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Theorem ang180lem3 22166
Description: Lemma for ang180 22169. Since ang180lem1 22164 shows that  N is an integer and ang180lem2 22165 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 10388 . . . . . . . . . 10  |-  2  e.  CC
2 picn 21881 . . . . . . . . . 10  |-  pi  e.  CC
31, 2mulcli 9387 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  CC
4 2ne0 10410 . . . . . . . . 9  |-  2  =/=  0
53, 1, 4divreci 10072 . . . . . . . 8  |-  ( ( 2  x.  pi )  /  2 )  =  ( ( 2  x.  pi )  x.  (
1  /  2 ) )
62, 1, 4divcan3i 10073 . . . . . . . 8  |-  ( ( 2  x.  pi )  /  2 )  =  pi
75, 6eqtr3i 2463 . . . . . . 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 22165 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 2  <  N  /\  N  <  1 ) )
1211simprd 460 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  <  1 )
13 1e0p1 10779 . . . . . . . . . . . . . 14  |-  1  =  ( 0  +  1 )
1412, 13syl6breq 4328 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  <  ( 0  +  1 ) )
159, 10, 8ang180lem1 22164 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  e.  ZZ  /\  ( T  /  _i )  e.  RR ) )
1615simpld 456 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  ZZ )
17 0z 10653 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
18 zleltp1 10691 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  ->  ( N  <_  0  <->  N  <  ( 0  +  1 ) ) )
1916, 17, 18sylancl 657 . . . . . . . . . . . . 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 462 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  N  <_  0 )
22 zlem1lt 10692 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  <_  N  <->  ( 0  -  1 )  <  N ) )
2317, 16, 22sylancr 658 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
0  <_  N  <->  ( 0  -  1 )  < 
N ) )
24 df-neg 9594 . . . . . . . . . . . . . 14  |-  -u 1  =  ( 0  -  1 )
2524breq1i 4296 . . . . . . . . . . . . 13  |-  ( -u
1  <  N  <->  ( 0  -  1 )  < 
N )
2623, 25syl6bbr 263 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
0  <_  N  <->  -u 1  < 
N ) )
2726biimpar 482 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  0  <_  N )
2816zred 10743 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  RR )
2928adantr 462 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  N  e.  RR )
30 0re 9382 . . . . . . . . . . . 12  |-  0  e.  RR
31 letri3 9456 . . . . . . . . . . . 12  |-  ( ( N  e.  RR  /\  0  e.  RR )  ->  ( N  =  0  <-> 
( N  <_  0  /\  0  <_  N ) ) )
3229, 30, 31sylancl 657 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( N  =  0  <->  ( N  <_  0  /\  0  <_  N ) ) )
3321, 27, 32mpbir2and 908 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  N  = 
0 )
348, 33syl5eqr 2487 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( (
( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  =  0 )
35 ax-1cn 9336 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
36 simp1 983 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
37 subcl 9605 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
3835, 36, 37sylancr 658 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  e.  CC )
39 simp3 985 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
4039necomd 2693 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
41 subeq0 9631 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
4235, 36, 41sylancr 658 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =  0  <->  1  =  A ) )
4342necon3bid 2641 . . . . . . . . . . . . . . . . . . 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 10096 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
4638, 44recne0d 10097 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
4745, 46logcld 21981 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
48 subcl 9605 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
4936, 35, 48sylancl 657 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  e.  CC )
50 simp2 984 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
5149, 36, 50divcld 10103 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
52 subeq0 9631 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
5336, 35, 52sylancl 657 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =  0  <->  A  =  1 ) )
5453necon3bid 2641 . . . . . . . . . . . . . . . . . . 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 10119 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
5751, 56logcld 21981 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
5847, 57addcld 9401 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
59 logcl 21979 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
60593adant3 1003 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
6158, 60addcld 9401 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  CC )
6210, 61syl5eqel 2525 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  CC )
63 ax-icn 9337 . . . . . . . . . . . . . 14  |-  _i  e.  CC
6463a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  e.  CC )
65 ine0 9776 . . . . . . . . . . . . . 14  |-  _i  =/=  0
6665a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  =/=  0 )
6762, 64, 66divcld 10103 . . . . . . . . . . . 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 21880 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
70 pipos 21882 . . . . . . . . . . . . . . 15  |-  0  <  pi
7169, 70gt0ne0ii 9872 . . . . . . . . . . . . . 14  |-  pi  =/=  0
721, 2, 4, 71mulne0i 9975 . . . . . . . . . . . . 13  |-  ( 2  x.  pi )  =/=  0
7372a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  =/=  0 )
7467, 68, 73divcld 10103 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  CC )
7574adantr 462 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( ( T  /  _i )  / 
( 2  x.  pi ) )  e.  CC )
