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Theorem cos01gt0 11769
Description: The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
Assertion
Ref Expression
cos01gt0  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  A
) )

Proof of Theorem cos01gt0
StepHypRef Expression
1 0xr 8003 . . . . . . . . . 10  |-  0  e.  RR*
2 1re 7955 . . . . . . . . . 10  |-  1  e.  RR
3 elioc2 9935 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
41, 2, 3mp2an 426 . . . . . . . . 9  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
54simp1bi 1012 . . . . . . . 8  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
65resqcld 10679 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  e.  RR )
76recnd 7985 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  e.  CC )
8 2cn 8989 . . . . . . 7  |-  2  e.  CC
9 3cn 8993 . . . . . . . 8  |-  3  e.  CC
10 3ap0 9014 . . . . . . . 8  |-  3 #  0
119, 10pm3.2i 272 . . . . . . 7  |-  ( 3  e.  CC  /\  3 #  0 )
12 div12ap 8650 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( A ^ 2 )  e.  CC  /\  (
3  e.  CC  /\  3 #  0 ) )  -> 
( 2  x.  (
( A ^ 2 )  /  3 ) )  =  ( ( A ^ 2 )  x.  ( 2  / 
3 ) ) )
138, 11, 12mp3an13 1328 . . . . . 6  |-  ( ( A ^ 2 )  e.  CC  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  =  ( ( A ^ 2 )  x.  ( 2  /  3
) ) )
147, 13syl 14 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  =  ( ( A ^ 2 )  x.  ( 2  /  3
) ) )
15 2z 9280 . . . . . . . . . 10  |-  2  e.  ZZ
16 expgt0 10552 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  2  e.  ZZ  /\  0  <  A )  ->  0  <  ( A ^ 2 ) )
1715, 16mp3an2 1325 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A ^ 2 ) )
18173adant3 1017 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  0  <  ( A ^ 2 ) )
194, 18sylbi 121 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A ^ 2 ) )
20 2lt3 9088 . . . . . . . . . 10  |-  2  <  3
21 2re 8988 . . . . . . . . . . 11  |-  2  e.  RR
22 3re 8992 . . . . . . . . . . 11  |-  3  e.  RR
23 3pos 9012 . . . . . . . . . . 11  |-  0  <  3
2421, 22, 22, 23ltdiv1ii 8885 . . . . . . . . . 10  |-  ( 2  <  3  <->  ( 2  /  3 )  < 
( 3  /  3
) )
2520, 24mpbi 145 . . . . . . . . 9  |-  ( 2  /  3 )  < 
( 3  /  3
)
269, 10dividapi 8701 . . . . . . . . 9  |-  ( 3  /  3 )  =  1
2725, 26breqtri 4028 . . . . . . . 8  |-  ( 2  /  3 )  <  1
2821, 22, 10redivclapi 8735 . . . . . . . . 9  |-  ( 2  /  3 )  e.  RR
29 ltmul2 8812 . . . . . . . . 9  |-  ( ( ( 2  /  3
)  e.  RR  /\  1  e.  RR  /\  (
( A ^ 2 )  e.  RR  /\  0  <  ( A ^
2 ) ) )  ->  ( ( 2  /  3 )  <  1  <->  ( ( A ^ 2 )  x.  ( 2  /  3
) )  <  (
( A ^ 2 )  x.  1 ) ) )
3028, 2, 29mp3an12 1327 . . . . . . . 8  |-  ( ( ( A ^ 2 )  e.  RR  /\  0  <  ( A ^
2 ) )  -> 
( ( 2  / 
3 )  <  1  <->  ( ( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) ) )
3127, 30mpbii 148 . . . . . . 7  |-  ( ( ( A ^ 2 )  e.  RR  /\  0  <  ( A ^
2 ) )  -> 
( ( A ^
2 )  x.  (
2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) )
326, 19, 31syl2anc 411 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) )
337mulridd 7973 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  1 )  =  ( A ^
2 ) )
3432, 33breqtrd 4029 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( A ^
2 ) )
3514, 34eqbrtrd 4025 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  ( A ^
2 ) )
36 0re 7956 . . . . . . . . 9  |-  0  e.  RR
37 ltle 8044 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
3836, 37mpan 424 . . . . . . . 8  |-  ( A  e.  RR  ->  (
0  <  A  ->  0  <_  A ) )
3938imdistani 445 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  RR  /\  0  <_  A )
)
40 le2sq2 10595 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( 1  e.  RR  /\  A  <_  1 ) )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
412, 40mpanr1 437 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  A  <_  1 )  ->  ( A ^
2 )  <_  (
1 ^ 2 ) )
4239, 41stoic3 1431 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
