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Theorem ang180lem1 22148
Description: Lemma for ang180 22153. Show that the "revolution number"  N is an integer, using efeq1 21928 to show that since the product of the three arguments  A ,  1  / 
( 1  -  A
) ,  ( A  -  1 )  /  A is  -u 1, the sum of the logarithms must be an integer multiple of  2
pi _i away from  pi _i  =  log ( -u 1 ). (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
ang180lem1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  e.  ZZ  /\  ( T  /  _i )  e.  RR ) )
Distinct variable group:    x, y, A
Allowed substitution hints:    T( x, y)    F( x, y)    N( x, y)

Proof of Theorem ang180lem1
StepHypRef Expression
1 picn 21865 . . . . . . 7  |-  pi  e.  CC
2 2re 10387 . . . . . . . . . 10  |-  2  e.  RR
3 pire 21864 . . . . . . . . . 10  |-  pi  e.  RR
42, 3remulcli 9396 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  RR
54recni 9394 . . . . . . . 8  |-  ( 2  x.  pi )  e.  CC
6 2pos 10409 . . . . . . . . . 10  |-  0  <  2
7 pipos 21866 . . . . . . . . . 10  |-  0  <  pi
82, 3, 6, 7mulgt0ii 9503 . . . . . . . . 9  |-  0  <  ( 2  x.  pi )
94, 8gt0ne0ii 9872 . . . . . . . 8  |-  ( 2  x.  pi )  =/=  0
105, 9pm3.2i 452 . . . . . . 7  |-  ( ( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 )
11 ax-icn 9337 . . . . . . . 8  |-  _i  e.  CC
12 ine0 9776 . . . . . . . 8  |-  _i  =/=  0
1311, 12pm3.2i 452 . . . . . . 7  |-  ( _i  e.  CC  /\  _i  =/=  0 )
14 divcan5 10029 . . . . . . 7  |-  ( ( pi  e.  CC  /\  ( ( 2  x.  pi )  e.  CC  /\  ( 2  x.  pi )  =/=  0 )  /\  ( _i  e.  CC  /\  _i  =/=  0 ) )  ->  ( (
_i  x.  pi )  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( pi  /  (
2  x.  pi ) ) )
151, 10, 13, 14mp3an 1309 . . . . . 6  |-  ( ( _i  x.  pi )  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( pi  /  (
2  x.  pi ) )
163, 7gt0ne0ii 9872 . . . . . . 7  |-  pi  =/=  0
17 recdiv 10033 . . . . . . 7  |-  ( ( ( ( 2  x.  pi )  e.  CC  /\  ( 2  x.  pi )  =/=  0 )  /\  ( pi  e.  CC  /\  pi  =/=  0 ) )  ->  ( 1  /  ( ( 2  x.  pi )  /  pi ) )  =  ( pi  /  ( 2  x.  pi ) ) )
185, 9, 1, 16, 17mp4an 668 . . . . . 6  |-  ( 1  /  ( ( 2  x.  pi )  /  pi ) )  =  ( pi  /  ( 2  x.  pi ) )
192recni 9394 . . . . . . . 8  |-  2  e.  CC
2019, 1, 16divcan4i 10074 . . . . . . 7  |-  ( ( 2  x.  pi )  /  pi )  =  2
2120oveq2i 6101 . . . . . 6  |-  ( 1  /  ( ( 2  x.  pi )  /  pi ) )  =  ( 1  /  2 )
2215, 18, 213eqtr2i 2467 . . . . 5  |-  ( ( _i  x.  pi )  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( 1  /  2
)
2322oveq2i 6101 . . . 4  |-  ( ( T  /  ( _i  x.  ( 2  x.  pi ) ) )  -  ( ( _i  x.  pi )  / 
( _i  x.  (
2  x.  pi ) ) ) )  =  ( ( T  / 
( _i  x.  (
2  x.  pi ) ) )  -  (
1  /  2 ) )
24 ang180lem1.2 . . . . . 6  |-  T  =  ( ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) )  +  ( log `  A ) )
25 ax-1cn 9336 . . . . . . . . . . 11  |-  1  e.  CC
26 simp1 983 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
27 subcl 9605 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
2825, 26, 27sylancr 658 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  e.  CC )
29 simp3 985 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
3029necomd 2693 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
31 subeq0 9631 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
3225, 26, 31sylancr 658 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =  0  <->  1  =  A ) )
3332necon3bid 2641 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =/=  0  <->  1  =/=  A ) )
3430, 33mpbird 232 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  =/=  0 )
3528, 34reccld 10096 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
3628, 34recne0d 10097 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
3735, 36logcld 21965 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
38 subcl 9605 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
