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Theorem sineq0ALT 37239
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 37239. The Virtual Deduction proof is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 23468. 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 23405 . . . . 5  |-  pi  e.  RR
2 pipos 23407 . . . . 5  |-  0  <  pi
31, 2elrpii 11307 . . . 4  |-  pi  e.  RR+
4 2ne0 10704 . . . . . 6  |-  2  =/=  0
54a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
2  =/=  0 )
6 2cn 10682 . . . . . . 7  |-  2  e.  CC
7 2re 10681 . . . . . . . 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 23 . . . . . 6  |-  ( A  e.  CC  ->  A  e.  CC )
1211adantr 467 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  A  e.  CC )
136a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  2  e.  CC )
1413, 11mulcld 9665 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  A )  e.  CC )
15 axicn 9576 . . . . . . . . . . . . . . 15  |-  _i  e.  CC
1615a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  _i  e.  CC )
1713, 16, 11mul12d 9844 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
2  x.  ( _i  x.  A ) )  =  ( _i  x.  ( 2  x.  A
) ) )
1816, 11mulcld 9665 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
19182timesd 10857 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
2  x.  ( _i  x.  A ) )  =  ( ( _i  x.  A )  +  ( _i  x.  A
) ) )
2017, 19eqtr3d 2466 . . . . . . . . . . . 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 14141 . . . . . . . . . . . 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 666 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( exp `  ( ( _i  x.  A )  +  ( _i  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( _i  x.  A
) ) ) )
2421, 23eqtrd 2464 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( 2  x.  A
) ) )  =  ( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( _i  x.  A
) ) ) )
2524adantr 467 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
_i  x.  ( 2  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A ) )  x.  ( exp `  (
_i  x.  A )
) ) )
26 sinval 14169 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
27 id 23 . . . . . . . . . . . . . . 15  |-  ( ( sin `  A )  =  0  ->  ( sin `  A )  =  0 )
2826, 27sylan9req 2485 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) )  =  0 )
29 efcl 14130 . . . . . . . . . . . . . . . . . 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 9878 . . . . . . . . . . . . . . . . . . . 20  |-  -u _i  e.  CC
3231a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  CC  ->  -u _i  e.  CC )
3332, 11mulcld 9665 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
34 efcl 14130 . . . . . . . . . . . . . . . . . 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 9988 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
37 2mulicn 10838 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  _i )  e.  CC
3837a1i 11 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
2  x.  _i )  e.  CC )
39 2muline0 10839 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  _i )  =/=  0
4039a1i 11 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  ->  (
2  x.  _i )  =/=  0 )
4136, 38, 40diveq0ad 10395 . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . 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 214 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  =  0 )
4430, 35subeq0ad 9998 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  =  0  <->  ( exp `  ( _i  x.  A
) )  =  ( exp `  ( -u _i  x.  A ) ) ) )
4544adantr 467 . . . . . . . . . . . . 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 214 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
_i  x.  A )
)  =  ( exp `  ( -u _i  x.  A ) ) )
4746oveq2d 6319 . . . . . . . . . . 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 14141 . . . . . . . . . . . . 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 666 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  ( exp `  ( ( _i  x.  A )  +  ( -u _i  x.  A ) ) )  =  ( ( exp `  ( _i  x.  A
) )  x.  ( exp `  ( -u _i  x.  A ) ) ) )
5049adantr 467 . . . . . . . . . . 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 2467 . . . . . . . . . 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 9945 . . . . . . . . . . . . . . 15  |-  ( _i  +  -u _i )  =  0
5352oveq1i 6313 . . . . . . . . . . . . . 14  |-  ( ( _i  +  -u _i )  x.  A )  =  ( 0  x.  A )
5416, 32, 11adddird 9670 . . . . . . . . . . . . . 14  |-  ( A  e.  CC  ->  (
( _i  +  -u _i )  x.  A
)  =  ( ( _i  x.  A )  +  ( -u _i  x.  A ) ) )
5553, 54syl5reqr 2479 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
( _i  x.  A
