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Theorem sinq12gt0 15544
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 8216 . . 3 |- 0 e. RR*
2 pire 15500 . . . 4 |- pi e. RR
32rexri 8227 . . 3 |- pi e. RR*
4 elioo2 10146 . . 3 |- ( ( 0 e. RR* /\ pi e. RR* ) -> ( A e. ( 0 (,) pi ) <-> ( A e. RR /\ 0 < A /\ A < pi ) ) )
51, 3, 4mp2an 426 . 2 |- ( A e. ( 0 (,) pi ) <-> ( A e. RR /\ 0 < A /\ A < pi ) )
6 rehalfcl 9361 . . . . . 6 |- ( A e. RR -> ( A / 2 ) e. RR )
763ad2ant1 1042 . . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( A / 2 ) e. RR )
8 halfpos2 9364 . . . . . . 7 |- ( A e. RR -> ( 0 < A <-> 0 < ( A / 2 ) ) )
98biimpa 296 . . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A / 2 ) )
1093adant3 1041 . . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( A / 2 ) )
11 2re 9203 . . . . . . . . 9 |- 2 e. RR
12 2pos 9224 . . . . . . . . 9 |- 0 < 2
1311, 12pm3.2i 272 . . . . . . . 8 |- ( 2 e. RR /\ 0 < 2 )
14 ltdiv1 9038 . . . . . . . 8 |- ( ( A e. RR /\ pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A < pi <-> ( A / 2 ) < ( pi / 2 ) ) )
152, 13, 14mp3an23 1363 . . . . . . 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 1379 . . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( A / 2 ) < ( pi / 2 ) )
18 sincosq1lem 15539 . . . . 5 |- ( ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) < ( pi / 2 ) ) -> 0 < ( sin ` ( A / 2 ) ) )
197, 10, 17, 18syl3anc 1271 . . . 4 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` ( A / 2 ) ) )
20 resubcl 8433 . . . . . . . . 9 |- ( ( pi e. RR /\ A e. RR ) -> ( pi - A ) e. RR )
212, 20mpan 424 . . . . . . . 8 |- ( A e. RR -> ( pi - A ) e. RR )
22 rehalfcl 9361 . . . . . . . 8 |- ( ( pi - A ) e. RR -> ( ( pi - A ) / 2 ) e. RR )
2321, 22syl 14 . . . . . . 7 |- ( A e. RR -> ( ( pi - A ) / 2 ) e. RR )
24233ad2ant1 1042 . . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( ( pi - A ) / 2 ) e. RR )
25 posdif 8625 . . . . . . . . . 10 |- ( ( A e. RR /\ pi e. RR ) -> ( A < pi <-> 0 < ( pi - A ) ) )
262, 25mpan2 425 . . . . . . . . 9 |- ( A e. RR -> ( A < pi <-> 0 < ( pi - A ) ) )
27 halfpos2 9364 . . . . . . . . . 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 1379 . . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( ( pi - A ) / 2 ) )
32 ltsubpos 8624 . . . . . . . . . 10 |- ( ( A e. RR /\ pi e. RR ) -> ( 0 < A <-> ( pi - A ) < pi ) )
332, 32mpan2 425 . . . . . . . . 9 |- ( A e. RR -> ( 0 < A <-> ( pi - A ) < pi ) )
34 ltdiv1 9038 . . . . . . . . . . 11 |- ( ( ( pi - A ) e. RR /\ pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( pi - A ) < pi <-> ( ( pi - A ) / 2 ) < ( pi / 2 ) ) )
352, 13, 34mp3an23 1363 . . . . . . . . . 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 1041 . . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( ( pi - A ) / 2 ) < ( pi / 2 ) )
40 sincosq1lem 15539 . . . . . 6 |- ( ( ( ( pi - A ) / 2 ) e. RR /\ 0 < ( ( pi - A ) / 2 ) /\ ( ( pi - A ) / 2 ) < ( pi / 2 ) ) -> 0 < ( sin ` ( ( pi - A ) / 2 ) ) )
4124, 31, 39, 40syl3anc 1271 . . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` ( ( pi - A ) / 2 ) ) )
42 recn 8155 . . . . . . . . 9 |- ( A e. RR -> A e. CC )
43 picn 15501 . . . . . . . . . 10 |- pi e. CC
44 2cn 9204 . . . . . . . . . . 11 |- 2 e. CC
45 2ap0 9226 . . . . . . . . . . 11 |- 2 # 0
4644, 45pm3.2i 272 . . . . . . . . . 10 |- ( 2 e. CC /\ 2 # 0 )
