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Theorem sinq12gt0 12927
Description: The sine of a number strictly between  0 and  pi is positive. (Contributed by Paul Chapman, 15-Mar-2008.)
Assertion
Ref Expression
sinq12gt0  |-  ( A  e.  ( 0 (,) pi )  ->  0  <  ( sin `  A
) )

Proof of Theorem sinq12gt0
StepHypRef Expression
1 0xr 7819 . . 3  |-  0  e.  RR*
2 pire 12883 . . . 4  |-  pi  e.  RR
32rexri 7830 . . 3  |-  pi  e.  RR*
4 elioo2 9711 . . 3  |-  ( ( 0  e.  RR*  /\  pi  e.  RR* )  ->  ( A  e.  ( 0 (,) pi )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  pi ) ) )
51, 3, 4mp2an 422 . 2  |-  ( A  e.  ( 0 (,) pi )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  pi ) )
6 rehalfcl 8954 . . . . . 6  |-  ( A  e.  RR  ->  ( A  /  2 )  e.  RR )
763ad2ant1 1002 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( A  /  2 )  e.  RR )
8 halfpos2 8957 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <  ( A  /  2 ) ) )
98biimpa 294 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A  /  2 ) )
1093adant3 1001 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( A  /  2
) )
11 2re 8797 . . . . . . . . 9  |-  2  e.  RR
12 2pos 8818 . . . . . . . . 9  |-  0  <  2
1311, 12pm3.2i 270 . . . . . . . 8  |-  ( 2  e.  RR  /\  0  <  2 )
14 ltdiv1 8633 . . . . . . . 8  |-  ( ( A  e.  RR  /\  pi  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( A  < 
pi 
<->  ( A  /  2
)  <  ( pi  /  2 ) ) )
152, 13, 14mp3an23 1307 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  <  pi  <->  ( A  /  2 )  < 
( pi  /  2
) ) )
1615adantr 274 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  <  pi  <->  ( A  /  2 )  <  ( pi  / 
2 ) ) )
1716biimp3a 1323 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( A  /  2 )  < 
( pi  /  2
) )
18 sincosq1lem 12922 . . . . 5  |-  ( ( ( A  /  2
)  e.  RR  /\  0  <  ( A  / 
2 )  /\  ( A  /  2 )  < 
( pi  /  2
) )  ->  0  <  ( sin `  ( A  /  2 ) ) )
197, 10, 17, 18syl3anc 1216 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( sin `  ( A  /  2 ) ) )
20 resubcl 8033 . . . . . . . . 9  |-  ( ( pi  e.  RR  /\  A  e.  RR )  ->  ( pi  -  A
)  e.  RR )
212, 20mpan 420 . . . . . . . 8  |-  ( A  e.  RR  ->  (
pi  -  A )  e.  RR )
22 rehalfcl 8954 . . . . . . . 8  |-  ( ( pi  -  A )  e.  RR  ->  (
( pi  -  A
)  /  2 )  e.  RR )
2321, 22syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  (
( pi  -  A
)  /  2 )  e.  RR )
24233ad2ant1 1002 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  (
( pi  -  A
)  /  2 )  e.  RR )
25 posdif 8224 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  pi  e.  RR )  -> 
( A  <  pi  <->  0  <  ( pi  -  A ) ) )
262, 25mpan2 421 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  <  pi  <->  0  <  ( pi  -  A ) ) )
27 halfpos2 8957 . . . . . . . . . 10  |-  ( ( pi  -  A )  e.  RR  ->  (
0  <  ( pi  -  A )  <->  0  <  ( ( pi  -  A
)  /  2 ) ) )
2821, 27syl 14 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
0  <  ( pi  -  A )  <->  0  <  ( ( pi  -  A
)  /  2 ) ) )
2926, 28bitrd 187 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  pi  <->  0  <  ( ( pi  -  A
)  /  2 ) ) )
3029adantr 274 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  <  pi  <->  0  <  ( ( pi 
-  A )  / 
2 ) ) )
3130biimp3a 1323 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( ( pi  -  A )  /  2
) )
32 ltsubpos 8223 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  pi  e.  RR )  -> 
