Step |
Hyp |
Ref |
Expression |
1 |
|
ang.1 |
|- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) ) |
2 |
|
ang180lem1.2 |
|- T = ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) ) |
3 |
|
ang180lem1.3 |
|- N = ( ( ( T / _i ) / ( 2 x. _pi ) ) - ( 1 / 2 ) ) |
4 |
|
2cn |
|- 2 e. CC |
5 |
|
picn |
|- _pi e. CC |
6 |
4 5
|
mulcli |
|- ( 2 x. _pi ) e. CC |
7 |
|
2ne0 |
|- 2 =/= 0 |
8 |
6 4 7
|
divreci |
|- ( ( 2 x. _pi ) / 2 ) = ( ( 2 x. _pi ) x. ( 1 / 2 ) ) |
9 |
5 4 7
|
divcan3i |
|- ( ( 2 x. _pi ) / 2 ) = _pi |
10 |
8 9
|
eqtr3i |
|- ( ( 2 x. _pi ) x. ( 1 / 2 ) ) = _pi |
11 |
1 2 3
|
ang180lem2 |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 2 < N /\ N < 1 ) ) |
12 |
11
|
simprd |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N < 1 ) |
13 |
|
1e0p1 |
|- 1 = ( 0 + 1 ) |
14 |
12 13
|
breqtrdi |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N < ( 0 + 1 ) ) |
15 |
1 2 3
|
ang180lem1 |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( N e. ZZ /\ ( T / _i ) e. RR ) ) |
16 |
15
|
simpld |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N e. ZZ ) |
17 |
|
0z |
|- 0 e. ZZ |
18 |
|
zleltp1 |
|- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N <_ 0 <-> N < ( 0 + 1 ) ) ) |
19 |
16 17 18
|
sylancl |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( N <_ 0 <-> N < ( 0 + 1 ) ) ) |
20 |
14 19
|
mpbird |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N <_ 0 ) |
21 |
20
|
adantr |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> N <_ 0 ) |
22 |
|
zlem1lt |
|- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 <_ N <-> ( 0 - 1 ) < N ) ) |
23 |
17 16 22
|
sylancr |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 0 <_ N <-> ( 0 - 1 ) < N ) ) |
24 |
|
df-neg |
|- -u 1 = ( 0 - 1 ) |
25 |
24
|
breq1i |
|- ( -u 1 < N <-> ( 0 - 1 ) < N ) |
26 |
23 25
|
bitr4di |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 0 <_ N <-> -u 1 < N ) ) |
27 |
26
|
biimpar |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> 0 <_ N ) |
28 |
16
|
zred |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> N e. RR ) |
29 |
28
|
adantr |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> N e. RR ) |
30 |
|
0re |
|- 0 e. RR |
31 |
|
letri3 |
|- ( ( N e. RR /\ 0 e. RR ) -> ( N = 0 <-> ( N <_ 0 /\ 0 <_ N ) ) ) |
32 |
29 30 31
|
sylancl |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( N = 0 <-> ( N <_ 0 /\ 0 <_ N ) ) ) |
33 |
21 27 32
|
mpbir2and |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> N = 0 ) |
34 |
3 33
|
eqtr3id |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( ( T / _i ) / ( 2 x. _pi ) ) - ( 1 / 2 ) ) = 0 ) |
35 |
|
ax-1cn |
|- 1 e. CC |
36 |
|
simp1 |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> A e. CC ) |
37 |
|
subcl |
|- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC ) |
38 |
35 36 37
|
sylancr |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 - A ) e. CC ) |
39 |
|
simp3 |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> A =/= 1 ) |
40 |
39
|
necomd |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> 1 =/= A ) |
41 |
|
subeq0 |
|- ( ( 1 e. CC /\ A e. CC ) -> ( ( 1 - A ) = 0 <-> 1 = A ) ) |
42 |
35 36 41
|
sylancr |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( 1 - A ) = 0 <-> 1 = A ) ) |
43 |
42
|
necon3bid |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( 1 - A ) =/= 0 <-> 1 =/= A ) ) |
44 |
