Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sineq0ALT Structured version   Unicode version

Theorem sineq0ALT 31490
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 31490. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 21926. 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 21864 . . . . 5  |-  pi  e.  RR
2 pipos 21866 . . . . 5  |-  0  <  pi
31, 2elrpii 10990 . . . 4  |-  pi  e.  RR+
4 2ne0 10410 . . . . . 6  |-  2  =/=  0
54a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
2  =/=  0 )
6 2cn 10388 . . . . . . 7  |-  2  e.  CC
7 2re 10387 . . . . . . . 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 462 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  A  e.  CC )
136a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  2  e.  CC )
1413, 11mulcld 9402 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  A )  e.  CC )
15 axicn 9313 . . . . . . . . . . . . . . 15  |-  _i  e.  CC
1615a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  _i  e.  CC )
1713, 16, 11mul12d 9574 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
2  x.  ( _i  x.  A ) )  =  ( _i  x.  ( 2  x.  A
) ) )
1816, 11mulcld 9402 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
19182timesd 10563 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
2  x.  ( _i  x.  A ) )  =  ( ( _i  x.  A )  +  ( _i  x.  A
) ) )
2017, 19eqtr3d 2475 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  A ) )  =  ( ( _i  x.  A )  +  ( _i  x.  A
) ) )
2120fveq2d 5692 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( 2  x.  A
) ) )  =  ( exp `  (
( _i  x.  A
)  +  ( _i  x.  A ) ) ) )
22 efadd 13375 . . . . . . . . . . . 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 656 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( exp `  ( ( _i  x.  A )  +  ( _i  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( _i  x.  A
) ) ) )
2421, 23eqtrd 2473 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( 2  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( _i  x.  A
) ) ) )
2524adantr 462 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
_i  x.  ( 2  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) )  x.  ( exp `  (
_i  x.  A )
) ) )
26 sinval 13402 . . . . . . . . . . . . . . 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 2494 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) )  =  0 )
29 efcl 13364 . . . . . . . . . . . . . . . . . 18  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
3018, 29syl 16 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
31 negicn 9607 . . . . . . . . . . . . . . . . . . . 20  |-  -u _i  e.  CC
3231a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  CC  ->  -u _i  e.  CC )
3332, 11mulcld 9402 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
34 efcl 13364 . . . . . . . . . . . . . . . . . 18  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
3533, 34syl 16 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
3630, 35subcld 9715 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
37 2mulicn 10544 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  _i )  e.  CC
3837a1i 11 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
2  x.  _i )  e.  CC )
39 2muline0 10545 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  _i )  =/=  0
4039a1i 11 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
2  x.  _i )  =/=  0 )
4136, 38, 40diveq0ad 10113 . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  =  0 )
4430, 35subeq0ad 9725 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  =  0  <->  ( exp `  ( _i  x.  A
) )  =  ( exp `  ( -u _i  x.  A ) ) ) )
4544adantr 462 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
_i  x.  A )
)  =  ( exp `  ( -u _i  x.  A ) ) )
4746oveq2d 6106 . . . . . . . . . . 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 13375 . . . . . . . . . . . . 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 656 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  ( exp `  ( ( _i  x.  A )  +  ( -u _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A
) )  x.  ( exp `  ( -u _i  x.  A ) ) ) )
5049adantr 462 . . . . . . . . . . 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 2476 . . . . . . . . . 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 9673 . . . . . . . . . . . . . . 15  |-  ( _i  +  -u _i )  =  0
5352oveq1i 6100 . . . . . . . . . . . . . 14  |-  ( ( _i  +  -u _i )  x.  A )  =  ( 0  x.  A )
