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Theorem ang180lem1 20653
Description: Lemma for ang180 20658. Show that the "revolution number"  N is an integer, using efeq1 20433 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 pire 20374 . . . . . . . 8  |-  pi  e.  RR
21recni 9104 . . . . . . 7  |-  pi  e.  CC
3 2re 10071 . . . . . . . . . 10  |-  2  e.  RR
43, 1remulcli 9106 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  RR
54recni 9104 . . . . . . . 8  |-  ( 2  x.  pi )  e.  CC
6 2pos 10084 . . . . . . . . . 10  |-  0  <  2
7 pipos 20375 . . . . . . . . . 10  |-  0  <  pi
83, 1, 6, 7mulgt0ii 9208 . . . . . . . . 9  |-  0  <  ( 2  x.  pi )
94, 8gt0ne0ii 9565 . . . . . . . 8  |-  ( 2  x.  pi )  =/=  0
105, 9pm3.2i 443 . . . . . . 7  |-  ( ( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 )
11 ax-icn 9051 . . . . . . . 8  |-  _i  e.  CC
12 ine0 9471 . . . . . . . 8  |-  _i  =/=  0
1311, 12pm3.2i 443 . . . . . . 7  |-  ( _i  e.  CC  /\  _i  =/=  0 )
14 divcan5 9718 . . . . . . 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 ) ) )
152, 10, 13, 14mp3an 1280 . . . . . 6  |-  ( ( _i  x.  pi )  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( pi  /  (
2  x.  pi ) )
161, 7gt0ne0ii 9565 . . . . . . 7  |-  pi  =/=  0
17 recdiv 9722 . . . . . . 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, 2, 16, 17mp4an 656 . . . . . 6  |-  ( 1  /  ( ( 2  x.  pi )  /  pi ) )  =  ( pi  /  ( 2  x.  pi ) )
193recni 9104 . . . . . . . 8  |-  2  e.  CC
2019, 2, 16divcan4i 9763 . . . . . . 7  |-  ( ( 2  x.  pi )  /  pi )  =  2
2120oveq2i 6094 . . . . . 6  |-  ( 1  /  ( ( 2  x.  pi )  /  pi ) )  =  ( 1  /  2 )
2215, 18, 213eqtr2i 2464 . . . . 5  |-  ( ( _i  x.  pi )  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( 1  /  2
)
2322oveq2i 6094 . . . 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 9050 . . . . . . . . . . 11  |-  1  e.  CC
26 simp1 958 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
27 subcl 9307 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
2825, 26, 27sylancr 646 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  e.  CC )
29 simp3 960 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
3029necomd 2689 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
31 subeq0 9329 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
3225, 26, 31sylancr 646 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =  0  <->  1  =  A ) )
3332necon3bid 2638 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =/=  0  <->  1  =/=  A ) )
3430, 33mpbird 225 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  =/=  0 )
3528, 34reccld 9785 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
3628, 34recne0d 9786 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
3735, 36logcld 20470 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
38 subcl 9307 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
3926, 25, 38sylancl 645 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  e.  CC )
40 simp2 959 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
4139, 26, 40divcld 9792 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
42 subeq0 9329 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
4326, 25, 42sylancl 645 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =  0  <->  A  =  1 ) )
4443necon3bid 2638 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =/=  0  <->  A  =/=  1 ) )
4529, 44mpbird 225 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  =/=  0 )
4639, 26, 45, 40divne0d 9808 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
4741, 46logcld 20470 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
4837, 47addcld 9109 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
4926, 40logcld 20470 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
5048, 49addcld 9109 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  CC )
5124, 50syl5eqel 2522 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  CC )
5211, 2mulcli 9097 . . . . . 6  |-  ( _i  x.  pi )  e.  CC
5352a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
_i  x.  pi )  e.  CC )
