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Theorem ang180lem1 22885
Description: Lemma for ang180 22890. Show that the "revolution number"  N is an integer, using efeq1 22665 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 22602 . . . . . . 7  |-  pi  e.  CC
2 2re 10604 . . . . . . . . . 10  |-  2  e.  RR
3 pire 22601 . . . . . . . . . 10  |-  pi  e.  RR
42, 3remulcli 9609 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  RR
54recni 9607 . . . . . . . 8  |-  ( 2  x.  pi )  e.  CC
6 2pos 10626 . . . . . . . . . 10  |-  0  <  2
7 pipos 22603 . . . . . . . . . 10  |-  0  <  pi
82, 3, 6, 7mulgt0ii 9716 . . . . . . . . 9  |-  0  <  ( 2  x.  pi )
94, 8gt0ne0ii 10088 . . . . . . . 8  |-  ( 2  x.  pi )  =/=  0
105, 9pm3.2i 455 . . . . . . 7  |-  ( ( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 )
11 ax-icn 9550 . . . . . . . 8  |-  _i  e.  CC
12 ine0 9991 . . . . . . . 8  |-  _i  =/=  0
1311, 12pm3.2i 455 . . . . . . 7  |-  ( _i  e.  CC  /\  _i  =/=  0 )
14 divcan5 10245 . . . . . . 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 1324 . . . . . 6  |-  ( ( _i  x.  pi )  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( pi  /  (
2  x.  pi ) )
163, 7gt0ne0ii 10088 . . . . . . 7  |-  pi  =/=  0
17 recdiv 10249 . . . . . . 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 673 . . . . . 6  |-  ( 1  /  ( ( 2  x.  pi )  /  pi ) )  =  ( pi  /  ( 2  x.  pi ) )
192recni 9607 . . . . . . . 8  |-  2  e.  CC
2019, 1, 16divcan4i 10290 . . . . . . 7  |-  ( ( 2  x.  pi )  /  pi )  =  2
2120oveq2i 6294 . . . . . 6  |-  ( 1  /  ( ( 2  x.  pi )  /  pi ) )  =  ( 1  /  2 )
2215, 18, 213eqtr2i 2502 . . . . 5  |-  ( ( _i  x.  pi )  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( 1  /  2
)
2322oveq2i 6294 . . . 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 9549 . . . . . . . . . . 11  |-  1  e.  CC
26 simp1 996 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
27 subcl 9818 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
2825, 26, 27sylancr 663 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  e.  CC )
29 simp3 998 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
3029necomd 2738 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
31 subeq0 9844 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
3225, 26, 31sylancr 663 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =  0  <->  1  =  A ) )
3332necon3bid 2725 . . . . . . . . . . 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 10312 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
3628, 34recne0d 10313 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
3735, 36logcld 22702 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
38 subcl 9818 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
3926, 25, 38sylancl 662 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  e.  CC )
40 simp2 997 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
4139, 26, 40divcld 10319 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
42 subeq0 9844 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
4326, 25, 42sylancl 662 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =  0  <->  A  =  1 ) )
4443necon3bid 2725 . . . . . . . . . . 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 10335 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
4741, 46logcld 22702 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
4837, 47addcld 9614 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
4926, 40logcld 22702 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
5048, 49addcld 9614 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  CC )
5124, 50syl5eqel 2559 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  CC )
5211, 1mulcli 9600 . . . . . 6  |-  ( _i  x.  pi )  e.  CC
5352a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
_i  x.  pi )  e.  CC )
5411, 5mulcli 9600 . . . . . 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 10191 . . . . . 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 10358 . . . 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 10254 . . . . . . 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 1228 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  =  ( T  /  (
_i  x.  ( 2  x.  pi ) ) ) )
6463oveq1d 6298 . . . . 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 2520 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  =  ( ( T  /  ( _i  x.  ( 2  x.  pi ) ) )  -  ( 1  /  2
) ) )
6623, 58, 653eqtr4a 2534 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  -  (
_i  x.  pi )
)  /  ( _i  x.  ( 2  x.  pi ) ) )  =  N )
67 efsub 13695 . . . . . 6  |-  ( ( T  e.  CC  /\  ( _i  x.  pi )  e.  CC )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  ( ( exp `  T )  /  ( exp `  ( _i  x.  pi ) ) ) )
6851, 52, 67sylancl 662 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  ( ( exp `  T
)  /  ( exp `  ( _i  x.  pi ) ) ) )
69 efipi 22615 . . . . . . 7  |-  ( exp `  ( _i  x.  pi ) )  =  -u
1
7069oveq2i 6294 . . . . . 6  |-  ( ( exp `  T )  /  ( exp `  (
_i  x.  pi )
) )  =  ( ( exp `  T
)  /  -u 1
)
7124fveq2i 5868 . . . . . . . . 9  |-  ( exp `  T )  =  ( exp `  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )
72 efadd 13690 . . . . . . . . . . 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 661 . . . . . . . . . 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 13690 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 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 22708 . . . . . . . . . . . . . 14  |-  ( ( ( 1  /  (
1  -  A ) )  e.  CC  /\  ( 1  /  (
1  -  A ) )  =/=  0 )  ->  ( exp `  ( log `  ( 1  / 
( 1  -  A
) ) ) )  =  ( 1  / 
( 1  -  A
) ) )
7735, 36, 76syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  (
1  /  ( 1  -  A ) ) ) )  =  ( 1  /  ( 1  -  A ) ) )
