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Theorem sinq12gt0 15931
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 8372 . . 3 |- 0 e. RR*
2 pire 15887 . . . 4 |- pi e. RR
32rexri 8383 . . 3 |- pi e. RR*
4 elioo2 10323 . . 3 |- ( ( 0 e. RR* /\ pi e. RR* ) -> ( A e. ( 0 (,) pi ) <-> ( A e. RR /\ 0 < A /\ A < pi ) ) )
51, 3, 4mp2an 430 . 2 |- ( A e. ( 0 (,) pi ) <-> ( A e. RR /\ 0 < A /\ A < pi ) )
6 rehalfcl 9532 . . . . . 6 |- ( A e. RR -> ( A / 2 ) e. RR )
763ad2ant1 1049 . . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( A / 2 ) e. RR )
8 halfpos2 9535 . . . . . . 7 |- ( A e. RR -> ( 0 < A <-> 0 < ( A / 2 ) ) )
98biimpa 296 . . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A / 2 ) )
1093adant3 1048 . . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( A / 2 ) )
11 2re 9374 . . . . . . . . 9 |- 2 e. RR
12 2pos 9395 . . . . . . . . 9 |- 0 < 2
1311, 12pm3.2i 272 . . . . . . . 8 |- ( 2 e. RR /\ 0 < 2 )
14 ltdiv1 9198 . . . . . . . 8 |- ( ( A e. RR /\ pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A < pi <-> ( A / 2 ) < ( pi / 2 ) ) )
152, 13, 14mp3an23 1370 . . . . . . 7 |- ( A e. RR -> ( A < pi <-> ( A / 2 ) < ( pi / 2 ) ) )
1615adantr 276 . . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> ( A < pi <-> ( A / 2 ) < ( pi / 2 ) ) )
1716biimp3a 1386 . . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( A / 2 ) < ( pi / 2 ) )
18 sincosq1lem 15926 . . . . 5 |- ( ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) < ( pi / 2 ) ) -> 0 < ( sin ` ( A / 2 ) ) )
197, 10, 17, 18syl3anc 1278 . . . 4 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` ( A / 2 ) ) )
20 resubcl 8590 . . . . . . . . 9 |- ( ( pi e. RR /\ A e. RR ) -> ( pi - A ) e. RR )
212, 20mpan 428 . . . . . . . 8 |- ( A e. RR -> ( pi - A ) e. RR )
22 rehalfcl 9532 . . . . . . . 8 |- ( ( pi - A ) e. RR -> ( ( pi - A ) / 2 ) e. RR )
2321, 22syl 14 . . . . . . 7 |- ( A e. RR -> ( ( pi - A ) / 2 ) e. RR )
24233ad2ant1 1049 . . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( ( pi - A ) / 2 ) e. RR )
25 posdif 8783 . . . . . . . . . 10 |- ( ( A e. RR /\ pi e. RR ) -> ( A < pi <-> 0 < ( pi - A ) ) )
262, 25mpan2 429 . . . . . . . . 9 |- ( A e. RR -> ( A < pi <-> 0 < ( pi - A ) ) )
27 halfpos2 9535 . . . . . . . . . 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 188 . . . . . . . 8 |- ( A e. RR -> ( A < pi <-> 0 < ( ( pi - A ) / 2 ) ) )
3029adantr 276 . . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> ( A < pi <-> 0 < ( ( pi - A ) / 2 ) ) )
3130biimp3a 1386 . . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( ( pi - A ) / 2 ) )
32 ltsubpos 8782 . . . . . . . . . 10 |- ( ( A e. RR /\ pi e. RR ) -> ( 0 < A <-> ( pi - A ) < pi ) )
332, 32mpan2 429 . . . . . . . . 9 |- ( A e. RR -> ( 0 < A <-> ( pi - A ) < pi ) )
34 ltdiv1 9198 . . . . . . . . . . 11 |- ( ( ( pi - A ) e. RR /\ pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( pi - A ) < pi <-> ( ( pi - A ) / 2 ) < ( pi / 2 ) ) )
352, 13, 34mp3an23 1370 . . . . . . . . . 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 188 . . . . . . . 8 |- ( A e. RR -> ( 0 < A <-> ( ( pi - A ) / 2 ) < ( pi / 2 ) ) )
3837biimpa 296 . . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> ( ( pi - A ) / 2 ) < ( pi / 2 ) )
39383adant3 1048 . . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( ( pi - A ) / 2 ) < ( pi / 2 ) )
40 sincosq1lem 15926 . . . . . 6 |- ( ( ( ( pi - A ) / 2 ) e. RR /\ 0 < ( ( pi - A ) / 2 ) /\ ( ( pi - A ) / 2 ) < ( pi / 2 ) ) -> 0 < ( sin ` ( ( pi - A ) / 2 ) ) )
4124, 31, 39, 40syl3anc 1278 . . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` ( ( pi - A ) / 2 ) ) )
