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Theorem cos2bnd 14220
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 10693 . . . . . 6  |-  7  e.  CC
2 9cn 10697 . . . . . 6  |-  9  e.  CC
3 9re 10696 . . . . . . 7  |-  9  e.  RR
4 9pos 10711 . . . . . . 7  |-  0  <  9
53, 4gt0ne0ii 10149 . . . . . 6  |-  9  =/=  0
6 divneg 10301 . . . . . 6  |-  ( ( 7  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
7  /  9 )  =  ( -u 7  /  9 ) )
71, 2, 5, 6mp3an 1360 . . . . 5  |-  -u (
7  /  9 )  =  ( -u 7  /  9 )
8 2cn 10680 . . . . . . 7  |-  2  e.  CC
92, 5pm3.2i 456 . . . . . . 7  |-  ( 9  e.  CC  /\  9  =/=  0 )
10 divsubdir 10302 . . . . . . 7  |-  ( ( 2  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 2  -  9 )  / 
9 )  =  ( ( 2  /  9
)  -  ( 9  /  9 ) ) )
118, 2, 9, 10mp3an 1360 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( ( 2  / 
9 )  -  (
9  /  9 ) )
122, 8negsubdi2i 9960 . . . . . . . 8  |-  -u (
9  -  2 )  =  ( 2  -  9 )
13 7p2e9 10754 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
142, 8, 1subadd2i 9962 . . . . . . . . . 10  |-  ( ( 9  -  2 )  =  7  <->  ( 7  +  2 )  =  9 )
1513, 14mpbir 212 . . . . . . . . 9  |-  ( 9  -  2 )  =  7
1615negeqi 9867 . . . . . . . 8  |-  -u (
9  -  2 )  =  -u 7
1712, 16eqtr3i 2460 . . . . . . 7  |-  ( 2  -  9 )  = 
-u 7
1817oveq1i 6315 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( -u 7  / 
9 )
1911, 18eqtr3i 2460 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( -u 7  / 
9 )
202, 5dividi 10339 . . . . . 6  |-  ( 9  /  9 )  =  1
2120oveq2i 6316 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 2  / 
9 )  -  1 )
227, 19, 213eqtr2ri 2465 . . . 4  |-  ( ( 2  /  9 )  -  1 )  = 
-u ( 7  / 
9 )
23 ax-1cn 9596 . . . . . . . 8  |-  1  e.  CC
248, 23, 2, 5divassi 10362 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  x.  (
1  /  9 ) )
25 2t1e2 10758 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
2625oveq1i 6315 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  /  9
)
2724, 26eqtr3i 2460 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  =  ( 2  /  9
)
28 3cn 10684 . . . . . . . . . 10  |-  3  e.  CC
29 3ne0 10704 . . . . . . . . . 10  |-  3  =/=  0
3023, 28, 29sqdivi 12356 . . . . . . . . 9  |-  ( ( 1  /  3 ) ^ 2 )  =  ( ( 1 ^ 2 )  /  (
3 ^ 2 ) )
31 sq1 12366 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
32 sq3 12369 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
3331, 32oveq12i 6317 . . . . . . . . 9  |-  ( ( 1 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 1  /  9
)
3430, 33eqtri 2458 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  =  ( 1  /  9
)
35 cos1bnd 14219 . . . . . . . . . 10  |-  ( ( 1  /  3 )  <  ( cos `  1
)  /\  ( cos `  1 )  <  (
2  /  3 ) )
3635simpli 459 . . . . . . . . 9  |-  ( 1  /  3 )  < 
( cos `  1
)
37 0le1 10136 . . . . . . . . . . 11  |-  0  <_  1
38 3pos 10703 . . . . . . . . . . 11  |-  0  <  3
39 1re 9641 . . . . . . . . . . . 12  |-  1  e.  RR
40 3re 10683 . . . . . . . . . . . 12  |-  3  e.  RR
4139, 40divge0i 10516 . . . . . . . . . . 11  |-  ( ( 0  <_  1  /\  0  <  3 )  -> 
0  <_  ( 1  /  3 ) )
4237, 38, 41mp2an 676 . . . . . . . . . 10  |-  0  <_  ( 1  /  3
)
43 0re 9642 . . . . . . . . . . 11  |-  0  e.  RR
44 recoscl 14173 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  ( cos `  1 )  e.  RR )
