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Theorem sin01gt0 11378
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 7776 . . . . . . . 8  |-  0  e.  RR*
2 1re 7729 . . . . . . . 8  |-  1  e.  RR
3 elioc2 9670 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
41, 2, 3mp2an 420 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
54simp1bi 979 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
6 3nn0 8949 . . . . . 6  |-  3  e.  NN0
7 reexpcl 10261 . . . . . 6  |-  ( ( A  e.  RR  /\  3  e.  NN0 )  -> 
( A ^ 3 )  e.  RR )
85, 6, 7sylancl 407 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  e.  RR )
9 3re 8754 . . . . . 6  |-  3  e.  RR
10 3ap0 8776 . . . . . 6  |-  3 #  0
11 redivclap 8454 . . . . . 6  |-  ( ( ( A ^ 3 )  e.  RR  /\  3  e.  RR  /\  3 #  0 )  ->  (
( A ^ 3 )  /  3 )  e.  RR )
129, 10, 11mp3an23 1290 . . . . 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 9037 . . . . . . . . 9  |-  3  e.  ZZ
15 expgt0 10277 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  3  e.  ZZ  /\  0  <  A )  ->  0  <  ( A ^ 3 ) )
1614, 15mp3an2 1286 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A ^ 3 ) )
17163adant3 984 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  0  <  ( A ^ 3 ) )
184, 17sylbi 120 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A ^ 3 ) )
19 0lt1 7853 . . . . . . . 8  |-  0  <  1
202, 19pm3.2i 268 . . . . . . 7  |-  ( 1  e.  RR  /\  0  <  1 )
21 3pos 8774 . . . . . . . 8  |-  0  <  3
229, 21pm3.2i 268 . . . . . . 7  |-  ( 3  e.  RR  /\  0  <  3 )
23 1lt3 8845 . . . . . . . 8  |-  1  <  3
24 ltdiv2 8605 . . . . . . . 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 147 . . . . . . 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 1288 . . . . . 6  |-  ( ( ( A ^ 3 )  e.  RR  /\  0  <  ( A ^
3 ) )  -> 
( ( A ^
3 )  /  3
)  <  ( ( A ^ 3 )  / 
1 ) )
278, 18, 26syl2anc 406 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  ( ( A ^ 3 )  / 
1 ) )
288recnd 7758 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  e.  CC )
2928div1d 8503 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  1 )  =  ( A ^
3 ) )
3027, 29breqtrd 3922 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  ( A ^
3 ) )
31 1nn0 8947 . . . . . . 7  |-  1  e.  NN0
3231a1i 9 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  1  e.  NN0 )
33 1le3 8885 . . . . . . . 8  |-  1  <_  3
34 1z 9034 . . . . . . . . 9  |-  1  e.  ZZ
3534eluz1i 9285 . . . . . . . 8  |-  ( 3  e.  ( ZZ>= `  1
)  <->  ( 3  e.  ZZ  /\  1  <_ 
3 ) )
3614, 33, 35mpbir2an 909 . . . . . . 7  |-  3  e.  ( ZZ>= `  1 )
3736a1i 9 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  3  e.  ( ZZ>= `  1 )
)
384simp2bi 980 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  A )
39 0re 7730 . . . . . . . 8  |-  0  e.  RR
40 ltle 7815 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
4139, 5, 40sylancr 408 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  (
0  <  A  ->  0  <_  A ) )
4238, 41mpd 13 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <_  A )
434simp3bi 981 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  <_  1 )
445, 32, 37, 42, 43leexp2rd 10405 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_ 
( A ^ 1 ) )
455recnd 7758 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  CC )
4645exp1d 10370 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 1 )  =  A )
4744, 46breqtrd 3922 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 3 )  <_  A )
4813, 8, 5, 30, 47ltletrd 8149 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 3 )  /  3 )  <  A )
4913, 5posdifd 8257 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( A ^
3 )  /  3
)  <  A  <->  0  <  ( A  -  ( ( A ^ 3 )  /  3 ) ) ) )
5048, 49mpbid 146 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A  -  (
( A ^ 3 )  /  3 ) ) )
51 sin01bnd 11374 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A  -  (
( A ^ 3 )  /  3 ) )  <  ( sin `  A )  /\  ( sin `  A )  < 
A ) )
5251simpld 111 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A  -  ( ( A ^ 3 )  / 
3 ) )  < 
( sin `  A
) )
535, 13resubcld 8107 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A  -  ( ( A ^ 3 )  / 
3 ) )  e.  RR )
545resincld 11340 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( sin `  A )  e.  RR )
55 lttr 7802 . . 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 1303 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 0  <  ( A  -  ( ( A ^ 3 )  / 
3 ) )  /\  ( A  -  (
( A ^ 3 )  /  3 ) )  <  ( sin `  A ) )  -> 
0  <  ( sin `  A ) ) )
5750, 52, 56mp2and 427 1  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( sin `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945    e. wcel 1463   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   RRcr 7583   0cc0 7584   1c1 7585   RR*cxr 7763    < clt 7764    <_ cle 7765    - cmin 7897   # cap 8306    / cdiv 8395   3c3 8732   NN0cn0 8931   ZZcz 9008   ZZ>=cuz 9278   (,]cioc 9623   ^cexp 10243   sincsin 11260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-5 8742  df-6 8743  df-7 8744  df-8 8745  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-ioc 9627  df-ico 9628  df-fz 9742  df-fzo 9871  df-seqfrec 10170  df-exp 10244  df-fac 10423  df-ihash 10473  df-shft 10538  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-clim 10999  df-sumdc 11074  df-ef 11264  df-sin 11266
This theorem is referenced by:  sin02gt0  11380  sincos1sgn  11381  sin24declemlt  13076
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