42 recn 8312 . . . . . . . . 9 |- ( A e. RR -> A e. CC )
43 picn 15888 . . . . . . . . . 10 |- pi e. CC
44 2cn 9375 . . . . . . . . . . 11 |- 2 e. CC
45 2ap0 9397 . . . . . . . . . . 11 |- 2 # 0
4644, 45pm3.2i 272 . . . . . . . . . 10 |- ( 2 e. CC /\ 2 # 0 )
47 divsubdirap 9038 . . . . . . . . . 10 |- ( ( pi e. CC /\ A e. CC /\ ( 2 e. CC /\ 2 # 0 ) ) -> ( ( pi - A ) / 2 ) = ( ( pi / 2 ) - ( A / 2 ) ) )
4843, 46, 47mp3an13 1369 . . . . . . . . 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 5699 . . . . . . 7 |- ( A e. RR -> ( sin ` ( ( pi - A ) / 2 ) ) = ( sin ` ( ( pi / 2 ) - ( A / 2 ) ) ) )
516recnd 8354 . . . . . . . 8 |- ( A e. RR -> ( A / 2 ) e. CC )
52 sinhalfpim 15922 . . . . . . . 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 2271 . . . . . 6 |- ( A e. RR -> ( sin ` ( ( pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) )
55543ad2ant1 1049 . . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( sin ` ( ( pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) )
5641, 55breqtrd 4156 . . . 4 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( cos ` ( A / 2 ) ) )
57 resincl 12487 . . . . . . . 8 |- ( ( A / 2 ) e. RR -> ( sin ` ( A / 2 ) ) e. RR )
58 recoscl 12488 . . . . . . . 8 |- ( ( A / 2 ) e. RR -> ( cos ` ( A / 2 ) ) e. RR )
5957, 58jca 306 . . . . . . 7 |- ( ( A / 2 ) e. RR -> ( ( sin ` ( A / 2 ) ) e. RR /\ ( cos ` ( A / 2 ) ) e. RR ) )
60 axmulgt0 8397 . . . . . . 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 8307 . . . . . . . . 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 8397 . . . . . . . 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 418 . . . . . . 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 434 . . . . . 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 1049 . . . 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 437 . . 3 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
70 divcanap2 9010 . . . . . . . 8 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( 2 x. ( A / 2 ) ) = A )
7144, 45, 70mp3an23 1370 . . . . . . 7 |- ( A e. CC -> ( 2 x. ( A / 2 ) ) = A )
7242, 71syl 14 . . . . . 6 |- ( A e. RR -> ( 2 x. ( A / 2 ) ) = A )
7372fveq2d 5699 . . . . 5 |- ( A e. RR -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( sin ` A ) )
74 sin2t 12516 . . . . . 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 2273 . . . 4 |- ( A e. RR -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
77763ad2ant1 1049 . . 3 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
7869, 77breqtrrd 4158 . 2 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` A ) )
795, 78sylbi 121 1 |- ( A e. ( 0 (,) pi ) -> 0 < ( sin ` A ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   x. cmul 8184   RR*cxr 8359   < clt 8360   - cmin 8497   # cap 8909   / cdiv 9002   2c2 9355   (,)cioo 10290   sincsin 12411   cosccos 12412   picpi 12414
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302
This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-xneg 10174  df-xadd 10175  df-ioo 10294  df-ioc 10295  df-ico 10296  df-icc 10297  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-bc 11186  df-ihash 11215  df-shft 11580  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120  df-ef 12415  df-sin 12417  df-cos 12418  df-pi 12420  df-rest 13595  df-topgen 13614  df-psmet 14880  df-xmet 14881  df-met 14882  df-bl 14883  df-mopn 14884  df-top 15099  df-topon 15112  df-bases 15144  df-ntr 15197  df-cn 15289  df-cnp 15290  df-tx 15354  df-cncf 15672  df-limced 15757  df-dvap 15758
This theorem is used by:  sinq34lt0t  15932  cosq14gt0  15933  cosordlem  15950
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