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Theorem cos2bnd 12468
Description: Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.)
Assertion
Ref Expression
cos2bnd  |-  ( -u ( 7  /  9
)  <  ( cos `  2 )  /\  ( cos `  2 )  <  -u ( 1  /  9
) )

Proof of Theorem cos2bnd
StepHypRef Expression
1 7re 9823 . . . . . . 7  |-  7  e.  RR
21recni 8849 . . . . . 6  |-  7  e.  CC
3 9re 9825 . . . . . . 7  |-  9  e.  RR
43recni 8849 . . . . . 6  |-  9  e.  CC
5 9pos 9837 . . . . . . 7  |-  0  <  9
63, 5gt0ne0ii 9309 . . . . . 6  |-  9  =/=  0
7 divneg 9455 . . . . . 6  |-  ( ( 7  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
7  /  9 )  =  ( -u 7  /  9 ) )
82, 4, 6, 7mp3an 1277 . . . . 5  |-  -u (
7  /  9 )  =  ( -u 7  /  9 )
9 2cn 9816 . . . . . . 7  |-  2  e.  CC
104, 6pm3.2i 441 . . . . . . 7  |-  ( 9  e.  CC  /\  9  =/=  0 )
11 divsubdir 9456 . . . . . . 7  |-  ( ( 2  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 2  -  9 )  / 
9 )  =  ( ( 2  /  9
)  -  ( 9  /  9 ) ) )
129, 4, 10, 11mp3an 1277 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( ( 2  / 
9 )  -  (
9  /  9 ) )
134, 9negsubdi2i 9132 . . . . . . . 8  |-  -u (
9  -  2 )  =  ( 2  -  9 )
14 7p2e9 9867 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
154, 9, 2subadd2i 9134 . . . . . . . . . 10  |-  ( ( 9  -  2 )  =  7  <->  ( 7  +  2 )  =  9 )
1614, 15mpbir 200 . . . . . . . . 9  |-  ( 9  -  2 )  =  7
1716negeqi 9045 . . . . . . . 8  |-  -u (
9  -  2 )  =  -u 7
1813, 17eqtr3i 2305 . . . . . . 7  |-  ( 2  -  9 )  = 
-u 7
1918oveq1i 5868 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( -u 7  / 
9 )
2012, 19eqtr3i 2305 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( -u 7  / 
9 )
214, 6dividi 9493 . . . . . 6  |-  ( 9  /  9 )  =  1
2221oveq2i 5869 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 2  / 
9 )  -  1 )
238, 20, 223eqtr2ri 2310 . . . 4  |-  ( ( 2  /  9 )  -  1 )  = 
-u ( 7  / 
9 )
24 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
259, 24, 4, 6divassi 9516 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  x.  (
1  /  9 ) )
269mulid1i 8839 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
2726oveq1i 5868 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  /  9
)
2825, 27eqtr3i 2305 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  =  ( 2  /  9
)
29 3cn 9818 . . . . . . . . . 10  |-  3  e.  CC
30 3ne0 9831 . . . . . . . . . 10  |-  3  =/=  0
3124, 29, 30sqdivi 11188 . . . . . . . . 9  |-  ( ( 1  /  3 ) ^ 2 )  =  ( ( 1 ^ 2 )  /  (
3 ^ 2 ) )
32 sq1 11198 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
33 sq3 11200 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
3432, 33oveq12i 5870 . . . . . . . . 9  |-  ( ( 1 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 1  /  9
)
3531, 34eqtri 2303 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  =  ( 1  /  9
)
36 cos1bnd 12467 . . . . . . . . . 10  |-  ( ( 1  /  3 )  <  ( cos `  1
)  /\  ( cos `  1 )  <  (
2  /  3 ) )
3736simpli 444 . . . . . . . . 9  |-  ( 1  /  3 )  < 
( cos `  1
)
38 0le1 9297 . . . . . . . . . . 11  |-  0  <_  1
39 3pos 9830 . . . . . . . . . . 11  |-  0  <  3
40 1re 8837 . . . . . . . . . . . 12  |-  1  e.  RR
41 3re 9817 . . . . . . . . . . . 12  |-  3  e.  RR
4240, 41divge0i 9666 . . . . . . . . . . 11  |-  ( ( 0  <_  1  /\  0  <  3 )  -> 
0  <_  ( 1  /  3 ) )
4338, 39, 42mp2an 653 . . . . . . . . . 10  |-  0  <_  ( 1  /  3
)
44 0re 8838 . . . . . . . . . . 11  |-  0  e.  RR
45 recoscl 12421 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  ( cos `  1 )  e.  RR )
4640, 45ax-mp 8 . . . . . . . . . . 11  |-  ( cos `  1 )  e.  RR
4741, 30rereccli 9525 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  RR
