Theorem sinq12gt0 15580

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

Step	Hyp	Ref	Expression
1		0xr 8228	. . 3 |- 0 e. RR*
2		pire 15536	. . . 4 |- pi e. RR
3	2	rexri 8239	. . 3 |- pi e. RR*
4		elioo2 10158	. . 3 |- ( ( 0 e. RR* /\ pi e. RR* ) -> ( A e. ( 0 (,) pi ) <-> ( A e. RR /\ 0 < A /\ A < pi ) ) )
5	1, 3, 4	mp2an 426	. 2 |- ( A e. ( 0 (,) pi ) <-> ( A e. RR /\ 0 < A /\ A < pi ) )
6		rehalfcl 9373	. . . . . 6 |- ( A e. RR -> ( A / 2 ) e. RR )
7	6	3ad2ant1 1044	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( A / 2 ) e. RR )
8		halfpos2 9376	. . . . . . 7 |- ( A e. RR -> ( 0 < A <-> 0 < ( A / 2 ) ) )
9	8	biimpa 296	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A / 2 ) )
10	9	3adant3 1043	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( A / 2 ) )
11		2re 9215	. . . . . . . . 9 |- 2 e. RR
12		2pos 9236	. . . . . . . . 9 |- 0 < 2
13	11, 12	pm3.2i 272	. . . . . . . 8 |- ( 2 e. RR /\ 0 < 2 )
14		ltdiv1 9050	. . . . . . . 8 |- ( ( A e. RR /\ pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A < pi <-> ( A / 2 ) < ( pi / 2 ) ) )
15	2, 13, 14	mp3an23 1365	. . . . . . 7 |- ( A e. RR -> ( A < pi <-> ( A / 2 ) < ( pi / 2 ) ) )
16	15	adantr 276	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> ( A < pi <-> ( A / 2 ) < ( pi / 2 ) ) )
17	16	biimp3a 1381	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( A / 2 ) < ( pi / 2 ) )
18		sincosq1lem 15575	. . . . 5 |- ( ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) < ( pi / 2 ) ) -> 0 < ( sin ` ( A / 2 ) ) )
19	7, 10, 17, 18	syl3anc 1273	. . . 4 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` ( A / 2 ) ) )
20		resubcl 8445	. . . . . . . . 9 |- ( ( pi e. RR /\ A e. RR ) -> ( pi - A ) e. RR )
21	2, 20	mpan 424	. . . . . . . 8 |- ( A e. RR -> ( pi - A ) e. RR )
22		rehalfcl 9373	. . . . . . . 8 |- ( ( pi - A ) e. RR -> ( ( pi - A ) / 2 ) e. RR )
23	21, 22	syl 14	. . . . . . 7 |- ( A e. RR -> ( ( pi - A ) / 2 ) e. RR )
24	23	3ad2ant1 1044	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( ( pi - A ) / 2 ) e. RR )
25		posdif 8637	. . . . . . . . . 10 |- ( ( A e. RR /\ pi e. RR ) -> ( A < pi <-> 0 < ( pi - A ) ) )
26	2, 25	mpan2 425	. . . . . . . . 9 |- ( A e. RR -> ( A < pi <-> 0 < ( pi - A ) ) )
27		halfpos2 9376	. . . . . . . . . 10 |- ( ( pi - A ) e. RR -> ( 0 < ( pi - A ) <-> 0 < ( ( pi - A ) / 2 ) ) )
28	21, 27	syl 14	. . . . . . . . 9 |- ( A e. RR -> ( 0 < ( pi - A ) <-> 0 < ( ( pi - A ) / 2 ) ) )
29	26, 28	bitrd 188	. . . . . . . 8 |- ( A e. RR -> ( A < pi <-> 0 < ( ( pi - A ) / 2 ) ) )
30	29	adantr 276	. . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> ( A < pi <-> 0 < ( ( pi - A ) / 2 ) ) )
31	30	biimp3a 1381	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( ( pi - A ) / 2 ) )
32		ltsubpos 8636	. . . . . . . . . 10 |- ( ( A e. RR /\ pi e. RR ) -> ( 0 < A <-> ( pi - A ) < pi ) )
33	2, 32	mpan2 425	. . . . . . . . 9 |- ( A e. RR -> ( 0 < A <-> ( pi - A ) < pi ) )
34		ltdiv1 9050	. . . . . . . . . . 11 |- ( ( ( pi - A ) e. RR /\ pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( pi - A ) < pi <-> ( ( pi - A ) / 2 ) < ( pi / 2 ) ) )
35	2, 13, 34	mp3an23 1365	. . . . . . . . . 10 |- ( ( pi - A ) e. RR -> ( ( pi - A ) < pi <-> ( ( pi - A ) / 2 ) < ( pi / 2 ) ) )
36	21, 35	syl 14	. . . . . . . . 9 |- ( A e. RR -> ( ( pi - A ) < pi <-> ( ( pi - A ) / 2 ) < ( pi / 2 ) ) )
37	33, 36	bitrd 188	. . . . . . . 8 |- ( A e. RR -> ( 0 < A <-> ( ( pi - A ) / 2 ) < ( pi / 2 ) ) )
38	37	biimpa 296	. . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> ( ( pi - A ) / 2 ) < ( pi / 2 ) )
39	38	3adant3 1043	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( ( pi - A ) / 2 ) < ( pi / 2 ) )
40		sincosq1lem 15575	. . . . . 6 |- ( ( ( ( pi - A ) / 2 ) e. RR /\ 0 < ( ( pi - A ) / 2 ) /\ ( ( pi - A ) / 2 ) < ( pi / 2 ) ) -> 0 < ( sin ` ( ( pi - A ) / 2 ) ) )
41	24, 31, 39, 40	syl3anc 1273	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` ( ( pi - A ) / 2 ) ) )
42		recn 8167	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
