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

Theorem cos2bnd 12781
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 10069 . . . . . . 7  |-  7  e.  RR
21recni 9094 . . . . . 6  |-  7  e.  CC
3 9re 10071 . . . . . . 7  |-  9  e.  RR
43recni 9094 . . . . . 6  |-  9  e.  CC
5 9pos 10083 . . . . . . 7  |-  0  <  9
63, 5gt0ne0ii 9555 . . . . . 6  |-  9  =/=  0
7 divneg 9701 . . . . . 6  |-  ( ( 7  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
7  /  9 )  =  ( -u 7  /  9 ) )
82, 4, 6, 7mp3an 1279 . . . . 5  |-  -u (
7  /  9 )  =  ( -u 7  /  9 )
9 2cn 10062 . . . . . . 7  |-  2  e.  CC
104, 6pm3.2i 442 . . . . . . 7  |-  ( 9  e.  CC  /\  9  =/=  0 )
11 divsubdir 9702 . . . . . . 7  |-  ( ( 2  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 2  -  9 )  / 
9 )  =  ( ( 2  /  9
)  -  ( 9  /  9 ) ) )
129, 4, 10, 11mp3an 1279 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( ( 2  / 
9 )  -  (
9  /  9 ) )
134, 9negsubdi2i 9378 . . . . . . . 8  |-  -u (
9  -  2 )  =  ( 2  -  9 )
14 7p2e9 10115 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
154, 9, 2subadd2i 9380 . . . . . . . . . 10  |-  ( ( 9  -  2 )  =  7  <->  ( 7  +  2 )  =  9 )
1614, 15mpbir 201 . . . . . . . . 9  |-  ( 9  -  2 )  =  7
1716negeqi 9291 . . . . . . . 8  |-  -u (
9  -  2 )  =  -u 7
1813, 17eqtr3i 2457 . . . . . . 7  |-  ( 2  -  9 )  = 
-u 7
1918oveq1i 6083 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( -u 7  / 
9 )
2012, 19eqtr3i 2457 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( -u 7  / 
9 )
214, 6dividi 9739 . . . . . 6  |-  ( 9  /  9 )  =  1
2221oveq2i 6084 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 2  / 
9 )  -  1 )
238, 20, 223eqtr2ri 2462 . . . 4  |-  ( ( 2  /  9 )  -  1 )  = 
-u ( 7  / 
9 )
24 ax-1cn 9040 . . . . . . . 8  |-  1  e.  CC
259, 24, 4, 6divassi 9762 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  x.  (
1  /  9 ) )
269mulid1i 9084 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
2726oveq1i 6083 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  /  9
)
2825, 27eqtr3i 2457 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  =  ( 2  /  9
)
29 3cn 10064 . . . . . . . . . 10  |-  3  e.  CC
30 3ne0 10077 . . . . . . . . . 10  |-  3  =/=  0
3124, 29, 30sqdivi 11458 . . . . . . . . 9  |-  ( ( 1  /  3 ) ^ 2 )  =  ( ( 1 ^ 2 )  /  (
3 ^ 2 ) )
32 sq1 11468 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
33 sq3 11470 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
3432, 33oveq12i 6085 . . . . . . . . 9  |-  ( ( 1 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 1  /  9
)
3531, 34eqtri 2455 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  =  ( 1  /  9
)
36 cos1bnd 12780 . . . . . . . . . 10  |-  ( ( 1  /  3 )  <  ( cos `  1
)  /\  ( cos `  1 )  <  (
2  /  3 ) )
3736simpli 445 . . . . . . . . 9  |-  ( 1  /  3 )  < 
( cos `  1
)
38 0le1 9543 . . . . . . . . . . 11  |-  0  <_  1
39 3pos 10076 . . . . . . . . . . 11  |-  0  <  3
40 1re 9082 . . . . . . . . . . . 12  |-  1  e.  RR
41 3re 10063 . . . . . . . . . . . 12  |-  3  e.  RR
4240, 41divge0i 9912 . . . . . . . . . . 11  |-  ( ( 0  <_  1  /\  0  <  3 )  -> 
0  <_  ( 1  /  3 ) )
4338, 39, 42mp2an 654 . . . . . . . . . 10  |-  0  <_  ( 1  /  3
)
44 0re 9083 . . . . . . . . . . 11  |-  0  e.  RR
45 recoscl 12734 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  ( cos `  1 )  e.  RR )
