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Theorem sineq0ALT 37397
Description: A complex number whose sine is zero is an integer multiple of  pi. The Virtual Deduction form of the proof is http://us.metamath.org/other/completeusersproof/sineq0altvd.html. The Metamath form of the proof is sineq0ALT 37397. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 23555. The Virtual Deduction proof is verified by automatically transforming it into the Metamath form of the proof using completeusersproof, which is verified by the Metamath program. The proof of http://us.metamath.org/other/completeusersproof/sineq0altro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sineq0ALT  |-  ( A  e.  CC  ->  (
( sin `  A
)  =  0  <->  ( A  /  pi )  e.  ZZ ) )

Proof of Theorem sineq0ALT
StepHypRef Expression
1 pire 23492 . . . . 5  |-  pi  e.  RR
2 pipos 23494 . . . . 5  |-  0  <  pi
31, 2elrpii 11328 . . . 4  |-  pi  e.  RR+
4 2ne0 10724 . . . . . 6  |-  2  =/=  0
54a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
2  =/=  0 )
6 2cn 10702 . . . . . . 7  |-  2  e.  CC
7 2re 10701 . . . . . . . 8  |-  2  e.  RR
87a1i 11 . . . . . . 7  |-  ( 2  e.  CC  ->  2  e.  RR )
96, 8ax-mp 5 . . . . . 6  |-  2  e.  RR
109a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
2  e.  RR )
11 id 22 . . . . . 6  |-  ( A  e.  CC  ->  A  e.  CC )
1211adantr 472 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  A  e.  CC )
136a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  2  e.  CC )
1413, 11mulcld 9681 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  A )  e.  CC )
15 axicn 9592 . . . . . . . . . . . . . . 15  |-  _i  e.  CC
1615a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  _i  e.  CC )
1713, 16, 11mul12d 9860 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
2  x.  ( _i  x.  A ) )  =  ( _i  x.  ( 2  x.  A
) ) )
1816, 11mulcld 9681 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
19182timesd 10878 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
2  x.  ( _i  x.  A ) )  =  ( ( _i  x.  A )  +  ( _i  x.  A
) ) )
2017, 19eqtr3d 2507 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  A ) )  =  ( ( _i  x.  A )  +  ( _i  x.  A
) ) )
2120fveq2d 5883 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( 2  x.  A
) ) )  =  ( exp `  (
( _i  x.  A
)  +  ( _i  x.  A ) ) ) )
22 efadd 14225 . . . . . . . . . . . 12  |-  ( ( ( _i  x.  A
)  e.  CC  /\  ( _i  x.  A
)  e.  CC )  ->  ( exp `  (
( _i  x.  A
)  +  ( _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) )  x.  ( exp `  (
_i  x.  A )
) ) )
2318, 18, 22syl2anc 673 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( exp `  ( ( _i  x.  A )  +  ( _i  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( _i  x.  A
) ) ) )
2421, 23eqtrd 2505 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( 2  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( _i  x.  A
) ) ) )
2524adantr 472 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
_i  x.  ( 2  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) )  x.  ( exp `  (
_i  x.  A )
) ) )
26 sinval 14253 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
27 id 22 . . . . . . . . . . . . . . 15  |-  ( ( sin `  A )  =  0  ->  ( sin `  A )  =  0 )
2826, 27sylan9req 2526 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) )  =  0 )
29 efcl 14214 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
3018, 29syl 17 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
31 negicn 9896 . . . . . . . . . . . . . . . . . . . 20  |-  -u _i  e.  CC
3231a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  CC  ->  -u _i  e.  CC )
3332, 11mulcld 9681 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
34 efcl 14214 . . . . . . . . . . . . . . . . . 18  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
3533, 34syl 17 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
3630, 35subcld 10005 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
37 2mulicn 10859 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  _i )  e.  CC
3837a1i 11 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
2  x.  _i )  e.  CC )
39 2muline0 10860 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  _i )  =/=  0