47 divsubdirap 8878 . . . . . . . . . 10 |- ( ( pi e. CC /\ A e. CC /\ ( 2 e. CC /\ 2 # 0 ) ) -> ( ( pi - A ) / 2 ) = ( ( pi / 2 ) - ( A / 2 ) ) )
4843, 46, 47mp3an13 1362 . . . . . . . . 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 5639 . . . . . . 7 |- ( A e. RR -> ( sin ` ( ( pi - A ) / 2 ) ) = ( sin ` ( ( pi / 2 ) - ( A / 2 ) ) ) )
516recnd 8198 . . . . . . . 8 |- ( A e. RR -> ( A / 2 ) e. CC )
52 sinhalfpim 15535 . . . . . . . 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 2262 . . . . . 6 |- ( A e. RR -> ( sin ` ( ( pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) )
55543ad2ant1 1042 . . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( sin ` ( ( pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) )
5641, 55breqtrd 4112 . . . 4 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( cos ` ( A / 2 ) ) )
57 resincl 12271 . . . . . . . 8 |- ( ( A / 2 ) e. RR -> ( sin ` ( A / 2 ) ) e. RR )
58 recoscl 12272 . . . . . . . 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 8241 . . . . . . 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 8150 . . . . . . . . 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 8241 . . . . . . . 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 414 . . . . . . 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 430 . . . . . 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 1042 . . . 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 433 . . 3 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
70 divcanap2 8850 . . . . . . . 8 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( 2 x. ( A / 2 ) ) = A )
7144, 45, 70mp3an23 1363 . . . . . . 7 |- ( A e. CC -> ( 2 x. ( A / 2 ) ) = A )
7242, 71syl 14 . . . . . 6 |- ( A e. RR -> ( 2 x. ( A / 2 ) ) = A )
7372fveq2d 5639 . . . . 5 |- ( A e. RR -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( sin ` A ) )
74 sin2t 12300 . . . . . 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 2264 . . . 4 |- ( A e. RR -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
77763ad2ant1 1042 . . 3 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
7869, 77breqtrrd 4114 . 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
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   x. cmul 8027   RR*cxr 8203   < clt 8204   - cmin 8340   # cap 8751   / cdiv 8842   2c2 9184   (,)cioo 10113   sincsin 12195   cosccos 12196   picpi 12198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142  ax-pre-suploc 8143  ax-addf 8144  ax-mulf 8145
This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-disj 4063  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-oadd 6581  df-er 6697  df-map 6814  df-pm 6815  df-en 6905  df-dom 6906  df-fin 6907  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-xneg 9997  df-xadd 9998  df-ioo 10117  df-ioc 10118  df-ico 10119  df-icc 10120  df-fz 10234  df-fzo 10368  df-seqfrec 10700  df-exp 10791  df-fac 10978  df-bc 11000  df-ihash 11028  df-shft 11366  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-clim 11830  df-sumdc 11905  df-ef 12199  df-sin 12201  df-cos 12202  df-pi 12204  df-rest 13314  df-topgen 13333  df-psmet 14547  df-xmet 14548  df-met 14549  df-bl 14550  df-mopn 14551  df-top 14712  df-topon 14725  df-bases 14757  df-ntr 14810  df-cn 14902  df-cnp 14903  df-tx 14967  df-cncf 15285  df-limced 15370  df-dvap 15371
This theorem is referenced by:  sinq34lt0t  15545  cosq14gt0  15546  cosordlem  15563
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