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Theorem ang180lem3 20606
Description: Lemma for ang180 20609. Since ang180lem1 20604 shows that  N is an integer and ang180lem2 20605 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 10026 . . . . . . . . . 10  |-  2  e.  CC
2 pire 20325 . . . . . . . . . . 11  |-  pi  e.  RR
32recni 9058 . . . . . . . . . 10  |-  pi  e.  CC
41, 3mulcli 9051 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  CC
5 2ne0 10039 . . . . . . . . 9  |-  2  =/=  0
64, 1, 5divreci 9715 . . . . . . . 8  |-  ( ( 2  x.  pi )  /  2 )  =  ( ( 2  x.  pi )  x.  (
1  /  2 ) )
73, 1, 5divcan3i 9716 . . . . . . . 8  |-  ( ( 2  x.  pi )  /  2 )  =  pi
86, 7eqtr3i 2426 . . . . . . 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 20605 . . . . . . . . . . . . . . 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 10366 . . . . . . . . . . . . . 14  |-  1  =  ( 0  +  1 )
1513, 14syl6breq 4211 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  <  ( 0  +  1 ) )
1610, 11, 9ang180lem1 20604 . . . . . . . . . . . . . . 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 10249 . . . . . . . . . . . . . 14  |-  0  e.  ZZ
19 zleltp1 10282 . . . . . . . . . . . . . 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 10283 . . . . . . . . . . . . . 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 9250 . . . . . . . . . . . . . 14  |-  -u 1  =  ( 0  -  1 )
2625breq1i 4179 . . . . . . . . . . . . 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 10331 . . . . . . . . . . . . 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 9047 . . . . . . . . . . . 12  |-  0  e.  RR
32 letri3 9116 . . . . . . . . . . . 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 2450 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  <  N
)  ->  ( (
( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  =  0 )
36 ax-1cn 9004 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
37 simp1 957 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
38 subcl 9261 . . . . . . . . . . . . . . . . . . 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 2650 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
42 subeq0 9283 . . . . . . . . . . . . . . . . . . . . 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 2602 . . . . . . . . . . . . . . . . . . 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 9739 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
4739, 45recne0d 9740 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
4846, 47logcld 20421 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
49 subcl 9261 . . . . . . . . . . . . . . . . . . 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 9746 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
53 subeq0 9283 . . . . . . . . . . . . . . . . . . . . 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 2602 . . . . . . . . . . . . . . . . . . 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 9762 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
5852, 57logcld 20421 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
5948, 58addcld 9063 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
60 logcl 20419 . . . . . . . . . . . . . . . 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 9063 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  CC )
6311, 62syl5eqel 2488 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  CC )
64 ax-icn 9005 . . . . . . . . . . . . . 14  |-  _i  e.  CC
6564a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  e.  CC )
66 ine0 9425 . . . . . . . . . . . . . 14  |-  _i  =/=  0
6766a1i 11 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  =/=  0 )
6863, 65, 67divcld 9746 . . . . . . . . . . . 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 20326 . . . . . . . . . . . . . . 15  |-  0  <  pi
712, 70gt0ne0ii 9519 . . . . . . . . . . . . . 14  |-  pi  =/=  0
721, 3, 5, 71mulne0i 9621 . . . . . . . . . . . . 13  |-  ( 2  x.  pi )  =/=  0
7372a1i 11 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  =/=  0 )
7468, 69, 73divcld 9746 . . . . . . . . . . 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 9046 . . . . . . . . . . . 12  |-  1  e.  RR
7776rehalfcli 10172 . . . . . . . . . . 11  |-  ( 1  /  2 )  e.  RR
7877recni 9058 . . . . . . . . . 10  |-  ( 1  /  2 )  e.  CC