4039a1i 11 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
2  x.  _i )  =/=  0 )
4136, 38, 40diveq0ad 10415 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  (
( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) )  =  0  <->  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  =  0 ) )
4241adantr 472 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  (
2  x.  _i ) )  =  0  <->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  =  0 ) )
4328, 42mpbid 215 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  =  0 )
4430, 35subeq0ad 10015 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  =  0  <->  ( exp `  ( _i  x.  A
) )  =  ( exp `  ( -u _i  x.  A ) ) ) )
4544adantr 472 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  =  0  <->  ( exp `  ( _i  x.  A
) )  =  ( exp `  ( -u _i  x.  A ) ) ) )
4643, 45mpbid 215 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
_i  x.  A )
)  =  ( exp `  ( -u _i  x.  A ) ) )
4746oveq2d 6324 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( _i  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( -u _i  x.  A ) ) ) )
48 efadd 14225 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  A
)  e.  CC  /\  ( -u _i  x.  A
)  e.  CC )  ->  ( exp `  (
( _i  x.  A
)  +  ( -u _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) )  x.  ( exp `  ( -u _i  x.  A ) ) ) )
4918, 33, 48syl2anc 673 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  ( exp `  ( ( _i  x.  A )  +  ( -u _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A
) )  x.  ( exp `  ( -u _i  x.  A ) ) ) )
5049adantr 472 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
( _i  x.  A
)  +  ( -u _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) )  x.  ( exp `  ( -u _i  x.  A ) ) ) )
5147, 50eqtr4d 2508 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( _i  x.  A
) ) )  =  ( exp `  (
( _i  x.  A
)  +  ( -u _i  x.  A ) ) ) )
5215negidi 9963 . . . . . . . . . . . . . . 15  |-  ( _i  +  -u _i )  =  0
5352oveq1i 6318 . . . . . . . . . . . . . 14  |-  ( ( _i  +  -u _i )  x.  A )  =  ( 0  x.  A )
5416, 32, 11adddird 9686 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  +  -u _i )  x.  A
)  =  ( ( _i  x.  A )  +  ( -u _i  x.  A ) ) )
5553, 54syl5reqr 2520 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( -u _i  x.  A ) )  =  ( 0  x.  A ) )
5611mul02d 9849 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
0  x.  A )  =  0 )
5755, 56eqtrd 2505 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( -u _i  x.  A ) )  =  0 )
5857fveq2d 5883 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( exp `  ( ( _i  x.  A )  +  ( -u _i  x.  A ) ) )  =  ( exp `  0
) )
5958adantr 472 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
( _i  x.  A
)  +  ( -u _i  x.  A ) ) )  =  ( exp `  0 ) )
60 ef0 14222 . . . . . . . . . . 11  |-  ( exp `  0 )  =  1
6160a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  0
)  =  1 )
6251, 59, 613eqtrd 2509 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( _i  x.  A
) ) )  =  1 )
6325, 62eqtrd 2505 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
_i  x.  ( 2  x.  A ) ) )  =  1 )
6463fveq2d 5883 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( exp `  ( _i  x.  ( 2  x.  A
) ) ) )  =  ( abs `  1
) )
65 abs1 13437 . . . . . . 7  |-  ( abs `  1 )  =  1
6664, 65syl6eq 2521 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( exp `  ( _i  x.  ( 2  x.  A
) ) ) )  =  1 )
67 absefib 14329 . . . . . . . 8  |-  ( ( 2  x.  A )  e.  CC  ->  (
( 2  x.  A
)  e.  RR  <->  ( abs `  ( exp `  (
_i  x.  ( 2  x.  A ) ) ) )  =  1 ) )
6867biimparc 495 . . . . . . 7  |-  ( ( ( abs `  ( exp `  ( _i  x.  ( 2  x.  A
) ) ) )  =  1  /\  (
2  x.  A )  e.  CC )  -> 
( 2  x.  A
)  e.  RR )
6968ancoms 460 . . . . . 6  |-  ( ( ( 2  x.  A
)  e.  CC  /\  ( abs `  ( exp `  ( _i  x.  (
2  x.  A ) ) ) )  =  1 )  ->  (
2  x.  A )  e.  RR )
7014, 66, 69syl2an2r 849 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( 2  x.  A
)  e.  RR )
71 mulre 13261 . . . . . . 7  |-  ( ( A  e.  CC  /\  2  e.  RR  /\  2  =/=  0 )  ->  ( A  e.  RR  <->  ( 2  x.  A )  e.  RR ) )
72714animp1 36923 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  2  e.  RR )  /\  2  =/=  0 )  /\  (
2  x.  A )  e.  RR )  ->  A  e.  RR )
