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Theorem ang180lem3 22887
Description: Lemma for ang180 22890. Since ang180lem1 22885 shows that  N is an integer and ang180lem2 22886 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 10605 . . . . . . . . . 10  |-  2  e.  CC
2 picn 22602 . . . . . . . . . 10  |-  pi  e.  CC
31, 2mulcli 9600 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  CC
4 2ne0 10627 . . . . . . . . 9  |-  2  =/=  0
53, 1, 4divreci 10288 . . . . . . . 8  |-  ( ( 2  x.  pi )  /  2 )  =  ( ( 2  x.  pi )  x.  (
1  /  2 ) )
62, 1, 4divcan3i 10289 . . . . . . . 8  |-  ( ( 2  x.  pi )  /  2 )  =  pi
75, 6eqtr3i 2498 . . . . . . 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 22886 . . . . . . . . . . . . . . 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 11003 . . . . . . . . . . . . . 14  |-  1  =  ( 0  +  1 )
1412, 13syl6breq 4486 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  <  ( 0  +  1 ) )
159, 10, 8ang180lem1 22885 . . . . . . . . . . . . . . 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 10874 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
18 zleltp1 10912 . . . . . . . . . . . . . 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 10913 . . . . . . . . . . . . . 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 9807 . . . . . . . . . . . . . 14  |-  -u 1  =  ( 0  -  1 )
2524breq1i 4454 . . . . . . . . . . . . 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 10965 . . . . . . . . . . . . 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 9595 . . . . . . . . . . . 12  |-  0  e.  RR
31 letri3 9669 . . . . . . . . . . . 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 920 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  N  = 
0 )
348, 33syl5eqr 2522 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( (
( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  =  0 )
35 ax-1cn 9549 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
36 simp1 996 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
37 subcl 9818 . . . . . . . . . . . . . . . . . . 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 998 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
4039necomd 2738 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
41 subeq0 9844 . . . . . . . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . . . . 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 10312 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
4638, 44recne0d 10313 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
4745, 46logcld 22702 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
48 subcl 9818 . . . . . . . . . . . . . . . . . . 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 997 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
5149, 36, 50divcld 10319 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
52 subeq0 9844 . . . . . . . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . . . . 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 10335 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
5751, 56logcld 22702 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
5847, 57addcld 9614 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
59 logcl 22700 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
60593adant3 1016 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
6158, 60addcld 9614 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  CC )
6210, 61syl5eqel 2559 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  CC )
63 ax-icn 9550 . . . . . . . . . . . . . 14  |-  _i  e.  CC
6463a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  e.  CC )
65 ine0 9991 . . . . . . . . . . . . . 14  |-  _i  =/=  0
6665a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  =/=  0 )
6762, 64, 66divcld 10319 . . . . . . . . . . . 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 22601 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
70 pipos 22603 . . . . . . . . . . . . . . 15  |-  0  <  pi
7169, 70gt0ne0ii 10088 . . . . . . . . . . . . . 14  |-  pi  =/=  0
721, 2, 4, 71mulne0i 10191 . . . . . . . . . . . . 13  |-  ( 2  x.  pi )  =/=  0
7372a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  =/=  0 )
7467, 68, 73divcld 10319 . . . . . . . . . . 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 10754 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  CC
77 subeq0 9844 . . . . . . . . . 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 10341 . . . . . . . 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 2523 . . . . . 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 10341 . . . . . 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 2475 . . . 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 10001 . . . . . . 7  |-  ( -u pi  x.  _i )  = 
