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Theorem sin01gt0 11750
Description: The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) (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 7991 . . . . . . . 8  |-  0  e.  RR*
2 1re 7944 . . . . . . . 8  |-  1  e.  RR
3 elioc2 9920 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
41, 2, 3mp2an 426 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
54simp1bi 1012 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
6 3nn0 9180 . . . . . 6  |-  3  e.  NN0
7 reexpcl 10520 . . . . . 6  |-  ( ( A  e.  RR  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  RR )
85, 6, 7sylancl 413 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  e.  RR )
9 3re 8979 . . . . . 6  |-  3  e.  RR
10 3ap0 9001 . . . . . 6  |-  3 #  0
11 redivclap 8674 . . . . . 6  |-  ( ( ( A ^ 3 )  e.  RR  /\  3  e.  RR  /\  3 #  0 )  ->  (
( A ^ 3 )  /  3 )  e.  RR )
129, 10, 11mp3an23 1329 . . . . 5  |-  ( ( A ^ 3 )  e.  RR  ->  (
( A ^ 3 )  /  3 )  e.  RR )
138, 12syl 14 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  e.  RR )
14 3z 9268 . . . . . . . . 9  |-  3  e.  ZZ
15 expgt0 10536 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  3  e.  ZZ  /\  0  <  A )  ->  0  <  ( A ^ 3 ) )
1614, 15mp3an2 1325 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A ^ 3 ) )
17163adant3 1017 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  0  <  ( A ^ 3 ) )
184, 17sylbi 121 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A ^ 3 ) )
19 0lt1 8071 . . . . . . . 8  |-  0  <  1
202, 19pm3.2i 272 . . . . . . 7  |-  ( 1  e.  RR  /\  0  <  1 )
21 3pos 8999 . . . . . . . 8  |-  0  <  3
229, 21pm3.2i 272 . . . . . . 7  |-  ( 3  e.  RR  /\  0  <  3 )
23 1lt3 9076 . . . . . . . 8  |-  1  <  3
24 ltdiv2 8830 . . . . . . . 8  |-  ( ( ( 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 148 . . . . . . 7  |-  ( ( ( 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 1327 . . . . . 6  |-  ( ( ( A ^ 3 )  e.  RR  /\  0  <  ( A ^
3 ) )  -> 
( ( A ^
3 )  /  3
)  <  ( ( A ^ 3 )  / 
1 ) )
278, 18, 26syl2anc 411 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  ( ( A ^ 3 )  / 
1 ) )
288recnd 7973 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  e.  CC )
2928div1d 8723 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  1 )  =  ( A ^
3 ) )
3027, 29breqtrd 4026 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  ( A ^
3 ) )
31 1nn0 9178 . . . . . . 7  |-  1  e.  NN0
3231a1i 9 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  1  e.  NN0 )
33 1le3 9116 . . . . . . . 8  |-  1  <_  3
34 1z 9265 . . . . . . . . 9  |-  1  e.  ZZ
3534eluz1i 9521 . . . . . . . 8  |-  ( 3  e.  ( ZZ>= `  1
)  <->  ( 3  e.  ZZ  /\  1  <_ 
3 ) )
3614, 33, 35mpbir2an 942 . . . . . . 7  |-  3  e.  ( ZZ>= `  1 )
3736a1i 9 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  3  e.  ( ZZ>= `  1 )
)
384simp2bi 1013 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  A )
39 0re 7945 . . . . . . . 8  |-  0  e.  RR
40 ltle 8032 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
4139, 5, 40sylancr 414 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  (
0  <  A  ->  0  <_  A ) )
4238, 41mpd 13 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <_  A )
434simp3bi 1014 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  <_  1 )
445, 32, 37, 42, 43leexp2rd 10666 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_ 
( A ^ 1 ) )
455recnd 7973 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  CC )
4645exp1d 10631 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 1 )  =  A )
4744, 46breqtrd 4026 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_  A )
4813, 8, 5, 30, 47ltletrd 8367 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  A )
4913, 5posdifd 8476 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( A ^
3 )  /  3
)  <  A  <->  0  <  ( A  -  ( ( A ^ 3 )  /  3 ) ) ) )
5048, 49mpbid 147 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A  -  (
( A ^ 3 )  /  3 ) ) )
51 sin01bnd 11746 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A  -  (
( A ^ 3 )  /  3 ) )  <  ( sin `  A )  /\  ( sin `  A )  < 
A ) )
5251simpld 112 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A  -  ( ( A ^ 3 )  / 
3 ) )  < 
( sin `  A
) )
535, 13resubcld 8325 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A  -  ( ( A ^ 3 )  / 
3 ) )  e.  RR )
545resincld 11712 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( sin `  A )  e.  RR )
55 lttr 8018 . . 3  |-  ( ( 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
) ) )
5639, 53, 54, 55mp3an2i 1342 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 0  <  ( A  -  ( ( A ^ 3 )  / 
3 ) )  /\  ( A  -  (
( A ^ 3 )  /  3 ) )  <  ( sin `  A ) )  -> 
0  <  ( sin `  A ) ) )
5750, 52, 56mp2and 433 1  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    e. wcel 2148   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   RRcr 7798   0cc0 7799   1c1 7800   RR*cxr 7978    < clt 7979    <_ cle 7980    - cmin 8115   # cap 8525    / cdiv 8615   3c3 8957   NN0cn0 9162   ZZcz 9239   ZZ>=cuz 9514   (,]cioc 9873   ^cexp 10502   sincsin 11633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7890  ax-resscn 7891  ax-1cn 7892  ax-1re 7893  ax-icn 7894  ax-addcl 7895  ax-addrcl 7896  ax-mulcl 7897  ax-mulrcl 7898  ax-addcom 7899  ax-mulcom 7900  ax-addass 7901  ax-mulass 7902  ax-distr 7903  ax-i2m1 7904  ax-0lt1 7905  ax-1rid 7906  ax-0id 7907  ax-rnegex 7908  ax-precex 7909  ax-cnre 7910  ax-pre-ltirr 7911  ax-pre-ltwlin 7912  ax-pre-lttrn 7913  ax-pre-apti 7914  ax-pre-ltadd 7915  ax-pre-mulgt0 7916  ax-pre-mulext 7917  ax-arch 7918  ax-caucvg 7919
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7981  df-mnf 7982  df-xr 7983  df-ltxr 7984  df-le 7985  df-sub 8117  df-neg 8118  df-reap 8519  df-ap 8526  df-div 8616  df-inn 8906  df-2 8964  df-3 8965  df-4 8966  df-5 8967  df-6 8968  df-7 8969  df-8 8970  df-n0 9163  df-z 9240  df-uz 9515  df-q 9606  df-rp 9638  df-ioc 9877  df-ico 9878  df-fz 9993  df-fzo 10126  df-seqfrec 10429  df-exp 10503  df-fac 10687  df-ihash 10737  df-shft 10805  df-cj 10832  df-re 10833  df-im 10834  df-rsqrt 10988  df-abs 10989  df-clim 11268  df-sumdc 11343  df-ef 11637  df-sin 11639
This theorem is referenced by:  sin02gt0  11752  sincos1sgn  11753  sincos4thpi  13921
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