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Theorem ang180lem3 20653
Description: Lemma for ang180 20656. Since ang180lem1 20651 shows that  N is an integer and ang180lem2 20652 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 10070 . . . . . . . . . 10  |-  2  e.  CC
2 pire 20372 . . . . . . . . . . 11  |-  pi  e.  RR
32recni 9102 . . . . . . . . . 10  |-  pi  e.  CC
41, 3mulcli 9095 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  CC
5 2ne0 10083 . . . . . . . . 9  |-  2  =/=  0
64, 1, 5divreci 9759 . . . . . . . 8  |-  ( ( 2  x.  pi )  /  2 )  =  ( ( 2  x.  pi )  x.  (
1  /  2 ) )
73, 1, 5divcan3i 9760 . . . . . . . 8  |-  ( ( 2  x.  pi )  /  2 )  =  pi
86, 7eqtr3i 2458 . . . . . . 7  |-  ( ( 2  x.  pi )  x.  ( 1  / 
2 ) )  =  pi
9 ang180lem1.3 . . . . . . . . . 10  |-  N  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )
10 ang.1 . . . . . . . . . . . . . . . 16  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
11 ang180lem1.2 . . . . . . . . . . . . . . . 16  |-  T  =  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) )
1210, 11, 9ang180lem2 20652 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 2  <  N  /\  N  <  1 ) )
1312simprd 450 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  <  1 )
14 1e0p1 10410 . . . . . . . . . . . . . 14  |-  1  =  ( 0  +  1 )
1513, 14syl6breq 4251 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  <  ( 0  +  1 ) )
1610, 11, 9ang180lem1 20651 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  e.  ZZ  /\  ( T  /  _i )  e.  RR ) )
1716simpld 446 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  ZZ )
18 0z 10293 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
19 zleltp1 10326 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  ->  ( N  <_  0  <->  N  <  ( 0  +  1 ) ) )
2017, 18, 19sylancl 644 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  <_  0  <->  N  <  ( 0  +  1 ) ) )
2115, 20mpbird 224 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  <_  0 )
2221adantr 452 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  N  <_  0 )
23 zlem1lt 10327 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  <_  N  <->  ( 0  -  1 )  <  N ) )
2418, 17, 23sylancr 645 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
0  <_  N  <->  ( 0  -  1 )  < 
N ) )
25 df-neg 9294 . . . . . . . . . . . . . 14  |-  -u 1  =  ( 0  -  1 )
2625breq1i 4219 . . . . . . . . . . . . 13  |-  ( -u
1  <  N  <->  ( 0  -  1 )  < 
N )
2724, 26syl6bbr 255 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
0  <_  N  <->  -u 1  < 
N ) )
2827biimpar 472 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  0  <_  N )
2917zred 10375 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  RR )
3029adantr 452 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  N  e.  RR )
31 0re 9091 . . . . . . . . . . . 12  |-  0  e.  RR
32 letri3 9160 . . . . . . . . . . . 12  |-  ( ( N  e.  RR  /\  0  e.  RR )  ->  ( N  =  0  <-> 
( N  <_  0  /\  0  <_  N ) ) )
3330, 31, 32sylancl 644 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( N  =  0  <->  ( N  <_  0  /\  0  <_  N ) ) )
3422, 28, 33mpbir2and 889 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  N  = 
0 )
359, 34syl5eqr 2482 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( (
( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  =  0 )
36 ax-1cn 9048 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
37 simp1 957 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
38 subcl 9305 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
3936, 37, 38sylancr 645 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  e.  CC )
40 simp3 959 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
4140necomd 2687 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
42 subeq0 9327 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
4336, 37, 42sylancr 645 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =  0  <->  1  =  A ) )
4443necon3bid 2636 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =/=  0  <->  1  =/=  A ) )
4541, 44mpbird 224 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  =/=  0 )
4639, 45reccld 9783 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
4739, 45recne0d 9784 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
4846, 47logcld 20468 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
49 subcl 9305 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
5037, 36, 49sylancl 644 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  e.  CC )
51 simp2 958 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
5250, 37, 51divcld 9790 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
53 subeq0 9327 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
5437, 36, 53sylancl 644 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =  0  <->  A  =  1 ) )
5554necon3bid 2636 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =/=  0  <->  A  =/=  1 ) )