3926, 25, 38sylancl 657 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  e.  CC )
40 simp2 984 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
4139, 26, 40divcld 10103 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
42 subeq0 9631 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
4326, 25, 42sylancl 657 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =  0  <->  A  =  1 ) )
4443necon3bid 2641 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =/=  0  <->  A  =/=  1 ) )
4529, 44mpbird 232 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  =/=  0 )
4639, 26, 45, 40divne0d 10119 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
4741, 46logcld 21965 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
4837, 47addcld 9401 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
4926, 40logcld 21965 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
5048, 49addcld 9401 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  CC )
5124, 50syl5eqel 2525 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  CC )
5211, 1mulcli 9387 . . . . . 6  |-  ( _i  x.  pi )  e.  CC
5352a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
_i  x.  pi )  e.  CC )
5411, 5mulcli 9387 . . . . . 6  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
5554a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
_i  x.  ( 2  x.  pi ) )  e.  CC )
5611, 5, 12, 9mulne0i 9975 . . . . . 6  |-  ( _i  x.  ( 2  x.  pi ) )  =/=  0
5756a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
_i  x.  ( 2  x.  pi ) )  =/=  0 )
5851, 53, 55, 57divsubdird 10142 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  -  (
_i  x.  pi )
)  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( ( T  /  ( _i  x.  ( 2  x.  pi ) ) )  -  ( ( _i  x.  pi )  /  (
_i  x.  ( 2  x.  pi ) ) ) ) )
59 ang180lem1.3 . . . . 5  |-  N  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )
6013a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
_i  e.  CC  /\  _i  =/=  0 ) )
6110a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 ) )
62 divdiv1 10038 . . . . . . 7  |-  ( ( T  e.  CC  /\  ( _i  e.  CC  /\  _i  =/=  0 )  /\  ( ( 2  x.  pi )  e.  CC  /\  ( 2  x.  pi )  =/=  0 ) )  -> 
( ( T  /  _i )  /  (
2  x.  pi ) )  =  ( T  /  ( _i  x.  ( 2  x.  pi ) ) ) )
6351, 60, 61, 62syl3anc 1213 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  =  ( T  /  (
_i  x.  ( 2  x.  pi ) ) ) )
6463oveq1d 6105 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  -  ( 1  /  2 ) )  =  ( ( T  /  ( _i  x.  ( 2  x.  pi ) ) )  -  ( 1  /  2
) ) )
6559, 64syl5eq 2485 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  =  ( ( T  /  ( _i  x.  ( 2  x.  pi ) ) )  -  ( 1  /  2
) ) )
6623, 58, 653eqtr4a 2499 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  -  (
_i  x.  pi )
)  /  ( _i  x.  ( 2  x.  pi ) ) )  =  N )
67 efsub 13380 . . . . . 6  |-  ( ( T  e.  CC  /\  ( _i  x.  pi )  e.  CC )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  ( ( exp `  T )  /  ( exp `  ( _i  x.  pi ) ) ) )
6851, 52, 67sylancl 657 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  ( ( exp `  T
)  /  ( exp `  ( _i  x.  pi ) ) ) )
69 efipi 21878 . . . . . . 7  |-  ( exp `  ( _i  x.  pi ) )  =  -u
1
7069oveq2i 6101 . . . . . 6  |-  ( ( exp `  T )  /  ( exp `  (
_i  x.  pi )
) )  =  ( ( exp `  T
)  /  -u 1
)
7124fveq2i 5691 . . . . . . . . 9  |-  ( exp `  T )  =  ( exp `  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )
72 efadd 13375 . . . . . . . . . . 11  |-  ( ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC  /\  ( log `  A )  e.  CC )  ->  ( exp `  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )  =  ( ( exp `  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) ) )  x.  ( exp `  ( log `  A ) ) ) )
7348, 49, 72syl2anc 656 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  ( ( exp `  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) ) )  x.  ( exp `  ( log `  A ) ) ) )
