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Theorem cos01gt0 11714
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 7955 . . . . . . . . . 10  |-  0  e.  RR*
2 1re 7908 . . . . . . . . . 10  |-  1  e.  RR
3 elioc2 9882 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
41, 2, 3mp2an 424 . . . . . . . . 9  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
54simp1bi 1007 . . . . . . . 8  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
65resqcld 10624 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  e.  RR )
76recnd 7937 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  e.  CC )
8 2cn 8938 . . . . . . 7  |-  2  e.  CC
9 3cn 8942 . . . . . . . 8  |-  3  e.  CC
10 3ap0 8963 . . . . . . . 8  |-  3 #  0
119, 10pm3.2i 270 . . . . . . 7  |-  ( 3  e.  CC  /\  3 #  0 )
12 div12ap 8600 . . . . . . 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 1323 . . . . . 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 9229 . . . . . . . . . 10  |-  2  e.  ZZ
16 expgt0 10498 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  2  e.  ZZ  /\  0  <  A )  ->  0  <  ( A ^ 2 ) )
1715, 16mp3an2 1320 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A ^ 2 ) )
18173adant3 1012 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  0  <  ( A ^ 2 ) )
194, 18sylbi 120 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A ^ 2 ) )
20 2lt3 9037 . . . . . . . . . 10  |-  2  <  3
21 2re 8937 . . . . . . . . . . 11  |-  2  e.  RR
22 3re 8941 . . . . . . . . . . 11  |-  3  e.  RR
23 3pos 8961 . . . . . . . . . . 11  |-  0  <  3
2421, 22, 22, 23ltdiv1ii 8834 . . . . . . . . . 10  |-  ( 2  <  3  <->  ( 2  /  3 )  < 
( 3  /  3
) )
2520, 24mpbi 144 . . . . . . . . 9  |-  ( 2  /  3 )  < 
( 3  /  3
)
269, 10dividapi 8651 . . . . . . . . 9  |-  ( 3  /  3 )  =  1
2725, 26breqtri 4012 . . . . . . . 8  |-  ( 2  /  3 )  <  1
2821, 22, 10redivclapi 8685 . . . . . . . . 9  |-  ( 2  /  3 )  e.  RR
29 ltmul2 8761 . . . . . . . . 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 1322 . . . . . . . 8  |-  ( ( ( A ^ 2 )  e.  RR  /\  0  <  ( A ^
2 ) )  -> 
( ( 2  / 
3 )  <  1  <->  ( ( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) ) )
3127, 30mpbii 147 . . . . . . 7  |-  ( ( ( A ^ 2 )  e.  RR  /\  0  <  ( A ^
2 ) )  -> 
( ( A ^
2 )  x.  (
2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) )
326, 19, 31syl2anc 409 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) )
337mulid1d 7926 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  1 )  =  ( A ^
2 ) )
3432, 33breqtrd 4013 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( A ^
2 ) )
3514, 34eqbrtrd 4009 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  ( A ^
2 ) )
36 0re 7909 . . . . . . . . 9  |-  0  e.  RR
37 ltle 7996 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
3836, 37mpan 422 . . . . . . . 8  |-  ( A  e.  RR  ->  (
0  <  A  ->  0  <_  A ) )
3938imdistani 443 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  RR  /\  0  <_  A )
)
40 le2sq2 10540 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( 1  e.  RR  /\  A  <_  1 ) )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
412, 40mpanr1 435 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  A  <_  1 )  ->  ( A ^
2 )  <_  (
1 ^ 2 ) )
4239, 41stoic3 1424 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
434, 42sylbi 120 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
44 sq1 10558 . . . . 5  |-  ( 1 ^ 2 )  =  1
4543, 44breqtrdi 4028 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  <_ 
1 )
46 redivclap 8637 . . . . . . . 8  |-  ( ( ( A ^ 2 )  e.  RR  /\  3  e.  RR  /\  3 #  0 )  ->  (
( A ^ 2 )  /  3 )  e.  RR )
4722, 10, 46mp3an23 1324 . . . . . . 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 7891 . . . . . 6  |-  ( ( 2  e.  RR  /\  ( ( A ^
2 )  /  3
)  e.  RR )  ->  ( 2  x.  ( ( A ^
2 )  /  3
) )  e.  RR )
5021, 48, 49sylancr 412 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  e.  RR )
51 ltletr 7998 . . . . . 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 1321 . . . . 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 409 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( 2  x.  ( ( A ^
2 )  /  3
) )  <  ( A ^ 2 )  /\  ( A ^ 2 )  <_  1 )  -> 
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1 ) )
5435, 45, 53mp2and 431 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  1 )
55 posdif 8363 . . . 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 411 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1  <->  0  <  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) ) ) )
5754, 56mpbid 146 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) ) )
58 cos01bnd 11710 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) )  <  ( cos `  A )  /\  ( cos `  A )  < 
( 1  -  (
( A ^ 2 )  /  3 ) ) ) )
5958simpld 111 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  <  ( cos `  A
) )
60 resubcl 8172 . . . 4  |-  ( ( 1  e.  RR  /\  ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR )  ->  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) )  e.  RR )
612, 50, 60sylancr 412 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  e.  RR )
625recoscld 11676 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( cos `  A )  e.  RR )
63 lttr 7982 . . 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 1337 . 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 431 1  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348    e. wcel 2141   class class class wbr 3987   ` cfv 5196  (class class class)co 5851   CCcc 7761   RRcr 7762   0cc0 7763   1c1 7764    x. cmul 7768   RR*cxr 7942    < clt 7943    <_ cle 7944    - cmin 8079   # cap 8489    / cdiv 8578   2c2 8918   3c3 8919   ZZcz 9201   (,]cioc 9835   ^cexp 10464   cosccos 11597
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880  ax-pre-mulext 7881  ax-arch 7882  ax-caucvg 7883
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-frec 6368  df-1o 6393  df-oadd 6397  df-er 6510  df-en 6716  df-dom 6717  df-fin 6718  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-div 8579  df-inn 8868  df-2 8926  df-3 8927  df-4 8928  df-5 8929  df-6 8930  df-7 8931  df-8 8932  df-n0 9125  df-z 9202  df-uz 9477  df-q 9568  df-rp 9600  df-ioc 9839  df-ico 9840  df-fz 9955  df-fzo 10088  df-seqfrec 10391  df-exp 10465  df-fac 10649  df-ihash 10699  df-shft 10768  df-cj 10795  df-re 10796  df-im 10797  df-rsqrt 10951  df-abs 10952  df-clim 11231  df-sumdc 11306  df-ef 11600  df-cos 11603
This theorem is referenced by:  sin02gt0  11715  sincos1sgn  11716  tangtx  13514
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