-u ( pi  x.  _i )
962, 63mulcomi 9601 . . . . . . . 8  |-  ( pi  x.  _i )  =  ( _i  x.  pi )
9796negeqi 9812 . . . . . . 7  |-  -u (
pi  x.  _i )  =  -u ( _i  x.  pi )
9895, 97eqtri 2496 . . . . . 6  |-  ( -u pi  x.  _i )  = 
-u ( _i  x.  pi )
9976, 3mulneg1i 10001 . . . . . . . . . 10  |-  ( -u ( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u ( ( 1  /  2 )  x.  ( 2  x.  pi ) )
10035, 1, 4divcan1i 10287 . . . . . . . . . . . . 13  |-  ( ( 1  /  2 )  x.  2 )  =  1
101100oveq1i 6293 . . . . . . . . . . . 12  |-  ( ( ( 1  /  2
)  x.  2 )  x.  pi )  =  ( 1  x.  pi )
10276, 1, 2mulassi 9604 . . . . . . . . . . . 12  |-  ( ( ( 1  /  2
)  x.  2 )  x.  pi )  =  ( ( 1  / 
2 )  x.  (
2  x.  pi ) )
1032mulid2i 9598 . . . . . . . . . . . 12  |-  ( 1  x.  pi )  =  pi
104101, 102, 1033eqtr3i 2504 . . . . . . . . . . 11  |-  ( ( 1  /  2 )  x.  ( 2  x.  pi ) )  =  pi
105104negeqi 9812 . . . . . . . . . 10  |-  -u (
( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u pi
10699, 105eqtri 2496 . . . . . . . . 9  |-  ( -u ( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u pi
10735, 76negsubdii 9903 . . . . . . . . . . . . 13  |-  -u (
1  -  ( 1  /  2 ) )  =  ( -u 1  +  ( 1  / 
2 ) )
108 1mhlfehlf 10757 . . . . . . . . . . . . . 14  |-  ( 1  -  ( 1  / 
2 ) )  =  ( 1  /  2
)
109108negeqi 9812 . . . . . . . . . . . . 13  |-  -u (
1  -  ( 1  /  2 ) )  =  -u ( 1  / 
2 )
110107, 109eqtr3i 2498 . . . . . . . . . . . 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 2524 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u 1  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) ) )
113112oveq1d 6298 . . . . . . . . . . . 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 2522 . . . . . . . . . . 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 9828 . . . . . . . . . . . . 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 2508 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( 1  /  2 )  =  ( ( T  /  _i )  /  (
2  x.  pi ) ) )
119118oveq1d 6298 . . . . . . . . 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 2522 . . . . . . . 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 10320 . . . . . . . . 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 2508 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u pi  =  ( T  /  _i ) )
124123oveq1d 6298 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( -u pi  x.  _i )  =  ( ( T  /  _i )  x.  _i )
)
12598, 124syl5eqr 2522 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( _i  x.  pi )  =  ( ( T  /  _i )  x.  _i ) )
12662, 64, 66divcan1d 10320 . . . . . 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 2509 . . . 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 10593 . . . . . . . 8  |-  2  =  ( 1  +  1 )
131130negeqi 9812 . . . . . . 7  |-  -u 2  =  -u ( 1  +  1 )
132 negdi2 9876 . . . . . . . 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 2496 . . . . . 6  |-  -u 2  =  ( -u 1  -  1 )
13511simpld 459 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 2  <  N )
136134, 135syl5eqbrr 4481 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  -  1 )  <  N )
137 neg1z 10898 . . . . . 6  |-  -u 1  e.  ZZ
138 zlem1lt 10913 . . . . . 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 10639 . . . . 5  |-  -u 1  e.  RR
142 leloe 9670 . . . . 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 6308 . . . 4  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  _V
14710, 146eqeltri 2551 . . 3  |-  T  e. 
_V
148147elpr 4045 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473   {csn 4027   {cpr 4029   class class class wbr 4447   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   CCcc 9489   RRcr 9490   0cc0 9491   1c1 9492   _ici 9493    + caddc 9494    x. cmul 9496    < clt 9627    <_ cle 9628    - cmin 9804   -ucneg 9805    / cdiv 10205   2c2 10584   ZZcz 10863   Imcim 12893   picpi 13663   logclog 22686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ioc 11533  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134  df-fac 12321  df-bc 12348  df-hash 12373  df-shft 12862  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-limsup 13256  df-clim 13273  df-rlim 13274  df-sum 13471  df-ef 13664  df-sin 13666  df-cos 13667  df-pi 13669  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-fbas 18203  df-fg 18204  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-ntr 19303  df-cls 19304  df-nei 19381  df-lp 19419  df-perf 19420  df-cn 19510  df-cnp 19511  df-haus 19598  df-tx 19814  df-hmeo 20007  df-fil 20098  df-fm 20190  df-flim 20191  df-flf 20192  df-xms 20574  df-ms 20575  df-tms 20576  df-cncf 21133  df-limc 22021  df-dv 22022  df-log 22688
This theorem is referenced by:  ang180lem4  22888
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