( 0  <  A  <->  ( pi  -  A )  <  pi ) )
332, 32mpan2 421 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
0  <  A  <->  ( pi  -  A )  <  pi ) )
34 ltdiv1 8633 . . . . . . . . . . 11  |-  ( ( ( pi  -  A
)  e.  RR  /\  pi  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( pi 
-  A )  < 
pi 
<->  ( ( pi  -  A )  /  2
)  <  ( pi  /  2 ) ) )
352, 13, 34mp3an23 1307 . . . . . . . . . 10  |-  ( ( pi  -  A )  e.  RR  ->  (
( pi  -  A
)  <  pi  <->  ( (
pi  -  A )  /  2 )  < 
( pi  /  2
) ) )
3621, 35syl 14 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( pi  -  A
)  <  pi  <->  ( (
pi  -  A )  /  2 )  < 
( pi  /  2
) ) )
3733, 36bitrd 187 . . . . . . . 8  |-  ( A  e.  RR  ->  (
0  <  A  <->  ( (
pi  -  A )  /  2 )  < 
( pi  /  2
) ) )
3837biimpa 294 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( pi  -  A )  /  2
)  <  ( pi  /  2 ) )
39383adant3 1001 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  (
( pi  -  A
)  /  2 )  <  ( pi  / 
2 ) )
40 sincosq1lem 12922 . . . . . 6  |-  ( ( ( ( pi  -  A )  /  2
)  e.  RR  /\  0  <  ( ( pi 
-  A )  / 
2 )  /\  (
( pi  -  A
)  /  2 )  <  ( pi  / 
2 ) )  -> 
0  <  ( sin `  ( ( pi  -  A )  /  2
) ) )
4124, 31, 39, 40syl3anc 1216 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( sin `  (
( pi  -  A
)  /  2 ) ) )
42 recn 7760 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
43 picn 12884 . . . . . . . . . 10  |-  pi  e.  CC
44 2cn 8798 . . . . . . . . . . 11  |-  2  e.  CC
45 2ap0 8820 . . . . . . . . . . 11  |-  2 #  0
4644, 45pm3.2i 270 . . . . . . . . . 10  |-  ( 2  e.  CC  /\  2 #  0 )
47 divsubdirap 8475 . . . . . . . . . 10  |-  ( ( pi  e.  CC  /\  A  e.  CC  /\  (
2  e.  CC  /\  2 #  0 ) )  -> 
( ( pi  -  A )  /  2
)  =  ( ( pi  /  2 )  -  ( A  / 
2 ) ) )
4843, 46, 47mp3an13 1306 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( pi  -  A
)  /  2 )  =  ( ( pi 
/  2 )  -  ( A  /  2
) ) )
4942, 48syl 14 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( pi  -  A
)  /  2 )  =  ( ( pi 
/  2 )  -  ( A  /  2
) ) )
5049fveq2d 5425 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
-  A )  / 
2 ) )  =  ( sin `  (
( pi  /  2
)  -  ( A  /  2 ) ) ) )
516recnd 7801 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  /  2 )  e.  CC )
52 sinhalfpim 12918 . . . . . . . 8  |-  ( ( A  /  2 )  e.  CC  ->  ( sin `  ( ( pi 
/  2 )  -  ( A  /  2
) ) )  =  ( cos `  ( A  /  2 ) ) )
5351, 52syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  -  ( A  /  2
) ) )  =  ( cos `  ( A  /  2 ) ) )
5450, 53eqtrd 2172 . . . . . 6  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
-  A )  / 
2 ) )  =  ( cos `  ( A  /  2 ) ) )
55543ad2ant1 1002 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( sin `  ( ( pi 
-  A )  / 
2 ) )  =  ( cos `  ( A  /  2 ) ) )
5641, 55breqtrd 3954 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( cos `  ( A  /  2 ) ) )
57 resincl 11433 . . . . . . . 8  |-  ( ( A  /  2 )  e.  RR  ->  ( sin `  ( A  / 
2 ) )  e.  RR )
58 recoscl 11434 . . . . . . . 8  |-  ( ( A  /  2 )  e.  RR  ->  ( cos `  ( A  / 
2 ) )  e.  RR )
5957, 58jca 304 . . . . . . 7  |-  ( ( A  /  2 )  e.  RR  ->  (
( sin `  ( A  /  2 ) )  e.  RR  /\  ( cos `  ( A  / 
2 ) )  e.  RR ) )
60 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 ) ) ) ) )
616, 59, 603syl 17 . . . . . 6  |-  ( A  e.  RR  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