4640, 45ax-mp 8 . . . . . . . . . . 11  |-  ( cos `  1 )  e.  RR
4741, 30rereccli 9771 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  RR
4844, 47, 46lelttri 9192 . . . . . . . . . . . 12  |-  ( ( 0  <_  ( 1  /  3 )  /\  ( 1  /  3
)  <  ( cos `  1 ) )  -> 
0  <  ( cos `  1 ) )
4943, 37, 48mp2an 654 . . . . . . . . . . 11  |-  0  <  ( cos `  1
)
5044, 46, 49ltleii 9188 . . . . . . . . . 10  |-  0  <_  ( cos `  1
)
5147, 46lt2sqi 11462 . . . . . . . . . 10  |-  ( ( 0  <_  ( 1  /  3 )  /\  0  <_  ( cos `  1
) )  ->  (
( 1  /  3
)  <  ( cos `  1 )  <->  ( (
1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 ) ) )
5243, 50, 51mp2an 654 . . . . . . . . 9  |-  ( ( 1  /  3 )  <  ( cos `  1
)  <->  ( ( 1  /  3 ) ^
2 )  <  (
( cos `  1
) ^ 2 ) )
5337, 52mpbi 200 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 )
5435, 53eqbrtrri 4225 . . . . . . 7  |-  ( 1  /  9 )  < 
( ( cos `  1
) ^ 2 )
55 2pos 10074 . . . . . . . 8  |-  0  <  2
563, 6rereccli 9771 . . . . . . . . 9  |-  ( 1  /  9 )  e.  RR
5746resqcli 11459 . . . . . . . . 9  |-  ( ( cos `  1 ) ^ 2 )  e.  RR
58 2re 10061 . . . . . . . . 9  |-  2  e.  RR
5956, 57, 58ltmul2i 9924 . . . . . . . 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 200 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6228, 61eqbrtrri 4225 . . . . 5  |-  ( 2  /  9 )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6358, 3, 6redivcli 9773 . . . . . 6  |-  ( 2  /  9 )  e.  RR
6458, 57remulcli 9096 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  e.  RR
65 ltsub1 9516 . . . . . 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 1279 . . . . 5  |-  ( ( 2  /  9 )  <  ( 2  x.  ( ( cos `  1
) ^ 2 ) )  <->  ( ( 2  /  9 )  - 
1 )  <  (
( 2  x.  (
( cos `  1
) ^ 2 ) )  -  1 ) )
6762, 66mpbi 200 . . . 4  |-  ( ( 2  /  9 )  -  1 )  < 
( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
6823, 67eqbrtrri 4225 . . 3  |-  -u (
7  /  9 )  <  ( ( 2  x.  ( ( cos `  1 ) ^
2 ) )  - 
1 )
6926fveq2i 5723 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( cos `  2 )
70 cos2t 12771 . . . . 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 2457 . . 3  |-  ( cos `  2 )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
7368, 72breqtrri 4229 . 2  |-  -u (
7  /  9 )  <  ( cos `  2
)
7436simpri 449 . . . . . . . . 9  |-  ( cos `  1 )  < 
( 2  /  3
)
75 2nn0 10230 . . . . . . . . . . . 12  |-  2  e.  NN0
7675nn0ge0i 10241 . . . . . . . . . . 11  |-  0  <_  2
7758, 41divge0i 9912 . . . . . . . . . . 11  |-  ( ( 0  <_  2  /\  0  <  3 )  -> 
0  <_  ( 2  /  3 ) )
7876, 39, 77mp2an 654 . . . . . . . . . 10  |-  0  <_  ( 2  /  3
)
7958, 41, 30redivcli 9773 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  RR
8046, 79lt2sqi 11462 . . . . . . . . . 10  |-  ( ( 0  <_  ( cos `  1 )  /\  0  <_  ( 2  /  3
) )  ->  (
( cos `  1
)  <  ( 2  /  3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) ) )
8150, 78, 80mp2an 654 . . . . . . . . 9  |-  ( ( cos `  1 )  <  ( 2  / 
3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) )
8274, 81mpbi 200 . . . . . . . 8  |-  ( ( cos `  1 ) ^ 2 )  < 
( ( 2  / 
3 ) ^ 2 )
839, 29, 30sqdivi 11458 . . . . . . . . 9  |-  ( ( 2  /  3 ) ^ 2 )  =  ( ( 2 ^ 2 )  /  (
3 ^ 2 ) )
84 sq2 11469 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
8584, 33oveq12i 6085 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 4  /  9