73724an31 36924 . . . . 5  |-  ( ( ( ( 2  =/=  0  /\  2  e.  RR )  /\  A  e.  CC )  /\  (
2  x.  A )  e.  RR )  ->  A  e.  RR )
745, 10, 12, 70, 73eel1111 37170 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  A  e.  RR )
753a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  pi  e.  RR+ )
7674, 75modcld 12135 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  e.  RR )
7776recnd 9687 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  e.  CC )
7877sincld 14261 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( sin `  ( A  mod  pi ) )  e.  CC )
791a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  pi  e.  RR )
80 0re 9661 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  RR
8180, 1, 2ltleii 9775 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <_  pi
82 gt0ne0 10100 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( pi  e.  RR  /\  0  <  pi )  ->  pi  =/=  0 )
83823adant3 1050 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( pi  e.  RR  /\  0  <  pi  /\  0  <_  pi )  ->  pi  =/=  0 )
84833com23 1237 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( pi  e.  RR  /\  0  <_  pi  /\  0  <  pi )  ->  pi  =/=  0 )
851, 81, 2, 84mp3an 1390 . . . . . . . . . . . . . . . . . . . 20  |-  pi  =/=  0
8685a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  pi  =/=  0 )
8774, 79, 86redivcld 10457 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  /  pi )  e.  RR )
8887flcld 12067 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( |_ `  ( A  /  pi ) )  e.  ZZ )
8988znegcld 11065 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( |_ `  ( A  /  pi ) )  e.  ZZ )
90 abssinper 23552 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  -u ( |_ `  ( A  /  pi ) )  e.  ZZ )  -> 
( abs `  ( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) )  =  ( abs `  ( sin `  A ) ) )
9190eqcomd 2477 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  -u ( |_ `  ( A  /  pi ) )  e.  ZZ )  -> 
( abs `  ( sin `  A ) )  =  ( abs `  ( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) ) )
9291ex 441 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( -u ( |_ `  ( A  /  pi ) )  e.  ZZ  ->  ( abs `  ( sin `  A
) )  =  ( abs `  ( sin `  ( A  +  (
-u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) ) ) )
9392adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( -u ( |_ `  ( A  /  pi ) )  e.  ZZ  ->  ( abs `  ( sin `  A ) )  =  ( abs `  ( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) ) ) )
9489, 93mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  A ) )  =  ( abs `  ( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) ) )
9588zcnd 11064 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( |_ `  ( A  /  pi ) )  e.  CC )
9695negcld 9992 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( |_ `  ( A  /  pi ) )  e.  CC )
971recni 9673 . . . . . . . . . . . . . . . . . . . . 21  |-  pi  e.  CC
9897a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  pi  e.  CC )
9996, 98mulcld 9681 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( -u ( |_ `  ( A  /  pi ) )  x.  pi )  e.  CC )
10098, 95mulcld 9681 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( pi  x.  ( |_ `  ( A  /  pi ) ) )  e.  CC )
101100negcld 9992 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) )  e.  CC )
10295, 98mulneg1d 10092 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( -u ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( ( |_ `  ( A  /  pi ) )  x.  pi ) )
10395, 98mulcomd 9682 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( |_ `  ( A  /  pi ) )  x.  pi )  =  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )
104103negeqd 9889 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )
105102, 104eqtrd 2505 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( -u ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )
106 oveq2 6316 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
-u ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) )  -> 
( A  +  (
-u ( |_ `  ( A  /  pi ) )  x.  pi ) )  =  ( A  +  -u (
pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
107106ad3antrrr 744 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( -u ( |_ `  ( A  /  pi ) )  x.  pi )  = 
