Theorem sin02gt0 12405

Description: The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Assertion

Ref	Expression
sin02gt0	|- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` A ) )

Proof of Theorem sin02gt0

Step	Hyp	Ref	Expression
1		0xr 8285	. . . . . . 7 |- 0 e. RR*
2		2re 9272	. . . . . . 7 |- 2 e. RR
3		elioc2 10232	. . . . . . 7 |- ( ( 0 e. RR* /\ 2 e. RR ) -> ( A e. ( 0 (,] 2 ) <-> ( A e. RR /\ 0 < A /\ A <_ 2 ) ) )
4	1, 2, 3	mp2an 426	. . . . . 6 |- ( A e. ( 0 (,] 2 ) <-> ( A e. RR /\ 0 < A /\ A <_ 2 ) )
5		rehalfcl 9430	. . . . . . 7 |- ( A e. RR -> ( A / 2 ) e. RR )
6	5	3ad2ant1 1045	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( A / 2 ) e. RR )
7	4, 6	sylbi 121	. . . . 5 |- ( A e. ( 0 (,] 2 ) -> ( A / 2 ) e. RR )
8		resincl 12361	. . . . . 6 |- ( ( A / 2 ) e. RR -> ( sin ` ( A / 2 ) ) e. RR )
9		recoscl 12362	. . . . . 6 |- ( ( A / 2 ) e. RR -> ( cos ` ( A / 2 ) ) e. RR )
10	8, 9	remulcld 8269	. . . . 5 |- ( ( A / 2 ) e. RR -> ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) e. RR )
11	7, 10	syl 14	. . . 4 |- ( A e. ( 0 (,] 2 ) -> ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) e. RR )
12		2pos 9293	. . . . . . . . . 10 |- 0 < 2
13		divgt0 9111	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( A / 2 ) )
14	2, 12, 13	mpanr12 439	. . . . . . . . 9 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A / 2 ) )
15	14	3adant3 1044	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> 0 < ( A / 2 ) )
16	2, 12	pm3.2i 272	. . . . . . . . . . . 12 |- ( 2 e. RR /\ 0 < 2 )
17		lediv1 9108	. . . . . . . . . . . 12 |- ( ( A e. RR /\ 2 e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A <_ 2 <-> ( A / 2 ) <_ ( 2 / 2 ) ) )
18	2, 16, 17	mp3an23 1366	. . . . . . . . . . 11 |- ( A e. RR -> ( A <_ 2 <-> ( A / 2 ) <_ ( 2 / 2 ) ) )
19	18	biimpa 296	. . . . . . . . . 10 |- ( ( A e. RR /\ A <_ 2 ) -> ( A / 2 ) <_ ( 2 / 2 ) )
20		2div2e1 9335	. . . . . . . . . 10 |- ( 2 / 2 ) = 1
21	19, 20	breqtrdi 4134	. . . . . . . . 9 |- ( ( A e. RR /\ A <_ 2 ) -> ( A / 2 ) <_ 1 )
22	21	3adant2 1043	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( A / 2 ) <_ 1 )
23	6, 15, 22	3jca 1204	. . . . . . 7 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) <_ 1 ) )
24		1re 8238	. . . . . . . 8 |- 1 e. RR
25		elioc2 10232	. . . . . . . 8 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( ( A / 2 ) e. ( 0 (,] 1 ) <-> ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) <_ 1 ) ) )
26	1, 24, 25	mp2an 426	. . . . . . 7 |- ( ( A / 2 ) e. ( 0 (,] 1 ) <-> ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) <_ 1 ) )
27	23, 4, 26	3imtr4i 201	. . . . . 6 |- ( A e. ( 0 (,] 2 ) -> ( A / 2 ) e. ( 0 (,] 1 ) )
28		sin01gt0 12403	. . . . . 6 |- ( ( A / 2 ) e. ( 0 (,] 1 ) -> 0 < ( sin ` ( A / 2 ) ) )
29	27, 28	syl 14	. . . . 5 |- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` ( A / 2 ) ) )
30		cos01gt0 12404	. . . . . 6 |- ( ( A / 2 ) e. ( 0 (,] 1 ) -> 0 < ( cos ` ( A / 2 ) ) )
31	27, 30	syl 14	. . . . 5 |- ( A e. ( 0 (,] 2 ) -> 0 < ( cos ` ( A / 2 ) ) )
32		axmulgt0 8310	. . . . . . 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 ) ) ) ) )
33	8, 9, 32	syl2anc 411	. . . . . 6 |- ( ( A / 2 ) e. RR -> ( ( 0 < ( sin ` ( A / 2 ) ) /\ 0 < ( cos ` ( A / 2 ) ) ) -> 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
34	7, 33	syl 14	. . . . 5 |- ( A e. ( 0 (,] 2 ) -> ( ( 0 < ( sin ` ( A / 2 ) ) /\ 0 < ( cos ` ( A / 2 ) ) ) -> 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
35	29, 31, 34	mp2and 433	. . . 4 |- ( A e. ( 0 (,] 2 ) -> 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) )
36		axmulgt0 8310	. . . . . 6 |- ( ( 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 ) ) ) ) ) )
37	2, 36	mpan 424	. . . . 5 |- ( ( ( 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 ) ) ) ) ) )
38	12, 37	mpani 430	. . . 4 |- ( ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) e. RR -> ( 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) ) )
39	11, 35, 38	sylc 62	. . 3 |- ( A e. ( 0 (,] 2 ) -> 0 < ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
40	7	recnd 8267	. . . 4 |- ( A e. ( 0 (,] 2 ) -> ( A / 2 ) e. CC )
41		sin2t 12390	. . . 4 |- ( ( A / 2 ) e. CC -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
42	40, 41	syl 14	. . 3 |- ( A e. ( 0 (,] 2 ) -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( 2 x. ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) ) )
43	39, 42	breqtrrd 4121	. 2 |- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` ( 2 x. ( A / 2 ) ) ) )
44	4	simp1bi 1039	. . . . 5 |- ( A e. ( 0 (,] 2 ) -> A e. RR )
45	44	recnd 8267	. . . 4 |- ( A e. ( 0 (,] 2 ) -> A e. CC )
46		2cn 9273	. . . . 5 |- 2 e. CC
47		2ap0 9295	. . . . 5 |- 2 # 0
48		divcanap2 8919	. . . . 5 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( 2 x. ( A / 2 ) ) = A )
49	46, 47, 48	mp3an23 1366	. . . 4 |- ( A e. CC -> ( 2 x. ( A / 2 ) ) = A )
50	45, 49	syl 14	. . 3 |- ( A e. ( 0 (,] 2 ) -> ( 2 x. ( A / 2 ) ) = A )
51	50	fveq2d 5652	. 2 |- ( A e. ( 0 (,] 2 ) -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( sin ` A ) )
52	43, 51	breqtrd 4119	1 |- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   RRcr 8091   0cc0 8092   1c1 8093   x. cmul 8097   RR*cxr 8272   < clt 8273   <_ cle 8274   # cap 8820   / cdiv 8911   2c2 9253   (,]cioc 10185   sincsin 12285   cosccos 12286

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-disj 4070  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-sup 7243  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-ioc 10189  df-ico 10190  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-fac 11051  df-bc 11073  df-ihash 11101  df-shft 11455  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-sumdc 11994  df-ef 12289  df-sin 12291  df-cos 12292

This theorem is referenced by:  sincos2sgn  12407  cos12dec  12409  sin0pilem1  15592  sin0pilem2  15593  sinhalfpilem  15602  sincosq1lem  15636