Theorem sin02gt0 11726

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 7966	. . . . . . 7 |- 0 e. RR*
2		2re 8948	. . . . . . 7 |- 2 e. RR
3		elioc2 9893	. . . . . . 7 |- ( ( 0 e. RR* /\ 2 e. RR ) -> ( A e. ( 0 (,] 2 ) <-> ( A e. RR /\ 0 < A /\ A <_ 2 ) ) )
4	1, 2, 3	mp2an 424	. . . . . 6 |- ( A e. ( 0 (,] 2 ) <-> ( A e. RR /\ 0 < A /\ A <_ 2 ) )
5		rehalfcl 9105	. . . . . . 7 |- ( A e. RR -> ( A / 2 ) e. RR )
6	5	3ad2ant1 1013	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( A / 2 ) e. RR )
7	4, 6	sylbi 120	. . . . 5 |- ( A e. ( 0 (,] 2 ) -> ( A / 2 ) e. RR )
8		resincl 11683	. . . . . 6 |- ( ( A / 2 ) e. RR -> ( sin ` ( A / 2 ) ) e. RR )
9		recoscl 11684	. . . . . 6 |- ( ( A / 2 ) e. RR -> ( cos ` ( A / 2 ) ) e. RR )
10	8, 9	remulcld 7950	. . . . 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 8969	. . . . . . . . . 10 |- 0 < 2
13		divgt0 8788	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( A / 2 ) )
14	2, 12, 13	mpanr12 437	. . . . . . . . 9 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A / 2 ) )
15	14	3adant3 1012	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> 0 < ( A / 2 ) )
16	2, 12	pm3.2i 270	. . . . . . . . . . . 12 |- ( 2 e. RR /\ 0 < 2 )
17		lediv1 8785	. . . . . . . . . . . 12 |- ( ( A e. RR /\ 2 e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A <_ 2 <-> ( A / 2 ) <_ ( 2 / 2 ) ) )
18	2, 16, 17	mp3an23 1324	. . . . . . . . . . 11 |- ( A e. RR -> ( A <_ 2 <-> ( A / 2 ) <_ ( 2 / 2 ) ) )
19	18	biimpa 294	. . . . . . . . . 10 |- ( ( A e. RR /\ A <_ 2 ) -> ( A / 2 ) <_ ( 2 / 2 ) )
20		2div2e1 9010	. . . . . . . . . 10 |- ( 2 / 2 ) = 1
21	19, 20	breqtrdi 4030	. . . . . . . . 9 |- ( ( A e. RR /\ A <_ 2 ) -> ( A / 2 ) <_ 1 )
22	21	3adant2 1011	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( A / 2 ) <_ 1 )
23	6, 15, 22	3jca 1172	. . . . . . 7 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) <_ 1 ) )
24		1re 7919	. . . . . . . 8 |- 1 e. RR
25		elioc2 9893	. . . . . . . 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 424	. . . . . . 7 |- ( ( A / 2 ) e. ( 0 (,] 1 ) <-> ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) <_ 1 ) )
27	23, 4, 26	3imtr4i 200	. . . . . 6 |- ( A e. ( 0 (,] 2 ) -> ( A / 2 ) e. ( 0 (,] 1 ) )
28		sin01gt0 11724	. . . . . 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 11725	. . . . . 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 7991	. . . . . . 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 409	. . . . . 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 431	. . . 4 |- ( A e. ( 0 (,] 2 ) -> 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) )
36		axmulgt0 7991	. . . . . 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 422	. . . . 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 428	. . . 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 7948	. . . 4 |- ( A e. ( 0 (,] 2 ) -> ( A / 2 ) e. CC )
41		sin2t 11712	. . . 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 4017	. 2 |- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` ( 2 x. ( A / 2 ) ) ) )
44	4	simp1bi 1007	. . . . 5 |- ( A e. ( 0 (,] 2 ) -> A e. RR )
45	44	recnd 7948	. . . 4 |- ( A e. ( 0 (,] 2 ) -> A e. CC )
46		2cn 8949	. . . . 5 |- 2 e. CC
47		2ap0 8971	. . . . 5 |- 2 # 0
48		divcanap2 8597	. . . . 5 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( 2 x. ( A / 2 ) ) = A )
49	46, 47, 48	mp3an23 1324	. . . 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 5500	. 2 |- ( A e. ( 0 (,] 2 ) -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( sin ` A ) )
52	43, 51	breqtrd 4015	1 |- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775   x. cmul 7779   RR*cxr 7953   < clt 7954   <_ cle 7955   # cap 8500   / cdiv 8589   2c2 8929   (,]cioc 9846   sincsin 11607   cosccos 11608

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-disj 3967  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ioc 9850  df-ico 9851  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-bc 10682  df-ihash 10710  df-shft 10779  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611  df-sin 11613  df-cos 11614

This theorem is referenced by:  sincos2sgn  11728  cos12dec  11730  sin0pilem1  13496  sin0pilem2  13497  sinhalfpilem  13506  sincosq1lem  13540