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Theorem cos2bnd 12484
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 9839 . . . . . . 7  |-  7  e.  RR
21recni 8865 . . . . . 6  |-  7  e.  CC
3 9re 9841 . . . . . . 7  |-  9  e.  RR
43recni 8865 . . . . . 6  |-  9  e.  CC
5 9pos 9853 . . . . . . 7  |-  0  <  9
63, 5gt0ne0ii 9325 . . . . . 6  |-  9  =/=  0
7 divneg 9471 . . . . . 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 9832 . . . . . . 7  |-  2  e.  CC
104, 6pm3.2i 441 . . . . . . 7  |-  ( 9  e.  CC  /\  9  =/=  0 )
11 divsubdir 9472 . . . . . . 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 9148 . . . . . . . 8  |-  -u (
9  -  2 )  =  ( 2  -  9 )
14 7p2e9 9883 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
154, 9, 2subadd2i 9150 . . . . . . . . . 10  |-  ( ( 9  -  2 )  =  7  <->  ( 7  +  2 )  =  9 )
1614, 15mpbir 200 . . . . . . . . 9  |-  ( 9  -  2 )  =  7
1716negeqi 9061 . . . . . . . 8  |-  -u (
9  -  2 )  =  -u 7
1813, 17eqtr3i 2318 . . . . . . 7  |-  ( 2  -  9 )  = 
-u 7
1918oveq1i 5884 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( -u 7  / 
9 )
2012, 19eqtr3i 2318 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( -u 7  / 
9 )
214, 6dividi 9509 . . . . . 6  |-  ( 9  /  9 )  =  1
2221oveq2i 5885 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 2  / 
9 )  -  1 )
238, 20, 223eqtr2ri 2323 . . . 4  |-  ( ( 2  /  9 )  -  1 )  = 
-u ( 7  / 
9 )
24 ax-1cn 8811 . . . . . . . 8  |-  1  e.  CC
259, 24, 4, 6divassi 9532 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  x.  (
1  /  9 ) )
269mulid1i 8855 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
2726oveq1i 5884 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  /  9
)
2825, 27eqtr3i 2318 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  =  ( 2  /  9
)
29 3cn 9834 . . . . . . . . . 10  |-  3  e.  CC
30 3ne0 9847 . . . . . . . . . 10  |-  3  =/=  0
3124, 29, 30sqdivi 11204 . . . . . . . . 9  |-  ( ( 1  /  3 ) ^ 2 )  =  ( ( 1 ^ 2 )  /  (
3 ^ 2 ) )
32 sq1 11214 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
33 sq3 11216 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
3432, 33oveq12i 5886 . . . . . . . . 9  |-  ( ( 1 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 1  /  9
)
3531, 34eqtri 2316 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  =  ( 1  /  9
)
36 cos1bnd 12483 . . . . . . . . . 10  |-  ( ( 1  /  3 )  <  ( cos `  1
)  /\  ( cos `  1 )  <  (
2  /  3 ) )
3736simpli 444 . . . . . . . . 9  |-  ( 1  /  3 )  < 
( cos `  1
)
38 0le1 9313 . . . . . . . . . . 11  |-  0  <_  1
39 3pos 9846 . . . . . . . . . . 11  |-  0  <  3
40 1re 8853 . . . . . . . . . . . 12  |-  1  e.  RR
41 3re 9833 . . . . . . . . . . . 12  |-  3  e.  RR
4240, 41divge0i 9682 . . . . . . . . . . 11  |-  ( ( 0  <_  1  /\  0  <  3 )  -> 
0  <_  ( 1  /  3 ) )
4338, 39, 42mp2an 653 . . . . . . . . . 10  |-  0  <_  ( 1  /  3
)
44 0re 8854 . . . . . . . . . . 11  |-  0  e.  RR
45 recoscl 12437 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  ( cos `  1 )  e.  RR )
4640, 45ax-mp 8 . . . . . . . . . . 11  |-  ( cos `  1 )  e.  RR
4741, 30rereccli 9541 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  RR
4844, 47, 46lelttri 8962 . . . . . . . . . . . 12  |-  ( ( 0  <_  ( 1  /  3 )  /\  ( 1  /  3
)  <  ( cos `  1 ) )  -> 
0  <  ( cos `  1 ) )
4943, 37, 48mp2an 653 . . . . . . . . . . 11  |-  0  <  ( cos `  1
)
5044, 46, 49ltleii 8957 . . . . . . . . . 10  |-  0  <_  ( cos `  1
)
5147, 46lt2sqi 11208 . . . . . . . . . 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 4060 . . . . . . 7  |-  ( 1  /  9 )  < 
( ( cos `  1
) ^ 2 )
55 2pos 9844 . . . . . . . 8  |-  0  <  2
563, 6rereccli 9541 . . . . . . . . 9  |-  ( 1  /  9 )  e.  RR
5746resqcli 11205 . . . . . . . . 9  |-  ( ( cos `  1 ) ^ 2 )  e.  RR
58 2re 9831 . . . . . . . . 9  |-  2  e.  RR