78 eflog 22708 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  - 
1 )  /  A
)  e.  CC  /\  ( ( A  - 
1 )  /  A
)  =/=  0 )  ->  ( exp `  ( log `  ( ( A  -  1 )  /  A ) ) )  =  ( ( A  -  1 )  /  A ) )
7941, 46, 78syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  (
( A  -  1 )  /  A ) ) )  =  ( ( A  -  1 )  /  A ) )
8077, 79oveq12d 6301 . . . . . . . . . . . 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 9616 . . . . . . . . . . . . 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 10334 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  /  -u (
1  -  A ) )  =  ( 1  /  ( 1  -  A ) ) )
84 negsubdi2 9877 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  e.  CC  /\  A  e.  CC )  -> 
-u ( 1  -  A )  =  ( A  -  1 ) )
8525, 26, 84sylancr 663 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u (
1  -  A )  =  ( A  - 
1 ) )
8685oveq2d 6299 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  /  -u (
1  -  A ) )  =  ( -u
1  /  ( A  -  1 ) ) )
8783, 86eqtr3d 2510 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =  ( -u 1  /  ( A  - 
1 ) ) )
8887oveq2d 6299 . . . . . . . . . . . . 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 10638 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
9089a1i 11 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 1  e.  CC )
9190, 39, 26, 45, 40dmdcand 10348 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( A  - 
1 )  /  A
)  x.  ( -u
1  /  ( A  -  1 ) ) )  =  ( -u
1  /  A ) )
9281, 88, 913eqtrd 2512 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  /  (
1  -  A ) )  x.  ( ( A  -  1 )  /  A ) )  =  ( -u 1  /  A ) )
9375, 80, 923eqtrd 2512 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) ) )  =  (
-u 1  /  A
) )
94 eflog 22708 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
9526, 40, 94syl2anc 661 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  A
) )  =  A )
9693, 95oveq12d 6301 . . . . . . . . . 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 10320 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( -u 1  /  A
)  x.  A )  =  -u 1 )
9873, 96, 973eqtrd 2512 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  -u
1 )
9971, 98syl5eq 2520 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  T )  = 
-u 1 )
10099oveq1d 6298 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  -u 1
)  =  ( -u
1  /  -u 1
) )
101 neg1ne0 10640 . . . . . . . 8  |-  -u 1  =/=  0
10289, 101dividi 10276 . . . . . . 7  |-  ( -u
1  /  -u 1
)  =  1
103100, 102syl6eq 2524 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  -u 1
)  =  1 )
10470, 103syl5eq 2520 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  ( exp `  ( _i  x.  pi ) ) )  =  1 )
10568, 104eqtrd 2508 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  1 )
106 subcl 9818 . . . . . 6  |-  ( ( T  e.  CC  /\  ( _i  x.  pi )  e.  CC )  ->  ( T  -  (
_i  x.  pi )
)  e.  CC )
10751, 52, 106sylancl 662 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  -  ( _i  x.  pi ) )  e.  CC )
108 efeq1 22665 . . . . 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 2556 . 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 10319 . . . 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 10320 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  =  ( T  /  _i ) )
11859oveq1i 6293 . . . . . 6  |-  ( N  +  ( 1  / 
2 ) )  =  ( ( ( ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  +  ( 1  /  2 ) )
119114, 115, 116divcld 10319 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  CC )
120 halfre 10753 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
121120recni 9607 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
122 npcan 9828 . . . . . . 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 662 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
124118, 123syl5eq 2520 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
125111zred 10965 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  RR )
126 readdcl 9574 . . . . . 6  |-  ( ( N  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( N  +  ( 1  /  2
) )  e.  RR )
127125, 120, 126sylancl 662 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  +  ( 1  /  2 ) )  e.  RR )
128124, 127eqeltrrd 2556 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  RR )
129 remulcl 9576 . . . 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 662 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  e.  RR )
131117, 130eqeltrrd 2556 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  /  _i )  e.  RR )
132111, 131jca 532 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473   {csn 4027   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   CCcc 9489   RRcr 9490   0cc0 9491   1c1 9492   _ici 9493    + caddc 9494    x. cmul 9496    - cmin 9804   -ucneg 9805    / cdiv 10205   2c2 10584   ZZcz 10863   Imcim 12893   expce 13658   picpi 13663   logclog 22686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ioc 11533  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134  df-fac 12321  df-bc 12348  df-hash 12373  df-shft 12862  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-limsup 13256  df-clim 13273  df-rlim 13274  df-sum 13471  df-ef 13664  df-sin 13666  df-cos 13667  df-pi 13669  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-fbas 18203  df-fg 18204  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-ntr 19303  df-cls 19304  df-nei 19381  df-lp 19419  df-perf 19420  df-cn 19510  df-cnp 19511  df-haus 19598  df-tx 19814  df-hmeo 20007  df-fil 20098  df-fm 20190  df-flim 20191  df-flf 20192  df-xms 20574  df-ms 20575  df-tms 20576  df-cncf 21133  df-limc 22021  df-dv 22022  df-log 22688
This theorem is referenced by:  ang180lem2  22886  ang180lem3  22887
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