Theorem sin02gt0 12531

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 8372	. . . . . . 7 |- 0 e. RR*
2		2re 9374	. . . . . . 7 |- 2 e. RR
3		elioc2 10338	. . . . . . 7 |- ( ( 0 e. RR* /\ 2 e. RR ) -> ( A e. ( 0 (,] 2 ) <-> ( A e. RR /\ 0 < A /\ A <_ 2 ) ) )
4	1, 2, 3	mp2an 430	. . . . . 6 |- ( A e. ( 0 (,] 2 ) <-> ( A e. RR /\ 0 < A /\ A <_ 2 ) )
5		rehalfcl 9532	. . . . . . 7 |- ( A e. RR -> ( A / 2 ) e. RR )
6	5	3ad2ant1 1049	. . . . . 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 12487	. . . . . 6 |- ( ( A / 2 ) e. RR -> ( sin ` ( A / 2 ) ) e. RR )
9		recoscl 12488	. . . . . 6 |- ( ( A / 2 ) e. RR -> ( cos ` ( A / 2 ) ) e. RR )
10	8, 9	remulcld 8356	. . . . 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 9395	. . . . . . . . . 10 |- 0 < 2
13		divgt0 9202	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( A / 2 ) )
14	2, 12, 13	mpanr12 443	. . . . . . . . 9 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A / 2 ) )
15	14	3adant3 1048	. . . . . . . 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 9199	. . . . . . . . . . . 12 |- ( ( A e. RR /\ 2 e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A <_ 2 <-> ( A / 2 ) <_ ( 2 / 2 ) ) )
18	2, 16, 17	mp3an23 1370	. . . . . . . . . . 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 9437	. . . . . . . . . 10 |- ( 2 / 2 ) = 1
21	19, 20	breqtrdi 4171	. . . . . . . . 9 |- ( ( A e. RR /\ A <_ 2 ) -> ( A / 2 ) <_ 1 )
22	21	3adant2 1047	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( A / 2 ) <_ 1 )
23	6, 15, 22	3jca 1208	. . . . . . 7 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) <_ 1 ) )
24		1re 8325	. . . . . . . 8 |- 1 e. RR
25		elioc2 10338	. . . . . . . 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 430	. . . . . . 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 12529	. . . . . 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 12530	. . . . . 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 8397	. . . . . . 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 415	. . . . . 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 437	. . . 4 |- ( A e. ( 0 (,] 2 ) -> 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) )
36		axmulgt0 8397	. . . . . 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 428	. . . . 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 434	. . . 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 8354	. . . 4 |- ( A e. ( 0 (,] 2 ) -> ( A / 2 ) e. CC )
41		sin2t 12516	. . . 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 4158	. 2 |- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` ( 2 x. ( A / 2 ) ) ) )
44	4	simp1bi 1043	. . . . 5 |- ( A e. ( 0 (,] 2 ) -> A e. RR )
45	44	recnd 8354	. . . 4 |- ( A e. ( 0 (,] 2 ) -> A e. CC )
46		2cn 9375	. . . . 5 |- 2 e. CC
47		2ap0 9397	. . . . 5 |- 2 # 0
48		divcanap2 9010	. . . . 5 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( 2 x. ( A / 2 ) ) = A )
49	46, 47, 48	mp3an23 1370	. . . 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 5699	. 2 |- ( A e. ( 0 (,] 2 ) -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( sin ` A ) )
52	43, 51	breqtrd 4156	1 |- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   x. cmul 8184   RR*cxr 8359   < clt 8360   <_ cle 8361   # cap 8909   / cdiv 9002   2c2 9355   (,]cioc 10291   sincsin 12411   cosccos 12412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-ioc 10295  df-ico 10296  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-bc 11186  df-ihash 11215  df-shft 11580  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120  df-ef 12415  df-sin 12417  df-cos 12418

This theorem is used by:  sincos2sgn  12533  cos12dec  12535  sin0pilem1  15882  sin0pilem2  15883  sinhalfpilem  15892  sincosq1lem  15926