5640, 55mpbird 224 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  =/=  0 )
5750, 37, 56, 51divne0d 9806 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
5852, 57logcld 20468 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
5948, 58addcld 9107 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
60 logcl 20466 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
61603adant3 977 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
6259, 61addcld 9107 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  CC )
6311, 62syl5eqel 2520 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  CC )
64 ax-icn 9049 . . . . . . . . . . . . . 14  |-  _i  e.  CC
6564a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  e.  CC )
66 ine0 9469 . . . . . . . . . . . . . 14  |-  _i  =/=  0
6766a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  =/=  0 )
6863, 65, 67divcld 9790 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  /  _i )  e.  CC )
694a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  e.  CC )
70 pipos 20373 . . . . . . . . . . . . . . 15  |-  0  <  pi
712, 70gt0ne0ii 9563 . . . . . . . . . . . . . 14  |-  pi  =/=  0
721, 3, 5, 71mulne0i 9665 . . . . . . . . . . . . 13  |-  ( 2  x.  pi )  =/=  0
7372a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  =/=  0 )
7468, 69, 73divcld 9790 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  CC )
7574adantr 452 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( ( T  /  _i )  / 
( 2  x.  pi ) )  e.  CC )
76 1re 9090 . . . . . . . . . . . 12  |-  1  e.  RR
7776rehalfcli 10216 . . . . . . . . . . 11  |-  ( 1  /  2 )  e.  RR
7877recni 9102 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  CC
79 subeq0 9327 . . . . . . . . . 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 ) ) )
8075, 78, 79sylancl 644 . . . . . . . . 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 ) ) )
8135, 80mpbid 202 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( ( T  /  _i )  / 
( 2  x.  pi ) )  =  ( 1  /  2 ) )
8268adantr 452 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( T  /  _i )  e.  CC )
834a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( 2  x.  pi )  e.  CC )
8478a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( 1  /  2 )  e.  CC )
8572a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( 2  x.  pi )  =/=  0 )
8682, 83, 84, 85divmuld 9812 . . . . . . . 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 ) ) )
8781, 86mpbid 202 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( (
2  x.  pi )  x.  ( 1  / 
2 ) )  =  ( T  /  _i ) )
888, 87syl5reqr 2483 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( T  /  _i )  =  pi )
8963adantr 452 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  T  e.  CC )
9064a1i 11 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  _i  e.  CC )
913a1i 11 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  pi  e.  CC )
9266a1i 11 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  _i  =/=  0 )
9389, 90, 91, 92divmuld 9812 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( ( T  /  _i )  =  pi  <->  ( _i  x.  pi )  =  T
) )
9488, 93mpbid 202 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( _i  x.  pi )  =  T )
9594eqcomd 2441 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  T  =  ( _i  x.  pi ) )
9695olcd 383 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( T  =  -u ( _i  x.  pi )  \/  T  =  ( _i  x.  pi ) ) )
973, 64mulneg1i 9479 . . . . . . 7  |-  ( -u pi  x.  _i )  = 
-u ( pi  x.  _i )
983, 64mulcomi 9096 . . . . . . . 8  |-  ( pi  x.  _i )  =  ( _i  x.  pi )
9998negeqi 9299 . . . . . . 7  |-  -u (
pi  x.  _i )  =  -u ( _i  x.  pi )
10097, 99eqtri 2456 . . . . . 6  |-  ( -u pi  x.  _i )  = 
-u ( _i  x.  pi )
10178, 4mulneg1i 9479 . . . . . . . . . 10  |-  ( -u ( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u ( ( 1  /  2 )  x.  ( 2  x.  pi ) )
10236, 1, 5divcan1i 9758 . . . . . . . . . . . . 13  |-  ( ( 1  /  2 )  x.  2 )  =  1
103102oveq1i 6091 . . . . . . . . . . . 12  |-  ( ( ( 1  /  2
)  x.  2 )  x.  pi )  =  ( 1  x.  pi )
10478, 1, 3mulassi 9099 . . . . . . . . . . . 12  |-  ( ( ( 1  /  2
)  x.  2 )  x.  pi )  =  ( ( 1  / 
2 )  x.  (
2  x.  pi ) )
1053mulid2i 9093 . . . . . . . . . . . 12  |-  ( 1  x.  pi )  =  pi
106103, 104, 1053eqtr3i 2464 . . . . . . . . . . 11  |-  ( ( 1  /  2 )  x.  ( 2  x.  pi ) )  =  pi
107106negeqi 9299 . . . . . . . . . 10  |-  -u (
( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u pi
108101, 107eqtri 2456 . . . . . . . . 9  |-  ( -u ( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u pi
10936, 78negsubdii 9385 . . . . . . . . . . . . 13  |-  -u (
1  -  ( 1  /  2 ) )  =  ( -u 1  +  ( 1  / 
2 ) )
110 1mhlfehlf 10190 . . . . . . . . . . . . . 14  |-  ( 1  -  ( 1  / 
2 ) )  =  ( 1  /  2
)
111110negeqi 9299 . . . . . . . . . . . . 13  |-  -u (
1  -  ( 1  /  2 ) )  =  -u ( 1  / 
2 )