4539, 44ax-mp 5 . . . . . . . . . . 11  |-  ( cos `  1 )  e.  RR
4640, 29rereccli 10371 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  RR
4743, 46, 45lelttri 9760 . . . . . . . . . . . 12  |-  ( ( 0  <_  ( 1  /  3 )  /\  ( 1  /  3
)  <  ( cos `  1 ) )  -> 
0  <  ( cos `  1 ) )
4842, 36, 47mp2an 676 . . . . . . . . . . 11  |-  0  <  ( cos `  1
)
4943, 45, 48ltleii 9756 . . . . . . . . . 10  |-  0  <_  ( cos `  1
)
5046, 45lt2sqi 12360 . . . . . . . . . 10  |-  ( ( 0  <_  ( 1  /  3 )  /\  0  <_  ( cos `  1
) )  ->  (
( 1  /  3
)  <  ( cos `  1 )  <->  ( (
1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 ) ) )
5142, 49, 50mp2an 676 . . . . . . . . 9  |-  ( ( 1  /  3 )  <  ( cos `  1
)  <->  ( ( 1  /  3 ) ^
2 )  <  (
( cos `  1
) ^ 2 ) )
5236, 51mpbi 211 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 )
5334, 52eqbrtrri 4447 . . . . . . 7  |-  ( 1  /  9 )  < 
( ( cos `  1
) ^ 2 )
54 2pos 10701 . . . . . . . 8  |-  0  <  2
553, 5rereccli 10371 . . . . . . . . 9  |-  ( 1  /  9 )  e.  RR
5645resqcli 12357 . . . . . . . . 9  |-  ( ( cos `  1 ) ^ 2 )  e.  RR
57 2re 10679 . . . . . . . . 9  |-  2  e.  RR
5855, 56, 57ltmul2i 10528 . . . . . . . 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 211 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6127, 60eqbrtrri 4447 . . . . 5  |-  ( 2  /  9 )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6257, 3, 5redivcli 10373 . . . . . 6  |-  ( 2  /  9 )  e.  RR
6357, 56remulcli 9656 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  e.  RR
64 ltsub1 10109 . . . . . 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 1360 . . . . 5  |-  ( ( 2  /  9 )  <  ( 2  x.  ( ( cos `  1
) ^ 2 ) )  <->  ( ( 2  /  9 )  - 
1 )  <  (
( 2  x.  (
( cos `  1
) ^ 2 ) )  -  1 ) )
6661, 65mpbi 211 . . . 4  |-  ( ( 2  /  9 )  -  1 )  < 
( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
6722, 66eqbrtrri 4447 . . 3  |-  -u (
7  /  9 )  <  ( ( 2  x.  ( ( cos `  1 ) ^
2 ) )  - 
1 )
6825fveq2i 5884 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( cos `  2 )
69 cos2t 14210 . . . . 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 2460 . . 3  |-  ( cos `  2 )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
7267, 71breqtrri 4451 . 2  |-  -u (
7  /  9 )  <  ( cos `  2
)
7335simpri 463 . . . . . . . . 9  |-  ( cos `  1 )  < 
( 2  /  3
)
74 0le2 10700 . . . . . . . . . . 11  |-  0  <_  2
7557, 40divge0i 10516 . . . . . . . . . . 11  |-  ( ( 0  <_  2  /\  0  <  3 )  -> 
0  <_  ( 2  /  3 ) )
7674, 38, 75mp2an 676 . . . . . . . . . 10  |-  0  <_  ( 2  /  3
)
7757, 40, 29redivcli 10373 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  RR
7845, 77lt2sqi 12360 . . . . . . . . . 10  |-  ( ( 0  <_  ( cos `  1 )  /\  0  <_  ( 2  /  3
) )  ->  (
( cos `  1
)  <  ( 2  /  3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) ) )
7949, 76, 78mp2an 676 . . . . . . . . 9  |-  ( ( cos `  1 )  <  ( 2  / 
3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) )
8073, 79mpbi 211 . . . . . . . 8  |-  ( ( cos `  1 ) ^ 2 )  < 
( ( 2  / 
3 ) ^ 2 )
818, 28, 29sqdivi 12356 . . . . . . . . 9  |-  ( ( 2  /  3 ) ^ 2 )  =  ( ( 2 ^ 2 )  /  (
3 ^ 2 ) )
82 sq2 12368 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
8382, 32oveq12i 6317 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 4  /  9
)
