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Theorem ang180lem3 22207
Description: Lemma for ang180 22210. Since ang180lem1 22205 shows that  N is an integer and ang180lem2 22206 shows that  N is strictly between  -u 2 and  1, it follows that  N  e.  { -u 1 ,  0 }, and these two cases correspond to the two possible values for  T. (Contributed by Mario Carneiro, 23-Sep-2014.)
Hypotheses
Ref Expression
ang.1  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
ang180lem1.2  |-  T  =  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) )
ang180lem1.3  |-  N  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )
Assertion
Ref Expression
ang180lem3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  { -u ( _i  x.  pi ) ,  ( _i  x.  pi ) } )
Distinct variable group:    x, y, A
Allowed substitution hints:    T( x, y)    F( x, y)    N( x, y)

Proof of Theorem ang180lem3
StepHypRef Expression
1 2cn 10392 . . . . . . . . . 10  |-  2  e.  CC
2 picn 21922 . . . . . . . . . 10  |-  pi  e.  CC
31, 2mulcli 9391 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  CC
4 2ne0 10414 . . . . . . . . 9  |-  2  =/=  0
53, 1, 4divreci 10076 . . . . . . . 8  |-  ( ( 2  x.  pi )  /  2 )  =  ( ( 2  x.  pi )  x.  (
1  /  2 ) )
62, 1, 4divcan3i 10077 . . . . . . . 8  |-  ( ( 2  x.  pi )  /  2 )  =  pi
75, 6eqtr3i 2465 . . . . . . 7  |-  ( ( 2  x.  pi )  x.  ( 1  / 
2 ) )  =  pi
8 ang180lem1.3 . . . . . . . . . 10  |-  N  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )
9 ang.1 . . . . . . . . . . . . . . . 16  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
10 ang180lem1.2 . . . . . . . . . . . . . . . 16  |-  T  =  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) )
119, 10, 8ang180lem2 22206 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 2  <  N  /\  N  <  1 ) )
1211simprd 463 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  <  1 )
13 1e0p1 10783 . . . . . . . . . . . . . 14  |-  1  =  ( 0  +  1 )
1412, 13syl6breq 4331 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  <  ( 0  +  1 ) )
159, 10, 8ang180lem1 22205 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  e.  ZZ  /\  ( T  /  _i )  e.  RR ) )
1615simpld 459 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  ZZ )
17 0z 10657 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
18 zleltp1 10695 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  ->  ( N  <_  0  <->  N  <  ( 0  +  1 ) ) )
1916, 17, 18sylancl 662 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  <_  0  <->  N  <  ( 0  +  1 ) ) )
2014, 19mpbird 232 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  <_  0 )
2120adantr 465 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  N  <_  0 )
22 zlem1lt 10696 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  <_  N  <->  ( 0  -  1 )  <  N ) )
2317, 16, 22sylancr 663 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
0  <_  N  <->  ( 0  -  1 )  < 
N ) )
24 df-neg 9598 . . . . . . . . . . . . . 14  |-  -u 1  =  ( 0  -  1 )
2524breq1i 4299 . . . . . . . . . . . . 13  |-  ( -u
1  <  N  <->  ( 0  -  1 )  < 
N )
2623, 25syl6bbr 263 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
0  <_  N  <->  -u 1  < 
N ) )
2726biimpar 485 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  0  <_  N )
2816zred 10747 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  RR )
2928adantr 465 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  N  e.  RR )
30 0re 9386 . . . . . . . . . . . 12  |-  0  e.  RR
31 letri3 9460 . . . . . . . . . . . 12  |-  ( ( N  e.  RR  /\  0  e.  RR )  ->  ( N  =  0  <-> 
( N  <_  0  /\  0  <_  N ) ) )
3229, 30, 31sylancl 662 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( N  =  0  <->  ( N  <_  0  /\  0  <_  N ) ) )
3321, 27, 32mpbir2and 913 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  N  = 
0 )
348, 33syl5eqr 2489 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( (
( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  =  0 )
35 ax-1cn 9340 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
36 simp1 988 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
37 subcl 9609 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
3835, 36, 37sylancr 663 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  e.  CC )
39 simp3 990 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
4039necomd 2695 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
41 subeq0 9635 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
4235, 36, 41sylancr 663 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =  0  <->  1  =  A ) )
4342necon3bid 2643 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =/=  0  <->  1  =/=  A ) )
4440, 43mpbird 232 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  =/=  0 )
4538, 44reccld 10100 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
4638, 44recne0d 10101 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
4745, 46logcld 22022 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
48 subcl 9609 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
