A mod _pi ) ) e. RR ) |
114 |
|
ltle |
|- ( ( 0 e. RR /\ ( sin ` ( A mod _pi ) ) e. RR ) -> ( 0 < ( sin ` ( A mod _pi ) ) -> 0 <_ ( sin ` ( A mod _pi ) ) ) ) |
115 |
102 113 114
|
sylancr |
|- ( ( A mod _pi ) e. ( 0 (,) _pi ) -> ( 0 < ( sin ` ( A mod _pi ) ) -> 0 <_ ( sin ` ( A mod _pi ) ) ) ) |
116 |
111 115
|
mpd |
|- ( ( A mod _pi ) e. ( 0 (,) _pi ) -> 0 <_ ( sin ` ( A mod _pi ) ) ) |
117 |
113 116
|
absidd |
|- ( ( A mod _pi ) e. ( 0 (,) _pi ) -> ( abs ` ( sin ` ( A mod _pi ) ) ) = ( sin ` ( A mod _pi ) ) ) |
118 |
111 117
|
breqtrrd |
|- ( ( A mod _pi ) e. ( 0 (,) _pi ) -> 0 < ( abs ` ( sin ` ( A mod _pi ) ) ) ) |
119 |
110 118
|
syl6bi |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( 0 < ( A mod _pi ) -> 0 < ( abs ` ( sin ` ( A mod _pi ) ) ) ) ) |
120 |
|
ltne |
|- ( ( 0 e. RR /\ 0 < ( abs ` ( sin ` ( A mod _pi ) ) ) ) -> ( abs ` ( sin ` ( A mod _pi ) ) ) =/= 0 ) |
121 |
102 120
|
mpan |
|- ( 0 < ( abs ` ( sin ` ( A mod _pi ) ) ) -> ( abs ` ( sin ` ( A mod _pi ) ) ) =/= 0 ) |
122 |
119 121
|
syl6 |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( 0 < ( A mod _pi ) -> ( abs ` ( sin ` ( A mod _pi ) ) ) =/= 0 ) ) |
123 |
122
|
necon2bd |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( ( abs ` ( sin ` ( A mod _pi ) ) ) = 0 -> -. 0 < ( A mod _pi ) ) ) |
124 |
95 123
|
mpd |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> -. 0 < ( A mod _pi ) ) |
125 |
|
modge0 |
|- ( ( A e. RR /\ _pi e. RR+ ) -> 0 <_ ( A mod _pi ) ) |
126 |
61 62 125
|
sylancl |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> 0 <_ ( A mod _pi ) ) |
127 |
|
leloe |
|- ( ( 0 e. RR /\ ( A mod _pi ) e. RR ) -> ( 0 <_ ( A mod _pi ) <-> ( 0 < ( A mod _pi ) \/ 0 = ( A mod _pi ) ) ) ) |
128 |
102 97 127
|
sylancr |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( 0 <_ ( A mod _pi ) <-> ( 0 < ( A mod _pi ) \/ 0 = ( A mod _pi ) ) ) ) |
129 |
126 128
|
mpbid |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( 0 < ( A mod _pi ) \/ 0 = ( A mod _pi ) ) ) |
130 |
129
|
ord |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( -. 0 < ( A mod _pi ) -> 0 = ( A mod _pi ) ) ) |
131 |
124 130
|
mpd |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> 0 = ( A mod _pi ) ) |
132 |
131
|
eqcomd |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( A mod _pi ) = 0 ) |
133 |
|
mod0 |
|- ( ( A e. RR /\ _pi e. RR+ ) -> ( ( A mod _pi ) = 0 <-> ( A / _pi ) e. ZZ ) ) |
134 |
61 62 133
|
sylancl |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( ( A mod _pi ) = 0 <-> ( A / _pi ) e. ZZ ) ) |
135 |
132 134
|
mpbid |
|- ( ( A e. CC /\ ( sin ` A ) = 0 ) -> ( A / _pi ) e. ZZ ) |
136 |
|
divcan1 |
|- ( ( A e. CC /\ _pi e. CC /\ _pi =/= 0 ) -> ( ( A / _pi ) x. _pi ) = A ) |
137 |
65 68 136
|
mp3an23 |
|- ( A e. CC -> ( ( A / _pi ) x. _pi ) = A ) |
138 |
137
|
fveq2d |
|- ( A e. CC -> ( sin ` ( ( A / _pi ) x. _pi ) ) = ( sin ` A ) ) |
139 |
|
sinkpi |
|- ( ( A / _pi ) e. ZZ -> ( sin ` ( ( A / _pi ) x. _pi ) ) = 0 ) |
140 |
138 139
|
sylan9req |
|- ( ( A e. CC /\ ( A / _pi ) e. ZZ ) -> ( sin ` A ) = 0 ) |
141 |
135 140
|
impbida |
|- ( A e. CC -> ( ( sin ` A ) = 0 <-> ( A / _pi ) e. ZZ ) ) |