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Theorem coseq00topi 21944
Description: Location of the zeroes of cosine in  ( 0 [,] pi ). (Contributed by David Moews, 28-Feb-2017.)
Assertion
Ref Expression
coseq00topi  |-  ( A  e.  ( 0 [,] pi )  ->  (
( cos `  A
)  =  0  <->  A  =  ( pi  / 
2 ) ) )

Proof of Theorem coseq00topi
StepHypRef Expression
1 simpl 457 . . . . . 6  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  ( 0 [,] pi ) )
2 0re 9378 . . . . . . 7  |-  0  e.  RR
3 pire 21901 . . . . . . 7  |-  pi  e.  RR
42, 3elicc2i 11353 . . . . . 6  |-  ( A  e.  ( 0 [,] pi )  <->  ( A  e.  RR  /\  0  <_  A  /\  A  <_  pi ) )
51, 4sylib 196 . . . . 5  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  e.  RR  /\  0  <_  A  /\  A  <_  pi ) )
65simp1d 1000 . . . 4  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  RR )
73a1i 11 . . . . 5  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  pi  e.  RR )
87rehalfcld 10563 . . . 4  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( pi  /  2
)  e.  RR )
96, 8lttri4d 9507 . . 3  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  <  (
pi  /  2 )  \/  A  =  ( pi  /  2 )  \/  ( pi  / 
2 )  <  A
) )
10 simplr 754 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  ( cos `  A )  =  0 )
116ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  A  e.  RR )
12 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  0  <  A )
13 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  A  <  ( pi  /  2
) )
142rexri 9428 . . . . . . . . . . 11  |-  0  e.  RR*
15 halfpire 21906 . . . . . . . . . . . 12  |-  ( pi 
/  2 )  e.  RR
1615rexri 9428 . . . . . . . . . . 11  |-  ( pi 
/  2 )  e. 
RR*
17 elioo2 11333 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  (
pi  /  2 )  e.  RR* )  ->  ( A  e.  ( 0 (,) ( pi  / 
2 ) )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  ( pi  /  2 ) ) ) )
1814, 16, 17mp2an 672 . . . . . . . . . 10  |-  ( A  e.  ( 0 (,) ( pi  /  2
) )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  ( pi  /  2 ) ) )
1911, 12, 13, 18syl3anbrc 1172 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  A  e.  ( 0 (,) (
pi  /  2 ) ) )
20 sincosq1sgn 21940 . . . . . . . . 9  |-  ( A  e.  ( 0 (,) ( pi  /  2
) )  ->  (
0  <  ( sin `  A )  /\  0  <  ( cos `  A
) ) )
2119, 20syl 16 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  (
0  <  ( sin `  A )  /\  0  <  ( cos `  A
) ) )
2221simprd 463 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  0  <  ( cos `  A
) )
2322gt0ne0d 9896 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  ( cos `  A )  =/=  0 )
24 cos0 13426 . . . . . . . 8  |-  ( cos `  0 )  =  1
25 simpr 461 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  0  =  A )
2625fveq2d 5690 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  0 )  =  ( cos `  A
) )
2724, 26syl5reqr 2485 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  A )  =  1 )
28 ax-1ne0 9343 . . . . . . . 8  |-  1  =/=  0
2928a1i 11 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  1  =/=  0 )
3027, 29eqnetrd 2621 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  A )  =/=  0 )
315simp2d 1001 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
0  <_  A )
32 0red 9379 . . . . . . . . 9  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
0  e.  RR )
3332, 6leloed 9509 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
3431, 33mpbid 210 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( 0  <  A  \/  0  =  A
) )
3534adantr 465 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  ( 0  <  A  \/  0  =  A ) )
3623, 30, 35mpjaodan 784 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  ( cos `  A )  =/=  0
)
