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Theorem cos2bnd 12709
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 10002 . . . . . . 7  |-  7  e.  RR
21recni 9028 . . . . . 6  |-  7  e.  CC
3 9re 10004 . . . . . . 7  |-  9  e.  RR
43recni 9028 . . . . . 6  |-  9  e.  CC
5 9pos 10016 . . . . . . 7  |-  0  <  9
63, 5gt0ne0ii 9488 . . . . . 6  |-  9  =/=  0
7 divneg 9634 . . . . . 6  |-  ( ( 7  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
7  /  9 )  =  ( -u 7  /  9 ) )
82, 4, 6, 7mp3an 1279 . . . . 5  |-  -u (
7  /  9 )  =  ( -u 7  /  9 )
9 2cn 9995 . . . . . . 7  |-  2  e.  CC
104, 6pm3.2i 442 . . . . . . 7  |-  ( 9  e.  CC  /\  9  =/=  0 )
11 divsubdir 9635 . . . . . . 7  |-  ( ( 2  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 2  -  9 )  / 
9 )  =  ( ( 2  /  9
)  -  ( 9  /  9 ) ) )
129, 4, 10, 11mp3an 1279 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( ( 2  / 
9 )  -  (
9  /  9 ) )
134, 9negsubdi2i 9311 . . . . . . . 8  |-  -u (
9  -  2 )  =  ( 2  -  9 )
14 7p2e9 10048 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
154, 9, 2subadd2i 9313 . . . . . . . . . 10  |-  ( ( 9  -  2 )  =  7  <->  ( 7  +  2 )  =  9 )
1614, 15mpbir 201 . . . . . . . . 9  |-  ( 9  -  2 )  =  7
1716negeqi 9224 . . . . . . . 8  |-  -u (
9  -  2 )  =  -u 7
1813, 17eqtr3i 2402 . . . . . . 7  |-  ( 2  -  9 )  = 
-u 7
1918oveq1i 6023 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( -u 7  / 
9 )
2012, 19eqtr3i 2402 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( -u 7  / 
9 )
214, 6dividi 9672 . . . . . 6  |-  ( 9  /  9 )  =  1
2221oveq2i 6024 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 2  / 
9 )  -  1 )
238, 20, 223eqtr2ri 2407 . . . 4  |-  ( ( 2  /  9 )  -  1 )  = 
-u ( 7  / 
9 )
24 ax-1cn 8974 . . . . . . . 8  |-  1  e.  CC
259, 24, 4, 6divassi 9695 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  x.  (
1  /  9 ) )
269mulid1i 9018 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
2726oveq1i 6023 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  /  9
)
2825, 27eqtr3i 2402 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  =  ( 2  /  9
)
29 3cn 9997 . . . . . . . . . 10  |-  3  e.  CC
30 3ne0 10010 . . . . . . . . . 10  |-  3  =/=  0
3124, 29, 30sqdivi 11386 . . . . . . . . 9  |-  ( ( 1  /  3 ) ^ 2 )  =  ( ( 1 ^ 2 )  /  (
3 ^ 2 ) )
32 sq1 11396 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
33 sq3 11398 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
3432, 33oveq12i 6025 . . . . . . . . 9  |-  ( ( 1 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 1  /  9
)
3531, 34eqtri 2400 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  =  ( 1  /  9
)
36 cos1bnd 12708 . . . . . . . . . 10  |-  ( ( 1  /  3 )  <  ( cos `  1
)  /\  ( cos `  1 )  <  (
2  /  3 ) )
3736simpli 445 . . . . . . . . 9  |-  ( 1  /  3 )  < 
( cos `  1
)
38 0le1 9476 . . . . . . . . . . 11  |-  0  <_  1
39 3pos 10009 . . . . . . . . . . 11  |-  0  <  3
40 1re 9016 . . . . . . . . . . . 12  |-  1  e.  RR
41 3re 9996 . . . . . . . . . . . 12  |-  3  e.  RR
4240, 41divge0i 9845 . . . . . . . . . . 11  |-  ( ( 0  <_  1  /\  0  <  3 )  -> 
0  <_  ( 1  /  3 ) )
4338, 39, 42mp2an 654 . . . . . . . . . 10  |-  0  <_  ( 1  /  3
)
44 0re 9017 . . . . . . . . . . 11  |-  0  e.  RR
45 recoscl 12662 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  ( cos `  1 )  e.  RR )
4640, 45ax-mp 8 . . . . . . . . . . 11  |-  ( cos `  1 )  e.  RR