74 efadd 13375 . . . . . . . . . . . . 13  |-  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  e.  CC  /\  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )  -> 
( exp `  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) ) )  =  ( ( exp `  ( log `  (
1  /  ( 1  -  A ) ) ) )  x.  ( exp `  ( log `  (
( A  -  1 )  /  A ) ) ) ) )
7537, 47, 74syl2anc 656 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) ) )  =  ( ( exp `  ( log `  ( 1  / 
( 1  -  A
) ) ) )  x.  ( exp `  ( log `  ( ( A  -  1 )  /  A ) ) ) ) )
76 eflog 21971 . . . . . . . . . . . . . 14  |-  ( ( ( 1  /  (
1  -  A ) )  e.  CC  /\  ( 1  /  (
1  -  A ) )  =/=  0 )  ->  ( exp `  ( log `  ( 1  / 
( 1  -  A
) ) ) )  =  ( 1  / 
( 1  -  A
) ) )
7735, 36, 76syl2anc 656 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  (
1  /  ( 1  -  A ) ) ) )  =  ( 1  /  ( 1  -  A ) ) )
78 eflog 21971 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  - 
1 )  /  A
)  e.  CC  /\  ( ( A  - 
1 )  /  A
)  =/=  0 )  ->  ( exp `  ( log `  ( ( A  -  1 )  /  A ) ) )  =  ( ( A  -  1 )  /  A ) )
7941, 46, 78syl2anc 656 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  (
( A  -  1 )  /  A ) ) )  =  ( ( A  -  1 )  /  A ) )
8077, 79oveq12d 6108 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  ( log `  ( 1  / 
( 1  -  A
) ) ) )  x.  ( exp `  ( log `  ( ( A  -  1 )  /  A ) ) ) )  =  ( ( 1  /  ( 1  -  A ) )  x.  ( ( A  -  1 )  /  A ) ) )
8135, 41mulcomd 9403 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  /  (
1  -  A ) )  x.  ( ( A  -  1 )  /  A ) )  =  ( ( ( A  -  1 )  /  A )  x.  ( 1  /  (
1  -  A ) ) ) )
8225a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  e.  CC )
8382, 28, 34div2negd 10118 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  /  -u (
1  -  A ) )  =  ( 1  /  ( 1  -  A ) ) )
84 negsubdi2 9664 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  e.  CC  /\  A  e.  CC )  -> 
-u ( 1  -  A )  =  ( A  -  1 ) )
8525, 26, 84sylancr 658 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u (
1  -  A )  =  ( A  - 
1 ) )
8685oveq2d 6106 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  /  -u (
1  -  A ) )  =  ( -u
1  /  ( A  -  1 ) ) )
8783, 86eqtr3d 2475 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =  ( -u 1  /  ( A  - 
1 ) ) )
8887oveq2d 6106 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( A  - 
1 )  /  A
)  x.  ( 1  /  ( 1  -  A ) ) )  =  ( ( ( A  -  1 )  /  A )  x.  ( -u 1  / 
( A  -  1 ) ) ) )
89 neg1cn 10421 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
9089a1i 11 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 1  e.  CC )
9190, 39, 26, 45, 40dmdcand 10132 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( A  - 
1 )  /  A
)  x.  ( -u
1  /  ( A  -  1 ) ) )  =  ( -u
1  /  A ) )
9281, 88, 913eqtrd 2477 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  /  (
1  -  A ) )  x.  ( ( A  -  1 )  /  A ) )  =  ( -u 1  /  A ) )
9375, 80, 923eqtrd 2477 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) ) )  =  (
-u 1  /  A
) )
94 eflog 21971 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
9526, 40, 94syl2anc 656 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  A
) )  =  A )
9693, 95oveq12d 6108 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) ) )  x.  ( exp `  ( log `  A ) ) )  =  ( (
-u 1  /  A
)  x.  A ) )
9790, 26, 40divcan1d 10104 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( -u 1  /  A
)  x.  A )  =  -u 1 )
9873, 96, 973eqtrd 2477 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  -u
1 )
9971, 98syl5eq 2485 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  T )  = 
-u 1 )
10099oveq1d 6105 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  -u 1
)  =  ( -u
1  /  -u 1
) )
101 neg1ne0 10423 . . . . . . . 8  |-  -u 1  =/=  0
10289, 101dividi 10060 . . . . . . 7  |-  ( -u
1  /  -u 1
)  =  1
103100, 102syl6eq 2489 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  -u 1