4844, 47, 46lelttri 8946 . . . . . . . . . . . 12  |-  ( ( 0  <_  ( 1  /  3 )  /\  ( 1  /  3
)  <  ( cos `  1 ) )  -> 
0  <  ( cos `  1 ) )
4943, 37, 48mp2an 653 . . . . . . . . . . 11  |-  0  <  ( cos `  1
)
5044, 46, 49ltleii 8941 . . . . . . . . . 10  |-  0  <_  ( cos `  1
)
5147, 46lt2sqi 11192 . . . . . . . . . 10  |-  ( ( 0  <_  ( 1  /  3 )  /\  0  <_  ( cos `  1
) )  ->  (
( 1  /  3
)  <  ( cos `  1 )  <->  ( (
1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 ) ) )
5243, 50, 51mp2an 653 . . . . . . . . 9  |-  ( ( 1  /  3 )  <  ( cos `  1
)  <->  ( ( 1  /  3 ) ^
2 )  <  (
( cos `  1
) ^ 2 ) )
5337, 52mpbi 199 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 )
5435, 53eqbrtrri 4044 . . . . . . 7  |-  ( 1  /  9 )  < 
( ( cos `  1
) ^ 2 )
55 2pos 9828 . . . . . . . 8  |-  0  <  2
563, 6rereccli 9525 . . . . . . . . 9  |-  ( 1  /  9 )  e.  RR
5746resqcli 11189 . . . . . . . . 9  |-  ( ( cos `  1 ) ^ 2 )  e.  RR
58 2re 9815 . . . . . . . . 9  |-  2  e.  RR
5956, 57, 58ltmul2i 9678 . . . . . . . 8  |-  ( 0  <  2  ->  (
( 1  /  9
)  <  ( ( cos `  1 ) ^
2 )  <->  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) ) ) )
6055, 59ax-mp 8 . . . . . . 7  |-  ( ( 1  /  9 )  <  ( ( cos `  1 ) ^
2 )  <->  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) ) )
6154, 60mpbi 199 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6228, 61eqbrtrri 4044 . . . . 5  |-  ( 2  /  9 )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6358, 3, 6redivcli 9527 . . . . . 6  |-  ( 2  /  9 )  e.  RR
6458, 57remulcli 8851 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  e.  RR
65 ltsub1 9270 . . . . . 6  |-  ( ( ( 2  /  9
)  e.  RR  /\  ( 2  x.  (
( cos `  1
) ^ 2 ) )  e.  RR  /\  1  e.  RR )  ->  ( ( 2  / 
9 )  <  (
2  x.  ( ( cos `  1 ) ^ 2 ) )  <-> 
( ( 2  / 
9 )  -  1 )  <  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 ) ) )
6663, 64, 40, 65mp3an 1277 . . . . 5  |-  ( ( 2  /  9 )  <  ( 2  x.  ( ( cos `  1
) ^ 2 ) )  <->  ( ( 2  /  9 )  - 
1 )  <  (
( 2  x.  (
( cos `  1
) ^ 2 ) )  -  1 ) )
6762, 66mpbi 199 . . . 4  |-  ( ( 2  /  9 )  -  1 )  < 
( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
6823, 67eqbrtrri 4044 . . 3  |-  -u (
7  /  9 )  <  ( ( 2  x.  ( ( cos `  1 ) ^
2 ) )  - 
1 )
6926fveq2i 5528 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( cos `  2 )
70 cos2t 12458 . . . . 5  |-  ( 1  e.  CC  ->  ( cos `  ( 2  x.  1 ) )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 ) )
7124, 70ax-mp 8 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( ( 2  x.  (
( cos `  1
) ^ 2 ) )  -  1 )
7269, 71eqtr3i 2305 . . 3  |-  ( cos `  2 )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
7368, 72breqtrri 4048 . 2  |-  -u (
7  /  9 )  <  ( cos `  2
)
7436simpri 448 . . . . . . . . 9  |-  ( cos `  1 )  < 
( 2  /  3
)
75 2nn0 9982 . . . . . . . . . . . 12  |-  2  e.  NN0
7675nn0ge0i 9993 . . . . . . . . . . 11  |-  0  <_  2
7758, 41divge0i 9666 . . . . . . . . . . 11  |-  ( ( 0  <_  2  /\  0  <  3 )  -> 
0  <_  ( 2  /  3 ) )
7876, 39, 77mp2an 653 . . . . . . . . . 10  |-  0  <_  ( 2  /  3
)
7958, 41, 30redivcli 9527 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  RR
8046, 79lt2sqi 11192 . . . . . . . . . 10  |-  ( ( 0  <_  ( cos `  1 )  /\  0  <_  ( 2  /  3
) )  ->  (
( cos `  1
)  <  ( 2  /  3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) ) )
8150, 78, 80mp2an 653 . . . . . . . . 9  |-  ( ( cos `  1 )  <  ( 2  / 
3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) )
8274, 81mpbi 199 . . . . . . . 8  |-  ( ( cos `  1 ) ^ 2 )  < 
( ( 2  / 
3 ) ^ 2 )
839, 29, 30sqdivi 11188 . . . . . . . . 9  |-  ( ( 2  /  3 ) ^ 2 )  =  ( ( 2 ^ 2 )  /  (
3 ^ 2 ) )
84 sq2 11199 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