76 halfcn 10537 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  CC
77 subeq0 9631 . . . . . . . . . 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 657 . . . . . . . . 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 462 . . . . . . . . 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 10125 . . . . . . . 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 2488 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( T  /  _i )  =  pi )
8762adantr 462 . . . . . . 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 10125 . . . . . 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 2446 . . . 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 9786 . . . . . . 7  |-  ( -u pi  x.  _i )  = 
-u ( pi  x.  _i )
962, 63mulcomi 9388 . . . . . . . 8  |-  ( pi  x.  _i )  =  ( _i  x.  pi )
9796negeqi 9599 . . . . . . 7  |-  -u (
pi  x.  _i )  =  -u ( _i  x.  pi )
9895, 97eqtri 2461 . . . . . 6  |-  ( -u pi  x.  _i )  = 
-u ( _i  x.  pi )
9976, 3mulneg1i 9786 . . . . . . . . . 10  |-  ( -u ( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u ( ( 1  /  2 )  x.  ( 2  x.  pi ) )
10035, 1, 4divcan1i 10071 . . . . . . . . . . . . 13  |-  ( ( 1  /  2 )  x.  2 )  =  1
101100oveq1i 6100 . . . . . . . . . . . 12  |-  ( ( ( 1  /  2
)  x.  2 )  x.  pi )  =  ( 1  x.  pi )
10276, 1, 2mulassi 9391 . . . . . . . . . . . 12  |-  ( ( ( 1  /  2
)  x.  2 )  x.  pi )  =  ( ( 1  / 
2 )  x.  (
2  x.  pi ) )
1032mulid2i 9385 . . . . . . . . . . . 12  |-  ( 1  x.  pi )  =  pi
104101, 102, 1033eqtr3i 2469 . . . . . . . . . . 11  |-  ( ( 1  /  2 )  x.  ( 2  x.  pi ) )  =  pi
105104negeqi 9599 . . . . . . . . . 10  |-  -u (
( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u pi
10699, 105eqtri 2461 . . . . . . . . 9  |-  ( -u ( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u pi
10735, 76negsubdii 9689 . . . . . . . . . . . . 13  |-  -u (
1  -  ( 1  /  2 ) )  =  ( -u 1  +  ( 1  / 
2 ) )
108 1mhlfehlf 10540 . . . . . . . . . . . . . 14  |-  ( 1  -  ( 1  / 
2 ) )  =  ( 1  /  2
)
109108negeqi 9599 . . . . . . . . . . . . 13  |-  -u (
1  -  ( 1  /  2 ) )  =  -u ( 1  / 
2 )
110107, 109eqtr3i 2463 . . . . . . . . . . . 12  |-  ( -u
1  +  ( 1  /  2 ) )  =  -u ( 1  / 
2 )
111 simpr 458 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u 1  =  N )
112111, 8syl6eq 2489 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u 1  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) ) )
113112oveq1d 6105 . . . . . . . . . . . 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 2487 . . . . . . . . . . 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 9615 . . . . . . . . . . . . 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 657 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
117116adantr 462 . . . . . . . . . . 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 2473 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( 1  /  2 )  =  ( ( T  /  _i )  /  (
2  x.  pi ) ) )
119118oveq1d 6105 . . . . . . . . 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 2487 . . . . . . . 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 10104 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  =  ( T  /  _i ) )
122121adantr 462 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( (
( T  /  _i )  /  ( 2  x.  pi ) )  x.  ( 2  x.  pi ) )  =  ( T  /  _i ) )
123120, 122eqtrd 2473 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u pi  =  ( T  /  _i ) )
124123oveq1d 6105 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( -u pi  x.  _i )  =  ( ( T  /  _i )  x.  _i )
)
12598, 124syl5eqr 2487 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( _i  x.  pi )  =  ( ( T  /  _i )  x.  _i ) )
12662, 64, 66divcan1d 10104 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  x.  _i )  =  T )
127126adantr 462 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( ( T  /  _i )  x.  _i )  =  T )
128125, 127eqtr2d 2474 . . . 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 10376 . . . . . . . 8  |-  2  =  ( 1  +  1 )
131130negeqi 9599 . . . . . . 7  |-  -u 2  =  -u ( 1  +  1 )
132 negdi2 9663 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  1  e.  CC )  -> 
-u ( 1  +  1 )  =  (
-u 1  -  1 ) )
13335, 35, 132mp2an 667 . . . . . . 7  |-  -u (
1  +  1 )  =  ( -u 1  -  1 )
134131, 133eqtri 2461 . . . . . 6  |-  -u 2  =  ( -u 1  -  1 )
13511simpld 456 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 2  <  N )
136134, 135syl5eqbrr 4323 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  -  1 )  <  N )
137 neg1z 10677 . . . . . 6  |-  -u 1  e.  ZZ
138 zlem1lt 10692 . . . . . 6  |-  ( (
-u 1  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u 1  <_  N  <->  ( -u 1  -  1 )  < 
N ) )
139137, 16, 138sylancr 658 . . . . 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 10422 . . . . 5  |-  -u 1  e.  RR
142 leloe 9457 . . . . 5  |-  ( (
-u 1  e.  RR  /\  N  e.  RR )  ->  ( -u 1  <_  N  <->  ( -u 1  <  N  \/  -u 1  =  N ) ) )
143141, 28, 142sylancr 658 . . . 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 779 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  =  -u ( _i  x.  pi )  \/  T  =  ( _i  x.  pi ) ) )
146 ovex 6115 . . . 4  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  _V
14710, 146eqeltri 2511 . . 3  |-  T  e. 
_V
148147elpr 3892 . 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    \ cdif 3322   {csn 3874   {cpr 3876   class class class wbr 4289   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279   _ici 9280    + caddc 9281    x. cmul 9283    < clt 9414    <_ cle 9415    - cmin 9591   -ucneg 9592    / cdiv 9989   2c2 10367   ZZcz 10642   Imcim 12583   picpi 13348   logclog 21965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967
This theorem is referenced by:  ang180lem4  22167
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