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Theorem sinq12gt0 20417
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 9133 . . 3  |-  0  e.  RR*
2 pire 20374 . . . 4  |-  pi  e.  RR
32rexri 9139 . . 3  |-  pi  e.  RR*
4 elioo2 10959 . . 3  |-  ( ( 0  e.  RR*  /\  pi  e.  RR* )  ->  ( A  e.  ( 0 (,) pi )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  pi ) ) )
51, 3, 4mp2an 655 . 2  |-  ( A  e.  ( 0 (,) pi )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  pi ) )
6 rehalfcl 10196 . . . . . 6  |-  ( A  e.  RR  ->  ( A  /  2 )  e.  RR )
763ad2ant1 979 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( A  /  2 )  e.  RR )
8 halfpos2 10199 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <  ( A  /  2 ) ) )
98biimpa 472 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A  /  2 ) )
1093adant3 978 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( A  /  2
) )
11 2re 10071 . . . . . . . . 9  |-  2  e.  RR
12 2pos 10084 . . . . . . . . 9  |-  0  <  2
1311, 12pm3.2i 443 . . . . . . . 8  |-  ( 2  e.  RR  /\  0  <  2 )
14 ltdiv1 9876 . . . . . . . 8  |-  ( ( A  e.  RR  /\  pi  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( A  < 
pi 
<->  ( A  /  2
)  <  ( pi  /  2 ) ) )
152, 13, 14mp3an23 1272 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  <  pi  <->  ( A  /  2 )  < 
( pi  /  2
) ) )
1615adantr 453 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  <  pi  <->  ( A  /  2 )  <  ( pi  / 
2 ) ) )
1716biimp3a 1284 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( A  /  2 )  < 
( pi  /  2
) )
18 sincosq1lem 20407 . . . . 5  |-  ( ( ( A  /  2
)  e.  RR  /\  0  <  ( A  / 
2 )  /\  ( A  /  2 )  < 
( pi  /  2
) )  ->  0  <  ( sin `  ( A  /  2 ) ) )
197, 10, 17, 18syl3anc 1185 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( sin `  ( A  /  2 ) ) )
20 resubcl 9367 . . . . . . . . 9  |-  ( ( pi  e.  RR  /\  A  e.  RR )  ->  ( pi  -  A
)  e.  RR )
212, 20mpan 653 . . . . . . . 8  |-  ( A  e.  RR  ->  (
pi  -  A )  e.  RR )
22 rehalfcl 10196 . . . . . . . 8  |-  ( ( pi  -  A )  e.  RR  ->  (
( pi  -  A
)  /  2 )  e.  RR )
2321, 22syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  (
( pi  -  A
)  /  2 )  e.  RR )
24233ad2ant1 979 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  (
( pi  -  A
)  /  2 )  e.  RR )
25 posdif 9523 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  pi  e.  RR )  -> 
( A  <  pi  <->  0  <  ( pi  -  A ) ) )
262, 25mpan2 654 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  <  pi  <->  0  <  ( pi  -  A ) ) )
27 halfpos2 10199 . . . . . . . . . 10  |-  ( ( pi  -  A )  e.  RR  ->  (
0  <  ( pi  -  A )  <->  0  <  ( ( pi  -  A
)  /  2 ) ) )
2821, 27syl 16 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
0  <  ( pi  -  A )  <->  0  <  ( ( pi  -  A
)  /  2 ) ) )
2926, 28bitrd 246 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  pi  <->  0  <  ( ( pi  -  A
)  /  2 ) ) )
3029adantr 453 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  <  pi  <->  0  <  ( ( pi 
-  A )  / 
2 ) ) )
3130biimp3a 1284 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( ( pi  -  A )  /  2
) )
32 ltsubpos 9522 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  pi  e.  RR )  -> 
( 0  <  A  <->  ( pi  -  A )  <  pi ) )
332, 32mpan2 654 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
0  <  A  <->  ( pi  -  A )  <  pi ) )
34 ltdiv1 9876 . . . . . . . . . . 11  |-  ( ( ( pi  -  A
)  e.  RR  /\  pi  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( pi 
-  A )  < 
pi 
<->  ( ( pi  -  A )  /  2
)  <  ( pi  /  2 ) ) )
352, 13, 34mp3an23 1272 . . . . . . . . . 10  |-  ( ( pi  -  A )  e.  RR  ->  (
( pi  -  A
)  <  pi  <->  ( (
pi  -  A )  /  2 )  < 
( pi  /  2
) ) )
3621, 35syl 16 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( pi  -  A
)  <  pi  <->  ( (
pi  -  A )  /  2 )  < 
( pi  /  2
) ) )
3733, 36bitrd 246 . . . . . . . 8  |-  ( A  e.  RR  ->  (
0  <  A  <->  ( (
pi  -  A )  /  2 )  < 
( pi  /  2
) ) )
3837biimpa 472 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( pi  -  A )  /  2
)  <  ( pi  /  2 ) )
39383adant3 978 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  (
( pi  -  A
)  /  2 )  <  ( pi  / 
2 ) )
40 sincosq1lem 20407 . . . . . 6  |-  ( ( ( ( pi  -  A )  /  2
)  e.  RR  /\  0  <  ( ( pi 
-  A )  / 
2 )  /\  (
( pi  -  A
)  /  2 )  <  ( pi  / 
2 ) )  -> 
0  <  ( sin `  ( ( pi  -  A )  /  2
) ) )
4124, 31, 39, 40syl3anc 1185 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( sin `  (
( pi  -  A
)  /  2 ) ) )
42 recn 9082 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
