MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cos2bnd Structured version   Unicode version

Theorem cos2bnd 13493
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 10426 . . . . . 6  |-  7  e.  CC
2 9cn 10430 . . . . . 6  |-  9  e.  CC
3 9re 10429 . . . . . . 7  |-  9  e.  RR
4 9pos 10444 . . . . . . 7  |-  0  <  9
53, 4gt0ne0ii 9897 . . . . . 6  |-  9  =/=  0
6 divneg 10047 . . . . . 6  |-  ( ( 7  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
7  /  9 )  =  ( -u 7  /  9 ) )
71, 2, 5, 6mp3an 1314 . . . . 5  |-  -u (
7  /  9 )  =  ( -u 7  /  9 )
8 2cn 10413 . . . . . . 7  |-  2  e.  CC
92, 5pm3.2i 455 . . . . . . 7  |-  ( 9  e.  CC  /\  9  =/=  0 )
10 divsubdir 10048 . . . . . . 7  |-  ( ( 2  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 2  -  9 )  / 
9 )  =  ( ( 2  /  9
)  -  ( 9  /  9 ) ) )
118, 2, 9, 10mp3an 1314 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( ( 2  / 
9 )  -  (
9  /  9 ) )
122, 8negsubdi2i 9715 . . . . . . . 8  |-  -u (
9  -  2 )  =  ( 2  -  9 )
13 7p2e9 10487 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
142, 8, 1subadd2i 9717 . . . . . . . . . 10  |-  ( ( 9  -  2 )  =  7  <->  ( 7  +  2 )  =  9 )
1513, 14mpbir 209 . . . . . . . . 9  |-  ( 9  -  2 )  =  7
1615negeqi 9624 . . . . . . . 8  |-  -u (
9  -  2 )  =  -u 7
1712, 16eqtr3i 2465 . . . . . . 7  |-  ( 2  -  9 )  = 
-u 7
1817oveq1i 6122 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( -u 7  / 
9 )
1911, 18eqtr3i 2465 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( -u 7  / 
9 )
202, 5dividi 10085 . . . . . 6  |-  ( 9  /  9 )  =  1
2120oveq2i 6123 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 2  / 
9 )  -  1 )
227, 19, 213eqtr2ri 2470 . . . 4  |-  ( ( 2  /  9 )  -  1 )  = 
-u ( 7  / 
9 )
23 ax-1cn 9361 . . . . . . . 8  |-  1  e.  CC
248, 23, 2, 5divassi 10108 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  x.  (
1  /  9 ) )
25 2t1e2 10491 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
2625oveq1i 6122 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  /  9
)
2724, 26eqtr3i 2465 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  =  ( 2  /  9
)
28 3cn 10417 . . . . . . . . . 10  |-  3  e.  CC
29 3ne0 10437 . . . . . . . . . 10  |-  3  =/=  0
3023, 28, 29sqdivi 11971 . . . . . . . . 9  |-  ( ( 1  /  3 ) ^ 2 )  =  ( ( 1 ^ 2 )  /  (
3 ^ 2 ) )
31 sq1 11981 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
32 sq3 11984 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
3331, 32oveq12i 6124 . . . . . . . . 9  |-  ( ( 1 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 1  /  9
)
3430, 33eqtri 2463 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  =  ( 1  /  9
)
35 cos1bnd 13492 . . . . . . . . . 10  |-  ( ( 1  /  3 )  <  ( cos `  1
)  /\  ( cos `  1 )  <  (
2  /  3 ) )
3635simpli 458 . . . . . . . . 9  |-  ( 1  /  3 )  < 
( cos `  1
)
37 0le1 9884 . . . . . . . . . . 11  |-  0  <_  1
38 3pos 10436 . . . . . . . . . . 11  |-  0  <  3
39 1re 9406 . . . . . . . . . . . 12  |-  1  e.  RR
40 3re 10416 . . . . . . . . . . . 12  |-  3  e.  RR
4139, 40divge0i 10263 . . . . . . . . . . 11  |-  ( ( 0  <_  1  /\  0  <  3 )  -> 
0  <_  ( 1  /  3 ) )
4237, 38, 41mp2an 672 . . . . . . . . . 10  |-  0  <_  ( 1  /  3
)
43 0re 9407 . . . . . . . . . . 11  |-  0  e.  RR
44 recoscl 13446 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  ( cos `  1 )  e.  RR )
4539, 44ax-mp 5 . . . . . . . . . . 11  |-  ( cos `  1 )  e.  RR
4640, 29rereccli 10117 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  RR
4743, 46, 45lelttri 9522 . . . . . . . . . . . 12  |-  ( ( 0  <_  ( 1  /  3 )  /\  ( 1  /  3
)  <  ( cos `  1 ) )  -> 
0  <  ( cos `  1 ) )
4842, 36, 47mp2an 672 . . . . . . . . . . 11  |-  0  <  ( cos `  1
)
4943, 45, 48ltleii 9518 . . . . . . . . . 10  |-  0  <_  ( cos `  1
)
5046, 45lt2sqi 11975 . . . . . . . . . 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 4334 . . . . . . 7  |-  ( 1  /  9 )  < 
( ( cos `  1
) ^ 2 )
54 2pos 10434 . . . . . . . 8  |-  0  <  2
553, 5rereccli 10117 . . . . . . . . 9  |-  ( 1  /  9 )  e.  RR
5645resqcli 11972 . . . . . . . . 9  |-  ( ( cos `  1 ) ^ 2 )  e.  RR
57 2re 10412 . . . . . . . . 9  |-  2  e.  RR
5855, 56, 57ltmul2i 10275 . . . . . . . 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 4334 . . . . 5  |-  ( 2  /  9 )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6257, 3, 5redivcli 10119 . . . . . 6  |-  ( 2  /  9 )  e.  RR
6357, 56remulcli 9421 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  e.  RR