4936, 35, 48sylancl 662 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  e.  CC )
50 simp2 989 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
5149, 36, 50divcld 10107 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
52 subeq0 9635 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
5336, 35, 52sylancl 662 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =  0  <->  A  =  1 ) )
5453necon3bid 2643 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =/=  0  <->  A  =/=  1 ) )
5539, 54mpbird 232 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  =/=  0 )
5649, 36, 55, 50divne0d 10123 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
5751, 56logcld 22022 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
5847, 57addcld 9405 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
59 logcl 22020 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
60593adant3 1008 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
6158, 60addcld 9405 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  CC )
6210, 61syl5eqel 2527 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  CC )
63 ax-icn 9341 . . . . . . . . . . . . . 14  |-  _i  e.  CC
6463a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  e.  CC )
65 ine0 9780 . . . . . . . . . . . . . 14  |-  _i  =/=  0
6665a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  =/=  0 )
6762, 64, 66divcld 10107 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  /  _i )  e.  CC )
683a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  e.  CC )
69 pire 21921 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
70 pipos 21923 . . . . . . . . . . . . . . 15  |-  0  <  pi
7169, 70gt0ne0ii 9876 . . . . . . . . . . . . . 14  |-  pi  =/=  0
721, 2, 4, 71mulne0i 9979 . . . . . . . . . . . . 13  |-  ( 2  x.  pi )  =/=  0
7372a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  =/=  0 )
7467, 68, 73divcld 10107 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  CC )
7574adantr 465 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( ( T  /  _i )  / 
( 2  x.  pi ) )  e.  CC )
76 halfcn 10541 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  CC
77 subeq0 9635 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )
7875, 76, 77sylancl 662 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
7934, 78mpbid 210 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( ( T  /  _i )  / 
( 2  x.  pi ) )  =  ( 1  /  2 ) )
8067adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( T  /  _i )  e.  CC )
813a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( 2  x.  pi )  e.  CC )
8276a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( 1  /  2 )  e.  CC )
8372a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( 2  x.  pi )  =/=  0 )
8480, 81, 82, 83divmuld 10129 . . . . . . . 8  |-  ( ( ( 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 ) ) )
8579, 84mpbid 210 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( (
2  x.  pi )  x.  ( 1  / 
2 ) )  =  ( T  /  _i ) )
867, 85syl5reqr 2490 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( T  /  _i )  =  pi )
8762adantr 465 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  T  e.  CC )
8863a1i 11 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  _i  e.  CC )
892a1i 11 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  pi  e.  CC )
9065a1i 11 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  _i  =/=  0 )
9187, 88, 89, 90divmuld 10129 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( ( T  /  _i )  =  pi  <->  ( _i  x.  pi )  =  T
) )
9286, 91mpbid 210 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( _i  x.  pi )  =  T )
9392eqcomd 2448 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  T  =  ( _i  x.  pi ) )
9493olcd 393 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( T  =  -u ( _i  x.  pi )  \/  T  =  ( _i  x.  pi ) ) )
952, 63mulneg1i 9790 . . . . . . 7  |-  ( -u pi  x.  _i )  = 
-u ( pi  x.  _i )
962, 63mulcomi 9392 . . . . . . . 8  |-  ( pi  x.  _i )  =  ( _i  x.  pi )
9796negeqi 9603 . . . . . . 7  |-  -u (
pi  x.  _i )  =  -u ( _i  x.  pi )
9895, 97eqtri 2463 . . . . . 6  |-  ( -u pi  x.  _i )  = 
-u ( _i  x.  pi )
9976, 3mulneg1i 9790 . . . . . . . . . 10  |-  ( -u ( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u ( ( 1  /  2 )  x.  ( 2  x.  pi ) )
10035, 1, 4divcan1i 10075 . . . . . . . . . . . . 13  |-  ( ( 1  /  2 )  x.  2 )  =  1
101100oveq1i 6101 . . . . . . . . . . . 12  |-  ( ( ( 1  /  2
)  x.  2 )  x.  pi )  =  ( 1  x.  pi )
10276, 1, 2mulassi 9395 . . . . . . . . . . . 12  |-  ( ( ( 1  /  2
)  x.  2 )  x.  pi )  =  ( ( 1  / 
2 )  x.  (
2  x.  pi ) )
1032mulid2i 9389 . . . . . . . . . . . 12  |-  ( 1  x.  pi )  =  pi
104101, 102, 1033eqtr3i 2471 . . . . . . . . . . 11  |-  ( ( 1  /  2 )  x.  ( 2  x.  pi ) )  =  pi
105104negeqi 9603 . . . . . . . . . 10  |-  -u (
( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u pi
10699, 105eqtri 2463 . . . . . . . . 9  |-  ( -u ( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u pi
10735, 76negsubdii 9693 . . . . . . . . . . . . 13  |-  -u (
