Theorem sinq12gt0 12911

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 7812	. . 3 |- 0 e. RR*
2		pire 12867	. . . 4 |- pi e. RR
3	2	rexri 7823	. . 3 |- pi e. RR*
4		elioo2 9704	. . 3 |- ( ( 0 e. RR* /\ pi e. RR* ) -> ( A e. ( 0 (,) pi ) <-> ( A e. RR /\ 0 < A /\ A < pi ) ) )
5	1, 3, 4	mp2an 422	. 2 |- ( A e. ( 0 (,) pi ) <-> ( A e. RR /\ 0 < A /\ A < pi ) )
6		rehalfcl 8947	. . . . . 6 |- ( A e. RR -> ( A / 2 ) e. RR )
7	6	3ad2ant1 1002	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( A / 2 ) e. RR )
8		halfpos2 8950	. . . . . . 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 1001	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( A / 2 ) )
11		2re 8790	. . . . . . . . 9 |- 2 e. RR
12		2pos 8811	. . . . . . . . 9 |- 0 < 2
13	11, 12	pm3.2i 270	. . . . . . . 8 |- ( 2 e. RR /\ 0 < 2 )
14		ltdiv1 8626	. . . . . . . 8 |- ( ( A e. RR /\ pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A < pi <-> ( A / 2 ) < ( pi / 2 ) ) )
15	2, 13, 14	mp3an23 1307	. . . . . . 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 1323	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( A / 2 ) < ( pi / 2 ) )
18		sincosq1lem 12906	. . . . 5 |- ( ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) < ( pi / 2 ) ) -> 0 < ( sin ` ( A / 2 ) ) )
19	7, 10, 17, 18	syl3anc 1216	. . . 4 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` ( A / 2 ) ) )
20		resubcl 8026	. . . . . . . . 9 |- ( ( pi e. RR /\ A e. RR ) -> ( pi - A ) e. RR )
21	2, 20	mpan 420	. . . . . . . 8 |- ( A e. RR -> ( pi - A ) e. RR )
22		rehalfcl 8947	. . . . . . . 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 1002	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( ( pi - A ) / 2 ) e. RR )
25		posdif 8217	. . . . . . . . . 10 |- ( ( A e. RR /\ pi e. RR ) -> ( A < pi <-> 0 < ( pi - A ) ) )
26	2, 25	mpan2 421	. . . . . . . . 9 |- ( A e. RR -> ( A < pi <-> 0 < ( pi - A ) ) )
27		halfpos2 8950	. . . . . . . . . 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 1323	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( ( pi - A ) / 2 ) )
32		ltsubpos 8216	. . . . . . . . . 10 |- ( ( A e. RR /\ pi e. RR ) -> ( 0 < A <-> ( pi - A ) < pi ) )
33	2, 32	mpan2 421	. . . . . . . . 9 |- ( A e. RR -> ( 0 < A <-> ( pi - A ) < pi ) )
34		ltdiv1 8626	. . . . . . . . . . 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 1307	. . . . . . . . . 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 1001	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( ( pi - A ) / 2 ) < ( pi / 2 ) )
40		sincosq1lem 12906	. . . . . 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 1216	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( sin ` ( ( pi - A ) / 2 ) ) )
42		recn 7753	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
43		picn 12868	. . . . . . . . . 10 |- pi e. CC
44		2cn 8791	. . . . . . . . . . 11 |- 2 e. CC
45		2ap0 8813	. . . . . . . . . . 11 |- 2 # 0
46	44, 45	pm3.2i 270	. . . . . . . . . 10 |- ( 2 e. CC /\ 2 # 0 )
47		divsubdirap 8468	. . . . . . . . . 10 |- ( ( pi e. CC /\ A e. CC /\ ( 2 e. CC /\ 2 # 0 ) ) -> ( ( pi - A ) / 2 ) = ( ( pi / 2 ) - ( A / 2 ) ) )
48	43, 46, 47	mp3an13 1306	. . . . . . . . 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 5425	. . . . . . 7 |- ( A e. RR -> ( sin ` ( ( pi - A ) / 2 ) ) = ( sin ` ( ( pi / 2 ) - ( A / 2 ) ) ) )
51	6	recnd 7794	. . . . . . . 8 |- ( A e. RR -> ( A / 2 ) e. CC )
52		sinhalfpim 12902	. . . . . . . 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 2172	. . . . . 6 |- ( A e. RR -> ( sin ` ( ( pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) )
55	54	3ad2ant1 1002	. . . . 5 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( sin ` ( ( pi - A ) / 2 ) ) = ( cos ` ( A / 2 ) ) )
56	41, 55	breqtrd 3954	. . . 4 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( cos ` ( A / 2 ) ) )
57		resincl 11427	. . . . . . . 8 |- ( ( A / 2 ) e. RR -> ( sin ` ( A / 2 ) ) e. RR )
58		recoscl 11428	. . . . . . . 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 7836	. . . . . . 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 7748	. . . . . . . . 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 7836	. . . . . . . 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 410	. . . . . . 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 426	. . . . . 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 1002	. . . 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 429	. . 3 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
70		divcanap2 8440	. . . . . . . 8 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( 2 x. ( A / 2 ) ) = A )
71	44, 45, 70	mp3an23 1307	. . . . . . 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 5425	. . . . 5 |- ( A e. RR -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( sin ` A ) )
74		sin2t 11456	. . . . . 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 2174	. . . 4 |- ( A e. RR -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
77	76	3ad2ant1 1002	. . 3 |- ( ( A e. RR /\ 0 < A /\ A < pi ) -> ( sin ` A ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
78	69, 77	breqtrrd 3956	. 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 962   = wceq 1331   e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   x. cmul 7625   RR*cxr 7799   < clt 7800   - cmin 7933   # cap 8343   / cdiv 8432   2c2 8771   (,)cioo 9671   sincsin 11350   cosccos 11351   picpi 11353

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740  ax-pre-suploc 7741  ax-addf 7742  ax-mulf 7743

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-map 6544  df-pm 6545  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-5 8782  df-6 8783  df-7 8784  df-8 8785  df-9 8786  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-xneg 9559  df-xadd 9560  df-ioo 9675  df-ioc 9676  df-ico 9677  df-icc 9678  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-fac 10472  df-bc 10494  df-ihash 10522  df-shft 10587  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123  df-ef 11354  df-sin 11356  df-cos 11357  df-pi 11359  df-rest 12122  df-topgen 12141  df-psmet 12156  df-xmet 12157  df-met 12158  df-bl 12159  df-mopn 12160  df-top 12165  df-topon 12178  df-bases 12210  df-ntr 12265  df-cn 12357  df-cnp 12358  df-tx 12422  df-cncf 12727  df-limced 12794  df-dvap 12795

This theorem is referenced by:  sinq34lt0t  12912  cosq14gt0  12913  cosordlem  12930