)  +  ( -u _i  x.  A ) )  =  ( 0  x.  A ) )
5611mul02d 9833 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
0  x.  A )  =  0 )
5755, 56eqtrd 2464 . . . . . . . . . . . 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 467 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  (
( _i  x.  A
)  +  ( -u _i  x.  A ) ) )  =  ( exp `  0 ) )
60 ef0 14138 . . . . . . . . . . 11  |-  ( exp `  0 )  =  1
6160a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( exp `  0
)  =  1 )
6251, 59, 613eqtrd 2468 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( exp `  (
_i  x.  A )
)  x.  ( exp `  ( _i  x.  A
) ) )  =  1 )
6325, 62eqtrd 2464 . . . . . . . 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 13354 . . . . . . 7  |-  ( abs `  1 )  =  1
6664, 65syl6eq 2480 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( exp `  ( _i  x.  ( 2  x.  A
) ) ) )  =  1 )
67 absefib 14245 . . . . . . . 8  |-  ( ( 2  x.  A )  e.  CC  ->  (
( 2  x.  A
)  e.  RR  <->  ( abs `  ( exp `  (
_i  x.  ( 2  x.  A ) ) ) )  =  1 ) )
6867biimparc 490 . . . . . . 7  |-  ( ( ( abs `  ( exp `  ( _i  x.  ( 2  x.  A
) ) ) )  =  1  /\  (
2  x.  A )  e.  CC )  -> 
( 2  x.  A
)  e.  RR )
6968ancoms 455 . . . . . 6  |-  ( ( ( 2  x.  A
)  e.  CC  /\  ( abs `  ( exp `  ( _i  x.  (
2  x.  A ) ) ) )  =  1 )  ->  (
2  x.  A )  e.  RR )
7014, 66, 69eel121 37001 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( 2  x.  A
)  e.  RR )
71 mulre 13178 . . . . . . 7  |-  ( ( A  e.  CC  /\  2  e.  RR  /\  2  =/=  0 )  ->  ( A  e.  RR  <->  ( 2  x.  A )  e.  RR ) )
72714animp1 36755 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  2  e.  RR )  /\  2  =/=  0 )  /\  (
2  x.  A )  e.  RR )  ->  A  e.  RR )
73724an31 36756 . . . . 5  |-  ( ( ( ( 2  =/=  0  /\  2  e.  RR )  /\  A  e.  CC )  /\  (
2  x.  A )  e.  RR )  ->  A  e.  RR )
745, 10, 12, 70, 73eel1111 37012 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  A  e.  RR )
753a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  pi  e.  RR+ )
7674, 75modcld 12103 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  e.  RR )
7776recnd 9671 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  e.  CC )
7877sincld 14177 . . . . . . . . . . . 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 9645 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  RR
8180, 1, 2ltleii 9759 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <_  pi
82 gt0ne0 10081 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( pi  e.  RR  /\  0  <  pi )  ->  pi  =/=  0 )
83823adant3 1026 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( pi  e.  RR  /\  0  <  pi  /\  0  <_  pi )  ->  pi  =/=  0 )
84833com23 1212 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( pi  e.  RR  /\  0  <_  pi  /\  0  <  pi )  ->  pi  =/=  0 )
851, 81, 2, 84mp3an 1361 . . . . . . . . . . . . . . . . . . . 20  |-  pi  =/=  0
8685a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  pi  =/=  0 )
8774, 79, 86redivcld 10437 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  /  pi )  e.  RR )
8887flcld 12035 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( |_ `  ( A  /  pi ) )  e.  ZZ )
8988znegcld 11044 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( |_ `  ( A  /  pi ) )  e.  ZZ )
90 abssinper 23465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  -u ( |_ `  ( A  /  pi ) )  e.  ZZ )  -> 
( abs `  ( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) )  =  ( abs `  ( sin `  A ) ) )
9190eqcomd 2431 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  -u ( |_ `  ( A  /  pi ) )  e.  ZZ )  -> 
( abs `  ( sin `  A ) )  =  ( abs `  ( sin `  ( A  +  ( -u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) ) )
9291ex 436 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  CC  ->  ( -u ( |_ `  ( A  /  pi ) )  e.  ZZ  ->  ( abs `  ( sin `  A
) )  =  ( abs `  ( sin `  ( A  +  (
-u ( |_ `  ( A  /  pi ) )  x.  pi ) ) ) ) ) )
9392adantr 467 . . . . . . . . . . . . . . . 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 11043 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( |_ `  ( A  /  pi ) )  e.  CC )
9695negcld 9975 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( |_ `  ( A  /  pi ) )  e.  CC )
971recni 9657 . . . . . . . . . . . . . . . . . . . . 21  |-  pi  e.  CC
9897a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  pi  e.  CC )
9996, 98mulcld 9665 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( -u ( |_ `  ( A  /  pi ) )  x.  pi )  e.  CC )
10098, 95mulcld 9665 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( pi  x.  ( |_ `  ( A  /  pi ) ) )  e.  CC )
101100negcld 9975 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) )  e.  CC )
