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Theorem cos2bnd 11377
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 7cn 8764 . . . . . 6  |-  7  e.  CC
2 9cn 8768 . . . . . 6  |-  9  e.  CC
3 9re 8767 . . . . . . 7  |-  9  e.  RR
4 9pos 8784 . . . . . . 7  |-  0  <  9
53, 4gt0ap0ii 8353 . . . . . 6  |-  9 #  0
6 divnegap 8429 . . . . . 6  |-  ( ( 7  e.  CC  /\  9  e.  CC  /\  9 #  0 )  ->  -u (
7  /  9 )  =  ( -u 7  /  9 ) )
71, 2, 5, 6mp3an 1298 . . . . 5  |-  -u (
7  /  9 )  =  ( -u 7  /  9 )
8 2cn 8751 . . . . . . 7  |-  2  e.  CC
92, 5pm3.2i 268 . . . . . . 7  |-  ( 9  e.  CC  /\  9 #  0 )
10 divsubdirap 8431 . . . . . . 7  |-  ( ( 2  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9 #  0 ) )  -> 
( ( 2  -  9 )  /  9
)  =  ( ( 2  /  9 )  -  ( 9  / 
9 ) ) )
118, 2, 9, 10mp3an 1298 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( ( 2  / 
9 )  -  (
9  /  9 ) )
122, 8negsubdi2i 8012 . . . . . . . 8  |-  -u (
9  -  2 )  =  ( 2  -  9 )
13 7p2e9 8825 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
142, 8, 1subadd2i 8014 . . . . . . . . . 10  |-  ( ( 9  -  2 )  =  7  <->  ( 7  +  2 )  =  9 )
1513, 14mpbir 145 . . . . . . . . 9  |-  ( 9  -  2 )  =  7
1615negeqi 7920 . . . . . . . 8  |-  -u (
9  -  2 )  =  -u 7
1712, 16eqtr3i 2138 . . . . . . 7  |-  ( 2  -  9 )  = 
-u 7
1817oveq1i 5750 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( -u 7  / 
9 )
1911, 18eqtr3i 2138 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( -u 7  / 
9 )
202, 5dividapi 8468 . . . . . 6  |-  ( 9  /  9 )  =  1
2120oveq2i 5751 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 2  / 
9 )  -  1 )
227, 19, 213eqtr2ri 2143 . . . 4  |-  ( ( 2  /  9 )  -  1 )  = 
-u ( 7  / 
9 )
23 ax-1cn 7677 . . . . . . . 8  |-  1  e.  CC
248, 23, 2, 5divassapi 8491 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  x.  (
1  /  9 ) )
25 2t1e2 8827 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
2625oveq1i 5750 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  /  9
)
2724, 26eqtr3i 2138 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  =  ( 2  /  9
)
28 3cn 8755 . . . . . . . . . 10  |-  3  e.  CC
29 3ap0 8776 . . . . . . . . . 10  |-  3 #  0
3023, 28, 29sqdivapi 10327 . . . . . . . . 9  |-  ( ( 1  /  3 ) ^ 2 )  =  ( ( 1 ^ 2 )  /  (
3 ^ 2 ) )
31 sq1 10337 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
32 sq3 10340 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
3331, 32oveq12i 5752 . . . . . . . . 9  |-  ( ( 1 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 1  /  9
)
3430, 33eqtri 2136 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  =  ( 1  /  9
)
35 cos1bnd 11376 . . . . . . . . . 10  |-  ( ( 1  /  3 )  <  ( cos `  1
)  /\  ( cos `  1 )  <  (
2  /  3 ) )
3635simpli 110 . . . . . . . . 9  |-  ( 1  /  3 )  < 
( cos `  1
)
37 0le1 8207 . . . . . . . . . . 11  |-  0  <_  1
38 3pos 8774 . . . . . . . . . . 11  |-  0  <  3
39 1re 7729 . . . . . . . . . . . 12  |-  1  e.  RR
40 3re 8754 . . . . . . . . . . . 12  |-  3  e.  RR
4139, 40divge0i 8629 . . . . . . . . . . 11  |-  ( ( 0  <_  1  /\  0  <  3 )  -> 
0  <_  ( 1  /  3 ) )
4237, 38, 41mp2an 420 . . . . . . . . . 10  |-  0  <_  ( 1  /  3
)
43 0re 7730 . . . . . . . . . . 11  |-  0  e.  RR
44 recoscl 11338 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  ( cos `  1 )  e.  RR )
4539, 44ax-mp 5 . . . . . . . . . . 11  |-  ( cos `  1 )  e.  RR
4640, 29rerecclapi 8500 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  RR
4743, 46, 45lelttri 7833 . . . . . . . . . . . 12  |-  ( ( 0  <_  ( 1  /  3 )  /\  ( 1  /  3
)  <  ( cos `  1 ) )  -> 
0  <  ( cos `  1 ) )
4842, 36, 47mp2an 420 . . . . . . . . . . 11  |-  0  <  ( cos `  1
)
4943, 45, 48ltleii 7830 . . . . . . . . . 10  |-  0  <_  ( cos `  1