5416, 32, 11adddird 9407 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  +  -u _i )  x.  A
)  =  ( ( _i  x.  A )  +  ( -u _i  x.  A ) ) )
5553, 54syl5reqr 2488 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( -u _i  x.  A ) )  =  ( 0  x.  A ) )
5611mul02d 9563 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
0  x.  A )  =  0 )
5755, 56eqtrd 2473 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( -u _i  x.  A ) )  =  0 )
5857fveq2d 5692 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( exp `  ( ( _i  x.  A )  +  ( -u _i  x.  A ) ) )  =  ( exp `  0
) )
5958adantr 462 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
( _i  x.  A
)  +  ( -u _i  x.  A ) ) )  =  ( exp `  0 ) )
60 ef0 13372 . . . . . . . . . . 11  |-  ( exp `  0 )  =  1
6160a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  0
)  =  1 )
6251, 59, 613eqtrd 2477 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( _i  x.  A
) ) )  =  1 )
6325, 62eqtrd 2473 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
_i  x.  ( 2  x.  A ) ) )  =  1 )
6463fveq2d 5692 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( exp `  ( _i  x.  ( 2  x.  A
) ) ) )  =  ( abs `  1
) )
65 abs1 12782 . . . . . . 7  |-  ( abs `  1 )  =  1
6664, 65syl6eq 2489 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( exp `  ( _i  x.  ( 2  x.  A
) ) ) )  =  1 )
67 absefib 13478 . . . . . . . 8  |-  ( ( 2  x.  A )  e.  CC  ->  (
( 2  x.  A
)  e.  RR  <->  ( abs `  ( exp `  (
_i  x.  ( 2  x.  A ) ) ) )  =  1 ) )
6867biimparc 484 . . . . . . 7  |-  ( ( ( abs `  ( exp `  ( _i  x.  ( 2  x.  A
) ) ) )  =  1  /\  (
2  x.  A )  e.  CC )  -> 
( 2  x.  A
)  e.  RR )
6968ancoms 450 . . . . . 6  |-  ( ( ( 2  x.  A
)  e.  CC  /\  ( abs `  ( exp `  ( _i  x.  (
2  x.  A ) ) ) )  =  1 )  ->  (
2  x.  A )  e.  RR )
7014, 66, 69eel121 31258 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( 2  x.  A
)  e.  RR )
71 mulre 12606 . . . . . . 7  |-  ( ( A  e.  CC  /\  2  e.  RR  /\  2  =/=  0 )  ->  ( A  e.  RR  <->  ( 2  x.  A )  e.  RR ) )
72714animp1 31017 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  2  e.  RR )  /\  2  =/=  0 )  /\  (
2  x.  A )  e.  RR )  ->  A  e.  RR )
73724an31 31018 . . . . 5  |-  ( ( ( ( 2  =/=  0  /\  2  e.  RR )  /\  A  e.  CC )  /\  (
2  x.  A )  e.  RR )  ->  A  e.  RR )
745, 10, 12, 70, 73eel1111 31270 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  A  e.  RR )
753a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  pi  e.  RR+ )
7674, 75modcld 11710 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  e.  RR )
7776recnd 9408 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  e.  CC )
7877sincld 13410 . . . . . . . . . . . 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 9382 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  RR
8180, 1, 2ltleii 9493 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <_  pi
82 gt0ne0 9800 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( pi  e.  RR  /\  0  <  pi )  ->  pi  =/=  0 )
83823adant3 1003 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( pi  e.  RR  /\  0  <  pi  /\  0  <_  pi )  ->  pi  =/=  0 )
84833com23 1188 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( pi  e.  RR  /\  0  <_  pi  /\  0  <  pi )  ->  pi  =/=  0 )
851, 81, 2, 84mp3an 1309 . . . . . . . . . . . . . . . . . . . 20  |-  pi  =/=  0
8685a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  pi  =/=  0 )
8774, 79, 86redivcld 10155 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  /  pi )  e.  RR )
8887flcld 11644 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( |_ `  ( A  /  pi ) )  e.  ZZ )
8988znegcld 10745 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( |_ `  ( A  /  pi ) )  e.  ZZ )
90 abssinper 21923 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  -u ( |_ `  ( A  /  pi ) )  e.  ZZ )  -> 
( abs `  ( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) )  =  ( abs `  ( sin `  A ) ) )
9190eqcomd 2446 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  -u ( |_ `  ( A  /  pi ) )  e.  ZZ )  -> 
( abs `  ( sin `  A ) )  =  ( abs `  ( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) ) )
9291ex 434 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( -u ( |_ `  ( A  /  pi ) )  e.  ZZ  ->  ( abs `  ( sin `  A
) )  =  ( abs `  ( sin `  ( A  +  (
-u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) ) ) )