10295, 98mulneg1d 10073 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( -u ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( ( |_ `  ( A  /  pi ) )  x.  pi ) )
10395, 98mulcomd 9666 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( |_ `  ( A  /  pi ) )  x.  pi )  =  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )
104103negeqd 9871 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -u ( ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )
105102, 104eqtrd 2464 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( -u ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )
106 oveq2 6311 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
-u ( |_ `  ( A  /  pi ) )  x.  pi )  =  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) )  -> 
( A  +  (
-u ( |_ `  ( A  /  pi ) )  x.  pi ) )  =  ( A  +  -u (
pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
107106ad3antrrr 735 . . . . . . . . . . . . . . . . . . . 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 36757 . . . . . . . . . . . . . . . . . . 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 37012 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  +  (
-u ( |_ `  ( A  /  pi ) )  x.  pi ) )  =  ( A  +  -u (
pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
11012, 100negsubd 9994 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  +  -u ( pi  x.  ( |_ `  ( A  /  pi ) ) ) )  =  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) )
111109, 110eqtrd 2464 . . . . . . . . . . . . . . . . 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 2464 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  A ) )  =  ( abs `  ( sin `  ( A  -  ( pi  x.  ( |_ `  ( A  /  pi ) ) ) ) ) ) )
115 modval 12099 . . . . . . . . . . . . . . . . . 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 676 . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . 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 13342 . . . . . . . . . . . . . . 15  |-  ( abs `  0 )  =  0
123121, 122syl6eq 2480 . . . . . . . . . . . . . 14  |-  ( ( sin `  A )  =  0  ->  ( abs `  ( sin `  A
) )  =  0 )
124123adantl 468 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  A ) )  =  0 )
125120, 124eqtr3d 2466 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( abs `  ( sin `  ( A  mod  pi ) ) )  =  0 )
12678, 125abs00d 13501 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( sin `  ( A  mod  pi ) )  =  0 )
127 notnot 293 . . . . . . . . . . . . 13  |-  ( ( sin `  ( A  mod  pi ) )  =  0  <->  -.  -.  ( sin `  ( A  mod  pi ) )  =  0 )
128127bicomi 206 . . . . . . . . . . . 12  |-  ( -. 
-.  ( sin `  ( A  mod  pi ) )  =  0  <->  ( sin `  ( A  mod  pi ) )  =  0 )
129 ltne 9732 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR  /\  0  <  ( sin `  ( A  mod  pi ) ) )  ->  ( sin `  ( A  mod  pi ) )  =/=  0
)
130129neneqd 2626 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  0  <  ( sin `  ( A  mod  pi ) ) )  ->  -.  ( sin `  ( A  mod  pi ) )  =  0 )
131130expcom 437 . . . . . . . . . . . . . 14  |-  ( 0  <  ( sin `  ( A  mod  pi ) )  ->  ( 0  e.  RR  ->  -.  ( sin `  ( A  mod  pi ) )  =  0 ) )
13280, 131mpi 21 . . . . . . . . . . . . 13  |-  ( 0  <  ( sin `  ( A  mod  pi ) )  ->  -.  ( sin `  ( A  mod  pi ) )  =  0 )
133132con3i 141 . . . . . . . . . . . 12  |-  ( -. 
-.  ( sin `  ( A  mod  pi ) )  =  0  ->  -.  0  <  ( sin `  ( A  mod  pi ) ) )
134128, 133sylbir 217 . . . . . . . . . . 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 23454 . . . . . . . . . 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 9695 . . . . . . . . . . 11  |-  0  e.  RR*
1391rexri 9695 . . . . . . . . . . 11  |-  pi  e.  RR*
140 elioo2 11679 . . . . . . . . . . 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 677 . . . . . . . . . 10  |-  ( ( A  mod  pi )  e.  ( 0 (,) pi )  <->  ( ( A  mod  pi )  e.  RR  /\  0  < 
( A  mod  pi )  /\  ( A  mod  pi )  <  pi ) )
142141notbii 298 . . . . . . . . 9  |-  ( -.  ( A  mod  pi )  e.  ( 0 (,) pi )  <->  -.  (
( A  mod  pi )  e.  RR  /\  0  <  ( A  mod  pi )  /\  ( A  mod  pi )  <  pi ) )
143137, 142sylib 200 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  ( ( A  mod  pi )  e.  RR  /\  0  <  ( A  mod  pi )  /\  ( A  mod  pi )  <  pi ) )
144 3anan12 996 . . . . . . . . 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 298 . . . . . . . 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 200 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  ( 0  <  ( A  mod  pi )  /\  ( ( A  mod  pi )  e.  RR  /\  ( A  mod  pi )  <  pi ) ) )
147 modlt 12108 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  ( A  mod  pi )  < 