4741, 30rereccli 9704 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  RR
4844, 47, 46lelttri 9125 . . . . . . . . . . . 12  |-  ( ( 0  <_  ( 1  /  3 )  /\  ( 1  /  3
)  <  ( cos `  1 ) )  -> 
0  <  ( cos `  1 ) )
4943, 37, 48mp2an 654 . . . . . . . . . . 11  |-  0  <  ( cos `  1
)
5044, 46, 49ltleii 9120 . . . . . . . . . 10  |-  0  <_  ( cos `  1
)
5147, 46lt2sqi 11390 . . . . . . . . . 10  |-  ( ( 0  <_  ( 1  /  3 )  /\  0  <_  ( cos `  1
) )  ->  (
( 1  /  3
)  <  ( cos `  1 )  <->  ( (
1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 ) ) )
5243, 50, 51mp2an 654 . . . . . . . . 9  |-  ( ( 1  /  3 )  <  ( cos `  1
)  <->  ( ( 1  /  3 ) ^
2 )  <  (
( cos `  1
) ^ 2 ) )
5337, 52mpbi 200 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 )
5435, 53eqbrtrri 4167 . . . . . . 7  |-  ( 1  /  9 )  < 
( ( cos `  1
) ^ 2 )
55 2pos 10007 . . . . . . . 8  |-  0  <  2
563, 6rereccli 9704 . . . . . . . . 9  |-  ( 1  /  9 )  e.  RR
5746resqcli 11387 . . . . . . . . 9  |-  ( ( cos `  1 ) ^ 2 )  e.  RR
58 2re 9994 . . . . . . . . 9  |-  2  e.  RR
5956, 57, 58ltmul2i 9857 . . . . . . . 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 200 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6228, 61eqbrtrri 4167 . . . . 5  |-  ( 2  /  9 )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6358, 3, 6redivcli 9706 . . . . . 6  |-  ( 2  /  9 )  e.  RR
6458, 57remulcli 9030 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  e.  RR
65 ltsub1 9449 . . . . . 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 1279 . . . . 5  |-  ( ( 2  /  9 )  <  ( 2  x.  ( ( cos `  1
) ^ 2 ) )  <->  ( ( 2  /  9 )  - 
1 )  <  (
( 2  x.  (
( cos `  1
) ^ 2 ) )  -  1 ) )
6762, 66mpbi 200 . . . 4  |-  ( ( 2  /  9 )  -  1 )  < 
( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
6823, 67eqbrtrri 4167 . . 3  |-  -u (
7  /  9 )  <  ( ( 2  x.  ( ( cos `  1 ) ^
2 ) )  - 
1 )
6926fveq2i 5664 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( cos `  2 )
70 cos2t 12699 . . . . 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 2402 . . 3  |-  ( cos `  2 )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
7368, 72breqtrri 4171 . 2  |-  -u (
7  /  9 )  <  ( cos `  2
)
7436simpri 449 . . . . . . . . 9  |-  ( cos `  1 )  < 
( 2  /  3
)
75 2nn0 10163 . . . . . . . . . . . 12  |-  2  e.  NN0
7675nn0ge0i 10174 . . . . . . . . . . 11  |-  0  <_  2
7758, 41divge0i 9845 . . . . . . . . . . 11  |-  ( ( 0  <_  2  /\  0  <  3 )  -> 
0  <_  ( 2  /  3 ) )
7876, 39, 77mp2an 654 . . . . . . . . . 10  |-  0  <_  ( 2  /  3
)
7958, 41, 30redivcli 9706 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  RR
8046, 79lt2sqi 11390 . . . . . . . . . 10  |-  ( ( 0  <_  ( cos `  1 )  /\  0  <_  ( 2  /  3
) )  ->  (
( cos `  1
)  <  ( 2  /  3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) ) )
8150, 78, 80mp2an 654 . . . . . . . . 9  |-  ( ( cos `  1 )  <  ( 2  / 
3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) )
8274, 81mpbi 200 . . . . . . . 8  |-  ( ( cos `  1 ) ^ 2 )  < 
( ( 2  / 
3 ) ^ 2 )
839, 29, 30sqdivi 11386 . . . . . . . . 9  |-  ( ( 2  /  3 ) ^ 2 )  =  ( ( 2 ^ 2 )  /  (
3 ^ 2 ) )
84 sq2 11397 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
8584, 33oveq12i 6025 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 4  /  9
)
8683, 85eqtri 2400 . . . . . . . 8  |-  ( ( 2  /  3 ) ^ 2 )  =  ( 4  /  9
)
8782, 86breqtri 4169 . . . . . . 7  |-  ( ( cos `  1 ) ^ 2 )  < 
( 4  /  9
)
88 4re 9998 . . . . . . . . . 10  |-  4  e.  RR
