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Theorem sinq12gt0 23066
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 9629 . . 3  |-  0  e.  RR*
2 pire 23017 . . . 4  |-  pi  e.  RR
32rexri 9635 . . 3  |-  pi  e.  RR*
4 elioo2 11573 . . 3  |-  ( ( 0  e.  RR*  /\  pi  e.  RR* )  ->  ( A  e.  ( 0 (,) pi )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  pi ) ) )
51, 3, 4mp2an 670 . 2  |-  ( A  e.  ( 0 (,) pi )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  pi ) )
6 rehalfcl 10761 . . . . . 6  |-  ( A  e.  RR  ->  ( A  /  2 )  e.  RR )
763ad2ant1 1015 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( A  /  2 )  e.  RR )
8 halfpos2 10764 . . . . . . 7  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <  ( A  /  2 ) ) )
98biimpa 482 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A  /  2 ) )
1093adant3 1014 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( A  /  2
) )
11 2re 10601 . . . . . . . . 9  |-  2  e.  RR
12 2pos 10623 . . . . . . . . 9  |-  0  <  2
1311, 12pm3.2i 453 . . . . . . . 8  |-  ( 2  e.  RR  /\  0  <  2 )
14 ltdiv1 10402 . . . . . . . 8  |-  ( ( A  e.  RR  /\  pi  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( A  < 
pi 
<->  ( A  /  2
)  <  ( pi  /  2 ) ) )
152, 13, 14mp3an23 1314 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  <  pi  <->  ( A  /  2 )  < 
( pi  /  2
) ) )
1615adantr 463 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  <  pi  <->  ( A  /  2 )  <  ( pi  / 
2 ) ) )
1716biimp3a 1326 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( A  /  2 )  < 
( pi  /  2
) )
18 sincosq1lem 23056 . . . . 5  |-  ( ( ( A  /  2
)  e.  RR  /\  0  <  ( A  / 
2 )  /\  ( A  /  2 )  < 
( pi  /  2
) )  ->  0  <  ( sin `  ( A  /  2 ) ) )
197, 10, 17, 18syl3anc 1226 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( sin `  ( A  /  2 ) ) )
20 resubcl 9874 . . . . . . . . 9  |-  ( ( pi  e.  RR  /\  A  e.  RR )  ->  ( pi  -  A
)  e.  RR )
212, 20mpan 668 . . . . . . . 8  |-  ( A  e.  RR  ->  (
pi  -  A )  e.  RR )
22 rehalfcl 10761 . . . . . . . 8  |-  ( ( pi  -  A )  e.  RR  ->  (
( pi  -  A
)  /  2 )  e.  RR )
2321, 22syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  (
( pi  -  A
)  /  2 )  e.  RR )
24233ad2ant1 1015 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  (
( pi  -  A
)  /  2 )  e.  RR )
25 posdif 10041 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  pi  e.  RR )  -> 
( A  <  pi  <->  0  <  ( pi  -  A ) ) )
262, 25mpan2 669 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  <  pi  <->  0  <  ( pi  -  A ) ) )
27 halfpos2 10764 . . . . . . . . . 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 253 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <  pi  <->  0  <  ( ( pi  -  A
)  /  2 ) ) )
3029adantr 463 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  <  pi  <->  0  <  ( ( pi 
-  A )  / 
2 ) ) )
3130biimp3a 1326 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( ( pi  -  A )  /  2
) )
32 ltsubpos 10040 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  pi  e.  RR )  -> 
( 0  <  A  <->  ( pi  -  A )  <  pi ) )
332, 32mpan2 669 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
0  <  A  <->  ( pi  -  A )  <  pi ) )
34 ltdiv1 10402 . . . . . . . . . . 11  |-  ( ( ( pi  -  A
)  e.  RR  /\  pi  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( pi 
-  A )  < 
pi 
<->  ( ( pi  -  A )  /  2
)  <  ( pi  /  2 ) ) )
352, 13, 34mp3an23 1314 . . . . . . . . . 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 253 . . . . . . . 8  |-  ( A  e.  RR  ->  (
0  <  A  <->  ( (
pi  -  A )  /  2 )  < 
( pi  /  2
) ) )
3837biimpa 482 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( pi  -  A )  /  2
)  <  ( pi  /  2 ) )
39383adant3 1014 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  (
( pi  -  A
)  /  2 )  <  ( pi  / 
2 ) )
40 sincosq1lem 23056 . . . . . 6  |-  ( ( ( ( pi  -  A )  /  2
)  e.  RR  /\  0  <  ( ( pi 
-  A )  / 
2 )  /\  (
( pi  -  A
)  /  2 )  <  ( pi  / 
2 ) )  -> 
0  <  ( sin `  ( ( pi  -  A )  /  2
) ) )
4124, 31, 39, 40syl3anc 1226 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( sin `  (
( pi  -  A
)  /  2 ) ) )
42 recn 9571 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
43 picn 23018 . . . . . . . . . 10  |-  pi  e.  CC
44 2cnne0 10746 . . . . . . . . . 10  |-  ( 2  e.  CC  /\  2  =/=  0 )
45 divsubdir 10236 . . . . . . . . . 10  |-  ( ( pi  e.  CC  /\  A  e.  CC  /\  (