-u ( pi  x.  ( |_ `  ( A  /  pi ) ) )  /\  -u (
pi  x.  ( |_ `  ( A  /  pi ) ) )  e.  CC )  /\  ( -u ( |_ `  ( A  /  pi ) )  x.  pi )  e.  CC )  /\  A  e.  CC )  ->  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) )  =  ( A  +  -u (
pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
1081074an4132 36925 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  CC  /\  ( -u ( |_ `  ( A  /  pi ) )  x.  pi )  e.  CC )  /\  -u (
pi  x.  ( |_ `  ( A  /  pi ) ) )  e.  CC )  /\  ( -u ( |_ `  ( A  /  pi ) )  x.  pi )  = 
-u ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )  ->  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) )  =  ( A  +  -u (
pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
10912, 99, 101, 105, 108eel1111 37170 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  +  (
-u ( |_ `  ( A  /  pi ) )  x.  pi ) )  =  ( A  +  -u (
pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
11012, 100negsubd 10011 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  +  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )  =  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
111109, 110eqtrd 2505 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  +  (
-u ( |_ `  ( A  /  pi ) )  x.  pi ) )  =  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
112111fveq2d 5883 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) )  =  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) )
113112fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) )  =  ( abs `  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) ) )
11494, 113eqtrd 2505 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  A ) )  =  ( abs `  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) ) )
115 modval 12131 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  ( A  mod  pi )  =  ( A  -  (
pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
116115fveq2d 5883 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  ( sin `  ( A  mod  pi ) )  =  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) )
117116fveq2d 5883 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  ( abs `  ( sin `  ( A  mod  pi ) ) )  =  ( abs `  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) ) )
1183, 117mpan2 685 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR  ->  ( abs `  ( sin `  ( A  mod  pi ) ) )  =  ( abs `  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) ) )
11974, 118syl 17 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  ( A  mod  pi ) ) )  =  ( abs `  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) ) )
120114, 119eqtr4d 2508 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  A ) )  =  ( abs `  ( sin `  ( A  mod  pi ) ) ) )
12127fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( ( sin `  A )  =  0  ->  ( abs `  ( sin `  A
) )  =  ( abs `  0 ) )
122 abs0 13425 . . . . . . . . . . . . . . 15  |-  ( abs `  0 )  =  0
123121, 122syl6eq 2521 . . . . . . . . . . . . . 14  |-  ( ( sin `  A )  =  0  ->  ( abs `  ( sin `  A
) )  =  0 )
124123adantl 473 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  A ) )  =  0 )
125120, 124eqtr3d 2507 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  ( A  mod  pi ) ) )  =  0 )
12678, 125abs00d 13585 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( sin `  ( A  mod  pi ) )  =  0 )
127 notnot 297 . . . . . . . . . . . . 13  |-  ( ( sin `  ( A  mod  pi ) )  =  0  <->  -.  -.  ( sin `  ( A  mod  pi ) )  =  0 )
128127bicomi 207 . . . . . . . . . . . 12  |-  ( -. 
-.  ( sin `  ( A  mod  pi ) )  =  0  <->  ( sin `  ( A  mod  pi ) )  =  0 )
129 ltne 9748 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR  /\  0  <  ( sin `  ( A  mod  pi ) ) )  ->  ( sin `  ( A  mod  pi ) )  =/=  0
)
130129neneqd 2648 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  0  <  ( sin `  ( A  mod  pi ) ) )  ->  -.  ( sin `  ( A  mod  pi ) )  =  0 )
131130expcom 442 . . . . . . . . . . . . . 14  |-  ( 0  <  ( sin `  ( A  mod  pi ) )  ->  ( 0  e.  RR  ->  -.  ( sin `  ( A  mod  pi ) )  =  0 ) )
13280, 131mpi 20 . . . . . . . . . . . . 13  |-  ( 0  <  ( sin `  ( A  mod  pi ) )  ->  -.  ( sin `  ( A  mod  pi ) )  =  0 )
133132con3i 142 . . . . . . . . . . . 12  |-  ( -. 
