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Theorem ang180lem1 20107
Description: Lemma for ang180 20112. Show that the "revolution number"  N is an integer, using efeq1 19891 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 19832 . . . . . . . 8  |-  pi  e.  RR
21recni 8849 . . . . . . 7  |-  pi  e.  CC
3 2re 9815 . . . . . . . . . 10  |-  2  e.  RR
43, 1remulcli 8851 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  RR
54recni 8849 . . . . . . . 8  |-  ( 2  x.  pi )  e.  CC
6 2pos 9828 . . . . . . . . . 10  |-  0  <  2
7 pipos 19833 . . . . . . . . . 10  |-  0  <  pi
83, 1, 6, 7mulgt0ii 8952 . . . . . . . . 9  |-  0  <  ( 2  x.  pi )
94, 8gt0ne0ii 9309 . . . . . . . 8  |-  ( 2  x.  pi )  =/=  0
105, 9pm3.2i 441 . . . . . . 7  |-  ( ( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 )
11 ax-icn 8796 . . . . . . . 8  |-  _i  e.  CC
12 ine0 9215 . . . . . . . 8  |-  _i  =/=  0
1311, 12pm3.2i 441 . . . . . . 7  |-  ( _i  e.  CC  /\  _i  =/=  0 )
14 divcan5 9462 . . . . . . 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 1277 . . . . . 6  |-  ( ( _i  x.  pi )  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( pi  /  (
2  x.  pi ) )
161, 7gt0ne0ii 9309 . . . . . . 7  |-  pi  =/=  0
17 recdiv 9466 . . . . . . 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 654 . . . . . 6  |-  ( 1  /  ( ( 2  x.  pi )  /  pi ) )  =  ( pi  /  ( 2  x.  pi ) )
193recni 8849 . . . . . . . 8  |-  2  e.  CC
2019, 2, 16divcan4i 9507 . . . . . . 7  |-  ( ( 2  x.  pi )  /  pi )  =  2
2120oveq2i 5869 . . . . . 6  |-  ( 1  /  ( ( 2  x.  pi )  /  pi ) )  =  ( 1  /  2 )
2215, 18, 213eqtr2i 2309 . . . . 5  |-  ( ( _i  x.  pi )  /  ( _i  x.  ( 2  x.  pi ) ) )  =  ( 1  /  2
)
2322oveq2i 5869 . . . 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 8795 . . . . . . . . . . 11  |-  1  e.  CC
26 simp1 955 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  e.  CC )
27 subcl 9051 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
2825, 26, 27sylancr 644 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  e.  CC )
29 simp3 957 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  1 )
3029necomd 2529 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  =/=  A )
31 subeq0 9073 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
3225, 26, 31sylancr 644 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =  0  <->  1  =  A ) )
3332necon3bid 2481 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  -  A
)  =/=  0  <->  1  =/=  A ) )
3430, 33mpbird 223 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  -  A )  =/=  0 )
3528, 34reccld 9529 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  e.  CC )
3628, 34recne0d 9530 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =/=  0 )
37 logcl 19926 . . . . . . . . 9  |-  ( ( ( 1  /  (
1  -  A ) )  e.  CC  /\  ( 1  /  (
1  -  A ) )  =/=  0 )  ->  ( log `  (
1  /  ( 1  -  A ) ) )  e.  CC )
3835, 36, 37syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( 1  / 
( 1  -  A
) ) )  e.  CC )
39 subcl 9051 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
4026, 25, 39sylancl 643 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  e.  CC )
41 simp2 956 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  A  =/=  0 )
4240, 26, 41divcld 9536 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  e.  CC )
43 subeq0 9073 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
4426, 25, 43sylancl 643 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =  0  <->  A  =  1 ) )
4544necon3bid 2481 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  =/=  0  <->  A  =/=  1 ) )
4629, 45mpbird 223 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( A  -  1 )  =/=  0 )
4740, 26, 46, 41divne0d 9552 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( A  -  1 )  /  A )  =/=  0 )
48 logcl 19926 . . . . . . . . 9  |-  ( ( ( ( A  - 
1 )  /  A
)  e.  CC  /\  ( ( A  - 
1 )  /  A
)  =/=  0 )  ->  ( log `  (
( A  -  1 )  /  A ) )  e.  CC )
4942, 47, 48syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  ( ( A  -  1 )  /  A ) )  e.  CC )
5038, 49addcld 8854 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  e.  CC )
51 logcl 19926 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
5226, 41, 51syl2anc 642 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( log `  A )  e.  CC )
5350, 52addcld 8854 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) )  e.  CC )
5424, 53syl5eqel 2367 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  T  e.  CC )
5511, 2mulcli 8842 . . . . . 6  |-  ( _i  x.  pi )  e.  CC
5655a1i 10 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
_i  x.  pi )  e.  CC )
5711, 5mulcli 8842 . . . . . 6  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
