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Theorem sin02gt0 11477
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 7819 . . . . . . 7  |-  0  e.  RR*
2 2re 8797 . . . . . . 7  |-  2  e.  RR
3 elioc2 9726 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  2  e.  RR )  ->  ( A  e.  ( 0 (,] 2 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  2 ) ) )
41, 2, 3mp2an 422 . . . . . 6  |-  ( A  e.  ( 0 (,] 2 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  2 ) )
5 rehalfcl 8954 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  /  2 )  e.  RR )
653ad2ant1 1002 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  ( A  /  2 )  e.  RR )
74, 6sylbi 120 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  RR )
8 resincl 11434 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  ( sin `  ( A  / 
2 ) )  e.  RR )
9 recoscl 11435 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  ( cos `  ( A  / 
2 ) )  e.  RR )
108, 9remulcld 7803 . . . . 5  |-  ( ( A  /  2 )  e.  RR  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
117, 10syl 14 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
12 2pos 8818 . . . . . . . . . 10  |-  0  <  2
13 divgt0 8637 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
0  <  ( A  /  2 ) )
142, 12, 13mpanr12 435 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A  /  2 ) )
15143adant3 1001 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  0  <  ( A  /  2
) )
162, 12pm3.2i 270 . . . . . . . . . . . 12  |-  ( 2  e.  RR  /\  0  <  2 )
17 lediv1 8634 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  2  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( A  <_ 
2  <->  ( A  / 
2 )  <_  (
2  /  2 ) ) )
182, 16, 17mp3an23 1307 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  <_  2  <->  ( A  /  2 )  <_ 
( 2  /  2
) ) )
1918biimpa 294 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  <_  2 )  -> 
( A  /  2
)  <_  ( 2  /  2 ) )
20 2div2e1 8859 . . . . . . . . . 10  |-  ( 2  /  2 )  =  1
2119, 20breqtrdi 3969 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <_  2 )  -> 
( A  /  2
)  <_  1 )
22213adant2 1000 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  ( A  /  2 )  <_ 
1 )
236, 15, 223jca 1161 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  (
( A  /  2
)  e.  RR  /\  0  <  ( A  / 
2 )  /\  ( A  /  2 )  <_ 
1 ) )
24 1re 7772 . . . . . . . 8  |-  1  e.  RR
25 elioc2 9726 . . . . . . . 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 422 . . . . . . 7  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  <->  ( ( A  /  2 )  e.  RR  /\  0  < 
( A  /  2
)  /\  ( A  /  2 )  <_ 
1 ) )
2723, 4, 263imtr4i 200 . . . . . 6  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  ( 0 (,] 1
) )
28 sin01gt0 11475 . . . . . 6  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  ( A  /  2 ) ) )
2927, 28syl 14 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  ( A  /  2 ) ) )
30 cos01gt0 11476 . . . . . 6  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  ( A  /  2 ) ) )
3127, 30syl 14 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( cos `  ( A  /  2 ) ) )
32 axmulgt0 7843 . . . . . . 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 408 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
347, 33syl 14 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
3529, 31, 34mp2and 429 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) )
36 axmulgt0 7843 . . . . . 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 420 . . . . 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 426 . . . 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 62 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
407recnd 7801 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  CC )
41 sin2t 11463 . . . 4  |-  ( ( A  /  2 )  e.  CC  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
4240, 41syl 14 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
4339, 42breqtrrd 3956 . 2  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  (
2  x.  ( A  /  2 ) ) ) )
444simp1bi 996 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  A  e.  RR )
4544recnd 7801 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  A  e.  CC )
46 2cn 8798 . . . . 5  |-  2  e.  CC
47 2ap0 8820 . . . . 5  |-  2 #  0
48 divcanap2 8447 . . . . 5  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2 #  0 )  ->  (
2  x.  ( A  /  2 ) )  =  A )
4946, 47, 48mp3an23 1307 . . . 4  |-  ( A  e.  CC  ->  (
2  x.  ( A  /  2 ) )  =  A )
5045, 49syl 14 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  (
2  x.  ( A  /  2 ) )  =  A )
5150fveq2d 5425 . 2  |-  ( A  e.  ( 0 (,] 2 )  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( sin `  A
) )
5243, 51breqtrd 3954 1  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7625   RRcr 7626   0cc0 7627   1c1 7628    x. cmul 7632   RR*cxr 7806    < clt 7807    <_ cle 7808   # cap 8350    / cdiv 8439   2c2 8778   (,]cioc 9679   sincsin 11357   cosccos 11358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-5 8789  df-6 8790  df-7 8791  df-8 8792  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-ioc 9683  df-ico 9684  df-fz 9798  df-fzo 9927  df-seqfrec 10226  df-exp 10300  df-fac 10479  df-bc 10501  df-ihash 10529  df-shft 10594  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055  df-sumdc 11130  df-ef 11361  df-sin 11363  df-cos 11364
This theorem is referenced by:  sincos2sgn  11479  cos12dec  11481  sin0pilem1  12875  sin0pilem2  12876  sinhalfpilem  12885  sincosq1lem  12919
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