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Theorem sin02gt0 7420
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
StepHypRef Expression
1 rehalfclt 5981 . . . . . . . . 9 |- (A e. RR -> (A / 2) e. RR)
213ad2ant1 798 . . . . . . . 8 |- ((A e. RR /\ 0 < A /\ A <_ 2) -> (A / 2) e. RR)
3 2re 5926 . . . . . . . . . 10 |- 2 e. RR
4 2pos 5936 . . . . . . . . . 10 |- 0 < 2
5 divgt0t 5809 . . . . . . . . . . 11 |- (((A e. RR /\ 0 < A) /\ (2 e. RR /\ 0 < 2)) -> 0 < (A / 2))
65expcom 374 . . . . . . . . . 10 |- ((2 e. RR /\ 0 < 2) -> ((A e. RR /\ 0 < A) -> 0 < (A / 2)))
73, 4, 6mp2an 695 . . . . . . . . 9 |- ((A e. RR /\ 0 < A) -> 0 < (A / 2))
873adant3 797 . . . . . . . 8 |- ((A e. RR /\ 0 < A /\ A <_ 2) -> 0 < (A / 2))
9 lediv1t 5806 . . . . . . . . . . . . 13 |- (((A e. RR /\ 2 e. RR /\ 2 e. RR) /\ 0 < 2) -> (A <_ 2 <-> (A / 2) <_ (2 / 2)))
104, 9mpan2 694 . . . . . . . . . . . 12 |- ((A e. RR /\ 2 e. RR /\ 2 e. RR) -> (A <_ 2 <-> (A / 2) <_ (2 / 2)))
113, 3, 10mp3an23 905 . . . . . . . . . . 11 |- (A e. RR -> (A <_ 2 <-> (A / 2) <_ (2 / 2)))
1211biimpa 416 . . . . . . . . . 10 |- ((A e. RR /\ A <_ 2) -> (A / 2) <_ (2 / 2))
13 2cn 5927 . . . . . . . . . . 11 |- 2 e. CC
14 2ne0 5937 . . . . . . . . . . 11 |- 2 =/= 0
1513, 14divid 5726 . . . . . . . . . 10 |- (2 / 2) = 1
1612, 15syl6breq 2644 . . . . . . . . 9 |- ((A e. RR /\ A <_ 2) -> (A / 2) <_ 1)
17163adant2 796 . . . . . . . 8 |- ((A e. RR /\ 0 < A /\ A <_ 2) -> (A / 2) <_ 1)
182, 8, 173jca 817 . . . . . . 7 |- ((A e. RR /\ 0 < A /\ A <_ 2) -> ((A / 2) e. RR /\ 0 < (A / 2) /\ (A / 2) <_ 1))
19 0re 5412 . . . . . . . 8 |- 0 e. RR
20 elioc2t 6322 . . . . . . . 8 |- ((0 e. RR /\ 2 e. RR) -> (A e. (0(,]2) <-> (A e. RR /\ 0 < A /\ A <_ 2)))
2119, 3, 20mp2an 695 . . . . . . 7 |- (A e. (0(,]2) <-> (A e. RR /\ 0 < A /\ A <_ 2))
22 1re 5407 . . . . . . . 8 |- 1 e. RR
23 elioc2t 6322 . . . . . . . 8 |- ((0 e. RR /\ 1 e. RR) -> ((A / 2) e. (0(,]1) <-> ((A / 2) e. RR /\ 0 < (A / 2) /\ (A / 2) <_ 1)))
2419, 22, 23mp2an 695 . . . . . . 7 |- ((A / 2) e. (0(,]1) <-> ((A / 2) e. RR /\ 0 < (A / 2) /\ (A / 2) <_ 1))
2518, 21, 243imtr4 219 . . . . . 6 |- (A e. (0(,]2) -> (A / 2) e. (0(,]1))
26 sin01gt0 7418 . . . . . 6 |- ((A / 2) e. (0(,]1) -> 0 < (sin` (A / 2)))
2725, 26syl 10 . . . . 5 |- (A e. (0(,]2) -> 0 < (sin` (A / 2)))
28 cos01gt0 7419 . . . . . 6 |- ((A / 2) e. (0(,]1) -> 0 < (cos` (A / 2)))
2925, 28syl 10 . . . . 5 |- (A e. (0(,]2) -> 0 < (cos` (A / 2)))
3021, 2sylbi 199 . . . . . 6 |- (A e. (0(,]2) -> (A / 2) e. RR)
31 axmulgt0 5478 . . . . . . 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)))))