9392adantr 462 . . . . . . . . . . . . . . . 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 10744 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( |_ `  ( A  /  pi ) )  e.  CC )
9695negcld 9702 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( |_ `  ( A  /  pi ) )  e.  CC )
971recni 9394 . . . . . . . . . . . . . . . . . . . . 21  |-  pi  e.  CC
9897a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  pi  e.  CC )
9996, 98mulcld 9402 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( -u ( |_ `  ( A  /  pi ) )  x.  pi )  e.  CC )
10098, 95mulcld 9402 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( pi  x.  ( |_ `  ( A  /  pi ) ) )  e.  CC )
101100negcld 9702 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) )  e.  CC )
10295, 98mulneg1d 9793 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( -u ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( ( |_ `  ( A  /  pi ) )  x.  pi ) )
10395, 98mulcomd 9403 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( |_ `  ( A  /  pi ) )  x.  pi )  =  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )
104103negeqd 9600 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )
105102, 104eqtrd 2473 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( -u ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )
106 oveq2 6098 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
-u ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) )  -> 
( A  +  (
-u ( |_ `  ( A  /  pi ) )  x.  pi ) )  =  ( A  +  -u (
pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
107106ad3antrrr 724 . . . . . . . . . . . . . . . . . . . 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 31019 . . . . . . . . . . . . . . . . . . 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 31270 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  +  (
-u ( |_ `  ( A  /  pi ) )  x.  pi ) )  =  ( A  +  -u (
pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
11012, 100negsubd 9721 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  +  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )  =  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
111109, 110eqtrd 2473 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  +  (
-u ( |_ `  ( A  /  pi ) )  x.  pi ) )  =  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
112111fveq2d 5692 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) )  =  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) )
113112fveq2d 5692 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) )  =  ( abs `  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) ) )
11494, 113eqtrd 2473 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  A ) )  =  ( abs `  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) ) )
115 modval 11706 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  ( A  mod  pi )  =  ( A  -  (
pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
116115fveq2d 5692 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  ( sin `  ( A  mod  pi ) )  =  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) )
117116fveq2d 5692 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  ( abs `  ( sin `  ( A  mod  pi ) ) )  =  ( abs `  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) ) )
1183, 117mpan2 666 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR  ->  ( abs `  ( sin `  ( A  mod  pi ) ) )  =  ( abs `  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) ) )
11974, 118syl 16 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  ( A  mod  pi ) ) )  =  ( abs `  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) ) )
120114, 119eqtr4d 2476 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  A ) )  =  ( abs `  ( sin `  ( A  mod  pi ) ) ) )
12127fveq2d 5692 . . . . . . . . . . . . . . 15  |-  ( ( sin `  A )  =  0  ->  ( abs `  ( sin `  A
) )  =  ( abs `  0 ) )
122 abs0 12770 . . . . . . . . . . . . . . 15  |-  ( abs `  0 )  =  0
123121, 122syl6eq 2489 . . . . . . . . . . . . . 14  |-  ( ( sin `  A )  =  0  ->  ( abs `  ( sin `  A
) )  =  0 )
124123adantl 463 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  A ) )  =  0 )
125120, 124eqtr3d 2475 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  ( A  mod  pi ) ) )  =  0 )
12678, 125abs00d 12928 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( sin `  ( A  mod  pi ) )  =  0 )
127 notnot 291 . . . . . . . . . . . . 13  |-  ( ( sin `  ( A  mod  pi ) )  =  0  <->  -.  -.  ( sin `  ( A  mod  pi ) )  =  0 )
128127bicomi 202 . . . . . . . . . . . 12  |-  ( -. 
