Theorem sinq12gt0 13391

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 7945	. . 3 |- 0 e. RR*
2		pire 13347	. . . 4 |- pi e. RR
3	2	rexri 7956	. . 3 |- pi e. RR*
4		elioo2 9857	. . 3 |- ( ( 0 e. RR* /\ pi e. RR* ) -> ( A e. ( 0 (,) pi ) <-> ( A e. RR /\ 0 < A /\ A < pi ) ) )
5	1, 3, 4	mp2an 423	. 2 |- ( A e. ( 0 (,) pi ) <-> ( A e. RR /\ 0 < A /\ A < pi ) )
6		rehalfcl 9084	. . . . . 6 |- ( A e. RR -> ( A / 2 ) e. RR )
7	6	3ad2ant1 1008	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( A / 2 ) e. RR )
8		halfpos2 9087	. . . . . . 7 |- ( A e. RR -> ( 0 < A <-> 0 < ( A / 2 ) ) )
9	8	biimpa 294	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A / 2 ) )
10	9	3adant3 1007	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( A / 2 ) )
11		2re 8927	. . . . . . . . 9 |- 2 e. RR
12		2pos 8948	. . . . . . . . 9 |- 0 < 2
13	11, 12	pm3.2i 270	. . . . . . . 8 |- ( 2 e. RR /\ 0 < 2 )
14		ltdiv1 8763	. . . . . . . 8 |- ( ( A e. RR /\ pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A < pi <-> ( A / 2 ) < ( pi / 2 ) ) )
15	2, 13, 14	mp3an23 1319	. . . . . . 7 |- ( A e. RR -> ( A < pi <-> ( A / 2 ) < ( pi / 2 ) ) )
16	15	adantr 274	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> ( A < pi <-> ( A / 2 ) < ( pi / 2 ) ) )
17	16	biimp3a 1335	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( A / 2 ) < ( pi / 2 ) )
18		sincosq1lem 13386	. . . . 5 |- ( ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) < ( pi / 2 ) ) -> 0 < ( sin ` ( A / 2 ) ) )
19	7, 10, 17, 18	syl3anc 1228	. . . 4 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` ( A / 2 ) ) )
20		resubcl 8162	. . . . . . . . 9 |- ( ( pi e. RR /\ A e. RR ) -> ( pi - A ) e. RR )
21	2, 20	mpan 421	. . . . . . . 8 |- ( A e. RR -> ( pi - A ) e. RR )
22		rehalfcl 9084	. . . . . . . 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 1008	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( ( pi - A ) / 2 ) e. RR )
25		posdif 8353	. . . . . . . . . 10 |- ( ( A e. RR /\ pi e. RR ) -> ( A < pi <-> 0 < ( pi - A ) ) )
26	2, 25	mpan2 422	. . . . . . . . 9 |- ( A e. RR -> ( A < pi <-> 0 < ( pi - A ) ) )
27		halfpos2 9087	. . . . . . . . . 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 187	. . . . . . . 8 |- ( A e. RR -> ( A < pi <-> 0 < ( ( pi - A ) / 2 ) ) )
30	29	adantr 274	. . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> ( A < pi <-> 0 < ( ( pi - A ) / 2 ) ) )
31	30	biimp3a 1335	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( ( pi - A ) / 2 ) )
32		ltsubpos 8352	. . . . . . . . . 10 |- ( ( A e. RR /\ pi e. RR ) -> ( 0 < A <-> ( pi - A ) < pi ) )
33	2, 32	mpan2 422	. . . . . . . . 9 |- ( A e. RR -> ( 0 < A <-> ( pi - A ) < pi ) )
34		ltdiv1 8763	. . . . . . . . . . 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 1319	. . . . . . . . . 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 187	. . . . . . . 8 |- ( A e. RR -> ( 0 < A <-> ( ( pi - A ) / 2 ) < ( pi / 2 ) ) )
38	37	biimpa 294	. . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> ( ( pi - A ) / 2 ) < ( pi / 2 ) )
39	38	3adant3 1007	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( ( pi - A ) / 2 ) < ( pi / 2 ) )
40		sincosq1lem 13386	. . . . . 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 1228	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` ( ( pi - A ) / 2 ) ) )
42		recn 7886	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
43		picn 13348	. . . . . . . . . 10 |- pi e. CC
44		2cn 8928	. . . . . . . . . . 11 |- 2 e. CC
45		2ap0 8950	. . . . . . . . . . 11 |- 2 # 0
