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Theorem sin01gt0 13782
Description: The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
Assertion
Ref Expression
sin01gt0  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  A
) )

Proof of Theorem sin01gt0
StepHypRef Expression
1 0xr 9636 . . . . . . . 8  |-  0  e.  RR*
2 1re 9591 . . . . . . . 8  |-  1  e.  RR
3 elioc2 11583 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
41, 2, 3mp2an 672 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
54simp1bi 1011 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
6 3nn0 10809 . . . . . 6  |-  3  e.  NN0
7 reexpcl 12147 . . . . . 6  |-  ( ( A  e.  RR  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  RR )
85, 6, 7sylancl 662 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  e.  RR )
9 3re 10605 . . . . . 6  |-  3  e.  RR
10 3ne0 10626 . . . . . 6  |-  3  =/=  0
11 redivcl 10259 . . . . . 6  |-  ( ( ( A ^ 3 )  e.  RR  /\  3  e.  RR  /\  3  =/=  0 )  ->  (
( A ^ 3 )  /  3 )  e.  RR )
129, 10, 11mp3an23 1316 . . . . 5  |-  ( ( A ^ 3 )  e.  RR  ->  (
( A ^ 3 )  /  3 )  e.  RR )
138, 12syl 16 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  e.  RR )
14 3z 10893 . . . . . . . . 9  |-  3  e.  ZZ
15 expgt0 12163 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  3  e.  ZZ  /\  0  <  A )  ->  0  <  ( A ^ 3 ) )
1614, 15mp3an2 1312 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A ^ 3 ) )
17163adant3 1016 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  0  <  ( A ^ 3 ) )
184, 17sylbi 195 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A ^ 3 ) )
19 0lt1 10071 . . . . . . . . 9  |-  0  <  1
20 3pos 10625 . . . . . . . . 9  |-  0  <  3
21 1lt3 10700 . . . . . . . . . . 11  |-  1  <  3
22 ltdiv2OLD 10427 . . . . . . . . . . 11  |-  ( ( ( 1  e.  RR  /\  3  e.  RR  /\  ( A ^ 3 )  e.  RR )  /\  ( 0  <  1  /\  0  <  3  /\  0  <  ( A ^ 3 ) ) )  ->  ( 1  <  3  <->  ( ( A ^ 3 )  / 
3 )  <  (
( A ^ 3 )  /  1 ) ) )
2321, 22mpbii 211 . . . . . . . . . 10  |-  ( ( ( 1  e.  RR  /\  3  e.  RR  /\  ( A ^ 3 )  e.  RR )  /\  ( 0  <  1  /\  0  <  3  /\  0  <  ( A ^ 3 ) ) )  ->  ( ( A ^ 3 )  / 
3 )  <  (
( A ^ 3 )  /  1 ) )
2423expcom 435 . . . . . . . . 9  |-  ( ( 0  <  1  /\  0  <  3  /\  0  <  ( A ^ 3 ) )  ->  ( ( 1  e.  RR  /\  3  e.  RR  /\  ( A ^ 3 )  e.  RR )  ->  (
( A ^ 3 )  /  3 )  <  ( ( A ^ 3 )  / 
1 ) ) )
2519, 20, 24mp3an12 1314 . . . . . . . 8  |-  ( 0  <  ( A ^
3 )  ->  (
( 1  e.  RR  /\  3  e.  RR  /\  ( A ^ 3 )  e.  RR )  -> 
( ( A ^
3 )  /  3
)  <  ( ( A ^ 3 )  / 
1 ) ) )
2625com12 31 . . . . . . 7  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  ( A ^ 3 )  e.  RR )  ->  (
0  <  ( A ^ 3 )  -> 
( ( A ^
3 )  /  3
)  <  ( ( A ^ 3 )  / 
1 ) ) )
272, 9, 26mp3an12 1314 . . . . . 6  |-  ( ( A ^ 3 )  e.  RR  ->  (
0  <  ( A ^ 3 )  -> 
( ( A ^
3 )  /  3
)  <  ( ( A ^ 3 )  / 
1 ) ) )
288, 18, 27sylc 60 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  ( ( A ^ 3 )  / 
1 ) )
298recnd 9618 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  e.  CC )
3029div1d 10308 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  1 )  =  ( A ^
3 ) )
3128, 30breqtrd 4471 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  ( A ^
3 ) )
32 1nn0 10807 . . . . . . 7  |-  1  e.  NN0