8481, 83eqtri 2458 . . . . . . . 8  |-  ( ( 2  /  3 ) ^ 2 )  =  ( 4  /  9
)
8580, 84breqtri 4449 . . . . . . 7  |-  ( ( cos `  1 ) ^ 2 )  < 
( 4  /  9
)
86 4re 10686 . . . . . . . . . 10  |-  4  e.  RR
8786, 3, 5redivcli 10373 . . . . . . . . 9  |-  ( 4  /  9 )  e.  RR
8856, 87, 57ltmul2i 10528 . . . . . . . 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 211 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) )
91 4cn 10687 . . . . . . . 8  |-  4  e.  CC
928, 91, 2, 5divassi 10362 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 2  x.  (
4  /  9 ) )
93 4t2e8 10763 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
9491, 8, 93mulcomli 9649 . . . . . . . 8  |-  ( 2  x.  4 )  =  8
9594oveq1i 6315 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 8  /  9
)
9692, 95eqtr3i 2460 . . . . . 6  |-  ( 2  x.  ( 4  / 
9 ) )  =  ( 8  /  9
)
9790, 96breqtri 4449 . . . . 5  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 8  /  9
)
98 8re 10694 . . . . . . 7  |-  8  e.  RR
9998, 3, 5redivcli 10373 . . . . . 6  |-  ( 8  /  9 )  e.  RR
100 ltsub1 10109 . . . . . 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 1360 . . . . 5  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  <  ( 8  / 
9 )  <->  ( (
2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 ) )
10297, 101mpbi 211 . . . 4  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 )
10320oveq2i 6316 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 8  / 
9 )  -  1 )
104 divneg 10301 . . . . . . 7  |-  ( ( 1  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
1  /  9 )  =  ( -u 1  /  9 ) )
10523, 2, 5, 104mp3an 1360 . . . . . 6  |-  -u (
1  /  9 )  =  ( -u 1  /  9 )
106 8cn 10695 . . . . . . . . 9  |-  8  e.  CC
1072, 106negsubdi2i 9960 . . . . . . . 8  |-  -u (
9  -  8 )  =  ( 8  -  9 )
108 8p1e9 10740 . . . . . . . . . 10  |-  ( 8  +  1 )  =  9
1092, 106, 23, 108subaddrii 9963 . . . . . . . . 9  |-  ( 9  -  8 )  =  1
110109negeqi 9867 . . . . . . . 8  |-  -u (
9  -  8 )  =  -u 1
111107, 110eqtr3i 2460 . . . . . . 7  |-  ( 8  -  9 )  = 
-u 1
112111oveq1i 6315 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( -u 1  / 
9 )
113 divsubdir 10302 . . . . . . 7  |-  ( ( 8  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 8  -  9 )  / 
9 )  =  ( ( 8  /  9
)  -  ( 9  /  9 ) ) )
114106, 2, 9, 113mp3an 1360 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( ( 8  / 
9 )  -  (
9  /  9 ) )
115105, 112, 1143eqtr2ri 2465 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  = 
-u ( 1  / 
9 )
116103, 115eqtr3i 2460 . . . 4  |-  ( ( 8  /  9 )  -  1 )  = 
-u ( 1  / 
9 )
117102, 116breqtri 4449 . . 3  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  <  -u ( 1  /  9
)
11871, 117eqbrtri 4445 . 2  |-  ( cos `  2 )  <  -u ( 1  /  9
)
11972, 118pm3.2i 456 1  |-  ( -u ( 7  /  9
)  <  ( cos `  2 )  /\  ( cos `  2 )  <  -u ( 1  /  9
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543    < clt 9674    <_ cle 9675    - cmin 9859   -ucneg 9860    / cdiv 10268   2c2 10659   3c3 10660   4c4 10661   7c7 10664   8c8 10665   9c9 10666   ^cexp 12269   cosccos 14095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ioc 11640  df-ico 11641  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099  df-sin 14101  df-cos 14102
This theorem is referenced by:  sincos2sgn  14226
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