8584, 33oveq12i 5870 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 4  /  9
)
8683, 85eqtri 2303 . . . . . . . 8  |-  ( ( 2  /  3 ) ^ 2 )  =  ( 4  /  9
)
8782, 86breqtri 4046 . . . . . . 7  |-  ( ( cos `  1 ) ^ 2 )  < 
( 4  /  9
)
88 4re 9819 . . . . . . . . . 10  |-  4  e.  RR
8988, 3, 6redivcli 9527 . . . . . . . . 9  |-  ( 4  /  9 )  e.  RR
9057, 89, 58ltmul2i 9678 . . . . . . . 8  |-  ( 0  <  2  ->  (
( ( cos `  1
) ^ 2 )  <  ( 4  / 
9 )  <->  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) ) ) )
9155, 90ax-mp 8 . . . . . . 7  |-  ( ( ( cos `  1
) ^ 2 )  <  ( 4  / 
9 )  <->  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) ) )
9287, 91mpbi 199 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) )
93 4cn 9820 . . . . . . . 8  |-  4  e.  CC
949, 93, 4, 6divassi 9516 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 2  x.  (
4  /  9 ) )
95 4t2e8 9874 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
9693, 9, 95mulcomli 8844 . . . . . . . 8  |-  ( 2  x.  4 )  =  8
9796oveq1i 5868 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 8  /  9
)
9894, 97eqtr3i 2305 . . . . . 6  |-  ( 2  x.  ( 4  / 
9 ) )  =  ( 8  /  9
)
9992, 98breqtri 4046 . . . . 5  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 8  /  9
)
100 8re 9824 . . . . . . 7  |-  8  e.  RR
101100, 3, 6redivcli 9527 . . . . . 6  |-  ( 8  /  9 )  e.  RR
102 ltsub1 9270 . . . . . 6  |-  ( ( ( 2  x.  (
( cos `  1
) ^ 2 ) )  e.  RR  /\  ( 8  /  9
)  e.  RR  /\  1  e.  RR )  ->  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  <  ( 8  /  9 )  <->  ( (
2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 ) ) )
10364, 101, 40, 102mp3an 1277 . . . . 5  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  <  ( 8  / 
9 )  <->  ( (
2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 ) )
10499, 103mpbi 199 . . . 4  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 )
10521oveq2i 5869 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 8  / 
9 )  -  1 )
106 divneg 9455 . . . . . . 7  |-  ( ( 1  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
1  /  9 )  =  ( -u 1  /  9 ) )
10724, 4, 6, 106mp3an 1277 . . . . . 6  |-  -u (
1  /  9 )  =  ( -u 1  /  9 )
108100recni 8849 . . . . . . . . 9  |-  8  e.  CC
1094, 108negsubdi2i 9132 . . . . . . . 8  |-  -u (
9  -  8 )  =  ( 8  -  9 )
110 8p1e9 9853 . . . . . . . . . 10  |-  ( 8  +  1 )  =  9
1114, 108, 24, 110subaddrii 9135 . . . . . . . . 9  |-  ( 9  -  8 )  =  1
112111negeqi 9045 . . . . . . . 8  |-  -u (
9  -  8 )  =  -u 1
113109, 112eqtr3i 2305 . . . . . . 7  |-  ( 8  -  9 )  = 
-u 1
114113oveq1i 5868 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( -u 1  / 
9 )
115 divsubdir 9456 . . . . . . 7  |-  ( ( 8  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 8  -  9 )  / 
9 )  =  ( ( 8  /  9
)  -  ( 9  /  9 ) ) )
116108, 4, 10, 115mp3an 1277 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( ( 8  / 
9 )  -  (
9  /  9 ) )
117107, 114, 1163eqtr2ri 2310 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  = 
-u ( 1  / 
9 )
118105, 117eqtr3i 2305 . . . 4  |-  ( ( 8  /  9 )  -  1 )  = 
-u ( 1  / 
9 )
119104, 118breqtri 4046 . . 3  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  <  -u ( 1  /  9
)
12072, 119eqbrtri 4042 . 2  |-  ( cos `  2 )  <  -u ( 1  /  9
)
12173, 120pm3.2i 441 1  |-  ( -u ( 7  /  9
)  <  ( cos `  2 )  /\  ( cos `  2 )  <  -u ( 1  /  9
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   3c3 9796   4c4 9797   7c7 9800   8c8 9801   9c9 9802   ^cexp 11104   cosccos 12346
This theorem is referenced by:  sincos2sgn  12474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ioc 10661  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352
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