)  =  1 )
10470, 103syl5eq 2485 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  ( exp `  ( _i  x.  pi ) ) )  =  1 )
10568, 104eqtrd 2473 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  1 )
106 subcl 9605 . . . . . 6  |-  ( ( T  e.  CC  /\  ( _i  x.  pi )  e.  CC )  ->  ( T  -  (
_i  x.  pi )
)  e.  CC )
10751, 52, 106sylancl 657 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  -  ( _i  x.  pi ) )  e.  CC )
108 efeq1 21928 . . . . 5  |-  ( ( T  -  ( _i  x.  pi ) )  e.  CC  ->  (
( exp `  ( T  -  ( _i  x.  pi ) ) )  =  1  <->  ( ( T  -  ( _i  x.  pi ) )  / 
( _i  x.  (
2  x.  pi ) ) )  e.  ZZ ) )
109107, 108syl 16 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  ( T  -  ( _i  x.  pi ) ) )  =  1  <->  ( ( T  -  ( _i  x.  pi ) )  / 
( _i  x.  (
2  x.  pi ) ) )  e.  ZZ ) )
110105, 109mpbid 210 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  -  (
_i  x.  pi )
)  /  ( _i  x.  ( 2  x.  pi ) ) )  e.  ZZ )
11166, 110eqeltrrd 2516 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  ZZ )
11211a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  e.  CC )
11312a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  =/=  0 )
11451, 112, 113divcld 10103 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  /  _i )  e.  CC )
1155a1i 11 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  e.  CC )
1169a1i 11 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  =/=  0 )
117114, 115, 116divcan1d 10104 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  =  ( T  /  _i ) )
11859oveq1i 6100 . . . . . 6  |-  ( N  +  ( 1  / 
2 ) )  =  ( ( ( ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  +  ( 1  /  2 ) )
119114, 115, 116divcld 10103 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  CC )
120 halfre 10536 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
121120recni 9394 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
122 npcan 9615 . . . . . . 7  |-  ( ( ( ( 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 ) ) )
123119, 121, 122sylancl 657 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
124118, 123syl5eq 2485 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
125111zred 10743 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  RR )
126 readdcl 9361 . . . . . 6  |-  ( ( N  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( N  +  ( 1  /  2
) )  e.  RR )
127125, 120, 126sylancl 657 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  +  ( 1  /  2 ) )  e.  RR )
128124, 127eqeltrrd 2516 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  RR )
129 remulcl 9363 . . . 4  |-  ( ( ( ( T  /  _i )  /  (
2  x.  pi ) )  e.  RR  /\  ( 2  x.  pi )  e.  RR )  ->  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  x.  (
2  x.  pi ) )  e.  RR )
130128, 4, 129sylancl 657 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  e.  RR )
131117, 130eqeltrrd 2516 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  /  _i )  e.  RR )
132111, 131jca 529 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  e.  ZZ  /\  ( T  /  _i )  e.  RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604    \ cdif 3322   {csn 3874   ` 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    - cmin 9591   -ucneg 9592    / cdiv 9989   2c2 10367   ZZcz 10642   Imcim 12583   expce 13343   picpi 13348   logclog 21949
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 2263  df-mo 2264  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 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-lp 18640  df-perf 18641  df-cn 18731  df-cnp 18732  df-haus 18819  df-tx 19035  df-hmeo 19228  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-xms 19795  df-ms 19796  df-tms 19797  df-cncf 20354  df-limc 21241  df-dv 21242  df-log 21951
This theorem is referenced by:  ang180lem2  22149  ang180lem3  22150
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