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Theorem cos01gt0 12792
Description: The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
Assertion
Ref Expression
cos01gt0  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  A
) )

Proof of Theorem cos01gt0
StepHypRef Expression
1 0xr 9131 . . . . . . . . . 10  |-  0  e.  RR*
2 1re 9090 . . . . . . . . . 10  |-  1  e.  RR
3 elioc2 10973 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
41, 2, 3mp2an 654 . . . . . . . . 9  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
54simp1bi 972 . . . . . . . 8  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
65resqcld 11549 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  e.  RR )
76recnd 9114 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  e.  CC )
8 2cn 10070 . . . . . . 7  |-  2  e.  CC
9 3cn 10072 . . . . . . . 8  |-  3  e.  CC
10 3ne0 10085 . . . . . . . 8  |-  3  =/=  0
119, 10pm3.2i 442 . . . . . . 7  |-  ( 3  e.  CC  /\  3  =/=  0 )
12 div12 9700 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( A ^ 2 )  e.  CC  /\  (
3  e.  CC  /\  3  =/=  0 ) )  ->  ( 2  x.  ( ( A ^
2 )  /  3
) )  =  ( ( A ^ 2 )  x.  ( 2  /  3 ) ) )
138, 11, 12mp3an13 1270 . . . . . 6  |-  ( ( A ^ 2 )  e.  CC  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  =  ( ( A ^ 2 )  x.  ( 2  /  3
) ) )
147, 13syl 16 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  =  ( ( A ^ 2 )  x.  ( 2  /  3
) ) )
15 2z 10312 . . . . . . . . . 10  |-  2  e.  ZZ
16 expgt0 11413 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  2  e.  ZZ  /\  0  <  A )  ->  0  <  ( A ^ 2 ) )
1715, 16mp3an2 1267 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A ^ 2 ) )
18173adant3 977 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  0  <  ( A ^ 2 ) )
194, 18sylbi 188 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A ^ 2 ) )
20 2lt3 10143 . . . . . . . . . 10  |-  2  <  3
21 2re 10069 . . . . . . . . . . 11  |-  2  e.  RR
22 3re 10071 . . . . . . . . . . 11  |-  3  e.  RR
23 3pos 10084 . . . . . . . . . . 11  |-  0  <  3
2421, 22, 22, 23ltdiv1ii 9940 . . . . . . . . . 10  |-  ( 2  <  3  <->  ( 2  /  3 )  < 
( 3  /  3
) )
2520, 24mpbi 200 . . . . . . . . 9  |-  ( 2  /  3 )  < 
( 3  /  3
)
269, 10dividi 9747 . . . . . . . . 9  |-  ( 3  /  3 )  =  1
2725, 26breqtri 4235 . . . . . . . 8  |-  ( 2  /  3 )  <  1
2821, 22, 10redivcli 9781 . . . . . . . . 9  |-  ( 2  /  3 )  e.  RR
29 ltmul2 9861 . . . . . . . . 9  |-  ( ( ( 2  /  3
)  e.  RR  /\  1  e.  RR  /\  (
( A ^ 2 )  e.  RR  /\  0  <  ( A ^
2 ) ) )  ->  ( ( 2  /  3 )  <  1  <->  ( ( A ^ 2 )  x.  ( 2  /  3
) )  <  (
( A ^ 2 )  x.  1 ) ) )
3028, 2, 29mp3an12 1269 . . . . . . . 8  |-  ( ( ( A ^ 2 )  e.  RR  /\  0  <  ( A ^
2 ) )  -> 
( ( 2  / 
3 )  <  1  <->  ( ( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) ) )
3127, 30mpbii 203 . . . . . . 7  |-  ( ( ( A ^ 2 )  e.  RR  /\  0  <  ( A ^
2 ) )  -> 
( ( A ^
2 )  x.  (
2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) )
326, 19, 31syl2anc 643 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) )
337mulid1d 9105 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  1 )  =  ( A ^
2 ) )
3432, 33breqtrd 4236 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( A ^
2 ) )
3514, 34eqbrtrd 4232 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  ( A ^
2 ) )
36 0re 9091 . . . . . . . . . 10  |-  0  e.  RR
37 ltle 9163 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
3836, 37mpan 652 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
0  <  A  ->  0  <_  A ) )
3938imdistani 672 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  RR  /\  0  <_  A )
)
40 le2sq2 11457 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( 1  e.  RR  /\  A  <_  1 ) )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
412, 40mpanr1 665 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  A  <_  1 )  ->  ( A ^
2 )  <_  (
1 ^ 2 ) )
4239, 41sylan 458 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  A  <_  1
)  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