3332a1i 11 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  1  e.  NN0 )
34 1le3 10748 . . . . . . . 8  |-  1  <_  3
35 1z 10890 . . . . . . . . 9  |-  1  e.  ZZ
3635eluz1i 11085 . . . . . . . 8  |-  ( 3  e.  ( ZZ>= `  1
)  <->  ( 3  e.  ZZ  /\  1  <_ 
3 ) )
3714, 34, 36mpbir2an 918 . . . . . . 7  |-  3  e.  ( ZZ>= `  1 )
3837a1i 11 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  3  e.  ( ZZ>= `  1 )
)
394simp2bi 1012 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  A )
40 0re 9592 . . . . . . . 8  |-  0  e.  RR
41 ltle 9669 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
4240, 5, 41sylancr 663 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  (
0  <  A  ->  0  <_  A ) )
4339, 42mpd 15 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <_  A )
444simp3bi 1013 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  <_  1 )
455, 33, 38, 43, 44leexp2rd 12307 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_ 
( A ^ 1 ) )
465recnd 9618 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  CC )
4746exp1d 12269 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 1 )  =  A )
4845, 47breqtrd 4471 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_  A )
4913, 8, 5, 31, 48ltletrd 9737 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  A )
5013, 5posdifd 10135 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( A ^
3 )  /  3
)  <  A  <->  0  <  ( A  -  ( ( A ^ 3 )  /  3 ) ) ) )
5149, 50mpbid 210 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A  -  (
( A ^ 3 )  /  3 ) ) )
52 sin01bnd 13777 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A  -  (
( A ^ 3 )  /  3 ) )  <  ( sin `  A )  /\  ( sin `  A )  < 
A ) )
5352simpld 459 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A  -  ( ( A ^ 3 )  / 
3 ) )  < 
( sin `  A
) )
545, 13resubcld 9983 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A  -  ( ( A ^ 3 )  / 
3 ) )  e.  RR )
555resincld 13735 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( sin `  A )  e.  RR )
56 lttr 9657 . . . 4  |-  ( ( 0  e.  RR  /\  ( A  -  (
( A ^ 3 )  /  3 ) )  e.  RR  /\  ( sin `  A )  e.  RR )  -> 
( ( 0  < 
( A  -  (
( A ^ 3 )  /  3 ) )  /\  ( A  -  ( ( A ^ 3 )  / 
3 ) )  < 
( sin `  A
) )  ->  0  <  ( sin `  A
) ) )
5740, 56mp3an1 1311 . . 3  |-  ( ( ( A  -  (
( A ^ 3 )  /  3 ) )  e.  RR  /\  ( sin `  A )  e.  RR )  -> 
( ( 0  < 
( A  -  (
( A ^ 3 )  /  3 ) )  /\  ( A  -  ( ( A ^ 3 )  / 
3 ) )  < 
( sin `  A
) )  ->  0  <  ( sin `  A
) ) )
5854, 55, 57syl2anc 661 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 0  <  ( A  -  ( ( A ^ 3 )  / 
3 ) )  /\  ( A  -  (
( A ^ 3 )  /  3 ) )  <  ( sin `  A ) )  -> 
0  <  ( sin `  A ) ) )
5951, 53, 58mp2and 679 1  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   RRcr 9487   0cc0 9488   1c1 9489   RR*cxr 9623    < clt 9624    <_ cle 9625    - cmin 9801    / cdiv 10202   3c3 10582   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   (,]cioc 11526   ^cexp 12130   sincsin 13657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-ioc 11530  df-ico 11531  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-fac 12318  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-sin 13663
This theorem is referenced by:  sin02gt0  13784  sincos1sgn  13785  sincos4thpi  22639
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