40 43
|
mpbird |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 - A ) =/= 0 ) |
45 |
38 44
|
reccld |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 / ( 1 - A ) ) e. CC ) |
46 |
38 44
|
recne0d |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 1 / ( 1 - A ) ) =/= 0 ) |
47 |
45 46
|
logcld |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( log ` ( 1 / ( 1 - A ) ) ) e. CC ) |
48 |
|
subcl |
|- ( ( A e. CC /\ 1 e. CC ) -> ( A - 1 ) e. CC ) |
49 |
36 35 48
|
sylancl |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( A - 1 ) e. CC ) |
50 |
|
simp2 |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> A =/= 0 ) |
51 |
49 36 50
|
divcld |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) / A ) e. CC ) |
52 |
|
subeq0 |
|- ( ( A e. CC /\ 1 e. CC ) -> ( ( A - 1 ) = 0 <-> A = 1 ) ) |
53 |
36 35 52
|
sylancl |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) = 0 <-> A = 1 ) ) |
54 |
53
|
necon3bid |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) =/= 0 <-> A =/= 1 ) ) |
55 |
39 54
|
mpbird |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( A - 1 ) =/= 0 ) |
56 |
49 36 55 50
|
divne0d |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( A - 1 ) / A ) =/= 0 ) |
57 |
51 56
|
logcld |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( log ` ( ( A - 1 ) / A ) ) e. CC ) |
58 |
47 57
|
addcld |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) e. CC ) |
59 |
|
logcl |
|- ( ( A e. CC /\ A =/= 0 ) -> ( log ` A ) e. CC ) |
60 |
59
|
3adant3 |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( log ` A ) e. CC ) |
61 |
58 60
|
addcld |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( log ` ( 1 / ( 1 - A ) ) ) + ( log ` ( ( A - 1 ) / A ) ) ) + ( log ` A ) ) e. CC ) |
62 |
2 61
|
eqeltrid |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> T e. CC ) |
63 |
|
ax-icn |
|- _i e. CC |
64 |
63
|
a1i |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> _i e. CC ) |
65 |
|
ine0 |
|- _i =/= 0 |
66 |
65
|
a1i |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> _i =/= 0 ) |
67 |
62 64 66
|
divcld |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( T / _i ) e. CC ) |
68 |
6
|
a1i |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 2 x. _pi ) e. CC ) |
69 |
|
pire |
|- _pi e. RR |
70 |
|
pipos |
|- 0 < _pi |
71 |
69 70
|
gt0ne0ii |
|- _pi =/= 0 |
72 |
4 5 7 71
|
mulne0i |
|- ( 2 x. _pi ) =/= 0 |
73 |
72
|
a1i |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( 2 x. _pi ) =/= 0 ) |
74 |
67 68 73
|
divcld |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( T / _i ) / ( 2 x. _pi ) ) e. CC ) |
75 |
74
|
adantr |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( T / _i ) / ( 2 x. _pi ) ) e. CC ) |
76 |
|
halfcn |
|- ( 1 / 2 ) e. CC |
77 |
|
subeq0 |
|- ( ( ( ( T / _i ) / ( 2 x. _pi ) ) e. CC /\ ( 1 / 2 ) e. CC ) -> ( ( ( ( T / _i ) / ( 2 x. _pi ) ) - ( 1 / 2 ) ) = 0 <-> ( ( T / _i ) / ( 2 x. _pi ) ) = ( 1 / 2 ) ) ) |
78 |
75 76 77
|
sylancl |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( ( ( T / _i ) / ( 2 x. _pi ) ) - ( 1 / 2 ) ) = 0 <-> ( ( T / _i ) / ( 2 x. _pi ) ) = ( 1 / 2 ) ) ) |
79 |
34 78
|
mpbid |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( T / _i ) / ( 2 x. _pi ) ) = ( 1 / 2 ) ) |
80 |
67
|
adantr |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( T / _i ) e. CC ) |
81 |
6
|