5857a1i 10 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
_i  x.  ( 2  x.  pi ) )  e.  CC )
5911, 5, 12, 9mulne0i 9411 . . . . . 6  |-  ( _i  x.  ( 2  x.  pi ) )  =/=  0
6059a1i 10 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
_i  x.  ( 2  x.  pi ) )  =/=  0 )
6154, 56, 58, 60divsubdird 9575 . . . 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 ) ) ) ) )
62 ang180lem1.3 . . . . 5  |-  N  =  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )
6313a1i 10 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
_i  e.  CC  /\  _i  =/=  0 ) )
6410a1i 10 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 2  x.  pi )  e.  CC  /\  (
2  x.  pi )  =/=  0 ) )
65 divdiv1 9471 . . . . . . 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 ) ) ) )
6654, 63, 64, 65syl3anc 1182 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  =  ( T  /  (
_i  x.  ( 2  x.  pi ) ) ) )
6766oveq1d 5873 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  -  ( 1  /  2 ) )  =  ( ( T  /  ( _i  x.  ( 2  x.  pi ) ) )  -  ( 1  /  2
) ) )
6862, 67syl5eq 2327 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  =  ( ( T  /  ( _i  x.  ( 2  x.  pi ) ) )  -  ( 1  /  2
) ) )
6923, 61, 683eqtr4a 2341 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  -  (
_i  x.  pi )
)  /  ( _i  x.  ( 2  x.  pi ) ) )  =  N )
70 efsub 12380 . . . . . 6  |-  ( ( T  e.  CC  /\  ( _i  x.  pi )  e.  CC )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  ( ( exp `  T )  /  ( exp `  ( _i  x.  pi ) ) ) )
7154, 55, 70sylancl 643 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  ( ( exp `  T
)  /  ( exp `  ( _i  x.  pi ) ) ) )
72 efipi 19841 . . . . . . 7  |-  ( exp `  ( _i  x.  pi ) )  =  -u
1
7372oveq2i 5869 . . . . . 6  |-  ( ( exp `  T )  /  ( exp `  (
_i  x.  pi )
) )  =  ( ( exp `  T
)  /  -u 1
)
7424fveq2i 5528 . . . . . . . . 9  |-  ( exp `  T )  =  ( exp `  ( ( ( log `  (
1  /  ( 1  -  A ) ) )  +  ( log `  ( ( A  - 
1 )  /  A
) ) )  +  ( log `  A
) ) )
75 efadd 12375 . . . . . . . . . . 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 ) ) ) )
7650, 52, 75syl2anc 642 . . . . . . . . . 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 ) ) ) )
77 efadd 12375 . . . . . . . . . . . . 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 ) ) ) ) )
7838, 49, 77syl2anc 642 . . . . . . . . . . . 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 ) ) ) ) )
79 eflog 19933 . . . . . . . . . . . . . 14  |-  ( ( ( 1  /  (
1  -  A ) )  e.  CC  /\  ( 1  /  (
1  -  A ) )  =/=  0 )  ->  ( exp `  ( log `  ( 1  / 
( 1  -  A
) ) ) )  =  ( 1  / 
( 1  -  A
) ) )
8035, 36, 79syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  (
1  /  ( 1  -  A ) ) ) )  =  ( 1  /  ( 1  -  A ) ) )
81 eflog 19933 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  - 
1 )  /  A
)  e.  CC  /\  ( ( A  - 
1 )  /  A
)  =/=  0 )  ->  ( exp `  ( log `  ( ( A  -  1 )  /  A ) ) )  =  ( ( A  -  1 )  /  A ) )
8242, 47, 81syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  (
( A  -  1 )  /  A ) ) )  =  ( ( A  -  1 )  /  A ) )
8380, 82oveq12d 5876 . . . . . . . . . . . 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 ) ) )
8435, 42mulcomd 8856 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  /  (
1  -  A ) )  x.  ( ( A  -  1 )  /  A ) )  =  ( ( ( A  -  1 )  /  A )  x.  ( 1  /  (
1  -  A ) ) ) )
8525a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  1  e.  CC )
8685, 28, 34div2negd 9551 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  /  -u (
1  -  A ) )  =  ( 1  /  ( 1  -  A ) ) )
87 negsubdi2 9106 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  e.  CC  /\  A  e.  CC )  -> 
-u ( 1  -  A )  =  ( A  -  1 ) )
8825, 26, 87sylancr 644 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u (
1  -  A )  =  ( A  - 
1 ) )
8988oveq2d 5874 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( -u 1  /  -u (
1  -  A ) )  =  ( -u
1  /  ( A  -  1 ) ) )
9086, 89eqtr3d 2317 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
1  /  ( 1  -  A ) )  =  ( -u 1  /  ( A  - 
1 ) ) )
9190oveq2d 5874 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( A  - 
1 )  /  A
)  x.  ( 1  /  ( 1  -  A ) ) )  =  ( ( ( A  -  1 )  /  A )  x.  ( -u 1  / 
( A  -  1 ) ) ) )
92 neg1cn 9813 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
9392a1i 10 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  -u 1  e.  CC )
9493, 40, 26, 46, 41dmdcand 9565 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( A  - 
1 )  /  A
)  x.  ( -u
1  /  ( A  -  1 ) ) )  =  ( -u
1  /  A ) )
9584, 91, 943eqtrd 2319 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( 1  /  (
1  -  A ) )  x.  ( ( A  -  1 )  /  A ) )  =  ( -u 1  /  A ) )
9678, 83, 953eqtrd 2319 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( log `  ( 1  /  (