5411, 5mulcli 9097 . . . . . 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 9667 . . . . . 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 9831 . . . 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 9727 . . . . . . 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 1185 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  =  ( T  /  (
_i  x.  ( 2  x.  pi ) ) ) )
6463oveq1d 6098 . . . . 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 2482 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  =  ( ( T  /  ( _i  x.  ( 2  x.  pi ) ) )  -  ( 1  /  2
) ) )
6623, 58, 653eqtr4a 2496 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  -  (
_i  x.  pi )
)  /  ( _i  x.  ( 2  x.  pi ) ) )  =  N )
67 efsub 12703 . . . . . 6  |-  ( ( T  e.  CC  /\  ( _i  x.  pi )  e.  CC )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  ( ( exp `  T )  /  ( exp `  ( _i  x.  pi ) ) ) )
6851, 52, 67sylancl 645 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  ( ( exp `  T
)  /  ( exp `  ( _i  x.  pi ) ) ) )
69 efipi 20383 . . . . . . 7  |-  ( exp `  ( _i  x.  pi ) )  =  -u
1
7069oveq2i 6094 . . . . . 6  |-  ( ( exp `  T )  /  ( exp `  (
_i  x.  pi )
) )  =  ( ( exp `  T
)  /  -u 1
)
7124fveq2i 5733 . . . . . . . . 9  |-  ( exp `  T )  =  ( exp `  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )
72 efadd 12698 . . . . . . . . . . 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 644 . . . . . . . . . 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 12698 . . . . . . . . . . . . 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 644 . . . . . . . . . . . 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 20476 . . . . . . . . . . . . . 14  |-  ( ( ( 1  /  (
1  -  A ) )  e.  CC  /\  ( 1  /  (
1  -  A ) )  =/=  0 )  ->  ( exp `  ( log `  ( 1  / 
( 1  -  A
) ) ) )  =  ( 1  / 
( 1  -  A
) ) )
7735, 36, 76syl2anc 644 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  (
1  /  ( 1  -  A ) ) ) )  =  ( 1  /  ( 1  -  A ) ) )
78 eflog 20476 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  - 
1 )  /  A
)  e.  CC  /\  ( ( A  - 
1 )  /  A
)  =/=  0 )  ->  ( exp `  ( log `  ( ( A  -  1 )  /  A ) ) )  =  ( ( A  -  1 )  /  A ) )
7941, 46, 78syl2anc 644 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  (
( A  -  1 )  /  A ) ) )  =  ( ( A  -  1 )  /  A ) )
8077, 79oveq12d 6101 . . . . . . . . . . . 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 9111 . . . . . . . . . . . . 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 9807 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  /  -u (
1  -  A ) )  =  ( 1  /  ( 1  -  A ) ) )
84 negsubdi2 9362 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  e.  CC  /\  A  e.  CC )  -> 
-u ( 1  -  A )  =  ( A  -  1 ) )
8525, 26, 84sylancr 646 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u (
1  -  A )  =  ( A  - 
1 ) )
8685oveq2d 6099 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  /  -u (
1  -  A ) )  =  ( -u
1  /  ( A  -  1 ) ) )
8783, 86eqtr3d 2472 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =  ( -u 1  /  ( A  - 
1 ) ) )
8887oveq2d 6099 . . . . . . . . . . . . 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 10069 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
9089a1i 11 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 1  e.  CC )
9190, 39, 26, 45, 40dmdcand 9821 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( A  - 
1 )  /  A
)  x.  ( -u
1  /  ( A  -  1 ) ) )  =  ( -u
1  /  A ) )
9281, 88, 913eqtrd 2474 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  /  (
1  -  A ) )  x.  ( ( A  -  1 )  /  A ) )  =  ( -u 1  /  A ) )
9375, 80, 923eqtrd 2474 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) ) )  =  (
-u 1  /  A
) )
94 eflog 20476 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
9526, 40, 94syl2anc 644 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  A
) )  =  A )
9693, 95oveq12d 6101 . . . . . . . . . 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 9793 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( -u 1  /  A
)  x.  A )  =  -u 1 )
9873, 96, 973eqtrd 2474 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  -u