a1i |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( 2 x. _pi ) e. CC ) |
82 |
76
|
a1i |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( 1 / 2 ) e. CC ) |
83 |
72
|
a1i |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( 2 x. _pi ) =/= 0 ) |
84 |
80 81 82 83
|
divmuld |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( ( T / _i ) / ( 2 x. _pi ) ) = ( 1 / 2 ) <-> ( ( 2 x. _pi ) x. ( 1 / 2 ) ) = ( T / _i ) ) ) |
85 |
79 84
|
mpbid |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( 2 x. _pi ) x. ( 1 / 2 ) ) = ( T / _i ) ) |
86 |
10 85
|
syl5reqr |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( T / _i ) = _pi ) |
87 |
62
|
adantr |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> T e. CC ) |
88 |
63
|
a1i |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> _i e. CC ) |
89 |
5
|
a1i |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> _pi e. CC ) |
90 |
65
|
a1i |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> _i =/= 0 ) |
91 |
87 88 89 90
|
divmuld |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( ( T / _i ) = _pi <-> ( _i x. _pi ) = T ) ) |
92 |
86 91
|
mpbid |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( _i x. _pi ) = T ) |
93 |
92
|
eqcomd |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> T = ( _i x. _pi ) ) |
94 |
93
|
olcd |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 < N ) -> ( T = -u ( _i x. _pi ) \/ T = ( _i x. _pi ) ) ) |
95 |
5 63
|
mulneg1i |
|- ( -u _pi x. _i ) = -u ( _pi x. _i ) |
96 |
5 63
|
mulcomi |
|- ( _pi x. _i ) = ( _i x. _pi ) |
97 |
96
|
negeqi |
|- -u ( _pi x. _i ) = -u ( _i x. _pi ) |
98 |
95 97
|
eqtri |
|- ( -u _pi x. _i ) = -u ( _i x. _pi ) |
99 |
76 6
|
mulneg1i |
|- ( -u ( 1 / 2 ) x. ( 2 x. _pi ) ) = -u ( ( 1 / 2 ) x. ( 2 x. _pi ) ) |
100 |
35 4 7
|
divcan1i |
|- ( ( 1 / 2 ) x. 2 ) = 1 |
101 |
100
|
oveq1i |
|- ( ( ( 1 / 2 ) x. 2 ) x. _pi ) = ( 1 x. _pi ) |
102 |
76 4 5
|
mulassi |
|- ( ( ( 1 / 2 ) x. 2 ) x. _pi ) = ( ( 1 / 2 ) x. ( 2 x. _pi ) ) |
103 |
5
|
mulid2i |
|- ( 1 x. _pi ) = _pi |
104 |
101 102 103
|
3eqtr3i |
|- ( ( 1 / 2 ) x. ( 2 x. _pi ) ) = _pi |
105 |
104
|
negeqi |
|- -u ( ( 1 / 2 ) x. ( 2 x. _pi ) ) = -u _pi |
106 |
99 105
|
eqtri |
|- ( -u ( 1 / 2 ) x. ( 2 x. _pi ) ) = -u _pi |
107 |
35 76
|
negsubdii |
|- -u ( 1 - ( 1 / 2 ) ) = ( -u 1 + ( 1 / 2 ) ) |
108 |
|
1mhlfehlf |
|- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 ) |
109 |
108
|
negeqi |
|- -u ( 1 - ( 1 / 2 ) ) = -u ( 1 / 2 ) |
110 |
107 109
|
eqtr3i |
|- ( -u 1 + ( 1 / 2 ) ) = -u ( 1 / 2 ) |
111 |
|
simpr |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u 1 = N ) |
112 |
111 3
|
eqtrdi |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u 1 = ( ( ( T / _i ) / ( 2 x. _pi ) ) - ( 1 / 2 ) ) ) |
113 |
112
|
oveq1d |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( -u 1 + ( 1 / 2 ) ) = ( ( ( ( T / _i ) / ( 2 x. _pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) ) ) |
114 |
110 113
|
eqtr3id |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u ( 1 / 2 ) = ( ( ( ( T / _i ) / ( 2 x. _pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) ) ) |
115 |
|
npcan |
|- ( ( ( ( T / _i ) / ( 2 x. _pi ) ) e. CC /\ ( 1 / 2 ) e. CC ) -> ( ( ( ( T / _i ) / ( 2 x. _pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) ) = ( ( T / _i ) / ( 2 x. _pi ) ) ) |
116 |
74 76 115
|