2  e.  CC  /\  2  =/=  0 ) )  ->  ( ( pi 
-  A )  / 
2 )  =  ( ( pi  /  2
)  -  ( A  /  2 ) ) )
4643, 44, 45mp3an13 1313 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( pi  -  A
)  /  2 )  =  ( ( pi 
/  2 )  -  ( A  /  2
) ) )
4742, 46syl 16 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( pi  -  A
)  /  2 )  =  ( ( pi 
/  2 )  -  ( A  /  2
) ) )
4847fveq2d 5852 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
-  A )  / 
2 ) )  =  ( sin `  (
( pi  /  2
)  -  ( A  /  2 ) ) ) )
496recnd 9611 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  /  2 )  e.  CC )
50 sinhalfpim 23052 . . . . . . . 8  |-  ( ( A  /  2 )  e.  CC  ->  ( sin `  ( ( pi 
/  2 )  -  ( A  /  2
) ) )  =  ( cos `  ( A  /  2 ) ) )
5149, 50syl 16 . . . . . . 7  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
/  2 )  -  ( A  /  2
) ) )  =  ( cos `  ( A  /  2 ) ) )
5248, 51eqtrd 2495 . . . . . 6  |-  ( A  e.  RR  ->  ( sin `  ( ( pi 
-  A )  / 
2 ) )  =  ( cos `  ( A  /  2 ) ) )
53523ad2ant1 1015 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( sin `  ( ( pi 
-  A )  / 
2 ) )  =  ( cos `  ( A  /  2 ) ) )
5441, 53breqtrd 4463 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( cos `  ( A  /  2 ) ) )
55 resincl 13957 . . . . . . . 8  |-  ( ( A  /  2 )  e.  RR  ->  ( sin `  ( A  / 
2 ) )  e.  RR )
56 recoscl 13958 . . . . . . . 8  |-  ( ( A  /  2 )  e.  RR  ->  ( cos `  ( A  / 
2 ) )  e.  RR )
5755, 56jca 530 . . . . . . 7  |-  ( ( A  /  2 )  e.  RR  ->  (
( sin `  ( A  /  2 ) )  e.  RR  /\  ( cos `  ( A  / 
2 ) )  e.  RR ) )
58 axmulgt0 9648 . . . . . . 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 ) ) ) ) )
596, 57, 583syl 20 . . . . . 6  |-  ( A  e.  RR  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
60 remulcl 9566 . . . . . . . . 9  |-  ( ( ( sin `  ( A  /  2 ) )  e.  RR  /\  ( cos `  ( A  / 
2 ) )  e.  RR )  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
616, 57, 603syl 20 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
62 axmulgt0 9648 . . . . . . . 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 ) ) ) ) ) )
6311, 61, 62sylancr 661 . . . . . . 7  |-  ( A  e.  RR  ->  (
( 0  <  2  /\  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) )  ->  0  <  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
6412, 63mpani 674 . . . . . 6  |-  ( A  e.  RR  ->  (
0  <  ( ( sin `  ( A  / 
2 ) )  x.  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( 2  x.  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
6559, 64syld 44 . . . . 5  |-  ( A  e.  RR  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( 2  x.  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) ) )
66653ad2ant1 1015 . . . 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 ) ) ) ) ) )
6719, 54, 66mp2and 677 . . 3  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
68 2cn 10602 . . . . . . . 8  |-  2  e.  CC
69 2ne0 10624 . . . . . . . 8  |-  2  =/=  0
70 divcan2 10211 . . . . . . . 8  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
2  x.  ( A  /  2 ) )  =  A )
7168, 69, 70mp3an23 1314 . . . . . . 7  |-  ( A  e.  CC  ->  (
2  x.  ( A  /  2 ) )  =  A )
7242, 71syl 16 . . . . . 6  |-  ( A  e.  RR  ->  (
2  x.  ( A  /  2 ) )  =  A )
7372fveq2d 5852 . . . . 5  |-  ( A  e.  RR  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( sin `  A
) )
74 sin2t 13994 . . . . . 6  |-  ( ( A  /  2 )  e.  CC  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
7549, 74syl 16 . . . . 5  |-  ( A  e.  RR  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
7673, 75eqtr3d 2497 . . . 4  |-  ( A  e.  RR  ->  ( sin `  A )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
77763ad2ant1 1015 . . 3  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  ( sin `  A )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
7867, 77breqtrrd 4465 . 2  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <  pi )  ->  0  <  ( sin `  A
) )
795, 78sylbi 195 1  |-  ( A  e.  ( 0 (,) pi )  ->  0  <  ( sin `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486   RR*cxr 9616    < clt 9617    - cmin 9796    / cdiv 10202   2c2 10581   (,)cioo 11532   sincsin 13881   cosccos 13882   picpi 13884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437
This theorem is referenced by:  sinq12ge0  23067  sinq34lt0t  23068  cosq14gt0  23069  sineq0  23080  cosordlem  23084  tan2h  30287  wallispilem1  32086  sineq0ALT  34138
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