1 )
9971, 98syl5eq 2482 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  T )  = 
-u 1 )
10099oveq1d 6098 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  -u 1
)  =  ( -u
1  /  -u 1
) )
101 ax-1ne0 9061 . . . . . . . . 9  |-  1  =/=  0
10225, 101negne0i 9377 . . . . . . . 8  |-  -u 1  =/=  0
10389, 102dividi 9749 . . . . . . 7  |-  ( -u
1  /  -u 1
)  =  1
104100, 103syl6eq 2486 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  -u 1
)  =  1 )
10570, 104syl5eq 2482 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  ( exp `  ( _i  x.  pi ) ) )  =  1 )
10668, 105eqtrd 2470 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  1 )
107 subcl 9307 . . . . . 6  |-  ( ( T  e.  CC  /\  ( _i  x.  pi )  e.  CC )  ->  ( T  -  (
_i  x.  pi )
)  e.  CC )
10851, 52, 107sylancl 645 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  -  ( _i  x.  pi ) )  e.  CC )
109 efeq1 20433 . . . . 5  |-  ( ( T  -  ( _i  x.  pi ) )  e.  CC  ->  (
( exp `  ( T  -  ( _i  x.  pi ) ) )  =  1  <->  ( ( T  -  ( _i  x.  pi ) )  / 
( _i  x.  (
2  x.  pi ) ) )  e.  ZZ ) )
110108, 109syl 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 ) )
111106, 110mpbid 203 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  -  (
_i  x.  pi )
)  /  ( _i  x.  ( 2  x.  pi ) ) )  e.  ZZ )
11266, 111eqeltrrd 2513 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  ZZ )
11311a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  e.  CC )
11412a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  =/=  0 )
11551, 113, 114divcld 9792 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  /  _i )  e.  CC )
1165a1i 11 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  e.  CC )
1179a1i 11 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  =/=  0 )
118115, 116, 117divcan1d 9793 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  =  ( T  /  _i ) )
11959oveq1i 6093 . . . . . 6  |-  ( N  +  ( 1  / 
2 ) )  =  ( ( ( ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  +  ( 1  /  2 ) )
120115, 116, 117divcld 9792 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  CC )
121 1re 9092 . . . . . . . . 9  |-  1  e.  RR
122121rehalfcli 10218 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
123122recni 9104 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
124 npcan 9316 . . . . . . 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 ) ) )
125120, 123, 124sylancl 645 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
126119, 125syl5eq 2482 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
127112zred 10377 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  RR )
128 readdcl 9075 . . . . . 6  |-  ( ( N  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( N  +  ( 1  /  2
) )  e.  RR )
129127, 122, 128sylancl 645 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  +  ( 1  /  2 ) )  e.  RR )
130126, 129eqeltrrd 2513 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  RR )
131 remulcl 9077 . . . 4  |-  ( ( ( ( T  /  _i )  /  (
2  x.  pi ) )  e.  RR  /\  ( 2  x.  pi )  e.  RR )  ->  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  x.  (
2  x.  pi ) )  e.  RR )
132130, 4, 131sylancl 645 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  e.  RR )
133118, 132eqeltrrd 2513 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  /  _i )  e.  RR )
134112, 133jca 520 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  e.  ZZ  /\  ( T  /  _i )  e.  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601    \ cdif 3319   {csn 3816   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993   _ici 8994    + caddc 8995    x. cmul 8997    - cmin 9293   -ucneg 9294    / cdiv 9679   2c2 10051   ZZcz 10284   Imcim 11905   expce 12666   picpi 12671   logclog 20454
This theorem is referenced by:  ang180lem2  20654  ang180lem3  20655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ioc 10923  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482  df-ef 12672  df-sin 12674  df-cos 12675  df-pi 12677  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756  df-log 20456
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