-.  ( sin `  ( A  mod  pi ) )  =  0  ->  -.  0  <  ( sin `  ( A  mod  pi ) ) )
134128, 133sylbir 218 . . . . . . . . . . 11  |-  ( ( sin `  ( A  mod  pi ) )  =  0  ->  -.  0  <  ( sin `  ( A  mod  pi ) ) )
135126, 134syl 17 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  0  <  ( sin `  ( A  mod  pi ) ) )
136 sinq12gt0 23541 . . . . . . . . . 10  |-  ( ( A  mod  pi )  e.  ( 0 (,) pi )  ->  0  <  ( sin `  ( A  mod  pi ) ) )
137135, 136nsyl 125 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  ( A  mod  pi )  e.  ( 0 (,) pi ) )
13880rexri 9711 . . . . . . . . . . 11  |-  0  e.  RR*
1391rexri 9711 . . . . . . . . . . 11  |-  pi  e.  RR*
140 elioo2 11702 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  pi  e.  RR* )  ->  (
( A  mod  pi )  e.  ( 0 (,) pi )  <->  ( ( A  mod  pi )  e.  RR  /\  0  < 
( A  mod  pi )  /\  ( A  mod  pi )  <  pi ) ) )
141138, 139, 140mp2an 686 . . . . . . . . . 10  |-  ( ( A  mod  pi )  e.  ( 0 (,) pi )  <->  ( ( A  mod  pi )  e.  RR  /\  0  < 
( A  mod  pi )  /\  ( A  mod  pi )  <  pi ) )
142141notbii 303 . . . . . . . . 9  |-  ( -.  ( A  mod  pi )  e.  ( 0 (,) pi )  <->  -.  (
( A  mod  pi )  e.  RR  /\  0  <  ( A  mod  pi )  /\  ( A  mod  pi )  <  pi ) )
143137, 142sylib 201 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  ( ( A  mod  pi )  e.  RR  /\  0  <  ( A  mod  pi )  /\  ( A  mod  pi )  <  pi ) )
144 3anan12 1020 . . . . . . . . 9  |-  ( ( ( A  mod  pi )  e.  RR  /\  0  <  ( A  mod  pi )  /\  ( A  mod  pi )  <  pi )  <-> 
( 0  <  ( A  mod  pi )  /\  ( ( A  mod  pi )  e.  RR  /\  ( A  mod  pi )  <  pi ) ) )
145144notbii 303 . . . . . . . 8  |-  ( -.  ( ( A  mod  pi )  e.  RR  /\  0  <  ( A  mod  pi )  /\  ( A  mod  pi )  <  pi )  <->  -.  (
0  <  ( A  mod  pi )  /\  (
( A  mod  pi )  e.  RR  /\  ( A  mod  pi )  < 
pi ) ) )
146143, 145sylib 201 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  ( 0  <  ( A  mod  pi )  /\  ( ( A  mod  pi )  e.  RR  /\  ( A  mod  pi )  <  pi ) ) )
147 modlt 12140 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  ( A  mod  pi )  < 
pi )
148147ancoms 460 . . . . . . . . 9  |-  ( ( pi  e.  RR+  /\  A  e.  RR )  ->  ( A  mod  pi )  < 
pi )
1493, 74, 148sylancr 676 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  <  pi )
15076, 149jca 541 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( A  mod  pi )  e.  RR  /\  ( A  mod  pi )  <  pi ) )
151 not12an2impnot1 37006 . . . . . . 7  |-  ( ( -.  ( 0  < 
( A  mod  pi )  /\  ( ( A  mod  pi )  e.  RR  /\  ( A  mod  pi )  < 
pi ) )  /\  ( ( A  mod  pi )  e.  RR  /\  ( A  mod  pi )  <  pi ) )  ->  -.  0  <  ( A  mod  pi ) )
152146, 150, 151syl2anc 673 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  0  <  ( A  mod  pi ) )
153 modge0 12139 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  0  <_  ( A  mod  pi ) )
154153ancoms 460 . . . . . . . 8  |-  ( ( pi  e.  RR+  /\  A  e.  RR )  ->  0  <_  ( A  mod  pi ) )
1553, 74, 154sylancr 676 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
0  <_  ( A  mod  pi ) )
156 leloe 9738 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( A  mod  pi )  e.  RR )  -> 
( 0  <_  ( A  mod  pi )  <->  ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) ) )
157156biimp3a 1397 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( A  mod  pi )  e.  RR  /\  0  <_  ( A  mod  pi ) )  ->  (
