Theorem sin02gt0 12275

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 8193	. . . . . . 7 |- 0 e. RR*
2		2re 9180	. . . . . . 7 |- 2 e. RR
3		elioc2 10132	. . . . . . 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 9338	. . . . . . 7 |- ( A e. RR -> ( A / 2 ) e. RR )
6	5	3ad2ant1 1042	. . . . . 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 12231	. . . . . 6 |- ( ( A / 2 ) e. RR -> ( sin ` ( A / 2 ) ) e. RR )
9		recoscl 12232	. . . . . 6 |- ( ( A / 2 ) e. RR -> ( cos ` ( A / 2 ) ) e. RR )
10	8, 9	remulcld 8177	. . . . 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 9201	. . . . . . . . . 10 |- 0 < 2
13		divgt0 9019	. . . . . . . . . 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 1041	. . . . . . . 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 9016	. . . . . . . . . . . 12 |- ( ( A e. RR /\ 2 e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A <_ 2 <-> ( A / 2 ) <_ ( 2 / 2 ) ) )
18	2, 16, 17	mp3an23 1363	. . . . . . . . . . 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 9243	. . . . . . . . . 10 |- ( 2 / 2 ) = 1
21	19, 20	breqtrdi 4124	. . . . . . . . 9 |- ( ( A e. RR /\ A <_ 2 ) -> ( A / 2 ) <_ 1 )
22	21	3adant2 1040	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( A / 2 ) <_ 1 )
23	6, 15, 22	3jca 1201	. . . . . . 7 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) <_ 1 ) )
24		1re 8145	. . . . . . . 8 |- 1 e. RR
25		elioc2 10132	. . . . . . . 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 12273	. . . . . 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 12274	. . . . . 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 8218	. . . . . . 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 8218	. . . . . 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 8175	. . . 4 |- ( A e. ( 0 (,] 2 ) -> ( A / 2 ) e. CC )
41		sin2t 12260	. . . 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 4111	. 2 |- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` ( 2 x. ( A / 2 ) ) ) )
44	4	simp1bi 1036	. . . . 5 |- ( A e. ( 0 (,] 2 ) -> A e. RR )
45	44	recnd 8175	. . . 4 |- ( A e. ( 0 (,] 2 ) -> A e. CC )
46		2cn 9181	. . . . 5 |- 2 e. CC
47		2ap0 9203	. . . . 5 |- 2 # 0
48		divcanap2 8827	. . . . 5 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( 2 x. ( A / 2 ) ) = A )
49	46, 47, 48	mp3an23 1363	. . . 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 5631	. 2 |- ( A e. ( 0 (,] 2 ) -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( sin ` A ) )
52	43, 51	breqtrd 4109	1 |- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   1c1 8000   x. cmul 8004   RR*cxr 8180   < clt 8181   <_ cle 8182   # cap 8728   / cdiv 8819   2c2 9161   (,]cioc 10085   sincsin 12155   cosccos 12156

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-disj 4060  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-sup 7151  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-ioc 10089  df-ico 10090  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-fac 10948  df-bc 10970  df-ihash 10998  df-shft 11326  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-sumdc 11865  df-ef 12159  df-sin 12161  df-cos 12162

This theorem is referenced by:  sincos2sgn  12277  cos12dec  12279  sin0pilem1  15455  sin0pilem2  15456  sinhalfpilem  15465  sincosq1lem  15499