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Theorem cos01gt0 13473
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 9428 . . . . . . . . . 10  |-  0  e.  RR*
2 1re 9383 . . . . . . . . . 10  |-  1  e.  RR
3 elioc2 11356 . . . . . . . . . 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 1003 . . . . . . . 8  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
65resqcld 12032 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  e.  RR )
76recnd 9410 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  e.  CC )
8 2cn 10390 . . . . . . 7  |-  2  e.  CC
9 3cn 10394 . . . . . . . 8  |-  3  e.  CC
10 3ne0 10414 . . . . . . . 8  |-  3  =/=  0
119, 10pm3.2i 455 . . . . . . 7  |-  ( 3  e.  CC  /\  3  =/=  0 )
12 div12 10014 . . . . . . 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 1305 . . . . . 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 10676 . . . . . . . . . 10  |-  2  e.  ZZ
16 expgt0 11895 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  2  e.  ZZ  /\  0  <  A )  ->  0  <  ( A ^ 2 ) )
1715, 16mp3an2 1302 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( A ^ 2 ) )
18173adant3 1008 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  0  <  ( A ^ 2 ) )
194, 18sylbi 195 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( A ^ 2 ) )
20 2lt3 10487 . . . . . . . . . 10  |-  2  <  3
21 2re 10389 . . . . . . . . . . 11  |-  2  e.  RR
22 3re 10393 . . . . . . . . . . 11  |-  3  e.  RR
23 3pos 10413 . . . . . . . . . . 11  |-  0  <  3
2421, 22, 22, 23ltdiv1ii 10260 . . . . . . . . . 10  |-  ( 2  <  3  <->  ( 2  /  3 )  < 
( 3  /  3
) )
2520, 24mpbi 208 . . . . . . . . 9  |-  ( 2  /  3 )  < 
( 3  /  3
)
269, 10dividi 10062 . . . . . . . . 9  |-  ( 3  /  3 )  =  1
2725, 26breqtri 4313 . . . . . . . 8  |-  ( 2  /  3 )  <  1
2821, 22, 10redivcli 10096 . . . . . . . . 9  |-  ( 2  /  3 )  e.  RR
29 ltmul2 10178 . . . . . . . . 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 1304 . . . . . . . 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 9401 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  1 )  =  ( A ^
2 ) )
3432, 33breqtrd 4314 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  x.  ( 2  /  3 ) )  <  ( A ^
2 ) )
3514, 34eqbrtrd 4310 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  ( A ^
2 ) )
36 0re 9384 . . . . . . . . . 10  |-  0  e.  RR
37 ltle 9461 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  ->  0  <_  A )
)
3836, 37mpan 670 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
0  <  A  ->  0  <_  A ) )
3938imdistani 690 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  RR  /\  0  <_  A )
)
40 le2sq2 11939 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( 1  e.  RR  /\  A  <_  1 ) )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
412, 40mpanr1 683 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  A  <_  1 )  ->  ( A ^
2 )  <_  (
1 ^ 2 ) )
4239, 41sylan 471 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  A  <_  1
)  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
43423impa 1182 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A  /\  A  <_  1 )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
444, 43sylbi 195 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  <_ 
( 1 ^ 2 ) )
45 sq1 11958 . . . . 5  |-  ( 1 ^ 2 )  =  1
4644, 45syl6breq 4329 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 2 )  <_ 
1 )
47 redivcl 10048 . . . . . . . 8  |-  ( ( ( A ^ 2 )  e.  RR  /\  3  e.  RR  /\  3  =/=  0 )  ->  (
( A ^ 2 )  /  3 )  e.  RR )
4822, 10, 47mp3an23 1306 . . . . . . 7  |-  ( ( A ^ 2 )  e.  RR  ->  (
( A ^ 2 )  /  3 )  e.  RR )
496, 48syl 16 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 2 )  /  3 )  e.  RR )
50 remulcl 9365 . . . . . 6  |-  ( ( 2  e.  RR  /\  ( ( A ^
2 )  /  3
)  e.  RR )  ->  ( 2  x.  ( ( A ^
2 )  /  3
) )  e.  RR )
5121, 49, 50sylancr 663 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  e.  RR )
52 ltletr 9464 . . . . . 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 ) )
532, 52mp3an3 1303 . . . . 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 ) )
5451, 6, 53syl2anc 661 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( 2  x.  ( ( A ^
2 )  /  3
) )  <  ( A ^ 2 )  /\  ( A ^ 2 )  <_  1 )  -> 
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1 ) )
5535, 46, 54mp2and 679 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
2  x.  ( ( A ^ 2 )  /  3 ) )  <  1 )
56 posdif 9830 . . . 4  |-  ( ( ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR  /\  1  e.  RR )  ->  ( ( 2  x.  ( ( A ^
2 )  /  3
) )  <  1  <->  0  <  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) ) ) )
5751, 2, 56sylancl 662 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 2  x.  (
( A ^ 2 )  /  3 ) )  <  1  <->  0  <  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) ) ) )
5855, 57mpbid 210 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) ) )
59 cos01bnd 13468 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 1  -  (
2  x.  ( ( A ^ 2 )  /  3 ) ) )  <  ( cos `  A )  /\  ( cos `  A )  < 
( 1  -  (
( A ^ 2 )  /  3 ) ) ) )
6059simpld 459 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  <  ( cos `  A
) )
61 resubcl 9671 . . . 4  |-  ( ( 1  e.  RR  /\  ( 2  x.  (
( A ^ 2 )  /  3 ) )  e.  RR )  ->  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) )  e.  RR )
622, 51, 61sylancr 663 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  e.  RR )
635recoscld 13426 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( cos `  A )  e.  RR )
64 lttr 9449 . . . 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
) ) )
6536, 64mp3an1 1301 . . 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
) ) )
6662, 63, 65syl2anc 661 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( 0  <  (
1  -  ( 2  x.  ( ( A ^ 2 )  / 
3 ) ) )  /\  ( 1  -  ( 2  x.  (
( A ^ 2 )  /  3 ) ) )  <  ( cos `  A ) )  ->  0  <  ( cos `  A ) ) )
6758, 60, 66mp2and 679 1  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  ( cos `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   CCcc 9278   RRcr 9279   0cc0 9280   1c1 9281    x. cmul 9285   RR*cxr 9415    < clt 9416    <_ cle 9417    - cmin 9593    / cdiv 9991   2c2 10369   3c3 10370   ZZcz 10644   (,]cioc 11299   ^cexp 11863   cosccos 13348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358  ax-addf 9359  ax-mulf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-oi 7722  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-ioc 11303  df-ico 11304  df-fz 11436  df-fzo 11547  df-fl 11640  df-seq 11805  df-exp 11864  df-fac 12050  df-hash 12102  df-shft 12554  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-limsup 12947  df-clim 12964  df-rlim 12965  df-sum 13162  df-ef 13351  df-cos 13354
This theorem is referenced by:  sin02gt0  13474  sincos1sgn  13475  tangtx  21965
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