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Theorem sin02gt0 12785
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 9123 . . . . . . 7  |-  0  e.  RR*
2 2re 10061 . . . . . . 7  |-  2  e.  RR
3 elioc2 10965 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  2  e.  RR )  ->  ( A  e.  ( 0 (,] 2 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  2 ) ) )
41, 2, 3mp2an 654 . . . . . 6  |-  ( A  e.  ( 0 (,] 2 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  2 ) )
5 rehalfcl 10186 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  /  2 )  e.  RR )
653ad2ant1 978 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  ( A  /  2 )  e.  RR )
74, 6sylbi 188 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  RR )
8 resincl 12733 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  ( sin `  ( A  / 
2 ) )  e.  RR )
9 recoscl 12734 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  ( cos `  ( A  / 
2 ) )  e.  RR )
108, 9remulcld 9108 . . . . 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 10074 . . . . . . . . . 10  |-  0  <  2
13 divgt0 9870 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
0  <  ( A  /  2 ) )
142, 12, 13mpanr12 667 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A  /  2 ) )
15143adant3 977 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  0  <  ( A  /  2
) )
162, 12pm3.2i 442 . . . . . . . . . . . 12  |-  ( 2  e.  RR  /\  0  <  2 )
17 lediv1 9867 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  2  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( A  <_ 
2  <->  ( A  / 
2 )  <_  (
2  /  2 ) ) )
182, 16, 17mp3an23 1271 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  <_  2  <->  ( A  /  2 )  <_ 
( 2  /  2
) ) )
1918biimpa 471 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  <_  2 )  -> 
( A  /  2
)  <_  ( 2  /  2 ) )
20 2cn 10062 . . . . . . . . . . 11  |-  2  e.  CC
21 2ne0 10075 . . . . . . . . . . 11  |-  2  =/=  0
2220, 21dividi 9739 . . . . . . . . . 10  |-  ( 2  /  2 )  =  1
2319, 22syl6breq 4243 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <_  2 )  -> 
( A  /  2
)  <_  1 )
24233adant2 976 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  ( A  /  2 )  <_ 
1 )
256, 15, 243jca 1134 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  (
( A  /  2
)  e.  RR  /\  0  <  ( A  / 
2 )  /\  ( A  /  2 )  <_ 
1 ) )
26 1re 9082 . . . . . . . 8  |-  1  e.  RR
27 elioc2 10965 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  (
( A  /  2
)  e.  ( 0 (,] 1 )  <->  ( ( A  /  2 )  e.  RR  /\  0  < 
( A  /  2
)  /\  ( A  /  2 )  <_ 
1 ) ) )
281, 26, 27mp2an 654 . . . . . . 7  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  <->  ( ( A  /  2 )  e.  RR  /\  0  < 
( A  /  2
)  /\  ( A  /  2 )  <_ 
1 ) )
2925, 4, 283imtr4i 258 . . . . . 6  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  ( 0 (,] 1
) )
30 sin01gt0 12783 . . . . . 6  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  ( A  /  2 ) ) )
3129, 30syl 16 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  ( A  /  2 ) ) )
32 cos01gt0 12784 . . . . . 6  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  ( A  /  2 ) ) )
3329, 32syl 16 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( cos `  ( A  /  2 ) ) )
34 axmulgt0 9142 . . . . . . 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 ) ) ) ) )
358, 9, 34syl2anc 643 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
367, 35syl 16 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
3731, 33, 36mp2and 661 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) )
38 axmulgt0 9142 . . . . . 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 ) ) ) ) ) )
392, 38mpan 652 . . . . 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 ) ) ) ) ) )
4012, 39mpani 658 . . . 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 ) ) ) ) ) )
4111, 37, 40sylc 58 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
427recnd 9106 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  CC )
43 sin2t 12770 . . . 4  |-  ( ( A  /  2 )  e.  CC  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
4442, 43syl 16 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
4541, 44breqtrrd 4230 . 2  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  (
2  x.  ( A  /  2 ) ) ) )
464simp1bi 972 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  A  e.  RR )
4746recnd 9106 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  A  e.  CC )
48 divcan2 9678 . . . . 5  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
2  x.  ( A  /  2 ) )  =  A )
4920, 21, 48mp3an23 1271 . . . 4  |-  ( A  e.  CC  ->  (
2  x.  ( A  /  2 ) )  =  A )
5047, 49syl 16 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  (
2  x.  ( A  /  2 ) )  =  A )
5150fveq2d 5724 . 2  |-  ( A  e.  ( 0 (,] 2 )  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( sin `  A
) )
5245, 51breqtrd 4228 1  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987   RR*cxr 9111    < clt 9112    <_ cle 9113    / cdiv 9669   2c2 10041   (,]cioc 10909   sincsin 12658   cosccos 12659
This theorem is referenced by:  sincos2sgn  12787  pilem2  20360  sinhalfpilem  20366  sincosq1lem  20397
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ioc 10913  df-ico 10914  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665
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