MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sin02gt0 Structured version   Visualization version   Unicode version

Theorem sin02gt0 14239
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 0xr 9684 . . . . . . 7  |-  0  e.  RR*
2 2re 10676 . . . . . . 7  |-  2  e.  RR
3 elioc2 11694 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  2  e.  RR )  ->  ( A  e.  ( 0 (,] 2 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  2 ) ) )
41, 2, 3mp2an 677 . . . . . 6  |-  ( A  e.  ( 0 (,] 2 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  2 ) )
5 rehalfcl 10836 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  /  2 )  e.  RR )
653ad2ant1 1028 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  ( A  /  2 )  e.  RR )
74, 6sylbi 199 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  RR )
8 resincl 14187 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  ( sin `  ( A  / 
2 ) )  e.  RR )
9 recoscl 14188 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  ( cos `  ( A  / 
2 ) )  e.  RR )
108, 9remulcld 9668 . . . . 5  |-  ( ( A  /  2 )  e.  RR  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
117, 10syl 17 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) )  e.  RR )
12 2pos 10698 . . . . . . . . . 10  |-  0  <  2
13 divgt0 10470 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
0  <  ( A  /  2 ) )
142, 12, 13mpanr12 690 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A  /  2 ) )
15143adant3 1027 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  0  <  ( A  /  2
) )
162, 12pm3.2i 457 . . . . . . . . . . . 12  |-  ( 2  e.  RR  /\  0  <  2 )
17 lediv1 10467 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  2  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( A  <_ 
2  <->  ( A  / 
2 )  <_  (
2  /  2 ) ) )
182, 16, 17mp3an23 1355 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  <_  2  <->  ( A  /  2 )  <_ 
( 2  /  2
) ) )
1918biimpa 487 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  <_  2 )  -> 
( A  /  2
)  <_  ( 2  /  2 ) )
20 2div2e1 10729 . . . . . . . . . 10  |-  ( 2  /  2 )  =  1
2119, 20syl6breq 4441 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <_  2 )  -> 
( A  /  2
)  <_  1 )
22213adant2 1026 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  ( A  /  2 )  <_ 
1 )
236, 15, 223jca 1187 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  2 )  ->  (
( A  /  2
)  e.  RR  /\  0  <  ( A  / 
2 )  /\  ( A  /  2 )  <_ 
1 ) )
24 1re 9639 . . . . . . . 8  |-  1  e.  RR
25 elioc2 11694 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  (
( A  /  2
)  e.  ( 0 (,] 1 )  <->  ( ( A  /  2 )  e.  RR  /\  0  < 
( A  /  2
)  /\  ( A  /  2 )  <_ 
1 ) ) )
261, 24, 25mp2an 677 . . . . . . 7  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  <->  ( ( A  /  2 )  e.  RR  /\  0  < 
( A  /  2
)  /\  ( A  /  2 )  <_ 
1 ) )
2723, 4, 263imtr4i 270 . . . . . 6  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  ( 0 (,] 1
) )
28 sin01gt0 14237 . . . . . 6  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  ( A  /  2 ) ) )
2927, 28syl 17 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  ( A  /  2 ) ) )
30 cos01gt0 14238 . . . . . 6  |-  ( ( A  /  2 )  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  ( A  /  2 ) ) )
3127, 30syl 17 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( cos `  ( A  /  2 ) ) )
32 axmulgt0 9705 . . . . . . 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 ) ) ) ) )
338, 9, 32syl2anc 666 . . . . . 6  |-  ( ( A  /  2 )  e.  RR  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
347, 33syl 17 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  (
( 0  <  ( sin `  ( A  / 
2 ) )  /\  0  <  ( cos `  ( A  /  2 ) ) )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
3529, 31, 34mp2and 684 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( ( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) )
36 axmulgt0 9705 . . . . . 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 ) ) ) ) ) )
372, 36mpan 675 . . . . 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 ) ) ) ) ) )
3812, 37mpani 681 . . . 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 ) ) ) ) ) )
3911, 35, 38sylc 62 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
407recnd 9666 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  ( A  /  2 )  e.  CC )
41 sin2t 14224 . . . 4  |-  ( ( A  /  2 )  e.  CC  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
4240, 41syl 17 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( 2  x.  (
( sin `  ( A  /  2 ) )  x.  ( cos `  ( A  /  2 ) ) ) ) )
4339, 42breqtrrd 4428 . 2  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  (
2  x.  ( A  /  2 ) ) ) )
444simp1bi 1022 . . . . 5  |-  ( A  e.  ( 0 (,] 2 )  ->  A  e.  RR )
4544recnd 9666 . . . 4  |-  ( A  e.  ( 0 (,] 2 )  ->  A  e.  CC )
46 2cn 10677 . . . . 5  |-  2  e.  CC
47 2ne0 10699 . . . . 5  |-  2  =/=  0
48 divcan2 10275 . . . . 5  |-  ( ( A  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
2  x.  ( A  /  2 ) )  =  A )
4946, 47, 48mp3an23 1355 . . . 4  |-  ( A  e.  CC  ->  (
2  x.  ( A  /  2 ) )  =  A )
5045, 49syl 17 . . 3  |-  ( A  e.  ( 0 (,] 2 )  ->  (
2  x.  ( A  /  2 ) )  =  A )
5150fveq2d 5867 . 2  |-  ( A  e.  ( 0 (,] 2 )  ->  ( sin `  ( 2  x.  ( A  /  2
) ) )  =  ( sin `  A
) )
5243, 51breqtrd 4426 1  |-  ( A  e.  ( 0 (,] 2 )  ->  0  <  ( sin `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537    x. cmul 9541   RR*cxr 9671    < clt 9672    <_ cle 9673    / cdiv 10266   2c2 10656   (,]cioc 11633   sincsin 14109   cosccos 14110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-ioc 11637  df-ico 11638  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13123  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117
This theorem is referenced by:  sincos2sgn  14241  pilem2  23400  pilem2OLD  23401  sinhalfpilem  23411  sincosq1lem  23445
  Copyright terms: Public domain W3C validator