43		picn 15537	. . . . . . . . . 10 |- pi e. CC
44		2cn 9216	. . . . . . . . . . 11 |- 2 e. CC
45		2ap0 9238	. . . . . . . . . . 11 |- 2 # 0
46	44, 45	pm3.2i 272	. . . . . . . . . 10 |- ( 2 e. CC /\ 2 # 0 )
47		divsubdirap 8890	. . . . . . . . . 10 |- ( ( pi e. CC /\ A e. CC /\ ( 2 e. CC /\ 2 # 0 ) ) -> ( ( pi - A ) / 2 ) = ( ( pi / 2 ) - ( A / 2 ) ) )
48	43, 46, 47	mp3an13 1364	. . . . . . . . 9 |- ( A e. CC -> ( ( pi - A ) / 2 ) = ( ( pi / 2 ) - ( A / 2 ) ) )
49	42, 48	syl 14	. . . . . . . 8 |- ( A e. RR -> ( ( pi - A ) / 2 ) = ( ( pi / 2 ) - ( A / 2 ) ) )
50	49	fveq2d 5643	. . . . . . 7 |- ( A e. RR -> ( sin ` ( ( pi - A ) / 2 ) ) = ( sin ` ( ( pi / 2 ) - ( A / 2 ) ) ) )
51	6	recnd 8210	. . . . . . . 8 |- ( A e. RR -> ( A / 2 ) e. CC )
52		sinhalfpim 15571	. . . . . . . 8 |- ( ( A / 2 ) e. CC -> ( sin ` ( ( pi / 2 ) - ( A / 2 ) ) ) = ( cos ` ( A / 2 ) ) )
53	51, 52	syl 14	. . . . . . 7 |- ( A e. RR -> ( sin ` ( ( pi / 2 ) - ( A / 2 ) ) ) = ( cos ` ( A / 2 ) ) )
54	50, 53	eqtrd 2263	. . . . . 6 |- ( A e. RR -> ( sin ` ( ( pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) )
55	54	3ad2ant1 1044	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( sin ` ( ( pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) )
56	41, 55	breqtrd 4113	. . . 4 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( cos ` ( A / 2 ) ) )
57		resincl 12301	. . . . . . . 8 |- ( ( A / 2 ) e. RR -> ( sin ` ( A / 2 ) ) e. RR )
58		recoscl 12302	. . . . . . . 8 |- ( ( A / 2 ) e. RR -> ( cos ` ( A / 2 ) ) e. RR )
59	57, 58	jca 306	. . . . . . 7 |- ( ( A / 2 ) e. RR -> ( ( sin ` ( A / 2 ) ) e. RR /\ ( cos ` ( A / 2 ) ) e. RR ) )
60		axmulgt0 8253	. . . . . . 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 ) ) ) ) )
61	6, 59, 60	3syl 17	. . . . . 6 |- ( A e. RR -> ( ( 0 < ( sin ` ( A / 2 ) ) /\ 0 < ( cos ` ( A / 2 ) ) ) -> 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
62		remulcl 8162	. . . . . . . . 9 |- ( ( ( sin ` ( A / 2 ) ) e. RR /\ ( cos ` ( A / 2 ) ) e. RR ) -> ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) e. RR )
63	6, 59, 62	3syl 17	. . . . . . . 8 |- ( A e. RR -> ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) e. RR )
64		axmulgt0 8253	. . . . . . . 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 ) ) ) ) ) )
65	11, 63, 64	sylancr 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 ) ) ) ) ) )
66	12, 65	mpani 430	. . . . . 6 |- ( A e. RR -> ( 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) )
67	61, 66	syld 45	. . . . 5 |- ( A e. RR -> ( ( 0 < ( sin ` ( A / 2 ) ) /\ 0 < ( cos ` ( A / 2 ) ) ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) )
68	67	3ad2ant1 1044	. . . 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 ) ) ) ) ) )
69	19, 56, 68	mp2and 433	. . 3 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
70		divcanap2 8862	. . . . . . . 8 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( 2 x. ( A / 2 ) ) = A )
71	44, 45, 70	mp3an23 1365	. . . . . . 7 |- ( A e. CC -> ( 2 x. ( A / 2 ) ) = A )
72	42, 71	syl 14	. . . . . 6 |- ( A e. RR -> ( 2 x. ( A / 2 ) ) = A )
73	72	fveq2d 5643	. . . . 5 |- ( A e. RR -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( sin ` A ) )
74		sin2t 12330	. . . . . 6 |- ( ( A / 2 ) e. CC -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
75	51, 74	syl 14	. . . . 5 |- ( A e. RR -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
76	73, 75	eqtr3d 2265	. . . 4 |- ( A e. RR -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
77	76	3ad2ant1 1044	. . 3 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
78	69, 77	breqtrrd 4115	. 2 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` A ) )
79	5, 78	sylbi 121	1 |- ( A e. ( 0 (,) pi ) -> 0 < ( sin ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2201   class class class wbr 4087   ` cfv 5325  (class class class)co 6020   CCcc 8032   RRcr 8033   0cc0 8034   x. cmul 8039   RR*cxr 8215   < clt 8216   - cmin 8352   # cap 8763   / cdiv 8854   2c2 9196   (,)cioo 10125   sincsin 12225   cosccos 12226   picpi 12228