432recni 9104 . . . . . . . . . 10  |-  pi  e.  CC
44 2cn 10072 . . . . . . . . . . 11  |-  2  e.  CC
45 2ne0 10085 . . . . . . . . . . 11  |-  2  =/=  0
4644, 45pm3.2i 443 . . . . . . . . . 10  |-  ( 2  e.  CC  /\  2  =/=  0 )
47 divsubdir 9712 . . . . . . . . . 10  |-  ( ( pi  e.  CC  /\  A  e.  CC  /\  (
2  e.  CC  /\  2  =/=  0 ) )  ->  ( ( pi 
-  A )  / 
2 )  =  ( ( pi  /  2
)  -  ( A  /  2 ) ) )
4843, 46, 47mp3an13 1271 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( pi  -  A
)  /  2 )  =  ( ( pi 
/  2 )  -  ( A  /  2
) ) )
4942, 48syl 16 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( pi  -  A
)  /  2 )  =  ( ( pi 
/  2 )  -  ( A  /  2
) ) )
5049fveq2d 5734 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
-  A )  / 
2 ) )  =  ( sin `  (
( pi  /  2
)  -  ( A  /  2 ) ) ) )
516recnd 9116 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  /  2 )  e.  CC )
52 sinhalfpim 20403 . . . . . . . 8  |-  ( ( A  /  2 )  e.  CC  ->  ( sin `  ( ( pi 
/  2 )  -  ( A  /  2
) ) )  =  ( cos `  ( A  /  2 ) ) )
5351, 52syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  -  ( A  /  2
) ) )  =  ( cos `  ( A  /  2 ) ) )
5450, 53eqtrd 2470 . . . . . 6  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
-  A )  / 
2 ) )  =  ( cos `  ( A  /  2 ) ) )
55543ad2ant1 979 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( sin `  ( ( pi 
-  A )  / 
2 ) )  =  ( cos `  ( A  /  2 ) ) )
5641, 55breqtrd 4238 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( cos `  ( A  /  2 ) ) )
57 resincl 12743 . . . . . . . 8  |-  ( ( A  /  2 )  e.  RR  ->  ( sin `  ( A  / 
2 ) )  e.  RR )
58 recoscl 12744 . . . . . . . 8  |-  ( ( A  /  2 )  e.  RR  ->  ( cos `  ( A  / 
2 ) )  e.  RR )
5957, 58jca 520 . . . . . . 7  |-  ( ( A  /  2 )  e.  RR  ->  (
( sin `  ( A  /  2 ) )  e.  RR  /\  ( cos `  ( A  / 
2 ) )  e.  RR ) )
60 axmulgt0 9152 . . . . . . 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 19 . . . . . 6  |-  ( A  e.  RR  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
62 remulcl 9077 . . . . . . . . 9  |-  ( ( ( sin `  ( A  /  2 ) )  e.  RR  /\  ( cos `  ( A  / 
2 ) )  e.  RR )  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
636, 59, 623syl 19 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
64 axmulgt0 9152 . . . . . . . 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 646 . . . . . . 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 659 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <  ( ( sin `  ( A  / 
2 ) )  x.  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( 2  x.  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
6761, 66syld 43 . . . . 5  |-  ( A  e.  RR  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( 2  x.  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
68673ad2ant1 979 . . . 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 662 . . 3  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
70 divcan2 9688 . . . . . . . 8  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
2  x.  ( A  /  2 ) )  =  A )
7144, 45, 70mp3an23 1272 . . . . . . 7  |-  ( A  e.  CC  ->  (
2  x.  ( A  /  2 ) )  =  A )
7242, 71syl 16 . . . . . 6  |-  ( A  e.  RR  ->  (
2  x.  ( A  /  2 ) )  =  A )
7372fveq2d 5734 . . . . 5  |-  ( A  e.  RR  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( sin `  A
) )
74 sin2t 12780 . . . . . 6  |-  ( ( A  /  2 )  e.  CC  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
7551, 74syl 16 . . . . 5  |-  ( A  e.  RR  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
7673, 75eqtr3d 2472 . . . 4  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
77763ad2ant1 979 . . 3  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( sin `  A )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
7869, 77breqtrrd 4240 . 2  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( sin `  A
) )
795, 78sylbi 189 1  |-  ( A  e.  ( 0 (,) pi )  ->  0  <  ( sin `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992    x. cmul 8997   RR*cxr 9121    < clt 9122    - cmin 9293    / cdiv 9679   2c2 10051   (,)cioo 10918   sincsin 12668   cosccos 12669   picpi 12671
This theorem is referenced by:  sinq12ge0  20418  sinq34lt0t  20419  cosq14gt0  20420  sineq0  20431  cosordlem  20435  wallispilem1  27792  sineq0ALT  29111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ioc 10923  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482  df-ef 12672  df-sin 12674  df-cos 12675  df-pi 12677  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756
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