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Theorem coseq00topi 22096
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 9496 . . . . . . 7  |-  0  e.  RR
3 pire 22053 . . . . . . 7  |-  pi  e.  RR
42, 3elicc2i 11471 . . . . . 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 10681 . . . 4  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( pi  /  2
)  e.  RR )
96, 8lttri4d 9625 . . 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 9546 . . . . . . . . . . 11  |-  0  e.  RR*
15 halfpire 22058 . . . . . . . . . . . 12  |-  ( pi 
/  2 )  e.  RR
1615rexri 9546 . . . . . . . . . . 11  |-  ( pi 
/  2 )  e. 
RR*
17 elioo2 11451 . . . . . . . . . . 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 22092 . . . . . . . . 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 10014 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  ( cos `  A )  =/=  0 )
24 cos0 13551 . . . . . . . 8  |-  ( cos `  0 )  =  1
25 simpr 461 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  0  =  A )
2625fveq2d 5802 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  0 )  =  ( cos `  A
) )
2724, 26syl5reqr 2510 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  A )  =  1 )
28 ax-1ne0 9461 . . . . . . . 8  |-  1  =/=  0
2928a1i 11 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  1  =/=  0 )
3027, 29eqnetrd 2744 . . . . . 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 9497 . . . . . . . . 9  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
0  e.  RR )
3332, 6leloed 9627 . . . . . . . 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 2765 . . . 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 9546 . . . . . . . . . . 11  |-  pi  e.  RR*
44 elioo2 11451 . . . . . . . . . . 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 22093 . . . . . . . . 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 10015 . . . . . 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 5802 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =  ( cos `  pi ) )
53 cospi 22066 . . . . . . . 8  |-  ( cos `  pi )  =  -u
1
5452, 53syl6eq 2511 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =  -u
1 )
55 neg1ne0 10537 . . . . . . . 8  |-  -u 1  =/=  0
5655a1i 11 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  -u 1  =/=  0 )
5754, 56eqnetrd 2744 . . . . . 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 9627 . . . . . . . 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 2765 . . . 4  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  A  =  ( pi  /  2
) )
6437, 38, 633jaodan 1285 . . 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 5798 . . . 4  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  ( cos `  (
pi  /  2 ) ) )
67 coshalfpi 22063 . . . 4  |-  ( cos `  ( pi  /  2
) )  =  0
6866, 67syl6eq 2511 . . 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 1370    e. wcel 1758    =/= wne 2647   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   RRcr 9391   0cc0 9392   1c1 9393   RR*cxr 9527    < clt 9528    <_ cle 9529   -ucneg 9706    / cdiv 10103   2c2 10481   (,)cioo 11410   [,]cicc 11413   sincsin 13466   cosccos 13467   picpi 13469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ioc 11415  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-seq 11923  df-exp 11982  df-fac 12168  df-bc 12195  df-hash 12220  df-shft 12673  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-limsup 13066  df-clim 13083  df-rlim 13084  df-sum 13281  df-ef 13470  df-sin 13472  df-cos 13473  df-pi 13475  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lp 18871  df-perf 18872  df-cn 18962  df-cnp 18963  df-haus 19050  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585  df-limc 21473  df-dv 21474
This theorem is referenced by:  coseq0negpitopi  22097
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