8988, 3, 6redivcli 9706 . . . . . . . . 9  |-  ( 4  /  9 )  e.  RR
9057, 89, 58ltmul2i 9857 . . . . . . . 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 200 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) )
93 4cn 9999 . . . . . . . 8  |-  4  e.  CC
949, 93, 4, 6divassi 9695 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 2  x.  (
4  /  9 ) )
95 4t2e8 10055 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
9693, 9, 95mulcomli 9023 . . . . . . . 8  |-  ( 2  x.  4 )  =  8
9796oveq1i 6023 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 8  /  9
)
9894, 97eqtr3i 2402 . . . . . 6  |-  ( 2  x.  ( 4  / 
9 ) )  =  ( 8  /  9
)
9992, 98breqtri 4169 . . . . 5  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 8  /  9
)
100 8re 10003 . . . . . . 7  |-  8  e.  RR
101100, 3, 6redivcli 9706 . . . . . 6  |-  ( 8  /  9 )  e.  RR
102 ltsub1 9449 . . . . . 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 1279 . . . . 5  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  <  ( 8  / 
9 )  <->  ( (
2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 ) )
10499, 103mpbi 200 . . . 4  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 )
10521oveq2i 6024 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 8  / 
9 )  -  1 )
106 divneg 9634 . . . . . . 7  |-  ( ( 1  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
1  /  9 )  =  ( -u 1  /  9 ) )
10724, 4, 6, 106mp3an 1279 . . . . . 6  |-  -u (
1  /  9 )  =  ( -u 1  /  9 )
108100recni 9028 . . . . . . . . 9  |-  8  e.  CC
1094, 108negsubdi2i 9311 . . . . . . . 8  |-  -u (
9  -  8 )  =  ( 8  -  9 )
110 8p1e9 10034 . . . . . . . . . 10  |-  ( 8  +  1 )  =  9
1114, 108, 24, 110subaddrii 9314 . . . . . . . . 9  |-  ( 9  -  8 )  =  1
112111negeqi 9224 . . . . . . . 8  |-  -u (
9  -  8 )  =  -u 1
113109, 112eqtr3i 2402 . . . . . . 7  |-  ( 8  -  9 )  = 
-u 1
114113oveq1i 6023 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( -u 1  / 
9 )
115 divsubdir 9635 . . . . . . 7  |-  ( ( 8  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 8  -  9 )  / 
9 )  =  ( ( 8  /  9
)  -  ( 9  /  9 ) ) )
116108, 4, 10, 115mp3an 1279 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( ( 8  / 
9 )  -  (
9  /  9 ) )
117107, 114, 1163eqtr2ri 2407 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  = 
-u ( 1  / 
9 )
118105, 117eqtr3i 2402 . . . 4  |-  ( ( 8  /  9 )  -  1 )  = 
-u ( 1  / 
9 )
119104, 118breqtri 4169 . . 3  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  <  -u ( 1  /  9
)
12072, 119eqbrtri 4165 . 2  |-  ( cos `  2 )  <  -u ( 1  /  9
)
12173, 120pm3.2i 442 1  |-  ( -u ( 7  /  9
)  <  ( cos `  2 )  /\  ( cos `  2 )  <  -u ( 1  /  9
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   CCcc 8914   RRcr 8915   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    < clt 9046    <_ cle 9047    - cmin 9216   -ucneg 9217    / cdiv 9602   2c2 9974   3c3 9975   4c4 9976   7c7 9979   8c8 9980   9c9 9981   ^cexp 11302   cosccos 12587
This theorem is referenced by:  sincos2sgn  12715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995  ax-mulf 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-ioc 10846  df-ico 10847  df-fz 10969  df-fzo 11059  df-fl 11122  df-seq 11244  df-exp 11303  df-fac 11487  df-bc 11514  df-hash 11539  df-shft 11802  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-limsup 12185  df-clim 12202  df-rlim 12203  df-sum 12400  df-ef 12590  df-sin 12592  df-cos 12593
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