5956, 57, 58ltmul2i 9694 . . . . . . . 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 4060 . . . . 5  |-  ( 2  /  9 )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6358, 3, 6redivcli 9543 . . . . . 6  |-  ( 2  /  9 )  e.  RR
6458, 57remulcli 8867 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  e.  RR
65 ltsub1 9286 . . . . . 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 4060 . . 3  |-  -u (
7  /  9 )  <  ( ( 2  x.  ( ( cos `  1 ) ^
2 ) )  - 
1 )
6926fveq2i 5544 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( cos `  2 )
70 cos2t 12474 . . . . 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 2318 . . 3  |-  ( cos `  2 )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
7368, 72breqtrri 4064 . 2  |-  -u (
7  /  9 )  <  ( cos `  2
)
7436simpri 448 . . . . . . . . 9  |-  ( cos `  1 )  < 
( 2  /  3
)
75 2nn0 9998 . . . . . . . . . . . 12  |-  2  e.  NN0
7675nn0ge0i 10009 . . . . . . . . . . 11  |-  0  <_  2
7758, 41divge0i 9682 . . . . . . . . . . 11  |-  ( ( 0  <_  2  /\  0  <  3 )  -> 
0  <_  ( 2  /  3 ) )
7876, 39, 77mp2an 653 . . . . . . . . . 10  |-  0  <_  ( 2  /  3
)
7958, 41, 30redivcli 9543 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  RR
8046, 79lt2sqi 11208 . . . . . . . . . 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 11204 . . . . . . . . 9  |-  ( ( 2  /  3 ) ^ 2 )  =  ( ( 2 ^ 2 )  /  (
3 ^ 2 ) )
84 sq2 11215 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
8584, 33oveq12i 5886 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 4  /  9
)
8683, 85eqtri 2316 . . . . . . . 8  |-  ( ( 2  /  3 ) ^ 2 )  =  ( 4  /  9
)
8782, 86breqtri 4062 . . . . . . 7  |-  ( ( cos `  1 ) ^ 2 )  < 
( 4  /  9
)
88 4re 9835 . . . . . . . . . 10  |-  4  e.  RR
8988, 3, 6redivcli 9543 . . . . . . . . 9  |-  ( 4  /  9 )  e.  RR
9057, 89, 58ltmul2i 9694 . . . . . . . 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 9836 . . . . . . . 8  |-  4  e.  CC
949, 93, 4, 6divassi 9532 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 2  x.  (
4  /  9 ) )
95 4t2e8 9890 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
9693, 9, 95mulcomli 8860 . . . . . . . 8  |-  ( 2  x.  4 )  =  8
9796oveq1i 5884 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 8  /  9
)
9894, 97eqtr3i 2318 . . . . . 6  |-  ( 2  x.  ( 4  / 
9 ) )  =  ( 8  /  9
)
9992, 98breqtri 4062 . . . . 5  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 8  /  9
)
100 8re 9840 . . . . . . 7  |-  8  e.  RR
101100, 3, 6redivcli 9543 . . . . . 6  |-  ( 8  /  9 )  e.  RR
102 ltsub1 9286 . . . . . 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 5885 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 8  / 
9 )  -  1 )
106 divneg 9471 . . . . . . 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 8865 . . . . . . . . 9  |-  8  e.  CC
1094, 108negsubdi2i 9148 . . . . . . . 8  |-  -u (
9  -  8 )  =  ( 8  -  9 )
110 8p1e9 9869 . . . . . . . . . 10  |-  ( 8  +  1 )  =  9
1114, 108, 24, 110subaddrii 9151 . . . . . . . . 9  |-  ( 9  -  8 )  =  1
112111negeqi 9061 . . . . . . . 8  |-  -u (
9  -  8 )  =  -u 1
113109, 112eqtr3i 2318 . . . . . . 7  |-  ( 8  -  9 )  = 
-u 1
114113oveq1i 5884 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( -u 1  / 
9 )
115 divsubdir 9472 . . . . . . 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 2323 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  = 
-u ( 1  / 
9 )
118105, 117eqtr3i 2318 . . . 4  |-  ( ( 8  /  9 )  -  1 )  = 
-u ( 1  / 
9 )
119104, 118breqtri 4062 . . 3  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  <  -u ( 1  /  9
)
12072, 119eqbrtri 4058 . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   3c3 9812   4c4 9813   7c7 9816   8c8 9817   9c9 9818   ^cexp 11120   cosccos 12362
This theorem is referenced by:  sincos2sgn  12490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ioc 10677  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368
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