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Theorem sin01gt0 13456
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 9417 . . . . . . . 8  |-  0  e.  RR*
2 1re 9372 . . . . . . . 8  |-  1  e.  RR
3 elioc2 11345 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
41, 2, 3mp2an 665 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
54simp1bi 996 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
6 3nn0 10584 . . . . . 6  |-  3  e.  NN0
7 reexpcl 11865 . . . . . 6  |-  ( ( A  e.  RR  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  RR )
85, 6, 7sylancl 655 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  e.  RR )
9 3re 10382 . . . . . 6  |-  3  e.  RR
10 3ne0 10403 . . . . . 6  |-  3  =/=  0
11 redivcl 10037 . . . . . 6  |-  ( ( ( A ^ 3 )  e.  RR  /\  3  e.  RR  /\  3  =/=  0 )  ->  (
( A ^ 3 )  /  3 )  e.  RR )
129, 10, 11mp3an23 1299 . . . . 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 10666 . . . . . . . . 9  |-  3  e.  ZZ
15 expgt0 11880 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  3  e.  ZZ  /\  0  <  A )  ->  0  <  ( A ^ 3 ) )
1614, 15mp3an2 1295 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A ^ 3 ) )
17163adant3 1001 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  0  <  ( A ^ 3 ) )
184, 17sylbi 195 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A ^ 3 ) )
19 0lt1 9849 . . . . . . . . 9  |-  0  <  1
20 3pos 10402 . . . . . . . . 9  |-  0  <  3
21 1lt3 10477 . . . . . . . . . . 11  |-  1  <  3
22 ltdiv2OLD 10205 . . . . . . . . . . 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 1297 . . . . . . . 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 1297 . . . . . 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 9399 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  e.  CC )
3029div1d 10086 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  1 )  =  ( A ^
3 ) )
3128, 30breqtrd 4304 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  ( A ^
3 ) )
32 1nn0 10582 . . . . . . 7  |-  1  e.  NN0
3332a1i 11 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  1  e.  NN0 )
34 1le3 10525 . . . . . . . 8  |-  1  <_  3
35 1z 10663 . . . . . . . . 9  |-  1  e.  ZZ
3635eluz1i 10855 . . . . . . . 8  |-  ( 3  e.  ( ZZ>= `  1
)  <->  ( 3  e.  ZZ  /\  1  <_ 
3 ) )
3714, 34, 36mpbir2an 904 . . . . . . 7  |-  3  e.  ( ZZ>= `  1 )
3837a1i 11 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  3  e.  ( ZZ>= `  1 )
)
394simp2bi 997 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  A )
40 0re 9373 . . . . . . . 8  |-  0  e.  RR
41 ltle 9450 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
4240, 5, 41sylancr 656 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  (
0  <  A  ->  0  <_  A ) )
4339, 42mpd 15 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <_  A )
444simp3bi 998 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  <_  1 )
455, 33, 38, 43, 44leexp2rd 12024 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_ 
( A ^ 1 ) )
465recnd 9399 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  CC )
4746exp1d 11986 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 1 )  =  A )
4845, 47breqtrd 4304 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_  A )
4913, 8, 5, 31, 48ltletrd 9518 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  A )
5013, 5posdifd 9913 . . 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 13451 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A  -  (
( A ^ 3 )  /  3 ) )  <  ( sin `  A )  /\  ( sin `  A )  < 
A ) )
5352simpld 456 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A  -  ( ( A ^ 3 )  / 
3 ) )  < 
( sin `  A
) )
545, 13resubcld 9763 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A  -  ( ( A ^ 3 )  / 
3 ) )  e.  RR )
555resincld 13409 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( sin `  A )  e.  RR )
56 lttr 9438 . . . 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 1294 . . 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 654 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 0  <  ( A  -  ( ( A ^ 3 )  / 
3 ) )  /\  ( A  -  (
( A ^ 3 )  /  3 ) )  <  ( sin `  A ) )  -> 
0  <  ( sin `  A ) ) )
5951, 53, 58mp2and 672 1  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    e. wcel 1755    =/= wne 2596   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   RRcr 9268   0cc0 9269   1c1 9270   RR*cxr 9404    < clt 9405    <_ cle 9406    - cmin 9582    / cdiv 9980   3c3 10359   NN0cn0 10566   ZZcz 10633   ZZ>=cuz 10848   (,]cioc 11288   ^cexp 11848   sincsin 13331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-n0 10567  df-z 10634  df-uz 10849  df-rp 10979  df-ioc 11292  df-ico 11293  df-fz 11424  df-fzo 11532  df-fl 11625  df-seq 11790  df-exp 11849  df-fac 12035  df-hash 12087  df-shft 12539  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-limsup 12932  df-clim 12949  df-rlim 12950  df-sum 13147  df-ef 13335  df-sin 13337
This theorem is referenced by:  sin02gt0  13458  sincos1sgn  13459  sincos4thpi  21859
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