46	44, 45	pm3.2i 270	. . . . . . . . . 10 |- ( 2 e. CC /\ 2 # 0 )
47		divsubdirap 8604	. . . . . . . . . 10 |- ( ( pi e. CC /\ A e. CC /\ ( 2 e. CC /\ 2 # 0 ) ) -> ( ( pi - A ) / 2 ) = ( ( pi / 2 ) - ( A / 2 ) ) )
48	43, 46, 47	mp3an13 1318	. . . . . . . . 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 5490	. . . . . . 7 |- ( A e. RR -> ( sin ` ( ( pi - A ) / 2 ) ) = ( sin ` ( ( pi / 2 ) - ( A / 2 ) ) ) )
51	6	recnd 7927	. . . . . . . 8 |- ( A e. RR -> ( A / 2 ) e. CC )
52		sinhalfpim 13382	. . . . . . . 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 2198	. . . . . 6 |- ( A e. RR -> ( sin ` ( ( pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) )
55	54	3ad2ant1 1008	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( sin ` ( ( pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) )
56	41, 55	breqtrd 4008	. . . 4 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( cos ` ( A / 2 ) ) )
57		resincl 11661	. . . . . . . 8 |- ( ( A / 2 ) e. RR -> ( sin ` ( A / 2 ) ) e. RR )
58		recoscl 11662	. . . . . . . 8 |- ( ( A / 2 ) e. RR -> ( cos ` ( A / 2 ) ) e. RR )
59	57, 58	jca 304	. . . . . . 7 |- ( ( A / 2 ) e. RR -> ( ( sin ` ( A / 2 ) ) e. RR /\ ( cos ` ( A / 2 ) ) e. RR ) )
60		axmulgt0 7970	. . . . . . 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 7881	. . . . . . . . 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 7970	. . . . . . . 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 411	. . . . . . 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 427	. . . . . 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 1008	. . . 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 430	. . 3 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
70		divcanap2 8576	. . . . . . . 8 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( 2 x. ( A / 2 ) ) = A )
71	44, 45, 70	mp3an23 1319	. . . . . . 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 5490	. . . . 5 |- ( A e. RR -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( sin ` A ) )
74		sin2t 11690	. . . . . 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 2200	. . . 4 |- ( A e. RR -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
77	76	3ad2ant1 1008	. . 3 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
78	69, 77	breqtrrd 4010	. 2 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` A ) )
79	5, 78	sylbi 120	1 |- ( A e. ( 0 (,) pi ) -> 0 < ( sin ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   x. cmul 7758   RR*cxr 7932   < clt 7933   - cmin 8069   # cap 8479   / cdiv 8568   2c2 8908   (,)cioo 9824   sincsin 11585   cosccos 11586   picpi 11588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-pre-suploc 7874  ax-addf 7875  ax-mulf 7876

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-map 6616  df-pm 6617  df-en 6707  df-dom 6708  df-fin 6709  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-5 8919  df-6 8920  df-7 8921  df-8 8922  df-9 8923  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-ioo 9828  df-ioc 9829  df-ico 9830  df-icc 9831  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-bc 10661  df-ihash 10689  df-shft 10757  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589  df-sin 11591  df-cos 11592  df-pi 11594  df-rest 12558  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-met 12629  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-ntr 12736  df-cn 12828  df-cnp 12829  df-tx 12893  df-cncf 13198  df-limced 13265  df-dvap 13266

This theorem is referenced by:  sinq34lt0t  13392  cosq14gt0  13393  cosordlem  13410