)
5046, 45lt2sqi 10331 . . . . . . . . . 10  |-  ( ( 0  <_  ( 1  /  3 )  /\  0  <_  ( cos `  1
) )  ->  (
( 1  /  3
)  <  ( cos `  1 )  <->  ( (
1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 ) ) )
5142, 49, 50mp2an 420 . . . . . . . . 9  |-  ( ( 1  /  3 )  <  ( cos `  1
)  <->  ( ( 1  /  3 ) ^
2 )  <  (
( cos `  1
) ^ 2 ) )
5236, 51mpbi 144 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 )
5334, 52eqbrtrri 3919 . . . . . . 7  |-  ( 1  /  9 )  < 
( ( cos `  1
) ^ 2 )
54 2pos 8771 . . . . . . . 8  |-  0  <  2
553, 5rerecclapi 8500 . . . . . . . . 9  |-  ( 1  /  9 )  e.  RR
5645resqcli 10328 . . . . . . . . 9  |-  ( ( cos `  1 ) ^ 2 )  e.  RR
57 2re 8750 . . . . . . . . 9  |-  2  e.  RR
5855, 56, 57ltmul2i 8641 . . . . . . . 8  |-  ( 0  <  2  ->  (
( 1  /  9
)  <  ( ( cos `  1 ) ^
2 )  <->  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) ) ) )
5954, 58ax-mp 5 . . . . . . 7  |-  ( ( 1  /  9 )  <  ( ( cos `  1 ) ^
2 )  <->  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) ) )
6053, 59mpbi 144 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6127, 60eqbrtrri 3919 . . . . 5  |-  ( 2  /  9 )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6257, 3, 5redivclapi 8502 . . . . . 6  |-  ( 2  /  9 )  e.  RR
6357, 56remulcli 7744 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  e.  RR
64 ltsub1 8184 . . . . . 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 ) ) )
6562, 63, 39, 64mp3an 1298 . . . . 5  |-  ( ( 2  /  9 )  <  ( 2  x.  ( ( cos `  1
) ^ 2 ) )  <->  ( ( 2  /  9 )  - 
1 )  <  (
( 2  x.  (
( cos `  1
) ^ 2 ) )  -  1 ) )
6661, 65mpbi 144 . . . 4  |-  ( ( 2  /  9 )  -  1 )  < 
( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
6722, 66eqbrtrri 3919 . . 3  |-  -u (
7  /  9 )  <  ( ( 2  x.  ( ( cos `  1 ) ^
2 ) )  - 
1 )
6825fveq2i 5390 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( cos `  2 )
69 cos2t 11367 . . . . 5  |-  ( 1  e.  CC  ->  ( cos `  ( 2  x.  1 ) )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 ) )
7023, 69ax-mp 5 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( ( 2  x.  (
( cos `  1
) ^ 2 ) )  -  1 )
7168, 70eqtr3i 2138 . . 3  |-  ( cos `  2 )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
7267, 71breqtrri 3923 . 2  |-  -u (
7  /  9 )  <  ( cos `  2
)
7335simpri 112 . . . . . . . . 9  |-  ( cos `  1 )  < 
( 2  /  3
)
74 0le2 8770 . . . . . . . . . . 11  |-  0  <_  2
7557, 40divge0i 8629 . . . . . . . . . . 11  |-  ( ( 0  <_  2  /\  0  <  3 )  -> 
0  <_  ( 2  /  3 ) )
7674, 38, 75mp2an 420 . . . . . . . . . 10  |-  0  <_  ( 2  /  3
)
7757, 40, 29redivclapi 8502 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  RR
7845, 77lt2sqi 10331 . . . . . . . . . 10  |-  ( ( 0  <_  ( cos `  1 )  /\  0  <_  ( 2  /  3
) )  ->  (
( cos `  1
)  <  ( 2  /  3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) ) )
7949, 76, 78mp2an 420 . . . . . . . . 9  |-  ( ( cos `  1 )  <  ( 2  / 
3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) )
8073, 79mpbi 144 . . . . . . . 8  |-  ( ( cos `  1 ) ^ 2 )  < 
( ( 2  / 
3 ) ^ 2 )
818, 28, 29sqdivapi 10327 . . . . . . . . 9  |-  ( ( 2  /  3 ) ^ 2 )  =  ( ( 2 ^ 2 )  /  (
3 ^ 2 ) )
82 sq2 10339 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
8382, 32oveq12i 5752 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 4  /  9
)
8481, 83eqtri 2136 . . . . . . . 8  |-  ( ( 2  /  3 ) ^ 2 )  =  ( 4  /  9
)
8580, 84breqtri 3921 . . . . . . 7  |-  ( ( cos `  1 ) ^ 2 )  < 
( 4  /  9
)
86 4re 8757 . . . . . . . . . 10  |-  4  e.  RR
8786, 3, 5redivclapi 8502 . . . . . . . . 9  |-  ( 4  /  9 )  e.  RR
8856, 87, 57ltmul2i 8641 . . . . . . . 8  |-  ( 0  <  2  ->  (
( ( cos `  1
) ^ 2 )  <  ( 4  / 
