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Theorem sin02gt0 12721
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 9065 . . . . . . 7  |-  0  e.  RR*
2 2re 10002 . . . . . . 7  |-  2  e.  RR
3 elioc2 10906 . . . . . . 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 10127 . . . . . . 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 12669 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  ( sin `  ( A  / 
2 ) )  e.  RR )
9 recoscl 12670 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  ( cos `  ( A  / 
2 ) )  e.  RR )
108, 9remulcld 9050 . . . . 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 10015 . . . . . . . . . 10  |-  0  <  2
13 divgt0 9811 . . . . . . . . . 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 9808 . . . . . . . . . . . 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 10003 . . . . . . . . . . 11  |-  2  e.  CC
21 2ne0 10016 . . . . . . . . . . 11  |-  2  =/=  0
2220, 21dividi 9680 . . . . . . . . . 10  |-  ( 2  /  2 )  =  1
2319, 22syl6breq 4193 . . . . . . . . 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 9024 . . . . . . . 8  |-  1  e.  RR
27 elioc2 10906 . . . . . . . 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 12719 . . . . . 6  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  ( A  /  2 ) ) )
3129, 30syl 16 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  ( A  /  2 ) ) )
32 cos01gt0 12720 . . . . . 6  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  ( A  /  2 ) ) )
3329, 32syl 16 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( cos `  ( A  /  2 ) ) )
34 axmulgt0 9084 . . . . . . 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 9084 . . . . . 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 9048 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  CC )
43 sin2t 12706 . . . 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 4180 . 2  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  (
2  x.  ( A  /  2 ) ) ) )
464simp1bi 972 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  A  e.  RR )
4746recnd 9048 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  A  e.  CC )
48 divcan2 9619 . . . . 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 5673 . 2  |-  ( A  e.  ( 0 (,] 2 )  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( sin `  A
) )
5245, 51breqtrd 4178 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 1649    e. wcel 1717    =/= wne 2551   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    x. cmul 8929   RR*cxr 9053    < clt 9054    <_ cle 9055    / cdiv 9610   2c2 9982   (,]cioc 10850   sincsin 12594   cosccos 12595
This theorem is referenced by:  sincos2sgn  12723  pilem2  20236  sinhalfpilem  20242  sincosq1lem  20273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-ioc 10854  df-ico 10855  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-ef 12598  df-sin 12600  df-cos 12601
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