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Theorem cos2bnd 13773
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 10608 . . . . . 6  |-  7  e.  CC
2 9cn 10612 . . . . . 6  |-  9  e.  CC
3 9re 10611 . . . . . . 7  |-  9  e.  RR
4 9pos 10626 . . . . . . 7  |-  0  <  9
53, 4gt0ne0ii 10078 . . . . . 6  |-  9  =/=  0
6 divneg 10228 . . . . . 6  |-  ( ( 7  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
7  /  9 )  =  ( -u 7  /  9 ) )
71, 2, 5, 6mp3an 1319 . . . . 5  |-  -u (
7  /  9 )  =  ( -u 7  /  9 )
8 2cn 10595 . . . . . . 7  |-  2  e.  CC
92, 5pm3.2i 455 . . . . . . 7  |-  ( 9  e.  CC  /\  9  =/=  0 )
10 divsubdir 10229 . . . . . . 7  |-  ( ( 2  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 2  -  9 )  / 
9 )  =  ( ( 2  /  9
)  -  ( 9  /  9 ) ) )
118, 2, 9, 10mp3an 1319 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( ( 2  / 
9 )  -  (
9  /  9 ) )
122, 8negsubdi2i 9894 . . . . . . . 8  |-  -u (
9  -  2 )  =  ( 2  -  9 )
13 7p2e9 10669 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
142, 8, 1subadd2i 9896 . . . . . . . . . 10  |-  ( ( 9  -  2 )  =  7  <->  ( 7  +  2 )  =  9 )
1513, 14mpbir 209 . . . . . . . . 9  |-  ( 9  -  2 )  =  7
1615negeqi 9802 . . . . . . . 8  |-  -u (
9  -  2 )  =  -u 7
1712, 16eqtr3i 2491 . . . . . . 7  |-  ( 2  -  9 )  = 
-u 7
1817oveq1i 6285 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( -u 7  / 
9 )
1911, 18eqtr3i 2491 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( -u 7  / 
9 )
202, 5dividi 10266 . . . . . 6  |-  ( 9  /  9 )  =  1
2120oveq2i 6286 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 2  / 
9 )  -  1 )
227, 19, 213eqtr2ri 2496 . . . 4  |-  ( ( 2  /  9 )  -  1 )  = 
-u ( 7  / 
9 )
23 ax-1cn 9539 . . . . . . . 8  |-  1  e.  CC
248, 23, 2, 5divassi 10289 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  x.  (
1  /  9 ) )
25 2t1e2 10673 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
2625oveq1i 6285 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  /  9
)
2724, 26eqtr3i 2491 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  =  ( 2  /  9
)
28 3cn 10599 . . . . . . . . . 10  |-  3  e.  CC
29 3ne0 10619 . . . . . . . . . 10  |-  3  =/=  0
3023, 28, 29sqdivi 12207 . . . . . . . . 9  |-  ( ( 1  /  3 ) ^ 2 )  =  ( ( 1 ^ 2 )  /  (
3 ^ 2 ) )
31 sq1 12217 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
32 sq3 12220 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
3331, 32oveq12i 6287 . . . . . . . . 9  |-  ( ( 1 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 1  /  9
)
3430, 33eqtri 2489 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  =  ( 1  /  9
)
35 cos1bnd 13772 . . . . . . . . . 10  |-  ( ( 1  /  3 )  <  ( cos `  1
)  /\  ( cos `  1 )  <  (
2  /  3 ) )
3635simpli 458 . . . . . . . . 9  |-  ( 1  /  3 )  < 
( cos `  1
)
37 0le1 10065 . . . . . . . . . . 11  |-  0  <_  1
38 3pos 10618 . . . . . . . . . . 11  |-  0  <  3
39 1re 9584 . . . . . . . . . . . 12  |-  1  e.  RR
40 3re 10598 . . . . . . . . . . . 12  |-  3  e.  RR
4139, 40divge0i 10444 . . . . . . . . . . 11  |-  ( ( 0  <_  1  /\  0  <  3 )  -> 
0  <_  ( 1  /  3 ) )
4237, 38, 41mp2an 672 . . . . . . . . . 10  |-  0  <_  ( 1  /  3
)
43 0re 9585 . . . . . . . . . . 11  |-  0  e.  RR
44 recoscl 13726 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  ( cos `  1 )  e.  RR )
4539, 44ax-mp 5 . . . . . . . . . . 11  |-  ( cos `  1 )  e.  RR
