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Theorem cos2bnd 11112
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 8567 . . . . . 6  |-  7  e.  CC
2 9cn 8571 . . . . . 6  |-  9  e.  CC
3 9re 8570 . . . . . . 7  |-  9  e.  RR
4 9pos 8587 . . . . . . 7  |-  0  <  9
53, 4gt0ap0ii 8165 . . . . . 6  |-  9 #  0
6 divnegap 8234 . . . . . 6  |-  ( ( 7  e.  CC  /\  9  e.  CC  /\  9 #  0 )  ->  -u (
7  /  9 )  =  ( -u 7  /  9 ) )
71, 2, 5, 6mp3an 1274 . . . . 5  |-  -u (
7  /  9 )  =  ( -u 7  /  9 )
8 2cn 8554 . . . . . . 7  |-  2  e.  CC
92, 5pm3.2i 267 . . . . . . 7  |-  ( 9  e.  CC  /\  9 #  0 )
10 divsubdirap 8236 . . . . . . 7  |-  ( ( 2  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9 #  0 ) )  -> 
( ( 2  -  9 )  /  9
)  =  ( ( 2  /  9 )  -  ( 9  / 
9 ) ) )
118, 2, 9, 10mp3an 1274 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( ( 2  / 
9 )  -  (
9  /  9 ) )
122, 8negsubdi2i 7829 . . . . . . . 8  |-  -u (
9  -  2 )  =  ( 2  -  9 )
13 7p2e9 8628 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
142, 8, 1subadd2i 7831 . . . . . . . . . 10  |-  ( ( 9  -  2 )  =  7  <->  ( 7  +  2 )  =  9 )
1513, 14mpbir 145 . . . . . . . . 9  |-  ( 9  -  2 )  =  7
1615negeqi 7737 . . . . . . . 8  |-  -u (
9  -  2 )  =  -u 7
1712, 16eqtr3i 2111 . . . . . . 7  |-  ( 2  -  9 )  = 
-u 7
1817oveq1i 5676 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( -u 7  / 
9 )
1911, 18eqtr3i 2111 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( -u 7  / 
9 )
202, 5dividapi 8273 . . . . . 6  |-  ( 9  /  9 )  =  1
2120oveq2i 5677 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 2  / 
9 )  -  1 )
227, 19, 213eqtr2ri 2116 . . . 4  |-  ( ( 2  /  9 )  -  1 )  = 
-u ( 7  / 
9 )
23 ax-1cn 7499 . . . . . . . 8  |-  1  e.  CC
248, 23, 2, 5divassapi 8296 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  x.  (
1  /  9 ) )
25 2t1e2 8630 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
2625oveq1i 5676 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  /  9
)
2724, 26eqtr3i 2111 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  =  ( 2  /  9
)
28 3cn 8558 . . . . . . . . . 10  |-  3  e.  CC
29 3ap0 8579 . . . . . . . . . 10  |-  3 #  0
3023, 28, 29sqdivapi 10099 . . . . . . . . 9  |-  ( ( 1  /  3 ) ^ 2 )  =  ( ( 1 ^ 2 )  /  (
3 ^ 2 ) )
31 sq1 10109 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
32 sq3 10112 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
3331, 32oveq12i 5678 . . . . . . . . 9  |-  ( ( 1 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 1  /  9
)
3430, 33eqtri 2109 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  =  ( 1  /  9
)
35 cos1bnd 11111 . . . . . . . . . 10  |-  ( ( 1  /  3 )  <  ( cos `  1
)  /\  ( cos `  1 )  <  (
2  /  3 ) )
3635simpli 110 . . . . . . . . 9  |-  ( 1  /  3 )  < 
( cos `  1
)
37 0le1 8020 . . . . . . . . . . 11  |-  0  <_  1
38 3pos 8577 . . . . . . . . . . 11  |-  0  <  3
39 1re 7548 . . . . . . . . . . . 12  |-  1  e.  RR
40 3re 8557 . . . . . . . . . . . 12  |-  3  e.  RR
4139, 40divge0i 8433 . . . . . . . . . . 11  |-  ( ( 0  <_  1  /\  0  <  3 )  -> 
0  <_  ( 1  /  3 ) )
4237, 38, 41mp2an 418 . . . . . . . . . 10  |-  0  <_  ( 1  /  3
)
43 0re 7549 . . . . . . . . . . 11  |-  0  e.  RR
44 recoscl 11073 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  ( cos `  1 )  e.  RR )
4539, 44ax-mp 7 . . . . . . . . . . 11  |-  ( cos `  1 )  e.  RR
4640, 29rerecclapi 8305 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  RR