434, 42sylbi 121 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
44 sq1 10613 . . . . 5  |-  ( 1 ^ 2 )  =  1
4543, 44breqtrdi 4044 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  <_ 
1 )
46 redivclap 8687 . . . . . . . 8  |-  ( ( ( A ^ 2 )  e.  RR  /\  3  e.  RR  /\  3 #  0 )  ->  (
( A ^ 2 )  /  3 )  e.  RR )
4722, 10, 46mp3an23 1329 . . . . . . 7  |-  ( ( A ^ 2 )  e.  RR  ->  (
( A ^ 2 )  /  3 )  e.  RR )
486, 47syl 14 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  /  3 )  e.  RR )
49 remulcl 7938 . . . . . 6  |-  ( ( 2  e.  RR  /\  ( ( A ^
2 )  /  3
)  e.  RR )  ->  ( 2  x.  ( ( A ^
2 )  /  3
) )  e.  RR )
5021, 48, 49sylancr 414 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  e.  RR )
51 ltletr 8046 . . . . . 6  |-  ( ( ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR  /\  ( A ^ 2 )  e.  RR  /\  1  e.  RR )  ->  (
( ( 2  x.  ( ( A ^
2 )  /  3
) )  <  ( A ^ 2 )  /\  ( A ^ 2 )  <_  1 )  -> 
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1 ) )
522, 51mp3an3 1326 . . . . 5  |-  ( ( ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR  /\  ( A ^ 2 )  e.  RR )  -> 
( ( ( 2  x.  ( ( A ^ 2 )  / 
3 ) )  < 
( A ^ 2 )  /\  ( A ^ 2 )  <_ 
1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  1 ) )
5350, 6, 52syl2anc 411 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( 2  x.  ( ( A ^
2 )  /  3
) )  <  ( A ^ 2 )  /\  ( A ^ 2 )  <_  1 )  -> 
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1 ) )
5435, 45, 53mp2and 433 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  1 )
55 posdif 8411 . . . 4  |-  ( ( ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR  /\  1  e.  RR )  ->  ( ( 2  x.  ( ( A ^
2 )  /  3
) )  <  1  <->  0  <  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) ) ) )
5650, 2, 55sylancl 413 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1  <->  0  <  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) ) ) )
5754, 56mpbid 147 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) ) )
58 cos01bnd 11765 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) )  <  ( cos `  A )  /\  ( cos `  A )  < 
( 1  -  (
( A ^ 2 )  /  3 ) ) ) )
5958simpld 112 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  <  ( cos `  A
) )
60 resubcl 8220 . . . 4  |-  ( ( 1  e.  RR  /\  ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR )  ->  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) )  e.  RR )
612, 50, 60sylancr 414 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  e.  RR )
625recoscld 11731 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( cos `  A )  e.  RR )
63 lttr 8030 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) )  e.  RR  /\  ( cos `  A )  e.  RR )  -> 
( ( 0  < 
( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) )  /\  ( 1  -  ( 2  x.  ( ( A ^
2 )  /  3
) ) )  < 
( cos `  A
) )  ->  0  <  ( cos `  A
) ) )
6436, 61, 62, 63mp3an2i 1342 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 0  <  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  /\  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) )  <  ( cos `  A ) )  ->  0  <  ( cos `  A ) ) )
6557, 59, 64mp2and 433 1  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   CCcc 7808   RRcr 7809   0cc0 7810   1c1 7811    x. cmul 7815   RR*cxr 7990    < clt 7991    <_ cle 7992    - cmin 8127   # cap 8537    / cdiv 8628   2c2 8969   3c3 8970   ZZcz 9252   (,]cioc 9888   ^cexp 10518   cosccos 11652
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-5 8980  df-6 8981  df-7 8982  df-8 8983  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-ioc 9892  df-ico 9893  df-fz 10008  df-fzo 10142  df-seqfrec 10445  df-exp 10519  df-fac 10705  df-ihash 10755  df-shft 10823  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-clim 11286  df-sumdc 11361  df-ef 11655  df-cos 11658
This theorem is referenced by:  sin02gt0  11770  sincos1sgn  11771  tangtx  14229
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