64 ltsub1 9856 . . . . . 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 1314 . . . . 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 4334 . . 3  |-  -u (
7  /  9 )  <  ( ( 2  x.  ( ( cos `  1 ) ^
2 ) )  - 
1 )
6825fveq2i 5715 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( cos `  2 )
69 cos2t 13483 . . . . 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 2465 . . 3  |-  ( cos `  2 )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
7267, 71breqtrri 4338 . 2  |-  -u (
7  /  9 )  <  ( cos `  2
)
7335simpri 462 . . . . . . . . 9  |-  ( cos `  1 )  < 
( 2  /  3
)
74 0le2 10433 . . . . . . . . . . 11  |-  0  <_  2
7557, 40divge0i 10263 . . . . . . . . . . 11  |-  ( ( 0  <_  2  /\  0  <  3 )  -> 
0  <_  ( 2  /  3 ) )
7674, 38, 75mp2an 672 . . . . . . . . . 10  |-  0  <_  ( 2  /  3
)
7757, 40, 29redivcli 10119 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  RR
7845, 77lt2sqi 11975 . . . . . . . . . 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 11971 . . . . . . . . 9  |-  ( ( 2  /  3 ) ^ 2 )  =  ( ( 2 ^ 2 )  /  (
3 ^ 2 ) )
82 sq2 11983 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
8382, 32oveq12i 6124 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 4  /  9
)
8481, 83eqtri 2463 . . . . . . . 8  |-  ( ( 2  /  3 ) ^ 2 )  =  ( 4  /  9
)
8580, 84breqtri 4336 . . . . . . 7  |-  ( ( cos `  1 ) ^ 2 )  < 
( 4  /  9
)
86 4re 10419 . . . . . . . . . 10  |-  4  e.  RR
8786, 3, 5redivcli 10119 . . . . . . . . 9  |-  ( 4  /  9 )  e.  RR
8856, 87, 57ltmul2i 10275 . . . . . . . 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 10420 . . . . . . . 8  |-  4  e.  CC
928, 91, 2, 5divassi 10108 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 2  x.  (
4  /  9 ) )
93 4t2e8 10496 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
9491, 8, 93mulcomli 9414 . . . . . . . 8  |-  ( 2  x.  4 )  =  8
9594oveq1i 6122 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 8  /  9
)
9692, 95eqtr3i 2465 . . . . . 6  |-  ( 2  x.  ( 4  / 
9 ) )  =  ( 8  /  9
)
9790, 96breqtri 4336 . . . . 5  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 8  /  9
)
98 8re 10427 . . . . . . 7  |-  8  e.  RR
9998, 3, 5redivcli 10119 . . . . . 6  |-  ( 8  /  9 )  e.  RR
100 ltsub1 9856 . . . . . 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 1314 . . . . 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 6123 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 8  / 
9 )  -  1 )
104 divneg 10047 . . . . . . 7  |-  ( ( 1  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
1  /  9 )  =  ( -u 1  /  9 ) )
10523, 2, 5, 104mp3an 1314 . . . . . 6  |-  -u (
1  /  9 )  =  ( -u 1  /  9 )
106 8cn 10428 . . . . . . . . 9  |-  8  e.  CC
1072, 106negsubdi2i 9715 . . . . . . . 8  |-  -u (
9  -  8 )  =  ( 8  -  9 )
108 8p1e9 10473 . . . . . . . . . 10  |-  ( 8  +  1 )  =  9
1092, 106, 23, 108subaddrii 9718 . . . . . . . . 9  |-  ( 9  -  8 )  =  1
110109negeqi 9624 . . . . . . . 8  |-  -u (
9  -  8 )  =  -u 1
111107, 110eqtr3i 2465 . . . . . . 7  |-  ( 8  -  9 )  = 
-u 1
112111oveq1i 6122 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( -u 1  / 
9 )
113 divsubdir 10048 . . . . . . 7  |-  ( ( 8  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 8  -  9 )  / 
9 )  =  ( ( 8  /  9
)  -  ( 9  /  9 ) ) )
114106, 2, 9, 113mp3an 1314 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( ( 8  / 
9 )  -  (
9  /  9 ) )
115105, 112, 1143eqtr2ri 2470 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  = 
-u ( 1  / 
9 )
116103, 115eqtr3i 2465 . . . 4  |-  ( ( 8  /  9 )  -  1 )  = 
-u ( 1  / 
9 )
117102, 116breqtri 4336 . . 3  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  <  -u ( 1  /  9
)
11871, 117eqbrtri 4332 . 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 1369    e. wcel 1756    =/= wne 2620   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   CCcc 9301   RRcr 9302   0cc0 9303   1c1 9304    + caddc 9306    x. cmul 9308    < clt 9439    <_ cle 9440    - cmin 9616   -ucneg 9617    / cdiv 10014   2c2 10392   3c3 10393   4c4 10394   7c7 10397   8c8 10398   9c9 10399   ^cexp 11886   cosccos 13371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-ioc 11326  df-ico 11327  df-fz 11459  df-fzo 11570  df-fl 11663  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-shft 12577  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-limsup 12970  df-clim 12987  df-rlim 12988  df-sum 13185  df-ef 13374  df-sin 13376  df-cos 13377
This theorem is referenced by:  sincos2sgn  13499
  Copyright terms: Public domain W3C validator