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Theorem sin01gt0 14136
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 9672 . . . . . . . 8  |-  0  e.  RR*
2 1re 9627 . . . . . . . 8  |-  1  e.  RR
3 elioc2 11643 . . . . . . . 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 1014 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
6 3nn0 10856 . . . . . 6  |-  3  e.  NN0
7 reexpcl 12229 . . . . . 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 10652 . . . . . 6  |-  3  e.  RR
10 3ne0 10673 . . . . . 6  |-  3  =/=  0
11 redivcl 10306 . . . . . 6  |-  ( ( ( A ^ 3 )  e.  RR  /\  3  e.  RR  /\  3  =/=  0 )  ->  (
( A ^ 3 )  /  3 )  e.  RR )
129, 10, 11mp3an23 1320 . . . . 5  |-  ( ( A ^ 3 )  e.  RR  ->  (
( A ^ 3 )  /  3 )  e.  RR )
138, 12syl 17 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  e.  RR )
14 3z 10940 . . . . . . . . 9  |-  3  e.  ZZ
15 expgt0 12245 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  3  e.  ZZ  /\  0  <  A )  ->  0  <  ( A ^ 3 ) )
1614, 15mp3an2 1316 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A ^ 3 ) )
17163adant3 1019 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  0  <  ( A ^ 3 ) )
184, 17sylbi 197 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A ^ 3 ) )
19 0lt1 10117 . . . . . . . . 9  |-  0  <  1
20 3pos 10672 . . . . . . . . 9  |-  0  <  3
21 1lt3 10747 . . . . . . . . . . 11  |-  1  <  3
22 ltdiv2OLD 10473 . . . . . . . . . . 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 213 . . . . . . . . . 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 1318 . . . . . . . 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 1318 . . . . . 6  |-  ( ( A ^ 3 )  e.  RR  ->  (
0  <  ( A ^ 3 )  -> 
( ( A ^
3 )  /  3
)  <  ( ( A ^ 3 )  / 
1 ) ) )
288, 18, 27sylc 61 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  ( ( A ^ 3 )  / 
1 ) )
298recnd 9654 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  e.  CC )
3029div1d 10355 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  1 )  =  ( A ^
3 ) )
3128, 30breqtrd 4421 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  ( A ^
3 ) )
32 1nn0 10854 . . . . . . 7  |-  1  e.  NN0
3332a1i 11 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  1  e.  NN0 )
34 1le3 10795 . . . . . . . 8  |-  1  <_  3
35 1z 10937 . . . . . . . . 9  |-  1  e.  ZZ
3635eluz1i 11136 . . . . . . . 8  |-  ( 3  e.  ( ZZ>= `  1
)  <->  ( 3  e.  ZZ  /\  1  <_ 
3 ) )
3714, 34, 36mpbir2an 923 . . . . . . 7  |-  3  e.  ( ZZ>= `  1 )
3837a1i 11 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  3  e.  ( ZZ>= `  1 )
)
394simp2bi 1015 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  A )
40 0re 9628 . . . . . . . 8  |-  0  e.  RR
41 ltle 9706 . . . . . . . 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 1016 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  <_  1 )
455, 33, 38, 43, 44leexp2rd 12389 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_ 
( A ^ 1 ) )
465recnd 9654 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  CC )
4746exp1d 12351 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 1 )  =  A )
4845, 47breqtrd 4421 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_  A )
4913, 8, 5, 31, 48ltletrd 9778 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  A )
5013, 5posdifd 10181 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( A ^
3 )  /  3
)  <  A  <->  0  <  ( A  -  ( ( A ^ 3 )  /  3 ) ) ) )
5149, 50mpbid 212 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A  -  (
( A ^ 3 )  /  3 ) ) )
52 sin01bnd 14131 . . 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 10030 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A  -  ( ( A ^ 3 )  / 
3 ) )  e.  RR )
555resincld 14089 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( sin `  A )  e.  RR )
56 lttr 9694 . . . 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 1315 . . 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 186    /\ wa 369    /\ w3a 976    e. wcel 1844    =/= wne 2600   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   RRcr 9523   0cc0 9524   1c1 9525   RR*cxr 9659    < clt 9660    <_ cle 9661    - cmin 9843    / cdiv 10249   3c3 10629   NN0cn0 10838   ZZcz 10907   ZZ>=cuz 11129   (,]cioc 11585   ^cexp 12212   sincsin 14010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602  ax-addf 9603  ax-mulf 9604
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-oi 7971  df-card 8354  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-n0 10839  df-z 10908  df-uz 11130  df-rp 11268  df-ioc 11589  df-ico 11590  df-fz 11729  df-fzo 11857  df-fl 11968  df-seq 12154  df-exp 12213  df-fac 12400  df-hash 12455  df-shft 13051  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-limsup 13445  df-clim 13462  df-rlim 13463  df-sum 13660  df-ef 14014  df-sin 14016
This theorem is referenced by:  sin02gt0  14138  sincos1sgn  14139  sincos4thpi  23200
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