-.  ( sin `  ( A  mod  pi ) )  =  0  <->  ( sin `  ( A  mod  pi ) )  =  0 )
129 ltne 9467 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR  /\  0  <  ( sin `  ( A  mod  pi ) ) )  ->  ( sin `  ( A  mod  pi ) )  =/=  0
)
130129neneqd 2622 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  0  <  ( sin `  ( A  mod  pi ) ) )  ->  -.  ( sin `  ( A  mod  pi ) )  =  0 )
131130expcom 435 . . . . . . . . . . . . . 14  |-  ( 0  <  ( sin `  ( A  mod  pi ) )  ->  ( 0  e.  RR  ->  -.  ( sin `  ( A  mod  pi ) )  =  0 ) )
13280, 131mpi 17 . . . . . . . . . . . . 13  |-  ( 0  <  ( sin `  ( A  mod  pi ) )  ->  -.  ( sin `  ( A  mod  pi ) )  =  0 )
133132con3i 135 . . . . . . . . . . . 12  |-  ( -. 
-.  ( sin `  ( A  mod  pi ) )  =  0  ->  -.  0  <  ( sin `  ( A  mod  pi ) ) )
134128, 133sylbir 213 . . . . . . . . . . 11  |-  ( ( sin `  ( A  mod  pi ) )  =  0  ->  -.  0  <  ( sin `  ( A  mod  pi ) ) )
135126, 134syl 16 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  0  <  ( sin `  ( A  mod  pi ) ) )
136 sinq12gt0 21912 . . . . . . . . . 10  |-  ( ( A  mod  pi )  e.  ( 0 (,) pi )  ->  0  <  ( sin `  ( A  mod  pi ) ) )
137135, 136nsyl 121 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  ( A  mod  pi )  e.  ( 0 (,) pi ) )
13880rexri 9432 . . . . . . . . . . 11  |-  0  e.  RR*
1391rexri 9432 . . . . . . . . . . 11  |-  pi  e.  RR*
140 elioo2 11337 . . . . . . . . . . 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 667 . . . . . . . . . 10  |-  ( ( A  mod  pi )  e.  ( 0 (,) pi )  <->  ( ( A  mod  pi )  e.  RR  /\  0  < 
( A  mod  pi )  /\  ( A  mod  pi )  <  pi ) )
142141notbii 296 . . . . . . . . 9  |-  ( -.  ( A  mod  pi )  e.  ( 0 (,) pi )  <->  -.  (
( A  mod  pi )  e.  RR  /\  0  <  ( A  mod  pi )  /\  ( A  mod  pi )  <  pi ) )
143137, 142sylib 196 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  ( ( A  mod  pi )  e.  RR  /\  0  <  ( A  mod  pi )  /\  ( A  mod  pi )  <  pi ) )
144 3anan12 973 . . . . . . . . 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 296 . . . . . . . 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 196 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  ( 0  <  ( A  mod  pi )  /\  ( ( A  mod  pi )  e.  RR  /\  ( A  mod  pi )  <  pi ) ) )
147 modlt 11714 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  ( A  mod  pi )  < 
pi )
148147ancoms 450 . . . . . . . . 9  |-  ( ( pi  e.  RR+  /\  A  e.  RR )  ->  ( A  mod  pi )  < 
pi )
1493, 74, 148sylancr 658 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  <  pi )
15076, 149jca 529 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( A  mod  pi )  e.  RR  /\  ( A  mod  pi )  <  pi ) )
151 not12an2impnot1 31097 . . . . . . 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 656 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  0  <  ( A  mod  pi ) )
153 modge0 11713 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  0  <_  ( A  mod  pi ) )
154153ancoms 450 . . . . . . . 8  |-  ( ( pi  e.  RR+  /\  A  e.  RR )  ->  0  <_  ( A  mod  pi ) )
1553, 74, 154sylancr 658 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
0  <_  ( A  mod  pi ) )
156 leloe 9457 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( A  mod  pi )  e.  RR )  -> 
( 0  <_  ( A  mod  pi )  <->  ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) ) )