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-iinf 4685  ax-cnex 8125  ax-resscn 8126  ax-1cn 8127  ax-1re 8128  ax-icn 8129  ax-addcl 8130  ax-addrcl 8131  ax-mulcl 8132  ax-mulrcl 8133  ax-addcom 8134  ax-mulcom 8135  ax-addass 8136  ax-mulass 8137  ax-distr 8138  ax-i2m1 8139  ax-0lt1 8140  ax-1rid 8141  ax-0id 8142  ax-rnegex 8143  ax-precex 8144  ax-cnre 8145  ax-pre-ltirr 8146  ax-pre-ltwlin 8147  ax-pre-lttrn 8148  ax-pre-apti 8149  ax-pre-ltadd 8150  ax-pre-mulgt0 8151  ax-pre-mulext 8152  ax-arch 8153  ax-caucvg 8154  ax-pre-suploc 8155  ax-addf 8156  ax-mulf 8157

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-disj 4064  df-br 4088  df-opab 4150  df-mpt 4151  df-tr 4187  df-id 4389  df-po 4392  df-iso 4393  df-iord 4462  df-on 4464  df-ilim 4465  df-suc 4467  df-iom 4688  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-isom 5334  df-riota 5973  df-ov 6023  df-oprab 6024  df-mpo 6025  df-of 6237  df-1st 6305  df-2nd 6306  df-recs 6473  df-irdg 6538  df-frec 6559  df-1o 6584  df-oadd 6588  df-er 6704  df-map 6821  df-pm 6822  df-en 6912  df-dom 6913  df-fin 6914  df-sup 7185  df-inf 7186  df-pnf 8218  df-mnf 8219  df-xr 8220  df-ltxr 8221  df-le 8222  df-sub 8354  df-neg 8355  df-reap 8757  df-ap 8764  df-div 8855  df-inn 9146  df-2 9204  df-3 9205  df-4 9206  df-5 9207  df-6 9208  df-7 9209  df-8 9210  df-9 9211  df-n0 9405  df-z 9482  df-uz 9758  df-q 9856  df-rp 9891  df-xneg 10009  df-xadd 10010  df-ioo 10129  df-ioc 10130  df-ico 10131  df-icc 10132  df-fz 10246  df-fzo 10380  df-seqfrec 10713  df-exp 10804  df-fac 10991  df-bc 11013  df-ihash 11041  df-shft 11395  df-cj 11422  df-re 11423  df-im 11424  df-rsqrt 11578  df-abs 11579  df-clim 11859  df-sumdc 11934  df-ef 12229  df-sin 12231  df-cos 12232  df-pi 12234  df-rest 13344  df-topgen 13363  df-psmet 14578  df-xmet 14579  df-met 14580  df-bl 14581  df-mopn 14582  df-top 14748  df-topon 14761  df-bases 14793  df-ntr 14846  df-cn 14938  df-cnp 14939  df-tx 15003  df-cncf 15321  df-limced 15406  df-dvap 15407

This theorem is referenced by:  sinq34lt0t  15581  cosq14gt0  15582  cosordlem  15599