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Theorem sin01gt0 14244
Description: The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) Replace OLD theorem. (Revised by Wolf Lammen, 25-Sep-2020.)
Assertion
Ref Expression
sin01gt0  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  A
) )

Proof of Theorem sin01gt0
StepHypRef Expression
1 0xr 9687 . . . . . . . 8  |-  0  e.  RR*
2 1re 9642 . . . . . . . 8  |-  1  e.  RR
3 elioc2 11697 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
41, 2, 3mp2an 678 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
54simp1bi 1023 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
6 3nn0 10887 . . . . . 6  |-  3  e.  NN0
7 reexpcl 12289 . . . . . 6  |-  ( ( A  e.  RR  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  RR )
85, 6, 7sylancl 668 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  e.  RR )
9 3re 10683 . . . . . 6  |-  3  e.  RR
10 3ne0 10704 . . . . . 6  |-  3  =/=  0
11 redivcl 10326 . . . . . 6  |-  ( ( ( A ^ 3 )  e.  RR  /\  3  e.  RR  /\  3  =/=  0 )  ->  (
( A ^ 3 )  /  3 )  e.  RR )
129, 10, 11mp3an23 1356 . . . . 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 10970 . . . . . . . . 9  |-  3  e.  ZZ
15 expgt0 12305 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  3  e.  ZZ  /\  0  <  A )  ->  0  <  ( A ^ 3 ) )
1614, 15mp3an2 1352 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A ^ 3 ) )
17163adant3 1028 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  0  <  ( A ^ 3 ) )
184, 17sylbi 199 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A ^ 3 ) )
19 0lt1 10136 . . . . . . . . 9  |-  0  <  1
202, 19pm3.2i 457 . . . . . . . 8  |-  ( 1  e.  RR  /\  0  <  1 )
21 3pos 10703 . . . . . . . . 9  |-  0  <  3
229, 21pm3.2i 457 . . . . . . . 8  |-  ( 3  e.  RR  /\  0  <  3 )
23 1lt3 10778 . . . . . . . . 9  |-  1  <  3
24 ltdiv2 10492 . . . . . . . . 9  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( 3  e.  RR  /\  0  <  3 )  /\  (
( A ^ 3 )  e.  RR  /\  0  <  ( A ^
3 ) ) )  ->  ( 1  <  3  <->  ( ( A ^ 3 )  / 
3 )  <  (
( A ^ 3 )  /  1 ) ) )
2523, 24mpbii 215 . . . . . . . 8  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( 3  e.  RR  /\  0  <  3 )  /\  (
( A ^ 3 )  e.  RR  /\  0  <  ( A ^
3 ) ) )  ->  ( ( A ^ 3 )  / 
3 )  <  (
( A ^ 3 )  /  1 ) )
2620, 22, 25mp3an12 1354 . . . . . . 7  |-  ( ( ( A ^ 3 )  e.  RR  /\  0  <  ( A ^
3 ) )  -> 
( ( A ^
3 )  /  3
)  <  ( ( A ^ 3 )  / 
1 ) )
2726ex 436 . . . . . 6  |-  ( ( A ^ 3 )  e.  RR  ->  (
0  <  ( A ^ 3 )  -> 
( ( A ^
3 )  /  3
)  <  ( ( A ^ 3 )  / 
1 ) ) )
288, 18, 27sylc 62 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  ( ( A ^ 3 )  / 
1 ) )
298recnd 9669 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  e.  CC )
3029div1d 10375 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  1 )  =  ( A ^
3 ) )
3128, 30breqtrd 4427 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  ( A ^
3 ) )
32 1nn0 10885 . . . . . . 7  |-  1  e.  NN0
3332a1i 11 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  1  e.  NN0 )
34 1le3 10826 . . . . . . . 8  |-  1  <_  3
35 1z 10967 . . . . . . . . 9  |-  1  e.  ZZ
3635eluz1i 11166 . . . . . . . 8  |-  ( 3  e.  ( ZZ>= `  1
)  <->  ( 3  e.  ZZ  /\  1  <_ 
3 ) )
3714, 34, 36mpbir2an 931 . . . . . . 7  |-  3  e.  ( ZZ>= `  1 )
3837a1i 11 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  3  e.  ( ZZ>= `  1 )
)
394simp2bi 1024 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  A )
40 0re 9643 . . . . . . . 8  |-  0  e.  RR
41 ltle 9722 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
4240, 5, 41sylancr 669 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  (
0  <  A  ->  0  <_  A ) )
4339, 42mpd 15 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <_  A )
444simp3bi 1025 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  <_  1 )
455, 33, 38, 43, 44leexp2rd 12449 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_ 
( A ^ 1 ) )
465recnd 9669 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  CC )
4746exp1d 12411 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 1 )  =  A )
4845, 47breqtrd 4427 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_  A )
4913, 8, 5, 31, 48ltletrd 9795 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  A )
5013, 5posdifd 10200 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( A ^
3 )  /  3
)  <  A  <->  0  <  ( A  -  ( ( A ^ 3 )  /  3 ) ) ) )
5149, 50mpbid 214 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A  -  (
( A ^ 3 )  /  3 ) ) )
52 sin01bnd 14239 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A  -  (
( A ^ 3 )  /  3 ) )  <  ( sin `  A )  /\  ( sin `  A )  < 
A ) )
5352simpld 461 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A  -  ( ( A ^ 3 )  / 
3 ) )  < 
( sin `  A
) )
545, 13resubcld 10047 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A  -  ( ( A ^ 3 )  / 
3 ) )  e.  RR )
555resincld 14197 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( sin `  A )  e.  RR )
56 lttr 9710 . . . 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 1351 . . 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 667 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 0  <  ( A  -  ( ( A ^ 3 )  / 
3 ) )  /\  ( A  -  (
( A ^ 3 )  /  3 ) )  <  ( sin `  A ) )  -> 
0  <  ( sin `  A ) ) )
5951, 53, 58mp2and 685 1  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    e. wcel 1887    =/= wne 2622   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   RR*cxr 9674    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   3c3 10660   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   (,]cioc 11636   ^cexp 12272   sincsin 14116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ioc 11640  df-ico 11641  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-fac 12460  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123
This theorem is referenced by:  sin02gt0  14246  sincos1sgn  14247  sincos4thpi  23468
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