157156biimp3a 1313 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( A  mod  pi )  e.  RR  /\  0  <_  ( A  mod  pi ) )  ->  (
0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) )
158157idi 2 . . . . . . 7  |-  ( ( 0  e.  RR  /\  ( A  mod  pi )  e.  RR  /\  0  <_  ( A  mod  pi ) )  ->  (
0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) )
15980, 76, 155, 158eel011 31248 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) )
160 pm2.53 373 . . . . . . . 8  |-  ( ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) )  ->  ( -.  0  <  ( A  mod  pi )  ->  0  =  ( A  mod  pi ) ) )
161160imp 429 . . . . . . 7  |-  ( ( ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) )  /\  -.  0  < 
( A  mod  pi ) )  ->  0  =  ( A  mod  pi ) )
162161ancoms 450 . . . . . 6  |-  ( ( -.  0  <  ( A  mod  pi )  /\  ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) )  ->  0  =  ( A  mod  pi ) )
163152, 159, 162syl2anc 656 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
0  =  ( A  mod  pi ) )
164163eqcomd 2446 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  =  0 )
165 mod0 11711 . . . . . 6  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  (
( A  mod  pi )  =  0  <->  ( A  /  pi )  e.  ZZ ) )
166165biimp3a 1313 . . . . 5  |-  ( ( A  e.  RR  /\  pi  e.  RR+  /\  ( A  mod  pi )  =  0 )  ->  ( A  /  pi )  e.  ZZ )
1671663com12 1186 . . . 4  |-  ( ( pi  e.  RR+  /\  A  e.  RR  /\  ( A  mod  pi )  =  0 )  ->  ( A  /  pi )  e.  ZZ )
1683, 74, 164, 167eel011 31248 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  /  pi )  e.  ZZ )
169168ex 434 . 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 10104 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  pi )  x.  pi )  =  A )
173172fveq2d 5692 . . . 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 21924 . . . . 5  |-  ( ( A  /  pi )  e.  ZZ  ->  ( sin `  ( ( A  /  pi )  x.  pi ) )  =  0 )
176174, 175syl 16 . . . 4  |-  ( ( A  /  pi )  e.  ZZ  ->  ( sin `  ( ( A  /  pi )  x.  pi ) )  =  0 )
177173, 176sylan9req 2494 . . 3  |-  ( ( A  e.  CC  /\  ( A  /  pi )  e.  ZZ )  ->  ( sin `  A
)  =  0 )
178177ex 434 . 2  |-  ( A  e.  CC  ->  (
( A  /  pi )  e.  ZZ  ->  ( sin `  A )  =  0 ) )
179169, 178impbid 191 1  |-  ( A  e.  CC  ->  (
( sin `  A
)  =  0  <->  ( A  /  pi )  e.  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279   _ici 9280    + caddc 9281    x. cmul 9283   RR*cxr 9413    < clt 9414    <_ cle 9415    - cmin 9591   -ucneg 9592    / cdiv 9989   2c2 10367   ZZcz 10642   RR+crp 10987   (,)cioo 11296   |_cfl 11636    mod cmo 11704   abscabs 12719   expce 13343   sincsin 13345   picpi 13348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-omul 6921  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-ni 9037  df-pli 9038  df-mi 9039  df-lti 9040  df-plpq 9073  df-mpq 9074  df-ltpq 9075  df-enq 9076  df-nq 9077  df-erq 9078  df-plq 9079  df-mq 9080  df-1nq 9081  df-rq 9082  df-ltnq 9083  df-np 9146  df-1p 9147  df-plp 9148  df-enr 9222  df-nr 9223  df-0r 9227  df-1r 9228  df-c 9284  df-i 9287  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-lp 18640  df-perf 18641  df-cn 18731  df-cnp 18732  df-haus 18819  df-tx 19035  df-hmeo 19228  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-xms 19795  df-ms 19796  df-tms 19797  df-cncf 20354  df-limc 21241  df-dv 21242
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator