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Theorem sin02gt0 13475
Description: The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
Assertion
Ref Expression
sin02gt0  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  A
) )

Proof of Theorem sin02gt0
StepHypRef Expression
1 0xr 9429 . . . . . . 7  |-  0  e.  RR*
2 2re 10390 . . . . . . 7  |-  2  e.  RR
3 elioc2 11357 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  2  e.  RR )  ->  ( A  e.  ( 0 (,] 2 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  2 ) ) )
41, 2, 3mp2an 672 . . . . . 6  |-  ( A  e.  ( 0 (,] 2 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  2 ) )
5 rehalfcl 10550 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  /  2 )  e.  RR )
653ad2ant1 1009 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  ( A  /  2 )  e.  RR )
74, 6sylbi 195 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  RR )
8 resincl 13423 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  ( sin `  ( A  / 
2 ) )  e.  RR )
9 recoscl 13424 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  ( cos `  ( A  / 
2 ) )  e.  RR )
108, 9remulcld 9413 . . . . 5  |-  ( ( A  /  2 )  e.  RR  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
117, 10syl 16 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
12 2pos 10412 . . . . . . . . . 10  |-  0  <  2
13 divgt0 10196 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
0  <  ( A  /  2 ) )
142, 12, 13mpanr12 685 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A  /  2 ) )
15143adant3 1008 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  0  <  ( A  /  2
) )
162, 12pm3.2i 455 . . . . . . . . . . . 12  |-  ( 2  e.  RR  /\  0  <  2 )
17 lediv1 10193 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  2  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( A  <_ 
2  <->  ( A  / 
2 )  <_  (
2  /  2 ) ) )
182, 16, 17mp3an23 1306 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  <_  2  <->  ( A  /  2 )  <_ 
( 2  /  2
) ) )
1918biimpa 484 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  <_  2 )  -> 
( A  /  2
)  <_  ( 2  /  2 ) )
20 2div2e1 10443 . . . . . . . . . 10  |-  ( 2  /  2 )  =  1
2119, 20syl6breq 4330 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <_  2 )  -> 
( A  /  2
)  <_  1 )
22213adant2 1007 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  ( A  /  2 )  <_ 
1 )
236, 15, 223jca 1168 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  (
( A  /  2
)  e.  RR  /\  0  <  ( A  / 
2 )  /\  ( A  /  2 )  <_ 
1 ) )
24 1re 9384 . . . . . . . 8  |-  1  e.  RR
25 elioc2 11357 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  (
( A  /  2
)  e.  ( 0 (,] 1 )  <->  ( ( A  /  2 )  e.  RR  /\  0  < 
( A  /  2
)  /\  ( A  /  2 )  <_ 
1 ) ) )
261, 24, 25mp2an 672 . . . . . . 7  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  <->  ( ( A  /  2 )  e.  RR  /\  0  < 
( A  /  2
)  /\  ( A  /  2 )  <_ 
1 ) )
2723, 4, 263imtr4i 266 . . . . . 6  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  ( 0 (,] 1
) )
28 sin01gt0 13473 . . . . . 6  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  ( A  /  2 ) ) )
2927, 28syl 16 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  ( A  /  2 ) ) )
30 cos01gt0 13474 . . . . . 6  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  ( A  /  2 ) ) )
3127, 30syl 16 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( cos `  ( A  /  2 ) ) )
32 axmulgt0 9448 . . . . . . 7  |-  ( ( ( sin `  ( A  /  2 ) )  e.  RR  /\  ( cos `  ( A  / 
2 ) )  e.  RR )  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
338, 9, 32syl2anc 661 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
347, 33syl 16 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
3529, 31, 34mp2and 679 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) )
36 axmulgt0 9448 . . . . . 6  |-  ( ( 2  e.  RR  /\  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )  ->  ( ( 0  <  2  /\  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) )  ->  0  <  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
372, 36mpan 670 . . . . 5  |-  ( ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR  ->  ( ( 0  <  2  /\  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) )  ->  0  <  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
3812, 37mpani 676 . . . 4  |-  ( ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR  ->  ( 0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( 2  x.  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
3911, 35, 38sylc 60 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
407recnd 9411 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  CC )
41 sin2t 13460 . . . 4  |-  ( ( A  /  2 )  e.  CC  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
4240, 41syl 16 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
4339, 42breqtrrd 4317 . 2  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  (
2  x.  ( A  /  2 ) ) ) )
444simp1bi 1003 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  A  e.  RR )
4544recnd 9411 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  A  e.  CC )
46 2cn 10391 . . . . 5  |-  2  e.  CC
47 2ne0 10413 . . . . 5  |-  2  =/=  0
48 divcan2 10001 . . . . 5  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
2  x.  ( A  /  2 ) )  =  A )
4946, 47, 48mp3an23 1306 . . . 4  |-  ( A  e.  CC  ->  (
2  x.  ( A  /  2 ) )  =  A )
5045, 49syl 16 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  (
2  x.  ( A  /  2 ) )  =  A )
5150fveq2d 5694 . 2  |-  ( A  e.  ( 0 (,] 2 )  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( sin `  A
) )
5243, 51breqtrd 4315 1  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    x. cmul 9286   RR*cxr 9416    < clt 9417    <_ cle 9418    / cdiv 9992   2c2 10370   (,]cioc 11300   sincsin 13348   cosccos 13349
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 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
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 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-ioc 11304  df-ico 11305  df-fz 11437  df-fzo 11548  df-fl 11641  df-seq 11806  df-exp 11865  df-fac 12051  df-bc 12078  df-hash 12103  df-shft 12555  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-limsup 12948  df-clim 12965  df-rlim 12966  df-sum 13163  df-ef 13352  df-sin 13354  df-cos 13355
This theorem is referenced by:  sincos2sgn  13477  pilem2  21916  sinhalfpilem  21924  sincosq1lem  21958
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