1  -  ( 1  /  2 ) )  =  ( -u 1  +  ( 1  / 
2 ) )
108 1mhlfehlf 10544 . . . . . . . . . . . . . 14  |-  ( 1  -  ( 1  / 
2 ) )  =  ( 1  /  2
)
109108negeqi 9603 . . . . . . . . . . . . 13  |-  -u (
1  -  ( 1  /  2 ) )  =  -u ( 1  / 
2 )
110107, 109eqtr3i 2465 . . . . . . . . . . . 12  |-  ( -u
1  +  ( 1  /  2 ) )  =  -u ( 1  / 
2 )
111 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u 1  =  N )
112111, 8syl6eq 2491 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u 1  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) ) )
113112oveq1d 6106 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( -u 1  +  ( 1  / 
2 ) )  =  ( ( ( ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  +  ( 1  /  2 ) ) )
114110, 113syl5eqr 2489 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( 1  /  2 )  =  ( ( ( ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  +  ( 1  /  2 ) ) )
115 npcan 9619 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 ) ) )
11674, 76, 115sylancl 662 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
117116adantr 465 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( (
( ( T  /  _i )  /  (
2  x.  pi ) )  -  ( 1  /  2 ) )  +  ( 1  / 
2 ) )  =  ( ( T  /  _i )  /  (
2  x.  pi ) ) )
118114, 117eqtrd 2475 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( 1  /  2 )  =  ( ( T  /  _i )  /  (
2  x.  pi ) ) )
119118oveq1d 6106 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
120106, 119syl5eqr 2489 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u pi  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  x.  (
2  x.  pi ) ) )
12167, 68, 73divcan1d 10108 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  =  ( T  /  _i ) )
122121adantr 465 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( (
( T  /  _i )  /  ( 2  x.  pi ) )  x.  ( 2  x.  pi ) )  =  ( T  /  _i ) )
123120, 122eqtrd 2475 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u pi  =  ( T  /  _i ) )
124123oveq1d 6106 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( -u pi  x.  _i )  =  ( ( T  /  _i )  x.  _i )
)
12598, 124syl5eqr 2489 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( _i  x.  pi )  =  ( ( T  /  _i )  x.  _i ) )
12662, 64, 66divcan1d 10108 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  x.  _i )  =  T )
127126adantr 465 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( ( T  /  _i )  x.  _i )  =  T )
128125, 127eqtr2d 2476 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  T  =  -u ( _i  x.  pi ) )
129128orcd 392 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( T  =  -u ( _i  x.  pi )  \/  T  =  ( _i  x.  pi ) ) )
130 df-2 10380 . . . . . . . 8  |-  2  =  ( 1  +  1 )
131130negeqi 9603 . . . . . . 7  |-  -u 2  =  -u ( 1  +  1 )
132 negdi2 9667 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  1  e.  CC )  -> 
-u ( 1  +  1 )  =  (
-u 1  -  1 ) )
13335, 35, 132mp2an 672 . . . . . . 7  |-  -u (
1  +  1 )  =  ( -u 1  -  1 )
134131, 133eqtri 2463 . . . . . 6  |-  -u 2  =  ( -u 1  -  1 )
13511simpld 459 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 2  <  N )
136134, 135syl5eqbrr 4326 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  -  1 )  <  N )
137 neg1z 10681 . . . . . 6  |-  -u 1  e.  ZZ
138 zlem1lt 10696 . . . . . 6  |-  ( (
-u 1  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u 1  <_  N  <->  ( -u 1  -  1 )  < 
N ) )
139137, 16, 138sylancr 663 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  <_  N  <->  ( -u 1  -  1 )  < 
N ) )
140136, 139mpbird 232 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 1  <_  N )
141 neg1rr 10426 . . . . 5  |-  -u 1  e.  RR
142 leloe 9461 . . . . 5  |-  ( (
-u 1  e.  RR  /\  N  e.  RR )  ->  ( -u 1  <_  N  <->  ( -u 1  <  N  \/  -u 1  =  N ) ) )
143141, 28, 142sylancr 663 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  <_  N  <->  ( -u 1  <  N  \/  -u 1  =  N ) ) )
144140, 143mpbid 210 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  <  N  \/  -u 1  =  N ) )
14594, 129, 144mpjaodan 784 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  =  -u ( _i  x.  pi )  \/  T  =  ( _i  x.  pi ) ) )
146 ovex 6116 . . . 4  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  _V
14710, 146eqeltri 2513 . . 3  |-  T  e. 
_V
148147elpr 3895 . 2  |-  ( T  e.  { -u (
_i  x.  pi ) ,  ( _i  x.  pi ) }  <->  ( T  =  -u ( _i  x.  pi )  \/  T  =  ( _i  x.  pi ) ) )
149145, 148sylibr 212 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  { -u ( _i  x.  pi ) ,  ( _i  x.  pi ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   _Vcvv 2972    \ cdif 3325   {csn 3877   {cpr 3879   class class class wbr 4292   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283   _ici 9284    + caddc 9285    x. cmul 9287    < clt 9418    <_ cle 9419    - cmin 9595   -ucneg 9596    / cdiv 9993   2c2 10371   ZZcz 10646   Imcim 12587   picpi 13352   logclog 22006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008
This theorem is referenced by:  ang180lem4  22208
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