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Theorem coseq0negpitopi 22721
Description: Location of the zeroes of cosine in  ( -u pi (,] pi ). (Contributed by David Moews, 28-Feb-2017.)
Assertion
Ref Expression
coseq0negpitopi  |-  ( A  e.  ( -u pi (,] pi )  ->  (
( cos `  A
)  =  0  <->  A  e.  { ( pi  / 
2 ) ,  -u ( pi  /  2
) } ) )

Proof of Theorem coseq0negpitopi
StepHypRef Expression
1 simpl 457 . . . . 5  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  ( -u pi (,] pi ) )
2 pire 22677 . . . . . . . 8  |-  pi  e.  RR
32renegcli 9881 . . . . . . 7  |-  -u pi  e.  RR
43rexri 9647 . . . . . 6  |-  -u pi  e.  RR*
5 elioc2 11588 . . . . . 6  |-  ( (
-u pi  e.  RR*  /\  pi  e.  RR )  ->  ( A  e.  ( -u pi (,] pi )  <->  ( A  e.  RR  /\  -u pi  <  A  /\  A  <_  pi ) ) )
64, 2, 5mp2an 672 . . . . 5  |-  ( A  e.  ( -u pi (,] pi )  <->  ( A  e.  RR  /\  -u pi  <  A  /\  A  <_  pi ) )
71, 6sylib 196 . . . 4  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  e.  RR  /\  -u pi  <  A  /\  A  <_  pi ) )
87simp1d 1008 . . 3  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  RR )
9 0re 9597 . . . 4  |-  0  e.  RR
109a1i 11 . . 3  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  -> 
0  e.  RR )
118adantr 465 . . . . . . 7  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  A  e.  RR )
1211recnd 9623 . . . . . 6  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  A  e.  CC )
138recnd 9623 . . . . . . . . . 10  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  CC )
1413adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  A  e.  CC )
15 cosneg 13746 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( cos `  -u A )  =  ( cos `  A
) )
1614, 15syl 16 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  ( cos `  -u A )  =  ( cos `  A ) )
17 simplr 754 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  ( cos `  A )  =  0 )
1816, 17eqtrd 2508 . . . . . . 7  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  ( cos `  -u A )  =  0 )
198renegcld 9987 . . . . . . . . . 10  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  -u A  e.  RR )
2019adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  -u A  e.  RR )
21 simpr 461 . . . . . . . . . 10  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  A  <_  0 )
2211le0neg1d 10125 . . . . . . . . . 10  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  ( A  <_  0  <->  0  <_  -u A
) )
2321, 22mpbid 210 . . . . . . . . 9  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  0  <_  -u A )
242a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  pi  e.  RR )
257simp2d 1009 . . . . . . . . . . . 12  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  -u pi  <  A )
2624, 8, 25ltnegcon1d 10133 . . . . . . . . . . 11  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  -u A  <  pi )
2719, 24, 26ltled 9733 . . . . . . . . . 10  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  -u A  <_  pi )
2827adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  -u A  <_  pi )
299, 2elicc2i 11591 . . . . . . . . 9  |-  ( -u A  e.  ( 0 [,] pi )  <->  ( -u A  e.  RR  /\  0  <_  -u A  /\  -u A  <_  pi ) )
3020, 23, 28, 29syl3anbrc 1180 . . . . . . . 8  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  -u A  e.  ( 0 [,] pi ) )
31 coseq00topi 22720 . . . . . . . 8  |-  ( -u A  e.  ( 0 [,] pi )  -> 
( ( cos `  -u A
)  =  0  <->  -u A  =  ( pi  / 
2 ) ) )
3230, 31syl 16 . . . . . . 7  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  ( ( cos `  -u A )  =  0  <->  -u A  =  ( pi  /  2 ) ) )
3318, 32mpbid 210 . . . . . 6  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  -u A  =  ( pi  /  2
) )
3412, 33negcon1ad 9926 . . . . 5  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  -u ( pi 
/  2 )  =  A )
3534eqcomd 2475 . . . 4  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  A  =  -u ( pi  /  2
) )
36 halfpire 22682 . . . . . 6  |-  ( pi 
/  2 )  e.  RR
3736renegcli 9881 . . . . 5  |-  -u (
pi  /  2 )  e.  RR
38 prid2g 4134 . . . . 5  |-  ( -u ( pi  /  2
)  e.  RR  ->  -u ( pi  /  2
)  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) } )
39 eleq1a 2550 . . . . 5  |-  ( -u ( pi  /  2
)  e.  { ( pi  /  2 ) ,  -u ( pi  / 
2 ) }  ->  ( A  =  -u (
pi  /  2 )  ->  A  e.  {
( pi  /  2
) ,  -u (
pi  /  2 ) } ) )
4037, 38, 39mp2b 10 . . . 4  |-  ( A  =  -u ( pi  / 
2 )  ->  A  e.  { ( pi  / 
2 ) ,  -u ( pi  /  2
) } )
4135, 40syl 16 . . 3  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <_  0
)  ->  A  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )
42 simplr 754 . . . . 5  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  0  <_  A
)  ->  ( cos `  A )  =  0 )
438adantr 465 . . . . . . 7  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  0  <_  A
)  ->  A  e.  RR )