0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) )
158157idiALT 36902 . . . . . . 7  |-  ( ( 0  e.  RR  /\  ( A  mod  pi )  e.  RR  /\  0  <_  ( A  mod  pi ) )  ->  (
0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) )
15980, 76, 155, 158mp3an2i 1395 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) )
160 pm2.53 380 . . . . . . . 8  |-  ( ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) )  ->  ( -.  0  <  ( A  mod  pi )  ->  0  =  ( A  mod  pi ) ) )
161160imp 436 . . . . . . 7  |-  ( ( ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) )  /\  -.  0  < 
( A  mod  pi ) )  ->  0  =  ( A  mod  pi ) )
162161ancoms 460 . . . . . 6  |-  ( ( -.  0  <  ( A  mod  pi )  /\  ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) )  ->  0  =  ( A  mod  pi ) )
163152, 159, 162syl2anc 673 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
0  =  ( A  mod  pi ) )
164163eqcomd 2477 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  =  0 )
165 mod0 12136 . . . . . 6  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  (
( A  mod  pi )  =  0  <->  ( A  /  pi )  e.  ZZ ) )
166165biimp3a 1397 . . . . 5  |-  ( ( A  e.  RR  /\  pi  e.  RR+  /\  ( A  mod  pi )  =  0 )  ->  ( A  /  pi )  e.  ZZ )
1671663com12 1235 . . . 4  |-  ( ( pi  e.  RR+  /\  A  e.  RR  /\  ( A  mod  pi )  =  0 )  ->  ( A  /  pi )  e.  ZZ )
1683, 74, 164, 167mp3an2i 1395 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  /  pi )  e.  ZZ )
169168ex 441 . 2  |-  ( A  e.  CC  ->  (
( sin `  A
)  =  0  -> 
( A  /  pi )  e.  ZZ )
)
17097a1i 11 . . . . . 6  |-  ( A  e.  CC  ->  pi  e.  CC )
17185a1i 11 . . . . . 6  |-  ( A  e.  CC  ->  pi  =/=  0 )
17211, 170, 171divcan1d 10406 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  pi )  x.  pi )  =  A )
173172fveq2d 5883 . . . 4  |-  ( A  e.  CC  ->  ( sin `  ( ( A  /  pi )  x.  pi ) )  =  ( sin `  A
) )
174 id 22 . . . . 5  |-  ( ( A  /  pi )  e.  ZZ  ->  ( A  /  pi )  e.  ZZ )
175 sinkpi 23553 . . . . 5  |-  ( ( A  /  pi )  e.  ZZ  ->  ( sin `  ( ( A  /  pi )  x.  pi ) )  =  0 )
176174, 175syl 17 . . . 4  |-  ( ( A  /  pi )  e.  ZZ  ->  ( sin `  ( ( A  /  pi )  x.  pi ) )  =  0 )
177173, 176sylan9req 2526 . . 3  |-  ( ( A  e.  CC  /\  ( A  /  pi )  e.  ZZ )  ->  ( sin `  A
)  =  0 )
178177ex 441 . 2  |-  ( A  e.  CC  ->  (
( A  /  pi )  e.  ZZ  ->  ( sin `  A )  =  0 ) )
179169, 178impbid 195 1  |-  ( A  e.  CC  ->  (
( sin `  A
)  =  0  <->  ( A  /  pi )  e.  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   _ici 9559    + caddc 9560    x. cmul 9562   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   -ucneg 9881    / cdiv 10291   2c2 10681   ZZcz 10961   RR+crp 11325   (,)cioo 11660   |_cfl 12059    mod cmo 12129   abscabs 13374   expce 14191   sincsin 14193   picpi 14196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-ni 9315  df-pli 9316  df-mi 9317  df-lti 9318  df-plpq 9351  df-mpq 9352  df-ltpq 9353  df-enq 9354  df-nq 9355  df-erq 9356  df-plq 9357  df-mq 9358  df-1nq 9359  df-rq 9360  df-ltnq 9361  df-np 9424  df-1p 9425  df-plp 9426  df-enr 9498  df-nr 9499  df-0r 9503  df-1r 9504  df-c 9563  df-i 9566  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901
This theorem is referenced by: (None)
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