79 subeq0 9283 . . . . . . . . . 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 9768 . . . . . . . 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 2451 . . . . . 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 9768 . . . . . 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 2409 . . . 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 9435 . . . . . . 7  |-  ( -u pi  x.  _i )  = 
-u ( pi  x.  _i )
983, 64mulcomi 9052 . . . . . . . 8  |-  ( pi  x.  _i )  =  ( _i  x.  pi )
9998negeqi 9255 . . . . . . 7  |-  -u (
pi  x.  _i )  =  -u ( _i  x.  pi )
10097, 99eqtri 2424 . . . . . 6  |-  ( -u pi  x.  _i )  = 
-u ( _i  x.  pi )
10178, 4mulneg1i 9435 . . . . . . . . . 10  |-  ( -u ( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u ( ( 1  /  2 )  x.  ( 2  x.  pi ) )
10236, 1, 5divcan1i 9714 . . . . . . . . . . . . 13  |-  ( ( 1  /  2 )  x.  2 )  =  1
103102oveq1i 6050 . . . . . . . . . . . 12  |-  ( ( ( 1  /  2
)  x.  2 )  x.  pi )  =  ( 1  x.  pi )
10478, 1, 3mulassi 9055 . . . . . . . . . . . 12  |-  ( ( ( 1  /  2
)  x.  2 )  x.  pi )  =  ( ( 1  / 
2 )  x.  (
2  x.  pi ) )
1053mulid2i 9049 . . . . . . . . . . . 12  |-  ( 1  x.  pi )  =  pi
106103, 104, 1053eqtr3i 2432 . . . . . . . . . . 11  |-  ( ( 1  /  2 )  x.  ( 2  x.  pi ) )  =  pi
107106negeqi 9255 . . . . . . . . . 10  |-  -u (
( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u pi
108101, 107eqtri 2424 . . . . . . . . 9  |-  ( -u ( 1  /  2
)  x.  ( 2  x.  pi ) )  =  -u pi
10936, 78negsubdii 9341 . . . . . . . . . . . . 13  |-  -u (
1  -  ( 1  /  2 ) )  =  ( -u 1  +  ( 1  / 
2 ) )
110 1mhlfehlf 10146 . . . . . . . . . . . . . 14  |-  ( 1  -  ( 1  / 
2 ) )  =  ( 1  /  2
)
111110negeqi 9255 . . . . . . . . . . . . 13  |-  -u (
1  -  ( 1  /  2 ) )  =  -u ( 1  / 
2 )
112109, 111eqtr3i 2426 . . . . . . . . . . . 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 2452 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u 1  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) ) )
115114oveq1d 6055 . . . . . . . . . . . 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 2450 . . . . . . . . . . 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 9270 . . . . . . . . . . . . 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 2436 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( 1  /  2 )  =  ( ( T  /  _i )  /  (
2  x.  pi ) ) )
121120oveq1d 6055 . . . . . . . . 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 2450 . . . . . . . 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 9747 . . . . . . . . 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 2436 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u pi  =  ( T  /  _i ) )
126125oveq1d 6055 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  ( -u pi  x.  _i )  =  ( ( T  /  _i )  x.  _i )
)
127100, 126syl5eqr 2450 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  /\  -u 1  =  N )  ->  -u ( _i  x.  pi )  =  ( ( T  /  _i )  x.  _i ) )
12863, 65, 67divcan1d 9747 . . . . . 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 2437 . . . 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 10014 . . . . . . . 8  |-  2  =  ( 1  +  1 )
133132negeqi 9255 . . . . . . 7  |-  -u 2  =  -u ( 1  +  1 )
134 negdi2 9315 . . . . . . . 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 2424 . . . . . 6  |-  -u 2  =  ( -u 1  -  1 )
13712simpld 446 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 2  <  N )
138136, 137syl5eqbrr 4206 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  -  1 )  <  N )
139 1z 10267 . . . . . . 7  |-  1  e.  ZZ
140 znegcl 10269 . . . . . . 7  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
141139, 140ax-mp 8 . . . . . 6  |-  -u 1  e.  ZZ
142 zlem1lt 10283 . . . . . 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 9318 . . . . 5  |-  -u 1  e.  RR
146 leloe 9117 . . . . 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 6065 . . . 4  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  _V
15111, 150eqeltri 2474 . . 3  |-  T  e. 
_V
152151elpr 3792 . 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 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    \ cdif 3277   {csn 3774   {cpr 3775   class class class wbr 4172   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   _ici 8948    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   2c2 10005   ZZcz 10238   Imcim 11858   picpi 12624   logclog 20405
This theorem is referenced by:  ang180lem4  20607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407
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