44 simpr 461 . . . . . . 7  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  0  <_  A
)  ->  0  <_  A )
457simp3d 1010 . . . . . . . 8  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  A  <_  pi )
4645adantr 465 . . . . . . 7  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  0  <_  A
)  ->  A  <_  pi )
479, 2elicc2i 11591 . . . . . . 7  |-  ( A  e.  ( 0 [,] pi )  <->  ( A  e.  RR  /\  0  <_  A  /\  A  <_  pi ) )
4843, 44, 46, 47syl3anbrc 1180 . . . . . 6  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  0  <_  A
)  ->  A  e.  ( 0 [,] pi ) )
49 coseq00topi 22720 . . . . . 6  |-  ( A  e.  ( 0 [,] pi )  ->  (
( cos `  A
)  =  0  <->  A  =  ( pi  / 
2 ) ) )
5048, 49syl 16 . . . . 5  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  0  <_  A
)  ->  ( ( cos `  A )  =  0  <->  A  =  (
pi  /  2 ) ) )
5142, 50mpbid 210 . . . 4  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  0  <_  A
)  ->  A  =  ( pi  /  2
) )
52 prid1g 4133 . . . . 5  |-  ( ( pi  /  2 )  e.  RR  ->  (
pi  /  2 )  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) } )
53 eleq1a 2550 . . . . 5  |-  ( ( pi  /  2 )  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) }  ->  ( A  =  ( pi 
/  2 )  ->  A  e.  { (
pi  /  2 ) ,  -u ( pi  / 
2 ) } ) )
5436, 52, 53mp2b 10 . . . 4  |-  ( A  =  ( pi  / 
2 )  ->  A  e.  { ( pi  / 
2 ) ,  -u ( pi  /  2
) } )
5551, 54syl 16 . . 3  |-  ( ( ( A  e.  (
-u pi (,] pi )  /\  ( cos `  A
)  =  0 )  /\  0  <_  A
)  ->  A  e.  { ( pi  /  2
) ,  -u (
pi  /  2 ) } )
568, 10, 41, 55lecasei 9691 . 2  |-  ( ( A  e.  ( -u pi (,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  { (
pi  /  2 ) ,  -u ( pi  / 
2 ) } )
57 elpri 4047 . . . 4  |-  ( A  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) }  ->  ( A  =  ( pi 
/  2 )  \/  A  =  -u (
pi  /  2 ) ) )
58 fveq2 5866 . . . . . 6  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  ( cos `  (
pi  /  2 ) ) )
59 coshalfpi 22687 . . . . . 6  |-  ( cos `  ( pi  /  2
) )  =  0
6058, 59syl6eq 2524 . . . . 5  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  0 )
61 fveq2 5866 . . . . . 6  |-  ( A  =  -u ( pi  / 
2 )  ->  ( cos `  A )  =  ( cos `  -u (
pi  /  2 ) ) )
62 cosneghalfpi 22688 . . . . . 6  |-  ( cos `  -u ( pi  / 
2 ) )  =  0
6361, 62syl6eq 2524 . . . . 5  |-  ( A  =  -u ( pi  / 
2 )  ->  ( cos `  A )  =  0 )
6460, 63jaoi 379 . . . 4  |-  ( ( A  =  ( pi 
/  2 )  \/  A  =  -u (
pi  /  2 ) )  ->  ( cos `  A )  =  0 )
6557, 64syl 16 . . 3  |-  ( A  e.  { ( pi 
/  2 ) , 
-u ( pi  / 
2 ) }  ->  ( cos `  A )  =  0 )
6665adantl 466 . 2  |-  ( ( A  e.  ( -u pi (,] pi )  /\  A  e.  { (
pi  /  2 ) ,  -u ( pi  / 
2 ) } )  ->  ( cos `  A
)  =  0 )
6756, 66impbida 830 1  |-  ( A  e.  ( -u pi (,] pi )  ->  (
( cos `  A
)  =  0  <->  A  e.  { ( pi  / 
2 ) ,  -u ( pi  /  2
) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cpr 4029   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   CCcc 9491   RRcr 9492   0cc0 9493   RR*cxr 9628    < clt 9629    <_ cle 9630   -ucneg 9807    / cdiv 10207   2c2 10586   (,]cioc 11531   [,]cicc 11533   cosccos 13665   picpi 13667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-addf 9572  ax-mulf 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-fi 7872  df-sup 7902  df-oi 7936  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-q 11184  df-rp 11222  df-xneg 11319  df-xadd 11320  df-xmul 11321  df-ioo 11534  df-ioc 11535  df-ico 11536  df-icc 11537  df-fz 11674  df-fzo 11794  df-fl 11898  df-seq 12077  df-exp 12136  df-fac 12323  df-bc 12350  df-hash 12375  df-shft 12866  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-limsup 13260  df-clim 13277  df-rlim 13278  df-sum 13475  df-ef 13668  df-sin 13670  df-cos 13671  df-pi 13673  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-starv 14573  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-unif 14581  df-hom 14582  df-cco 14583  df-rest 14681  df-topn 14682  df-0g 14700  df-gsum 14701  df-topgen 14702  df-pt 14703  df-prds 14706  df-xrs 14760  df-qtop 14765  df-imas 14766  df-xps 14768  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-submnd 15790  df-mulg 15874  df-cntz 16169  df-cmn 16615  df-psmet 18222  df-xmet 18223  df-met 18224  df-bl 18225  df-mopn 18226  df-fbas 18227  df-fg 18228  df-cnfld 18232  df-top 19206  df-bases 19208  df-topon 19209  df-topsp 19210  df-cld 19326  df-ntr 19327  df-cls 19328  df-nei 19405  df-lp 19443  df-perf 19444  df-cn 19534  df-cnp 19535  df-haus 19622  df-tx 19890  df-hmeo 20083  df-fil 20174  df-fm 20266  df-flim 20267  df-flf 20268  df-xms 20650  df-ms 20651  df-tms 20652  df-cncf 21209  df-limc 22097  df-dv 22098
This theorem is referenced by:  angrteqvd  22963  chordthmlem  22988
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