4743, 46, 45lelttri 7651 . . . . . . . . . . . 12  |-  ( ( 0  <_  ( 1  /  3 )  /\  ( 1  /  3
)  <  ( cos `  1 ) )  -> 
0  <  ( cos `  1 ) )
4842, 36, 47mp2an 418 . . . . . . . . . . 11  |-  0  <  ( cos `  1
)
4943, 45, 48ltleii 7648 . . . . . . . . . 10  |-  0  <_  ( cos `  1
)
5046, 45lt2sqi 10103 . . . . . . . . . 10  |-  ( ( 0  <_  ( 1  /  3 )  /\  0  <_  ( cos `  1
) )  ->  (
( 1  /  3
)  <  ( cos `  1 )  <->  ( (
1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 ) ) )
5142, 49, 50mp2an 418 . . . . . . . . 9  |-  ( ( 1  /  3 )  <  ( cos `  1
)  <->  ( ( 1  /  3 ) ^
2 )  <  (
( cos `  1
) ^ 2 ) )
5236, 51mpbi 144 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 )
5334, 52eqbrtrri 3872 . . . . . . 7  |-  ( 1  /  9 )  < 
( ( cos `  1
) ^ 2 )
54 2pos 8574 . . . . . . . 8  |-  0  <  2
553, 5rerecclapi 8305 . . . . . . . . 9  |-  ( 1  /  9 )  e.  RR
5645resqcli 10100 . . . . . . . . 9  |-  ( ( cos `  1 ) ^ 2 )  e.  RR
57 2re 8553 . . . . . . . . 9  |-  2  e.  RR
5855, 56, 57ltmul2i 8445 . . . . . . . 8  |-  ( 0  <  2  ->  (
( 1  /  9
)  <  ( ( cos `  1 ) ^
2 )  <->  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) ) ) )
5954, 58ax-mp 7 . . . . . . 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 3872 . . . . 5  |-  ( 2  /  9 )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6257, 3, 5redivclapi 8307 . . . . . 6  |-  ( 2  /  9 )  e.  RR
6357, 56remulcli 7563 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  e.  RR
64 ltsub1 7997 . . . . . 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 1274 . . . . 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 3872 . . 3  |-  -u (
7  /  9 )  <  ( ( 2  x.  ( ( cos `  1 ) ^
2 ) )  - 
1 )
6825fveq2i 5321 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( cos `  2 )
69 cos2t 11102 . . . . 5  |-  ( 1  e.  CC  ->  ( cos `  ( 2  x.  1 ) )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 ) )
7023, 69ax-mp 7 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( ( 2  x.  (
( cos `  1
) ^ 2 ) )  -  1 )
7168, 70eqtr3i 2111 . . 3  |-  ( cos `  2 )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
7267, 71breqtrri 3876 . 2  |-  -u (
7  /  9 )  <  ( cos `  2
)
7335simpri 112 . . . . . . . . 9  |-  ( cos `  1 )  < 
( 2  /  3
)
74 0le2 8573 . . . . . . . . . . 11  |-  0  <_  2
7557, 40divge0i 8433 . . . . . . . . . . 11  |-  ( ( 0  <_  2  /\  0  <  3 )  -> 
0  <_  ( 2  /  3 ) )
7674, 38, 75mp2an 418 . . . . . . . . . 10  |-  0  <_  ( 2  /  3
)
7757, 40, 29redivclapi 8307 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  RR
7845, 77lt2sqi 10103 . . . . . . . . . 10  |-  ( ( 0  <_  ( cos `  1 )  /\  0  <_  ( 2  /  3
) )  ->  (
( cos `  1
)  <  ( 2  /  3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) ) )
7949, 76, 78mp2an 418 . . . . . . . . 9  |-  ( ( cos `  1 )  <  ( 2  / 
3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) )
8073, 79mpbi 144 . . . . . . . 8  |-  ( ( cos `  1 ) ^ 2 )  < 
( ( 2  / 
3 ) ^ 2 )
818, 28, 29sqdivapi 10099 . . . . . . . . 9  |-  ( ( 2  /  3 ) ^ 2 )  =  ( ( 2 ^ 2 )  /  (
3 ^ 2 ) )
82 sq2 10111 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
8382, 32oveq12i 5678 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 4  /  9
)
8481, 83eqtri 2109 . . . . . . . 8  |-  ( ( 2  /  3 ) ^ 2 )  =  ( 4  /  9
)
8580, 84breqtri 3874 . . . . . . 7  |-  ( ( cos `  1 ) ^ 2 )  < 
( 4  /  9
)
86 4re 8560 . . . . . . . . . 10  |-  4  e.  RR
8786, 3, 5redivclapi 8307 . . . . . . . . 9  |-  ( 4  /  9 )  e.  RR
8856, 87, 57ltmul2i 8445 . . . . . . . 8  |-  ( 0  <  2  ->  (