32 resinclt 7380 . . . . . . 7 |- ((A / 2) e. RR -> (sin` (A / 2)) e. RR)
33 recosclt 7381 . . . . . . 7 |- ((A / 2) e. RR -> (cos` (A / 2)) e. RR)
3431, 32, 33sylanc 471 . . . . . 6 |- ((A / 2) e. RR -> ((0 < (sin` (A / 2)) /\ 0 < (cos` (A / 2))) -> 0 < ((sin` (A / 2)) x. (cos` (A / 2)))))
3530, 34syl 10 . . . . 5 |- (A e. (0(,]2) -> ((0 < (sin` (A / 2)) /\ 0 < (cos` (A / 2))) -> 0 < ((sin` (A / 2)) x. (cos` (A / 2)))))
3627, 29, 35mp2and 701 . . . 4 |- (A e. (0(,]2) -> 0 < ((sin` (A / 2)) x. (cos` (A / 2))))
37 axmulrcl 5246 . . . . . 6 |- (((sin` (A / 2)) e. RR /\ (cos` (A / 2)) e. RR) -> ((sin` (A / 2)) x. (cos` (A / 2))) e. RR)
3837, 32, 33sylanc 471 . . . . 5 |- ((A / 2) e. RR -> ((sin` (A / 2)) x. (cos` (A / 2))) e. RR)
39 axmulgt0 5478 . . . . . . 7 |- ((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))))))
403, 39mpan 693 . . . . . 6 |- (((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))))))
414, 40mpani 696 . . . . 5 |- (((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))))))
4230, 38, 413syl 20 . . . 4 |- (A e. (0(,]2) -> (0 < ((sin` (A / 2)) x. (cos` (A / 2))) -> 0 < (2 x. ((sin` (A / 2)) x. (cos` (A / 2))))))
4336, 42mpd 26 . . 3 |- (A e. (0(,]2) -> 0 < (2 x. ((sin` (A / 2)) x. (cos` (A / 2)))))
4430recnd 5287 . . . 4 |- (A e. (0(,]2) -> (A / 2) e. CC)
45 sin2tt 7404 . . . 4 |- ((A / 2) e. CC -> (sin` (2 x. (A / 2))) = (2 x. ((sin` (A / 2)) x. (cos` (A / 2)))))
4644, 45syl 10 . . 3 |- (A e. (0(,]2) -> (sin` (2 x. (A / 2))) = (2 x. ((sin` (A / 2)) x. (cos` (A / 2)))))
4743, 46breqtrrd 2631 . 2 |- (A e. (0(,]2) -> 0 < (sin` (2 x. (A / 2))))
4821biimp 151 . . . . . 6 |- (A e. (0(,]2) -> (A e. RR /\ 0 < A /\ A <_ 2))
49483simp1d 792 . . . . 5 |- (A e. (0(,]2) -> A e. RR)
5049recnd 5287 . . . 4 |- (A e. (0(,]2) -> A e. CC)
51 divcan2t 5690 . . . . 5 |- ((2 e. CC /\ A e. CC /\ 2 =/= 0) -> (2 x. (A / 2)) = A)
5213, 14, 51mp3an13 904 . . . 4 |- (A e. CC -> (2 x. (A / 2)) = A)
5350, 52syl 10 . . 3 |- (A e. (0(,]2) -> (2 x. (A / 2)) = A)
5453fveq2d 3713 . 2 |- (A e. (0(,]2) -> (sin` (2 x. (A / 2))) = (sin` A))
5547, 54breqtrd 2629 1 |- (A e. (0(,]2) -> 0 < (sin` A))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   =/= wne 1577   class class class wbr 2609  ` cfv 3172  (class class class)co 3948  CCcc 5204  RRcr 5205  0cc0 5206  1c1 5207   x. cmul 5211   / cdiv 5266   <_ cle 5267   < clt 5458  2c2 5908  (,]cioc 6295  sincsin 7237  cosccos 7238
This theorem is referenced by:  sincos2sgn 7422  pilem1 8590  sinhalfpilem 8598  sincosq1lem 8620
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-3 5918  df-4 5919  df-5 5920  df-6 5921  df-7 5922  df-8 5923  df-n0 6047  df-z 6083  df-fl 6172  df-seq1