MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ang180lem1 Structured version   Unicode version

Theorem ang180lem1 22205
Description: Lemma for ang180 22210. Show that the "revolution number"  N is an integer, using efeq1 21985 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 21922 . . . . . . 7  |-  pi  e.  CC
2 2re 10391 . . . . . . . . . 10  |-  2  e.  RR
3 pire 21921 . . . . . . . . . 10  |-  pi  e.  RR
42, 3remulcli 9400 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  RR
54recni 9398 . . . . . . . 8  |-  ( 2  x.  pi )  e.  CC
6 2pos 10413 . . . . . . . . . 10  |-  0  <  2
7 pipos 21923 . . . . . . . . . 10  |-  0  <  pi
82, 3, 6, 7mulgt0ii 9507 . . . . . . . . 9  |-  0  <  ( 2  x.  pi )
94, 8gt0ne0ii 9876 . . . . . . . 8  |-  ( 2  x.  pi )  =/=  0
105, 9pm3.2i 455 . . . . . . 7  |-  ( ( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 )
11 ax-icn 9341 . . . . . . . 8  |-  _i  e.  CC
12 ine0 9780 . . . . . . . 8  |-  _i  =/=  0
1311, 12pm3.2i 455 . . . . . . 7  |-  ( _i  e.  CC  /\  _i  =/=  0 )
14 divcan5 10033 . . . . . . 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 1314 . . . . . 6  |-  ( ( _i  x.  pi )  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( pi  /  (
2  x.  pi ) )
163, 7gt0ne0ii 9876 . . . . . . 7  |-  pi  =/=  0
17 recdiv 10037 . . . . . . 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 9398 . . . . . . . 8  |-  2  e.  CC
2019, 1, 16divcan4i 10078 . . . . . . 7  |-  ( ( 2  x.  pi )  /  pi )  =  2
2120oveq2i 6102 . . . . . 6  |-  ( 1  /  ( ( 2  x.  pi )  /  pi ) )  =  ( 1  /  2 )
2215, 18, 213eqtr2i 2469 . . . . 5  |-  ( ( _i  x.  pi )  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( 1  /  2
)
2322oveq2i 6102 . . . 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 9340 . . . . . . . . . . 11  |-  1  e.  CC
26 simp1 988 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
27 subcl 9609 . . . . . . . . . . 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 990 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
3029necomd 2695 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
31 subeq0 9635 . . . . . . . . . . . . 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 2643 . . . . . . . . . . 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 10100 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
3628, 34recne0d 10101 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
3735, 36logcld 22022 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
38 subcl 9609 . . . . . . . . . . 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 989 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
4139, 26, 40divcld 10107 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
42 subeq0 9635 . . . . . . . . . . . . 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 2643 . . . . . . . . . . 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 10123 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
4741, 46logcld 22022 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
4837, 47addcld 9405 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
4926, 40logcld 22022 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
5048, 49addcld 9405 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  CC )
5124, 50syl5eqel 2527 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  CC )
5211, 1mulcli 9391 . . . . . 6  |-  ( _i  x.  pi )  e.  CC
5352a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
_i  x.  pi )  e.  CC )
5411, 5mulcli 9391 . . . . . 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 9979 . . . . . 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 10146 . . . 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 10042 . . . . . . 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 1218 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  =  ( T  /  (
_i  x.  ( 2  x.  pi ) ) ) )
6463oveq1d 6106 . . . . 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 2487 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  =  ( ( T  /  ( _i  x.  ( 2  x.  pi ) ) )  -  ( 1  /  2
) ) )
6623, 58, 653eqtr4a 2501 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  -  (
_i  x.  pi )
)  /  ( _i  x.  ( 2  x.  pi ) ) )  =  N )
67 efsub 13384 . . . . . 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 21935 . . . . . . 7  |-  ( exp `  ( _i  x.  pi ) )  =  -u
1
7069oveq2i 6102 . . . . . 6  |-  ( ( exp `  T )  /  ( exp `  (
_i  x.  pi )
) )  =  ( ( exp `  T
)  /  -u 1
)
7124fveq2i 5694 . . . . . . . . 9  |-  ( exp `  T )  =  ( exp `  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )
72 efadd 13379 . . . . . . . . . . 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 13379 . . . . . . . . . . . . 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 22028 . . . . . . . . . . . . . 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 22028 . . . . . . . . . . . . . 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 6109 . . . . . . . . . . . 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 9407 . . . . . . . . . . . . 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 10122 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  /  -u (
1  -  A ) )  =  ( 1  /  ( 1  -  A ) ) )
84 negsubdi2 9668 . . . . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  /  -u (
1  -  A ) )  =  ( -u
1  /  ( A  -  1 ) ) )
8783, 86eqtr3d 2477 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =  ( -u 1  /  ( A  - 
1 ) ) )
8887oveq2d 6107 . . . . . . . . . . . . 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 10425 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
9089a1i 11 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 1  e.  CC )
9190, 39, 26, 45, 40dmdcand 10136 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( A  - 
1 )  /  A
)  x.  ( -u
1  /  ( A  -  1 ) ) )  =  ( -u
1  /  A ) )
9281, 88, 913eqtrd 2479 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  /  (
1  -  A ) )  x.  ( ( A  -  1 )  /  A ) )  =  ( -u 1  /  A ) )
9375, 80, 923eqtrd 2479 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) ) )  =  (
-u 1  /  A
) )
94 eflog 22028 . . . . . . . . . . . 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 6109 . . . . . . . . . 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 10108 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( -u 1  /  A
)  x.  A )  =  -u 1 )
9873, 96, 973eqtrd 2479 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  -u
1 )
9971, 98syl5eq 2487 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  T )  = 
-u 1 )
10099oveq1d 6106 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  -u 1
)  =  ( -u
1  /  -u 1
) )
101 neg1ne0 10427 . . . . . . . 8  |-  -u 1  =/=  0
10289, 101dividi 10064 . . . . . . 7  |-  ( -u
1  /  -u 1
)  =  1
103100, 102syl6eq 2491 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  -u 1
)  =  1 )
10470, 103syl5eq 2487 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  ( exp `  ( _i  x.  pi ) ) )  =  1 )
10568, 104eqtrd 2475 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  1 )
106 subcl 9609 . . . . . 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 21985 . . . . 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 2518 . 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 10107 . . . 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 10108 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  =  ( T  /  _i ) )
11859oveq1i 6101 . . . . . 6  |-  ( N  +  ( 1  / 
2 ) )  =  ( ( ( ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  +  ( 1  /  2 ) )
119114, 115, 116divcld 10107 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  CC )
120 halfre 10540 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
121120recni 9398 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
122 npcan 9619 . . . . . . 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 2487 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
125111zred 10747 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  RR )
126 readdcl 9365 . . . . . 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 2518 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  RR )
129 remulcl 9367 . . . 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 2518 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606    \ cdif 3325   {csn 3877   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283   _ici 9284    + caddc 9285    x. cmul 9287    - cmin 9595   -ucneg 9596    / cdiv 9993   2c2 10371   ZZcz 10646   Imcim 12587   expce 13347   picpi 13352   logclog 22006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008
This theorem is referenced by:  ang180lem2  22206  ang180lem3  22207
  Copyright terms: Public domain W3C validator