( ( cos `  1
) ^ 2 )  <  ( 4  / 
9 )  <->  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) ) ) )
8954, 88ax-mp 7 . . . . . . 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 8561 . . . . . . . 8  |-  4  e.  CC
928, 91, 2, 5divassapi 8296 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 2  x.  (
4  /  9 ) )
93 4t2e8 8635 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
9491, 8, 93mulcomli 7556 . . . . . . . 8  |-  ( 2  x.  4 )  =  8
9594oveq1i 5676 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 8  /  9
)
9692, 95eqtr3i 2111 . . . . . 6  |-  ( 2  x.  ( 4  / 
9 ) )  =  ( 8  /  9
)
9790, 96breqtri 3874 . . . . 5  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 8  /  9
)
98 8re 8568 . . . . . . 7  |-  8  e.  RR
9998, 3, 5redivclapi 8307 . . . . . 6  |-  ( 8  /  9 )  e.  RR
100 ltsub1 7997 . . . . . 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 1274 . . . . 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 5677 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 8  / 
9 )  -  1 )
104 divnegap 8234 . . . . . . 7  |-  ( ( 1  e.  CC  /\  9  e.  CC  /\  9 #  0 )  ->  -u (
1  /  9 )  =  ( -u 1  /  9 ) )
10523, 2, 5, 104mp3an 1274 . . . . . 6  |-  -u (
1  /  9 )  =  ( -u 1  /  9 )
106 8cn 8569 . . . . . . . . 9  |-  8  e.  CC
1072, 106negsubdi2i 7829 . . . . . . . 8  |-  -u (
9  -  8 )  =  ( 8  -  9 )
108 8p1e9 8617 . . . . . . . . . 10  |-  ( 8  +  1 )  =  9
1092, 106, 23, 108subaddrii 7832 . . . . . . . . 9  |-  ( 9  -  8 )  =  1
110109negeqi 7737 . . . . . . . 8  |-  -u (
9  -  8 )  =  -u 1
111107, 110eqtr3i 2111 . . . . . . 7  |-  ( 8  -  9 )  = 
-u 1
112111oveq1i 5676 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( -u 1  / 
9 )
113 divsubdirap 8236 . . . . . . 7  |-  ( ( 8  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9 #  0 ) )  -> 
( ( 8  -  9 )  /  9
)  =  ( ( 8  /  9 )  -  ( 9  / 
9 ) ) )
114106, 2, 9, 113mp3an 1274 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( ( 8  / 
9 )  -  (
9  /  9 ) )
115105, 112, 1143eqtr2ri 2116 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  = 
-u ( 1  / 
9 )
116103, 115eqtr3i 2111 . . . 4  |-  ( ( 8  /  9 )  -  1 )  = 
-u ( 1  / 
9 )
117102, 116breqtri 3874 . . 3  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  <  -u ( 1  /  9
)
11871, 117eqbrtri 3870 . 2  |-  ( cos `  2 )  <  -u ( 1  /  9
)
11972, 118pm3.2i 267 1  |-  ( -u ( 7  /  9
)  <  ( cos `  2 )  /\  ( cos `  2 )  <  -u ( 1  /  9
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   CCcc 7409   RRcr 7410   0cc0 7411   1c1 7412    + caddc 7414    x. cmul 7416    < clt 7583    <_ cle 7584    - cmin 7714   -ucneg 7715   # cap 8119    / cdiv 8200   2c2 8534   3c3 8535   4c4 8536   7c7 8539   8c8 8540   9c9 8541   ^cexp 10015   cosccos 10996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525  ax-caucvg 7526
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-disj 3829  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-sup 6733  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-5 8545  df-6 8546  df-7 8547  df-8 8548  df-9 8549  df-n0 8735  df-z 8812  df-uz 9081  df-q 9166  df-rp 9196  df-ioc 9372  df-ico 9373  df-fz 9486  df-fzo 9615  df-iseq 9914  df-seq3 9915  df-exp 10016  df-fac 10195  df-bc 10217  df-ihash 10245  df-shft 10310  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493  df-clim 10728  df-isum 10804  df-ef 10999  df-sin 11001  df-cos 11002
This theorem is referenced by:  sincos2sgn  11117
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