4640, 29rereccli 10298 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  RR
4743, 46, 45lelttri 9700 . . . . . . . . . . . 12  |-  ( ( 0  <_  ( 1  /  3 )  /\  ( 1  /  3
)  <  ( cos `  1 ) )  -> 
0  <  ( cos `  1 ) )
4842, 36, 47mp2an 672 . . . . . . . . . . 11  |-  0  <  ( cos `  1
)
4943, 45, 48ltleii 9696 . . . . . . . . . 10  |-  0  <_  ( cos `  1
)
5046, 45lt2sqi 12211 . . . . . . . . . 10  |-  ( ( 0  <_  ( 1  /  3 )  /\  0  <_  ( cos `  1
) )  ->  (
( 1  /  3
)  <  ( cos `  1 )  <->  ( (
1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 ) ) )
5142, 49, 50mp2an 672 . . . . . . . . 9  |-  ( ( 1  /  3 )  <  ( cos `  1
)  <->  ( ( 1  /  3 ) ^
2 )  <  (
( cos `  1
) ^ 2 ) )
5236, 51mpbi 208 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 )
5334, 52eqbrtrri 4461 . . . . . . 7  |-  ( 1  /  9 )  < 
( ( cos `  1
) ^ 2 )
54 2pos 10616 . . . . . . . 8  |-  0  <  2
553, 5rereccli 10298 . . . . . . . . 9  |-  ( 1  /  9 )  e.  RR
5645resqcli 12208 . . . . . . . . 9  |-  ( ( cos `  1 ) ^ 2 )  e.  RR
57 2re 10594 . . . . . . . . 9  |-  2  e.  RR
5855, 56, 57ltmul2i 10456 . . . . . . . 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 208 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6127, 60eqbrtrri 4461 . . . . 5  |-  ( 2  /  9 )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6257, 3, 5redivcli 10300 . . . . . 6  |-  ( 2  /  9 )  e.  RR
6357, 56remulcli 9599 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  e.  RR
64 ltsub1 10037 . . . . . 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 1319 . . . . 5  |-  ( ( 2  /  9 )  <  ( 2  x.  ( ( cos `  1
) ^ 2 ) )  <->  ( ( 2  /  9 )  - 
1 )  <  (
( 2  x.  (
( cos `  1
) ^ 2 ) )  -  1 ) )
6661, 65mpbi 208 . . . 4  |-  ( ( 2  /  9 )  -  1 )  < 
( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
6722, 66eqbrtrri 4461 . . 3  |-  -u (
7  /  9 )  <  ( ( 2  x.  ( ( cos `  1 ) ^
2 ) )  - 
1 )
6825fveq2i 5860 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( cos `  2 )
69 cos2t 13763 . . . . 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 2491 . . 3  |-  ( cos `  2 )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
7267, 71breqtrri 4465 . 2  |-  -u (
7  /  9 )  <  ( cos `  2
)
7335simpri 462 . . . . . . . . 9  |-  ( cos `  1 )  < 
( 2  /  3
)
74 0le2 10615 . . . . . . . . . . 11  |-  0  <_  2
7557, 40divge0i 10444 . . . . . . . . . . 11  |-  ( ( 0  <_  2  /\  0  <  3 )  -> 
0  <_  ( 2  /  3 ) )
7674, 38, 75mp2an 672 . . . . . . . . . 10  |-  0  <_  ( 2  /  3
)
7757, 40, 29redivcli 10300 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  RR
7845, 77lt2sqi 12211 . . . . . . . . . 10  |-  ( ( 0  <_  ( cos `  1 )  /\  0  <_  ( 2  /  3
) )  ->  (
( cos `  1
)  <  ( 2  /  3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) ) )
7949, 76, 78mp2an 672 . . . . . . . . 9  |-  ( ( cos `  1 )  <  ( 2  / 
3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) )
8073, 79mpbi 208 . . . . . . . 8  |-  ( ( cos `  1 ) ^ 2 )  < 
( ( 2  / 
3 ) ^ 2 )
818, 28, 29sqdivi 12207 . . . . . . . . 9  |-  ( ( 2  /  3 ) ^ 2 )  =  ( ( 2 ^ 2 )  /  (
3 ^ 2 ) )
82 sq2 12219 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
8382, 32oveq12i 6287 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 4  /  9
)
8481, 83eqtri 2489 . . . . . . . 8  |-  ( ( 2  /  3 ) ^ 2 )  =  ( 4  /  9
)
8580, 84breqtri 4463 . . . . . . 7  |-  ( ( cos `  1 ) ^ 2 )  < 