1  -  A ) ) )  +  ( log `  ( ( A  -  1 )  /  A ) ) ) )  =  (
-u 1  /  A
) )
97 eflog 19933 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
9826, 41, 97syl2anc 642 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( log `  A
) )  =  A )
9996, 98oveq12d 5876 . . . . . . . . . 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 ) )
10093, 26, 41divcan1d 9537 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( -u 1  /  A
)  x.  A )  =  -u 1 )
10176, 99, 1003eqtrd 2319 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( ( ( log `  ( 1  /  ( 1  -  A ) ) )  +  ( log `  (
( A  -  1 )  /  A ) ) )  +  ( log `  A ) ) )  =  -u
1 )
10274, 101syl5eq 2327 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  T )  = 
-u 1 )
103102oveq1d 5873 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  -u 1
)  =  ( -u
1  /  -u 1
) )
104 ax-1ne0 8806 . . . . . . . . 9  |-  1  =/=  0
10525, 104negne0i 9121 . . . . . . . 8  |-  -u 1  =/=  0
10692, 105dividi 9493 . . . . . . 7  |-  ( -u
1  /  -u 1
)  =  1
107103, 106syl6eq 2331 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  -u 1
)  =  1 )
10873, 107syl5eq 2327 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( exp `  T
)  /  ( exp `  ( _i  x.  pi ) ) )  =  1 )
10971, 108eqtrd 2315 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( exp `  ( T  -  ( _i  x.  pi ) ) )  =  1 )
110 subcl 9051 . . . . . 6  |-  ( ( T  e.  CC  /\  ( _i  x.  pi )  e.  CC )  ->  ( T  -  (
_i  x.  pi )
)  e.  CC )
11154, 55, 110sylancl 643 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  -  ( _i  x.  pi ) )  e.  CC )
112 efeq1 19891 . . . . 5  |-  ( ( T  -  ( _i  x.  pi ) )  e.  CC  ->  (
( exp `  ( T  -  ( _i  x.  pi ) ) )  =  1  <->  ( ( T  -  ( _i  x.  pi ) )  / 
( _i  x.  (
2  x.  pi ) ) )  e.  ZZ ) )
113111, 112syl 15 . . . 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 ) )
114109, 113mpbid 201 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  -  (
_i  x.  pi )
)  /  ( _i  x.  ( 2  x.  pi ) ) )  e.  ZZ )
11569, 114eqeltrrd 2358 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  ZZ )
11611a1i 10 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  e.  CC )
11712a1i 10 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  _i  =/=  0 )
11854, 116, 117divcld 9536 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  /  _i )  e.  CC )
1195a1i 10 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  e.  CC )
1209a1i 10 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
2  x.  pi )  =/=  0 )
121118, 119, 120divcan1d 9537 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  =  ( T  /  _i ) )
12262oveq1i 5868 . . . . . 6  |-  ( N  +  ( 1  / 
2 ) )  =  ( ( ( ( T  /  _i )  /  ( 2  x.  pi ) )  -  ( 1  /  2
) )  +  ( 1  /  2 ) )
123118, 119, 120divcld 9536 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  CC )
124 1re 8837 . . . . . . . . 9  |-  1  e.  RR
125 rehalfcl 9938 . . . . . . . . 9  |-  ( 1  e.  RR  ->  (
1  /  2 )  e.  RR )
126124, 125ax-mp 8 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
127126recni 8849 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
128 npcan 9060 . . . . . . 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 ) ) )
129123, 127, 128sylancl 643 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( ( T  /  _i )  / 
( 2  x.  pi ) )  -  (
1  /  2 ) )  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
130122, 129syl5eq 2327 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  +  ( 1  /  2 ) )  =  ( ( T  /  _i )  / 
( 2  x.  pi ) ) )
131115zred 10117 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  N  e.  RR )
132 readdcl 8820 . . . . . 6  |-  ( ( N  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( N  +  ( 1  /  2
) )  e.  RR )
133131, 126, 132sylancl 643 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( N  +  ( 1  /  2 ) )  e.  RR )
134130, 133eqeltrrd 2358 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( T  /  _i )  /  ( 2  x.  pi ) )  e.  RR )
135 remulcl 8822 . . . 4  |-  ( ( ( ( T  /  _i )  /  (
2  x.  pi ) )  e.  RR  /\  ( 2  x.  pi )  e.  RR )  ->  ( ( ( T  /  _i )  / 
( 2  x.  pi ) )  x.  (
2  x.  pi ) )  e.  RR )
136134, 4, 135sylancl 643 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  (
( ( T  /  _i )  /  (
2  x.  pi ) )  x.  ( 2  x.  pi ) )  e.  RR )
137121, 136eqeltrrd 2358 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  A  =/=  1 )  ->  ( T  /  _i )  e.  RR )
138115, 137jca 518 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   ZZcz 10024   Imcim 11583   expce 12343   picpi 12348   logclog 19912
This theorem is referenced by:  ang180lem2  20108  ang180lem3  20109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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