62 remulcl 7755 . . . . . . . . 9  |-  ( ( ( sin `  ( A  /  2 ) )  e.  RR  /\  ( cos `  ( A  / 
2 ) )  e.  RR )  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
636, 59, 623syl 17 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
64 axmulgt0 7843 . . . . . . . 8  |-  ( ( 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 ) ) ) ) ) )
6511, 63, 64sylancr 410 . . . . . . 7  |-  ( A  e.  RR  ->  (
( 0  <  2  /\  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) )  ->  0  <  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
6612, 65mpani 426 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <  ( ( sin `  ( A  / 
2 ) )  x.  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( 2  x.  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
6761, 66syld 45 . . . . 5  |-  ( A  e.  RR  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( 2  x.  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
68673ad2ant1 1002 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( 2  x.  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
6919, 56, 68mp2and 429 . . 3  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
70 divcanap2 8447 . . . . . . . 8  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2 #  0 )  ->  (
2  x.  ( A  /  2 ) )  =  A )
7144, 45, 70mp3an23 1307 . . . . . . 7  |-  ( A  e.  CC  ->  (
2  x.  ( A  /  2 ) )  =  A )
7242, 71syl 14 . . . . . 6  |-  ( A  e.  RR  ->  (
2  x.  ( A  /  2 ) )  =  A )
7372fveq2d 5425 . . . . 5  |-  ( A  e.  RR  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( sin `  A
) )
74 sin2t 11462 . . . . . 6  |-  ( ( A  /  2 )  e.  CC  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
7551, 74syl 14 . . . . 5  |-  ( A  e.  RR  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
7673, 75eqtr3d 2174 . . . 4  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
77763ad2ant1 1002 . . 3  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( sin `  A )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
7869, 77breqtrrd 3956 . 2  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( sin `  A
) )
795, 78sylbi 120 1  |-  ( A  e.  ( 0 (,) pi )  ->  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    x. cmul 7632   RR*cxr 7806    < clt 7807    - cmin 7940   # cap 8350    / cdiv 8439   2c2 8778   (,)cioo 9678   sincsin 11357   cosccos 11358   picpi 11360
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  ax-pre-suploc 7748  ax-addf 7749  ax-mulf 7750
This theorem depends on definitions:  df-bi 116  df-stab 816  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-of 5982  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-map 6544  df-pm 6545  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-inf 6872  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-9 8793  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-xneg 9566  df-xadd 9567  df-ioo 9682  df-ioc 9683  df-ico 9684  df-icc 9685  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  df-pi 11366  df-rest 12131  df-topgen 12150  df-psmet 12165  df-xmet 12166  df-met 12167  df-bl 12168  df-mopn 12169  df-top 12174  df-topon 12187  df-bases 12219  df-ntr 12274  df-cn 12366  df-cnp 12367  df-tx 12431  df-cncf 12736  df-limced 12803  df-dvap 12804
This theorem is referenced by:  sinq34lt0t  12928  cosq14gt0  12929  cosordlem  12946
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