pi )
148147ancoms 455 . . . . . . . . 9  |-  ( ( pi  e.  RR+  /\  A  e.  RR )  ->  ( A  mod  pi )  < 
pi )
1493, 74, 148sylancr 668 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  <  pi )
15076, 149jca 535 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( ( A  mod  pi )  e.  RR  /\  ( A  mod  pi )  <  pi ) )
151 not12an2impnot1 36841 . . . . . . 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 666 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  ->  -.  0  <  ( A  mod  pi ) )
153 modge0 12107 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  0  <_  ( A  mod  pi ) )
154153ancoms 455 . . . . . . . 8  |-  ( ( pi  e.  RR+  /\  A  e.  RR )  ->  0  <_  ( A  mod  pi ) )
1553, 74, 154sylancr 668 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
0  <_  ( A  mod  pi ) )
156 leloe 9722 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( A  mod  pi )  e.  RR )  -> 
( 0  <_  ( A  mod  pi )  <->  ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) ) )
157156biimp3a 1365 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( A  mod  pi )  e.  RR  /\  0  <_  ( A  mod  pi ) )  ->  (
0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) )
158157idiALT 36734 . . . . . . 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 36991 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) )
160 pm2.53 375 . . . . . . . 8  |-  ( ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) )  ->  ( -.  0  <  ( A  mod  pi )  ->  0  =  ( A  mod  pi ) ) )
161160imp 431 . . . . . . 7  |-  ( ( ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) )  /\  -.  0  < 
( A  mod  pi ) )  ->  0  =  ( A  mod  pi ) )
162161ancoms 455 . . . . . 6  |-  ( ( -.  0  <  ( A  mod  pi )  /\  ( 0  <  ( A  mod  pi )  \/  0  =  ( A  mod  pi ) ) )  ->  0  =  ( A  mod  pi ) )
163152, 159, 162syl2anc 666 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
0  =  ( A  mod  pi ) )
164163eqcomd 2431 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  mod  pi )  =  0 )
165 mod0 12104 . . . . . 6  |-  ( ( A  e.  RR  /\  pi  e.  RR+ )  ->  (
( A  mod  pi )  =  0  <->  ( A  /  pi )  e.  ZZ ) )
166165biimp3a 1365 . . . . 5  |-  ( ( A  e.  RR  /\  pi  e.  RR+  /\  ( A  mod  pi )  =  0 )  ->  ( A  /  pi )  e.  ZZ )
1671663com12 1210 . . . 4  |-  ( ( pi  e.  RR+  /\  A  e.  RR  /\  ( A  mod  pi )  =  0 )  ->  ( A  /  pi )  e.  ZZ )
1683, 74, 164, 167eel011 36991 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =  0 )  -> 
( A  /  pi )  e.  ZZ )
169168ex 436 . 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 10386 . . . . 5  |-  ( A  e.  CC  ->  (
( A  /  pi )  x.  pi )  =  A )
173172fveq2d 5883 . . . 4  |-  ( A  e.  CC  ->  ( sin `  ( ( A  /  pi )  x.  pi ) )  =  ( sin `  A
) )
174 id 23 . . . . 5  |-  ( ( A  /  pi )  e.  ZZ  ->  ( A  /  pi )  e.  ZZ )
175 sinkpi 23466 . . . . 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 2485 . . 3  |-  ( ( A  e.  CC  /\  ( A  /  pi )  e.  ZZ )  ->  ( sin `  A
)  =  0 )
178177ex 436 . 2  |-  ( A  e.  CC  ->  (
( A  /  pi )  e.  ZZ  ->  ( sin `  A )  =  0 ) )
179169, 178impbid 194 1  |-  ( A  e.  CC  ->  (
( sin `  A
)  =  0  <->  ( A  /  pi )  e.  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   class class class wbr 4421   ` cfv 5599  (class class class)co 6303   CCcc 9539   RRcr 9540   0cc0 9541   1c1 9542   _ici 9543    + caddc 9544    x. cmul 9546   RR*cxr 9676    < clt 9677    <_ cle 9678    - cmin 9862   -ucneg 9863    / cdiv 10271   2c2 10661   ZZcz 10939   RR+crp 11304   (,)cioo 11637   |_cfl 12027    mod cmo 12097   abscabs 13291   expce 14107   sincsin 14109   picpi 14112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-omul 7193  df-er 7369  df-ec 7371  df-qs 7375  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-cda 8600  df-ni 9299  df-pli 9300  df-mi 9301  df-lti 9302  df-plpq 9335  df-mpq 9336  df-ltpq 9337  df-enq 9338  df-nq 9339  df-erq 9340  df-plq 9341  df-mq 9342  df-1nq 9343  df-rq 9344  df-ltnq 9345  df-np 9408  df-1p 9409  df-plp 9410  df-enr 9482  df-nr 9483  df-0r 9487  df-1r 9488  df-c 9547  df-i 9550  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ioo 11641  df-ioc 11642  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274  df-fac 12461  df-bc 12489  df-hash 12517  df-shft 13124  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-lp 20144  df-perf 20145  df-cn 20235  df-cnp 20236  df-haus 20323  df-tx 20569  df-hmeo 20762  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-xms 21327  df-ms 21328  df-tms 21329  df-cncf 21902  df-limc 22813  df-dv 22814
This theorem is referenced by: (None)
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