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Theorem coseq00topi 19886
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 8854 . . . . . . 7  |-  0  e.  RR
3 pire 19848 . . . . . . 7  |-  pi  e.  RR
42, 3elicc2i 10732 . . . . . 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 9974 . . . 4  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( pi  /  2
)  e.  RR )
96, 8lttri4d 8976 . . 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 8900 . . . . . . . . . . 11  |-  0  e.  RR*
15 halfpire 19851 . . . . . . . . . . . 12  |-  ( pi 
/  2 )  e.  RR
1615rexri 8900 . . . . . . . . . . 11  |-  ( pi 
/  2 )  e. 
RR*
17 elioo2 10713 . . . . . . . . . . 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 19882 . . . . . . . . 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 9353 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  ( cos `  A )  =/=  0 )
24 cos0 12446 . . . . . . . 8  |-  ( cos `  0 )  =  1
25 simpr 447 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  0  =  A )
2625fveq2d 5545 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  0 )  =  ( cos `  A
) )
2724, 26syl5reqr 2343 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  A )  =  1 )
28 ax-1ne0 8822 . . . . . . . 8  |-  1  =/=  0
2928a1i 10 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  1  =/=  0 )
3027, 29eqnetrd 2477 . . . . . 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 8978 . . . . . . . 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 2533 . . . 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 8900 . . . . . . . . . . 11  |-  pi  e.  RR*
44 elioo2 10713 . . . . . . . . . . 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 19883 . . . . . . . . 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 9354 . . . . . 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 5545 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =  ( cos `  pi ) )
53 cospi 19856 . . . . . . . 8  |-  ( cos `  pi )  =  -u
1
5452, 53syl6eq 2344 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =  -u
1 )
55 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
5655, 28negne0i 9137 . . . . . . . 8  |-  -u 1  =/=  0
5756a1i 10 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  -u 1  =/=  0 )
5854, 57eqnetrd 2477 . . . . . 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 8978 . . . . . . . 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 2533 . . . 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 5541 . . . 4  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  ( cos `  (
pi  /  2 ) ) )
68 coshalfpi 19853 . . . 4  |-  ( cos `  ( pi  /  2
) )  =  0
6967, 68syl6eq 2344 . . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754   RR*cxr 8882    < clt 8883    <_ cle 8884   -ucneg 9054    / cdiv 9439   2c2 9811   (,)cioo 10672   [,]cicc 10675   sincsin 12361   cosccos 12362   picpi 12364
This theorem is referenced by:  coseq0negpitopi  19887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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