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Theorem cos01gt0 13798
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 9638 . . . . . . . . . 10  |-  0  e.  RR*
2 1re 9593 . . . . . . . . . 10  |-  1  e.  RR
3 elioc2 11591 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
41, 2, 3mp2an 672 . . . . . . . . 9  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
54simp1bi 1010 . . . . . . . 8  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
65resqcld 12310 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  e.  RR )
76recnd 9620 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  e.  CC )
8 2cn 10607 . . . . . . 7  |-  2  e.  CC
9 3cn 10611 . . . . . . . 8  |-  3  e.  CC
10 3ne0 10631 . . . . . . . 8  |-  3  =/=  0
119, 10pm3.2i 455 . . . . . . 7  |-  ( 3  e.  CC  /\  3  =/=  0 )
12 div12 10230 . . . . . . 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 1314 . . . . . 6  |-  ( ( A ^ 2 )  e.  CC  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  =  ( ( A ^ 2 )  x.  ( 2  /  3
) ) )
147, 13syl 16 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  =  ( ( A ^ 2 )  x.  ( 2  /  3
) ) )
15 2z 10897 . . . . . . . . . 10  |-  2  e.  ZZ
16 expgt0 12173 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  2  e.  ZZ  /\  0  <  A )  ->  0  <  ( A ^ 2 ) )
1715, 16mp3an2 1311 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A ^ 2 ) )
18173adant3 1015 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  0  <  ( A ^ 2 ) )
194, 18sylbi 195 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A ^ 2 ) )
20 2lt3 10704 . . . . . . . . . 10  |-  2  <  3
21 2re 10606 . . . . . . . . . . 11  |-  2  e.  RR
22 3re 10610 . . . . . . . . . . 11  |-  3  e.  RR
23 3pos 10630 . . . . . . . . . . 11  |-  0  <  3
2421, 22, 22, 23ltdiv1ii 10476 . . . . . . . . . 10  |-  ( 2  <  3  <->  ( 2  /  3 )  < 
( 3  /  3
) )
2520, 24mpbi 208 . . . . . . . . 9  |-  ( 2  /  3 )  < 
( 3  /  3
)
269, 10dividi 10278 . . . . . . . . 9  |-  ( 3  /  3 )  =  1
2725, 26breqtri 4456 . . . . . . . 8  |-  ( 2  /  3 )  <  1
2821, 22, 10redivcli 10312 . . . . . . . . 9  |-  ( 2  /  3 )  e.  RR
29 ltmul2 10394 . . . . . . . . 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 1313 . . . . . . . 8  |-  ( ( ( A ^ 2 )  e.  RR  /\  0  <  ( A ^
2 ) )  -> 
( ( 2  / 
3 )  <  1  <->  ( ( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) ) )
3127, 30mpbii 211 . . . . . . 7  |-  ( ( ( A ^ 2 )  e.  RR  /\  0  <  ( A ^
2 ) )  -> 
( ( A ^
2 )  x.  (
2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) )
326, 19, 31syl2anc 661 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( ( A ^ 2 )  x.  1 ) )
337mulid1d 9611 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  1 )  =  ( A ^
2 ) )
3432, 33breqtrd 4457 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( A ^
2 ) )
3514, 34eqbrtrd 4453 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  ( A ^
2 ) )
36 0re 9594 . . . . . . . . 9  |-  0  e.  RR
37 ltle 9671 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
3836, 37mpan 670 . . . . . . . 8  |-  ( A  e.  RR  ->  (
0  <  A  ->  0  <_  A ) )
3938imdistani 690 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  RR  /\  0  <_  A )
)
40 le2sq2 12217 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( 1  e.  RR  /\  A  <_  1 ) )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
412, 40mpanr1 683 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  A  <_  1 )  ->  ( A ^
2 )  <_  (
1 ^ 2 ) )
4239, 41stoic3 1594 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
434, 42sylbi 195 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
44 sq1 12236 . . . . 5  |-  ( 1 ^ 2 )  =  1
4543, 44syl6breq 4472 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  <_ 
1 )
46 redivcl 10264 . . . . . . . 8  |-  ( ( ( A ^ 2 )  e.  RR  /\  3  e.  RR  /\  3  =/=  0 )  ->  (
( A ^ 2 )  /  3 )  e.  RR )
4722, 10, 46mp3an23 1315 . . . . . . 7  |-  ( ( A ^ 2 )  e.  RR  ->  (
( A ^ 2 )  /  3 )  e.  RR )
486, 47syl 16 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  /  3 )  e.  RR )
49 remulcl 9575 . . . . . 6  |-  ( ( 2  e.  RR  /\  ( ( A ^
2 )  /  3
)  e.  RR )  ->  ( 2  x.  ( ( A ^
2 )  /  3
) )  e.  RR )
5021, 48, 49sylancr 663 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  e.  RR )
51 ltletr 9674 . . . . . 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 1312 . . . . 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 661 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( 2  x.  ( ( A ^
2 )  /  3
) )  <  ( A ^ 2 )  /\  ( A ^ 2 )  <_  1 )  -> 
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1 ) )
5435, 45, 53mp2and 679 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  1 )
55 posdif 10046 . . . 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 662 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1  <->  0  <  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) ) ) )
5754, 56mpbid 210 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) ) )
58 cos01bnd 13793 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) )  <  ( cos `  A )  /\  ( cos `  A )  < 
( 1  -  (
( A ^ 2 )  /  3 ) ) ) )
5958simpld 459 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  <  ( cos `  A
) )
60 resubcl 9883 . . . 4  |-  ( ( 1  e.  RR  /\  ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR )  ->  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) )  e.  RR )
612, 50, 60sylancr 663 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  e.  RR )
625recoscld 13751 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( cos `  A )  e.  RR )
63 lttr 9659 . . . 4  |-  ( ( 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, 63mp3an1 1310 . . 3  |-  ( ( ( 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
) ) )
6561, 62, 64syl2anc 661 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 0  <  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  /\  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) )  <  ( cos `  A ) )  ->  0  <  ( cos `  A ) ) )
6657, 59, 65mp2and 679 1  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    x. cmul 9495   RR*cxr 9625    < clt 9626    <_ cle 9627    - cmin 9805    / cdiv 10207   2c2 10586   3c3 10587   ZZcz 10865   (,]cioc 11534   ^cexp 12140   cosccos 13673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-ioc 11538  df-ico 11539  df-fz 11677  df-fzo 11799  df-fl 11903  df-seq 12082  df-exp 12141  df-fac 12328  df-hash 12380  df-shft 12874  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-limsup 13268  df-clim 13285  df-rlim 13286  df-sum 13483  df-ef 13676  df-cos 13679
This theorem is referenced by:  sin02gt0  13799  sincos1sgn  13800  tangtx  22763
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