43423impa 1148 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
444, 43sylbi 188 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
45 sq1 11476 . . . . 5  |-  ( 1 ^ 2 )  =  1
4644, 45syl6breq 4251 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  <_ 
1 )
47 redivcl 9733 . . . . . . . 8  |-  ( ( ( A ^ 2 )  e.  RR  /\  3  e.  RR  /\  3  =/=  0 )  ->  (
( A ^ 2 )  /  3 )  e.  RR )
4822, 10, 47mp3an23 1271 . . . . . . 7  |-  ( ( A ^ 2 )  e.  RR  ->  (
( A ^ 2 )  /  3 )  e.  RR )
496, 48syl 16 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  /  3 )  e.  RR )
50 remulcl 9075 . . . . . 6  |-  ( ( 2  e.  RR  /\  ( ( A ^
2 )  /  3
)  e.  RR )  ->  ( 2  x.  ( ( A ^
2 )  /  3
) )  e.  RR )
5121, 49, 50sylancr 645 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  e.  RR )
52 ltletr 9166 . . . . . 6  |-  ( ( ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR  /\  ( A ^ 2 )  e.  RR  /\  1  e.  RR )  ->  (
( ( 2  x.  ( ( A ^
2 )  /  3
) )  <  ( A ^ 2 )  /\  ( A ^ 2 )  <_  1 )  -> 
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1 ) )
532, 52mp3an3 1268 . . . . 5  |-  ( ( ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR  /\  ( A ^ 2 )  e.  RR )  -> 
( ( ( 2  x.  ( ( A ^ 2 )  / 
3 ) )  < 
( A ^ 2 )  /\  ( A ^ 2 )  <_ 
1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  1 ) )
5451, 6, 53syl2anc 643 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( 2  x.  ( ( A ^
2 )  /  3
) )  <  ( A ^ 2 )  /\  ( A ^ 2 )  <_  1 )  -> 
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1 ) )
5535, 46, 54mp2and 661 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  1 )
56 posdif 9521 . . . 4  |-  ( ( ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR  /\  1  e.  RR )  ->  ( ( 2  x.  ( ( A ^
2 )  /  3
) )  <  1  <->  0  <  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) ) ) )
5751, 2, 56sylancl 644 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1  <->  0  <  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) ) ) )
5855, 57mpbid 202 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) ) )
59 cos01bnd 12787 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) )  <  ( cos `  A )  /\  ( cos `  A )  < 
( 1  -  (
( A ^ 2 )  /  3 ) ) ) )
6059simpld 446 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  <  ( cos `  A
) )
61 resubcl 9365 . . . 4  |-  ( ( 1  e.  RR  /\  ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR )  ->  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) )  e.  RR )
622, 51, 61sylancr 645 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  e.  RR )
635recoscld 12745 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( cos `  A )  e.  RR )
64 lttr 9152 . . . 4  |-  ( ( 0  e.  RR  /\  ( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) )  e.  RR  /\  ( cos `  A )  e.  RR )  -> 
( ( 0  < 
( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) )  /\  ( 1  -  ( 2  x.  ( ( A ^
2 )  /  3
) ) )  < 
( cos `  A
) )  ->  0  <  ( cos `  A
) ) )
6536, 64mp3an1 1266 . . 3  |-  ( ( ( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) )  e.  RR  /\  ( cos `  A )  e.  RR )  -> 
( ( 0  < 
( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) )  /\  ( 1  -  ( 2  x.  ( ( A ^
2 )  /  3
) ) )  < 
( cos `  A
) )  ->  0  <  ( cos `  A
) ) )
6662, 63, 65syl2anc 643 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 0  <  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  /\  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) )  <  ( cos `  A ) )  ->  0  <  ( cos `  A ) ) )
6758, 60, 66mp2and 661 1  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995   RR*cxr 9119    < clt 9120    <_ cle 9121    - cmin 9291    / cdiv 9677   2c2 10049   3c3 10050   ZZcz 10282   (,]cioc 10917   ^cexp 11382   cosccos 12667
This theorem is referenced by:  sin02gt0  12793  sincos1sgn  12794  tangtx  20413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ioc 10921  df-ico 10922  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-fac 11567  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-cos 12673
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