)
8683, 85eqtri 2455 . . . . . . . 8  |-  ( ( 2  /  3 ) ^ 2 )  =  ( 4  /  9
)
8782, 86breqtri 4227 . . . . . . 7  |-  ( ( cos `  1 ) ^ 2 )  < 
( 4  /  9
)
88 4re 10065 . . . . . . . . . 10  |-  4  e.  RR
8988, 3, 6redivcli 9773 . . . . . . . . 9  |-  ( 4  /  9 )  e.  RR
9057, 89, 58ltmul2i 9924 . . . . . . . 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 200 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) )
93 4cn 10066 . . . . . . . 8  |-  4  e.  CC
949, 93, 4, 6divassi 9762 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 2  x.  (
4  /  9 ) )
95 4t2e8 10122 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
9693, 9, 95mulcomli 9089 . . . . . . . 8  |-  ( 2  x.  4 )  =  8
9796oveq1i 6083 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 8  /  9
)
9894, 97eqtr3i 2457 . . . . . 6  |-  ( 2  x.  ( 4  / 
9 ) )  =  ( 8  /  9
)
9992, 98breqtri 4227 . . . . 5  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 8  /  9
)
100 8re 10070 . . . . . . 7  |-  8  e.  RR
101100, 3, 6redivcli 9773 . . . . . 6  |-  ( 8  /  9 )  e.  RR
102 ltsub1 9516 . . . . . 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 1279 . . . . 5  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  <  ( 8  / 
9 )  <->  ( (
2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 ) )
10499, 103mpbi 200 . . . 4  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 )
10521oveq2i 6084 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 8  / 
9 )  -  1 )
106 divneg 9701 . . . . . . 7  |-  ( ( 1  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
1  /  9 )  =  ( -u 1  /  9 ) )
10724, 4, 6, 106mp3an 1279 . . . . . 6  |-  -u (
1  /  9 )  =  ( -u 1  /  9 )
108100recni 9094 . . . . . . . . 9  |-  8  e.  CC
1094, 108negsubdi2i 9378 . . . . . . . 8  |-  -u (
9  -  8 )  =  ( 8  -  9 )
110 8p1e9 10101 . . . . . . . . . 10  |-  ( 8  +  1 )  =  9
1114, 108, 24, 110subaddrii 9381 . . . . . . . . 9  |-  ( 9  -  8 )  =  1
112111negeqi 9291 . . . . . . . 8  |-  -u (
9  -  8 )  =  -u 1
113109, 112eqtr3i 2457 . . . . . . 7  |-  ( 8  -  9 )  = 
-u 1
114113oveq1i 6083 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( -u 1  / 
9 )
115 divsubdir 9702 . . . . . . 7  |-  ( ( 8  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 8  -  9 )  / 
9 )  =  ( ( 8  /  9
)  -  ( 9  /  9 ) ) )
116108, 4, 10, 115mp3an 1279 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( ( 8  / 
9 )  -  (
9  /  9 ) )
117107, 114, 1163eqtr2ri 2462 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  = 
-u ( 1  / 
9 )
118105, 117eqtr3i 2457 . . . 4  |-  ( ( 8  /  9 )  -  1 )  = 
-u ( 1  / 
9 )
119104, 118breqtri 4227 . . 3  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  <  -u ( 1  /  9
)
12072, 119eqbrtri 4223 . 2  |-  ( cos `  2 )  <  -u ( 1  /  9
)
12173, 120pm3.2i 442 1  |-  ( -u ( 7  /  9
)  <  ( cos `  2 )  /\  ( cos `  2 )  <  -u ( 1  /  9
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    <_ cle 9113    - cmin 9283   -ucneg 9284    / cdiv 9669   2c2 10041   3c3 10042   4c4 10043   7c7 10046   8c8 10047   9c9 10048   ^cexp 11374   cosccos 12659
This theorem is referenced by:  sincos2sgn  12787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ioc 10913  df-ico 10914  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665
  Copyright terms: Public domain W3C validator