112109, 111eqtr3i 2458 . . . . . . . . . . . 12  |-  ( -u
1  +  ( 1  /  2 ) )  =  -u ( 1  / 
2 )
113 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u 1  =  N )
114113, 9syl6eq 2484 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u 1  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) ) )
115114oveq1d 6096 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( -u 1  +  ( 1  / 
2 ) )  =  ( ( ( ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  +  ( 1  /  2 ) ) )
116112, 115syl5eqr 2482 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( 1  /  2 )  =  ( ( ( ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  +  ( 1  /  2 ) ) )
117 npcan 9314 . . . . . . . . . . . . 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 ) ) )
11874, 78, 117sylancl 644 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
119118adantr 452 . . . . . . . . . . 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 ) ) )
120116, 119eqtrd 2468 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( 1  /  2 )  =  ( ( T  /  _i )  /  (
2  x.  pi ) ) )
121120oveq1d 6096 . . . . . . . . 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 ) ) )
122108, 121syl5eqr 2482 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u pi  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  x.  (
2  x.  pi ) ) )
12368, 69, 73divcan1d 9791 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  =  ( T  /  _i ) )
124123adantr 452 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( (
( T  /  _i )  /  ( 2  x.  pi ) )  x.  ( 2  x.  pi ) )  =  ( T  /  _i ) )
125122, 124eqtrd 2468 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u pi  =  ( T  /  _i ) )
126125oveq1d 6096 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( -u pi  x.  _i )  =  ( ( T  /  _i )  x.  _i )
)
127100, 126syl5eqr 2482 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( _i  x.  pi )  =  ( ( T  /  _i )  x.  _i ) )
12863, 65, 67divcan1d 9791 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  x.  _i )  =  T )
129128adantr 452 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( ( T  /  _i )  x.  _i )  =  T )
130127, 129eqtr2d 2469 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  T  =  -u ( _i  x.  pi ) )
131130orcd 382 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( T  =  -u ( _i  x.  pi )  \/  T  =  ( _i  x.  pi ) ) )
132 df-2 10058 . . . . . . . 8  |-  2  =  ( 1  +  1 )
133132negeqi 9299 . . . . . . 7  |-  -u 2  =  -u ( 1  +  1 )
134 negdi2 9359 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  1  e.  CC )  -> 
-u ( 1  +  1 )  =  (
-u 1  -  1 ) )
13536, 36, 134mp2an 654 . . . . . . 7  |-  -u (
1  +  1 )  =  ( -u 1  -  1 )
136133, 135eqtri 2456 . . . . . 6  |-  -u 2  =  ( -u 1  -  1 )
13712simpld 446 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 2  <  N )
138136, 137syl5eqbrr 4246 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  -  1 )  <  N )
139 1z 10311 . . . . . . 7  |-  1  e.  ZZ
140 znegcl 10313 . . . . . . 7  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
141139, 140ax-mp 8 . . . . . 6  |-  -u 1  e.  ZZ
142 zlem1lt 10327 . . . . . 6  |-  ( (
-u 1  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u 1  <_  N  <->  ( -u 1  -  1 )  < 
N ) )
143141, 17, 142sylancr 645 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  <_  N  <->  ( -u 1  -  1 )  < 
N ) )
144138, 143mpbird 224 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 1  <_  N )
14576renegcli 9362 . . . . 5  |-  -u 1  e.  RR
146 leloe 9161 . . . . 5  |-  ( (
-u 1  e.  RR  /\  N  e.  RR )  ->  ( -u 1  <_  N  <->  ( -u 1  <  N  \/  -u 1  =  N ) ) )
147145, 29, 146sylancr 645 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  <_  N  <->  ( -u 1  <  N  \/  -u 1  =  N ) ) )
148144, 147mpbid 202 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  <  N  \/  -u 1  =  N ) )
14996, 131, 148mpjaodan 762 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  =  -u ( _i  x.  pi )  \/  T  =  ( _i  x.  pi ) ) )
150 ovex 6106 . . . 4  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  _V
15111, 150eqeltri 2506 . . 3  |-  T  e. 
_V
152151elpr 3832 . 2  |-  ( T  e.  { -u (
_i  x.  pi ) ,  ( _i  x.  pi ) }  <->  ( T  =  -u ( _i  x.  pi )  \/  T  =  ( _i  x.  pi ) ) )
153149, 152sylibr 204 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  { -u ( _i  x.  pi ) ,  ( _i  x.  pi ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    \ cdif 3317   {csn 3814   {cpr 3815   class class class wbr 4212   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991   _ici 8992    + caddc 8993    x. cmul 8995    < clt 9120    <_ cle 9121    - cmin 9291   -ucneg 9292    / cdiv 9677   2c2 10049   ZZcz 10282   Imcim 11903   picpi 12669   logclog 20452
This theorem is referenced by:  ang180lem4  20654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673  df-pi 12675  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754  df-log 20454
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