sylancl |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( ( T / _i ) / ( 2 x. _pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) ) = ( ( T / _i ) / ( 2 x. _pi ) ) ) |
117 |
116
|
adantr |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( ( ( ( T / _i ) / ( 2 x. _pi ) ) - ( 1 / 2 ) ) + ( 1 / 2 ) ) = ( ( T / _i ) / ( 2 x. _pi ) ) ) |
118 |
114 117
|
eqtrd |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u ( 1 / 2 ) = ( ( T / _i ) / ( 2 x. _pi ) ) ) |
119 |
118
|
oveq1d |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( -u ( 1 / 2 ) x. ( 2 x. _pi ) ) = ( ( ( T / _i ) / ( 2 x. _pi ) ) x. ( 2 x. _pi ) ) ) |
120 |
106 119
|
eqtr3id |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u _pi = ( ( ( T / _i ) / ( 2 x. _pi ) ) x. ( 2 x. _pi ) ) ) |
121 |
67 68 73
|
divcan1d |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( ( T / _i ) / ( 2 x. _pi ) ) x. ( 2 x. _pi ) ) = ( T / _i ) ) |
122 |
121
|
adantr |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( ( ( T / _i ) / ( 2 x. _pi ) ) x. ( 2 x. _pi ) ) = ( T / _i ) ) |
123 |
120 122
|
eqtrd |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u _pi = ( T / _i ) ) |
124 |
123
|
oveq1d |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( -u _pi x. _i ) = ( ( T / _i ) x. _i ) ) |
125 |
98 124
|
eqtr3id |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> -u ( _i x. _pi ) = ( ( T / _i ) x. _i ) ) |
126 |
62 64 66
|
divcan1d |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( ( T / _i ) x. _i ) = T ) |
127 |
126
|
adantr |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( ( T / _i ) x. _i ) = T ) |
128 |
125 127
|
eqtr2d |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> T = -u ( _i x. _pi ) ) |
129 |
128
|
orcd |
|- ( ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) /\ -u 1 = N ) -> ( T = -u ( _i x. _pi ) \/ T = ( _i x. _pi ) ) ) |
130 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
131 |
130
|
negeqi |
|- -u 2 = -u ( 1 + 1 ) |
132 |
|
negdi2 |
|- ( ( 1 e. CC /\ 1 e. CC ) -> -u ( 1 + 1 ) = ( -u 1 - 1 ) ) |
133 |
35 35 132
|
mp2an |
|- -u ( 1 + 1 ) = ( -u 1 - 1 ) |
134 |
131 133
|
eqtri |
|- -u 2 = ( -u 1 - 1 ) |
135 |
11
|
simpld |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> -u 2 < N ) |
136 |
134 135
|
eqbrtrrid |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 1 - 1 ) < N ) |
137 |
|
neg1z |
|- -u 1 e. ZZ |
138 |
|
zlem1lt |
|- ( ( -u 1 e. ZZ /\ N e. ZZ ) -> ( -u 1 <_ N <-> ( -u 1 - 1 ) < N ) ) |
139 |
137 16 138
|
sylancr |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 1 <_ N <-> ( -u 1 - 1 ) < N ) ) |
140 |
136 139
|
mpbird |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> -u 1 <_ N ) |
141 |
|
neg1rr |
|- -u 1 e. RR |
142 |
|
leloe |
|- ( ( -u 1 e. RR /\ N e. RR ) -> ( -u 1 <_ N <-> ( -u 1 < N \/ -u 1 = N ) ) ) |
143 |
141 28 142
|
sylancr |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 1 <_ N <-> ( -u 1 < N \/ -u 1 = N ) ) ) |
144 |
140 143
|
mpbid |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( -u 1 < N \/ -u 1 = N ) ) |
145 |
94 129 144
|
mpjaodan |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> ( T = -u ( _i x. _pi ) \/ T = ( _i x. _pi ) ) ) |
146 |
2
|
ovexi |
|- T e. _V |
147 |
146
|
elpr |
|- ( T e. { -u ( _i x. _pi ) , ( _i x. _pi ) } <-> ( T = -u ( _i x. _pi ) \/ T = ( _i x. _pi ) ) ) |
148 |
145 147
|
sylibr |
|- ( ( A e. CC /\ A =/= 0 /\ A =/= 1 ) -> T e. { -u ( _i x. _pi ) , ( _i x. _pi ) } ) |