( 4  /  9
)
86 4re 10601 . . . . . . . . . 10  |-  4  e.  RR
8786, 3, 5redivcli 10300 . . . . . . . . 9  |-  ( 4  /  9 )  e.  RR
8856, 87, 57ltmul2i 10456 . . . . . . . 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 208 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) )
91 4cn 10602 . . . . . . . 8  |-  4  e.  CC
928, 91, 2, 5divassi 10289 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 2  x.  (
4  /  9 ) )
93 4t2e8 10678 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
9491, 8, 93mulcomli 9592 . . . . . . . 8  |-  ( 2  x.  4 )  =  8
9594oveq1i 6285 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 8  /  9
)
9692, 95eqtr3i 2491 . . . . . 6  |-  ( 2  x.  ( 4  / 
9 ) )  =  ( 8  /  9
)
9790, 96breqtri 4463 . . . . 5  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 8  /  9
)
98 8re 10609 . . . . . . 7  |-  8  e.  RR
9998, 3, 5redivcli 10300 . . . . . 6  |-  ( 8  /  9 )  e.  RR
100 ltsub1 10037 . . . . . 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 1319 . . . . 5  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  <  ( 8  / 
9 )  <->  ( (
2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 ) )
10297, 101mpbi 208 . . . 4  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 )
10320oveq2i 6286 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 8  / 
9 )  -  1 )
104 divneg 10228 . . . . . . 7  |-  ( ( 1  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
1  /  9 )  =  ( -u 1  /  9 ) )
10523, 2, 5, 104mp3an 1319 . . . . . 6  |-  -u (
1  /  9 )  =  ( -u 1  /  9 )
106 8cn 10610 . . . . . . . . 9  |-  8  e.  CC
1072, 106negsubdi2i 9894 . . . . . . . 8  |-  -u (
9  -  8 )  =  ( 8  -  9 )
108 8p1e9 10655 . . . . . . . . . 10  |-  ( 8  +  1 )  =  9
1092, 106, 23, 108subaddrii 9897 . . . . . . . . 9  |-  ( 9  -  8 )  =  1
110109negeqi 9802 . . . . . . . 8  |-  -u (
9  -  8 )  =  -u 1
111107, 110eqtr3i 2491 . . . . . . 7  |-  ( 8  -  9 )  = 
-u 1
112111oveq1i 6285 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( -u 1  / 
9 )
113 divsubdir 10229 . . . . . . 7  |-  ( ( 8  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 8  -  9 )  / 
9 )  =  ( ( 8  /  9
)  -  ( 9  /  9 ) ) )
114106, 2, 9, 113mp3an 1319 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( ( 8  / 
9 )  -  (
9  /  9 ) )
115105, 112, 1143eqtr2ri 2496 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  = 
-u ( 1  / 
9 )
116103, 115eqtr3i 2491 . . . 4  |-  ( ( 8  /  9 )  -  1 )  = 
-u ( 1  / 
9 )
117102, 116breqtri 4463 . . 3  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  <  -u ( 1  /  9
)
11871, 117eqbrtri 4459 . 2  |-  ( cos `  2 )  <  -u ( 1  /  9
)
11972, 118pm3.2i 455 1  |-  ( -u ( 7  /  9
)  <  ( cos `  2 )  /\  ( cos `  2 )  <  -u ( 1  /  9
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    <_ cle 9618    - cmin 9794   -ucneg 9795    / cdiv 10195   2c2 10574   3c3 10575   4c4 10576   7c7 10579   8c8 10580   9c9 10581   ^cexp 12122   cosccos 13651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-ioc 11523  df-ico 11524  df-fz 11662  df-fzo 11782  df-fl 11886  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-shft 12850  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-limsup 13243  df-clim 13260  df-rlim 13261  df-sum 13458  df-ef 13654  df-sin 13656  df-cos 13657
This theorem is referenced by:  sincos2sgn  13779
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