9 )  <->  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) ) ) )
8954, 88ax-mp 5 . . . . . . 7  |-  ( ( ( cos `  1
) ^ 2 )  <  ( 4  / 
9 )  <->  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) ) )
9085, 89mpbi 144 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) )
91 4cn 8758 . . . . . . . 8  |-  4  e.  CC
928, 91, 2, 5divassapi 8491 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 2  x.  (
4  /  9 ) )
93 4t2e8 8832 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
9491, 8, 93mulcomli 7737 . . . . . . . 8  |-  ( 2  x.  4 )  =  8
9594oveq1i 5750 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 8  /  9
)
9692, 95eqtr3i 2138 . . . . . 6  |-  ( 2  x.  ( 4  / 
9 ) )  =  ( 8  /  9
)
9790, 96breqtri 3921 . . . . 5  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 8  /  9
)
98 8re 8765 . . . . . . 7  |-  8  e.  RR
9998, 3, 5redivclapi 8502 . . . . . 6  |-  ( 8  /  9 )  e.  RR
100 ltsub1 8184 . . . . . 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 ) ) )
10163, 99, 39, 100mp3an 1298 . . . . 5  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  <  ( 8  / 
9 )  <->  ( (
2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 ) )
10297, 101mpbi 144 . . . 4  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 )
10320oveq2i 5751 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 8  / 
9 )  -  1 )
104 divnegap 8429 . . . . . . 7  |-  ( ( 1  e.  CC  /\  9  e.  CC  /\  9 #  0 )  ->  -u (
1  /  9 )  =  ( -u 1  /  9 ) )
10523, 2, 5, 104mp3an 1298 . . . . . 6  |-  -u (
1  /  9 )  =  ( -u 1  /  9 )
106 8cn 8766 . . . . . . . . 9  |-  8  e.  CC
1072, 106negsubdi2i 8012 . . . . . . . 8  |-  -u (
9  -  8 )  =  ( 8  -  9 )
108 8p1e9 8814 . . . . . . . . . 10  |-  ( 8  +  1 )  =  9
1092, 106, 23, 108subaddrii 8015 . . . . . . . . 9  |-  ( 9  -  8 )  =  1
110109negeqi 7920 . . . . . . . 8  |-  -u (
9  -  8 )  =  -u 1
111107, 110eqtr3i 2138 . . . . . . 7  |-  ( 8  -  9 )  = 
-u 1
112111oveq1i 5750 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( -u 1  / 
9 )
113 divsubdirap 8431 . . . . . . 7  |-  ( ( 8  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9 #  0 ) )  -> 
( ( 8  -  9 )  /  9
)  =  ( ( 8  /  9 )  -  ( 9  / 
9 ) ) )
114106, 2, 9, 113mp3an 1298 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( ( 8  / 
9 )  -  (
9  /  9 ) )
115105, 112, 1143eqtr2ri 2143 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  = 
-u ( 1  / 
9 )
116103, 115eqtr3i 2138 . . . 4  |-  ( ( 8  /  9 )  -  1 )  = 
-u ( 1  / 
9 )
117102, 116breqtri 3921 . . 3  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  <  -u ( 1  /  9
)
11871, 117eqbrtri 3917 . 2  |-  ( cos `  2 )  <  -u ( 1  /  9
)
11972, 118pm3.2i 268 1  |-  ( -u ( 7  /  9
)  <  ( cos `  2 )  /\  ( cos `  2 )  <  -u ( 1  /  9
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   CCcc 7582   RRcr 7583   0cc0 7584   1c1 7585    + caddc 7587    x. cmul 7589    < clt 7764    <_ cle 7765    - cmin 7897   -ucneg 7898   # cap 8306    / cdiv 8395   2c2 8731   3c3 8732   4c4 8733   7c7 8736   8c8 8737   9c9 8738   ^cexp 10243   cosccos 11261
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-disj 3875  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-sup 6837  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-5 8742  df-6 8743  df-7 8744  df-8 8745  df-9 8746  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-ioc 9627  df-ico 9628  df-fz 9742  df-fzo 9871  df-seqfrec 10170  df-exp 10244  df-fac 10423  df-bc 10445  df-ihash 10473  df-shft 10538  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-clim 10999  df-sumdc 11074  df-ef 11264  df-sin 11266  df-cos 11267
This theorem is referenced by:  sincos2sgn  11382
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