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Theorem coseq00topi 19870
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 443 . . . . . 6  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  ( 0 [,] pi ) )
2 0re 8838 . . . . . . 7  |-  0  e.  RR
3 pire 19832 . . . . . . 7  |-  pi  e.  RR
42, 3elicc2i 10716 . . . . . 6  |-  ( A  e.  ( 0 [,] pi )  <->  ( A  e.  RR  /\  0  <_  A  /\  A  <_  pi ) )
51, 4sylib 188 . . . . 5  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  e.  RR  /\  0  <_  A  /\  A  <_  pi ) )
65simp1d 967 . . . 4  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  RR )
73a1i 10 . . . . 5  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  pi  e.  RR )
87rehalfcld 9958 . . . 4  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( pi  /  2
)  e.  RR )
96, 8lttri4d 8960 . . 3  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  <  (
pi  /  2 )  \/  A  =  ( pi  /  2 )  \/  ( pi  / 
2 )  <  A
) )
10 simplr 731 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  ( cos `  A )  =  0 )
116ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  A  e.  RR )
12 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  0  <  A )
13 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  A  <  ( pi  /  2
) )
142rexri 8884 . . . . . . . . . . 11  |-  0  e.  RR*
15 halfpire 19835 . . . . . . . . . . . 12  |-  ( pi 
/  2 )  e.  RR
1615rexri 8884 . . . . . . . . . . 11  |-  ( pi 
/  2 )  e. 
RR*
17 elioo2 10697 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  (
pi  /  2 )  e.  RR* )  ->  ( A  e.  ( 0 (,) ( pi  / 
2 ) )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  ( pi  /  2 ) ) ) )
1814, 16, 17mp2an 653 . . . . . . . . . 10  |-  ( A  e.  ( 0 (,) ( pi  /  2
) )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  ( pi  /  2 ) ) )
1911, 12, 13, 18syl3anbrc 1136 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  A  e.  ( 0 (,) (
pi  /  2 ) ) )
20 sincosq1sgn 19866 . . . . . . . . 9  |-  ( A  e.  ( 0 (,) ( pi  /  2
) )  ->  (
0  <  ( sin `  A )  /\  0  <  ( cos `  A
) ) )
2119, 20syl 15 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  (
0  <  ( sin `  A )  /\  0  <  ( cos `  A
) ) )
2221simprd 449 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  0  <  ( cos `  A
) )
2322gt0ne0d 9337 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  ( cos `  A )  =/=  0 )
24 cos0 12430 . . . . . . . 8  |-  ( cos `  0 )  =  1
25 simpr 447 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  0  =  A )
2625fveq2d 5529 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  0 )  =  ( cos `  A
) )
2724, 26syl5reqr 2330 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  A )  =  1 )
28 ax-1ne0 8806 . . . . . . . 8  |-  1  =/=  0
2928a1i 10 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  1  =/=  0 )
3027, 29eqnetrd 2464 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  A )  =/=  0 )
315simp2d 968 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
0  <_  A )
322a1i 10 . . . . . . . . 9  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
0  e.  RR )
3332, 6leloed 8962 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
3431, 33mpbid 201 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( 0  <  A  \/  0  =  A
) )
3534adantr 451 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  ( 0  <  A  \/  0  =  A ) )
3623, 30, 35mpjaodan 761 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  ( cos `  A )  =/=  0
)
3710, 36pm2.21ddne 2520 . . . 4  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  A  =  ( pi  /  2
) )
38 simpr 447 . . . 4  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  =  ( pi  /  2 ) )  ->  A  =  ( pi  /  2
) )
39 simplr 731 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  ( cos `  A )  =  0 )
406ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  A  e.  RR )
41 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( pi  /  2 )  <  A
)
42 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  A  <  pi )
433rexri 8884 . . . . . . . . . . 11  |-  pi  e.  RR*
44 elioo2 10697 . . . . . . . . . . 11  |-  ( ( ( pi  /  2
)  e.  RR*  /\  pi  e.  RR* )  ->  ( A  e.  ( (
pi  /  2 ) (,) pi )  <->  ( A  e.  RR  /\  ( pi 
/  2 )  < 
A  /\  A  <  pi ) ) )
4516, 43, 44mp2an 653 . . . . . . . . . 10  |-  ( A  e.  ( ( pi 
/  2 ) (,) pi )  <->  ( A  e.  RR  /\  ( pi 
/  2 )  < 
A  /\  A  <  pi ) )
4640, 41, 42, 45syl3anbrc 1136 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  A  e.  ( ( pi  / 
2 ) (,) pi ) )
47 sincosq2sgn 19867 . . . . . . . . 9  |-  ( A  e.  ( ( pi 
/  2 ) (,) pi )  ->  (
0  <  ( sin `  A )  /\  ( cos `  A )  <  0 ) )
4846, 47syl 15 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( 0  <  ( sin `  A
)  /\  ( cos `  A )  <  0
) )
4948simprd 449 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( cos `  A )  <  0
)
5049lt0ne0d 9338 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( cos `  A )  =/=  0
)
51 simpr 447 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  A  =  pi )
5251fveq2d 5529 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =  ( cos `  pi ) )
53 cospi 19840 . . . . . . . 8  |-  ( cos `  pi )  =  -u
1
5452, 53syl6eq 2331 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =  -u
1 )
55 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
5655, 28negne0i 9121 . . . . . . . 8  |-  -u 1  =/=  0
5756a1i 10 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  -u 1  =/=  0 )
5854, 57eqnetrd 2464 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =/=  0
)
595simp3d 969 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  <_  pi )
606, 7leloed 8962 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  <_  pi  <->  ( A  <  pi  \/  A  =  pi )
) )
6159, 60mpbid 201 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  <  pi  \/  A  =  pi ) )
6261adantr 451 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  ( A  <  pi  \/  A  =  pi ) )
6350, 58, 62mpjaodan 761 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  ( cos `  A )  =/=  0
)
6439, 63pm2.21ddne 2520 . . . 4  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  A  =  ( pi  /  2
) )
6537, 38, 643jaodan 1248 . . 3  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( A  < 
( pi  /  2
)  \/  A  =  ( pi  /  2
)  \/  ( pi 
/  2 )  < 
A ) )  ->  A  =  ( pi  /  2 ) )
669, 65mpdan 649 . 2  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  =  ( pi  /  2 ) )
67 fveq2 5525 . . . 4  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  ( cos `  (
pi  /  2 ) ) )
68 coshalfpi 19837 . . . 4  |-  ( cos `  ( pi  /  2
) )  =  0
6967, 68syl6eq 2331 . . 3  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  0 )
7069adantl 452 . 2  |-  ( ( A  e.  ( 0 [,] pi )  /\  A  =  ( pi  /  2 ) )  -> 
( cos `  A
)  =  0 )
7166, 70impbida 805 1  |-  ( A  e.  ( 0 [,] pi )  ->  (
( cos `  A
)  =  0  <->  A  =  ( pi  / 
2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738   RR*cxr 8866    < clt 8867    <_ cle 8868   -ucneg 9038    / cdiv 9423   2c2 9795   (,)cioo 10656   [,]cicc 10659   sincsin 12345   cosccos 12346   picpi 12348
This theorem is referenced by:  coseq0negpitopi  19871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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