Theorem sin02gt0 11115

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 7595	. . . . . . 7 |- 0 e. RR*
2		2re 8553	. . . . . . 7 |- 2 e. RR
3		elioc2 9415	. . . . . . 7 |- ( ( 0 e. RR* /\ 2 e. RR ) -> ( A e. ( 0 (,] 2 ) <-> ( A e. RR /\ 0 < A /\ A <_ 2 ) ) )
4	1, 2, 3	mp2an 418	. . . . . 6 |- ( A e. ( 0 (,] 2 ) <-> ( A e. RR /\ 0 < A /\ A <_ 2 ) )
5		rehalfcl 8704	. . . . . . 7 |- ( A e. RR -> ( A / 2 ) e. RR )
6	5	3ad2ant1 965	. . . . . 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 11072	. . . . . 6 |- ( ( A / 2 ) e. RR -> ( sin ` ( A / 2 ) ) e. RR )
9		recoscl 11073	. . . . . 6 |- ( ( A / 2 ) e. RR -> ( cos ` ( A / 2 ) ) e. RR )
10	8, 9	remulcld 7579	. . . . 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 8574	. . . . . . . . . 10 |- 0 < 2
13		divgt0 8394	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( A / 2 ) )
14	2, 12, 13	mpanr12 431	. . . . . . . . 9 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A / 2 ) )
15	14	3adant3 964	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> 0 < ( A / 2 ) )
16	2, 12	pm3.2i 267	. . . . . . . . . . . 12 |- ( 2 e. RR /\ 0 < 2 )
17		lediv1 8391	. . . . . . . . . . . 12 |- ( ( A e. RR /\ 2 e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( A <_ 2 <-> ( A / 2 ) <_ ( 2 / 2 ) ) )
18	2, 16, 17	mp3an23 1266	. . . . . . . . . . 11 |- ( A e. RR -> ( A <_ 2 <-> ( A / 2 ) <_ ( 2 / 2 ) ) )
19	18	biimpa 291	. . . . . . . . . 10 |- ( ( A e. RR /\ A <_ 2 ) -> ( A / 2 ) <_ ( 2 / 2 ) )
20		2div2e1 8609	. . . . . . . . . 10 |- ( 2 / 2 ) = 1
21	19, 20	syl6breq 3890	. . . . . . . . 9 |- ( ( A e. RR /\ A <_ 2 ) -> ( A / 2 ) <_ 1 )
22	21	3adant2 963	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( A / 2 ) <_ 1 )
23	6, 15, 22	3jca 1124	. . . . . . 7 |- ( ( A e. RR /\ 0 < A /\ A <_ 2 ) -> ( ( A / 2 ) e. RR /\ 0 < ( A / 2 ) /\ ( A / 2 ) <_ 1 ) )
24		1re 7548	. . . . . . . 8 |- 1 e. RR
25		elioc2 9415	. . . . . . . 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 418	. . . . . . 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 11113	. . . . . 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 11114	. . . . . 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 7619	. . . . . . 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 404	. . . . . 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 425	. . . 4 |- ( A e. ( 0 (,] 2 ) -> 0 < ( ( sin ` ( A / 2 ) ) x. ( cos ` ( A / 2 ) ) ) )
36		axmulgt0 7619	. . . . . 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 416	. . . . 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 422	. . . 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 7577	. . . 4 |- ( A e. ( 0 (,] 2 ) -> ( A / 2 ) e. CC )
41		sin2t 11101	. . . 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 3877	. 2 |- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` ( 2 x. ( A / 2 ) ) ) )
44	4	simp1bi 959	. . . . 5 |- ( A e. ( 0 (,] 2 ) -> A e. RR )
45	44	recnd 7577	. . . 4 |- ( A e. ( 0 (,] 2 ) -> A e. CC )
46		2cn 8554	. . . . 5 |- 2 e. CC
47		2ap0 8576	. . . . 5 |- 2 # 0
48		divcanap2 8208	. . . . 5 |- ( ( A e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( 2 x. ( A / 2 ) ) = A )
49	46, 47, 48	mp3an23 1266	. . . 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 5322	. 2 |- ( A e. ( 0 (,] 2 ) -> ( sin ` ( 2 x. ( A / 2 ) ) ) = ( sin ` A ) )
52	43, 51	breqtrd 3875	1 |- ( A e. ( 0 (,] 2 ) -> 0 < ( sin ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 925   = wceq 1290   e. wcel 1439   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   CCcc 7409   RRcr 7410   0cc0 7411   1c1 7412   x. cmul 7416   RR*cxr 7582   < clt 7583   <_ cle 7584   # cap 8119   / cdiv 8200   2c2 8534   (,]cioc 9368   sincsin 10995   cosccos 10996

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525  ax-caucvg 7526

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-disj 3829  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-sup 6733  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-5 8545  df-6 8546  df-7 8547  df-8 8548  df-n0 8735  df-z 8812  df-uz 9081  df-q 9166  df-rp 9196  df-ioc 9372  df-ico 9373  df-fz 9486  df-fzo 9615  df-iseq 9914  df-seq3 9915  df-exp 10016  df-fac 10195  df-bc 10217  df-ihash 10245  df-shft 10310  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493  df-clim 10728  df-isum 10804  df-ef 10999  df-sin 11001  df-cos 11002

This theorem is referenced by:  sincos2sgn  11117