3710, 36pm2.21ddne 2680 . . . 4  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  A  =  ( pi  /  2
) )
38 simpr 461 . . . 4  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  =  ( pi  /  2 ) )  ->  A  =  ( pi  /  2
) )
39 simplr 754 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  ( cos `  A )  =  0 )
406ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  A  e.  RR )
41 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( pi  /  2 )  <  A
)
42 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  A  <  pi )
433rexri 9428 . . . . . . . . . . 11  |-  pi  e.  RR*
44 elioo2 11333 . . . . . . . . . . 11  |-  ( ( ( pi  /  2
)  e.  RR*  /\  pi  e.  RR* )  ->  ( A  e.  ( (
pi  /  2 ) (,) pi )  <->  ( A  e.  RR  /\  ( pi 
/  2 )  < 
A  /\  A  <  pi ) ) )
4516, 43, 44mp2an 672 . . . . . . . . . 10  |-  ( A  e.  ( ( pi 
/  2 ) (,) pi )  <->  ( A  e.  RR  /\  ( pi 
/  2 )  < 
A  /\  A  <  pi ) )
4640, 41, 42, 45syl3anbrc 1172 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  A  e.  ( ( pi  / 
2 ) (,) pi ) )
47 sincosq2sgn 21941 . . . . . . . . 9  |-  ( A  e.  ( ( pi 
/  2 ) (,) pi )  ->  (
0  <  ( sin `  A )  /\  ( cos `  A )  <  0 ) )
4846, 47syl 16 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( 0  <  ( sin `  A
)  /\  ( cos `  A )  <  0
) )
4948simprd 463 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( cos `  A )  <  0
)
5049lt0ne0d 9897 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( cos `  A )  =/=  0
)
51 simpr 461 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  A  =  pi )
5251fveq2d 5690 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =  ( cos `  pi ) )
53 cospi 21914 . . . . . . . 8  |-  ( cos `  pi )  =  -u
1
5452, 53syl6eq 2486 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =  -u
1 )
55 neg1ne0 10419 . . . . . . . 8  |-  -u 1  =/=  0
5655a1i 11 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  -u 1  =/=  0 )
5754, 56eqnetrd 2621 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =/=  0
)
585simp3d 1002 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  <_  pi )
596, 7leloed 9509 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  <_  pi  <->  ( A  <  pi  \/  A  =  pi )
) )
6058, 59mpbid 210 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  <  pi  \/  A  =  pi ) )
6160adantr 465 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  ( A  <  pi  \/  A  =  pi ) )
6250, 57, 61mpjaodan 784 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  ( cos `  A )  =/=  0
)
6339, 62pm2.21ddne 2680 . . . 4  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  A  =  ( pi  /  2
) )
6437, 38, 633jaodan 1284 . . 3  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( A  < 
( pi  /  2
)  \/  A  =  ( pi  /  2
)  \/  ( pi 
/  2 )  < 
A ) )  ->  A  =  ( pi  /  2 ) )
659, 64mpdan 668 . 2  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  =  ( pi  /  2 ) )
66 fveq2 5686 . . . 4  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  ( cos `  (
pi  /  2 ) ) )
67 coshalfpi 21911 . . . 4  |-  ( cos `  ( pi  /  2
) )  =  0
6866, 67syl6eq 2486 . . 3  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  0 )
6968adantl 466 . 2  |-  ( ( A  e.  ( 0 [,] pi )  /\  A  =  ( pi  /  2 ) )  -> 
( cos `  A
)  =  0 )
7065, 69impbida 828 1  |-  ( A  e.  ( 0 [,] pi )  ->  (
( cos `  A
)  =  0  <->  A  =  ( pi  / 
2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   RRcr 9273   0cc0 9274   1c1 9275   RR*cxr 9409    < clt 9410    <_ cle 9411   -ucneg 9588    / cdiv 9985   2c2 10363   (,)cioo 11292   [,]cicc 11295   sincsin 13341   cosccos 13342   picpi 13344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434  df-limc 21321  df-dv 21322
This theorem is referenced by:  coseq0negpitopi  21945
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