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Theorem cos2bnd 12794
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 10082 . . . . . . 7  |-  7  e.  RR
21recni 9107 . . . . . 6  |-  7  e.  CC
3 9re 10084 . . . . . . 7  |-  9  e.  RR
43recni 9107 . . . . . 6  |-  9  e.  CC
5 9pos 10096 . . . . . . 7  |-  0  <  9
63, 5gt0ne0ii 9568 . . . . . 6  |-  9  =/=  0
7 divneg 9714 . . . . . 6  |-  ( ( 7  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
7  /  9 )  =  ( -u 7  /  9 ) )
82, 4, 6, 7mp3an 1280 . . . . 5  |-  -u (
7  /  9 )  =  ( -u 7  /  9 )
9 2cn 10075 . . . . . . 7  |-  2  e.  CC
104, 6pm3.2i 443 . . . . . . 7  |-  ( 9  e.  CC  /\  9  =/=  0 )
11 divsubdir 9715 . . . . . . 7  |-  ( ( 2  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 2  -  9 )  / 
9 )  =  ( ( 2  /  9
)  -  ( 9  /  9 ) ) )
129, 4, 10, 11mp3an 1280 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( ( 2  / 
9 )  -  (
9  /  9 ) )
134, 9negsubdi2i 9391 . . . . . . . 8  |-  -u (
9  -  2 )  =  ( 2  -  9 )
14 7p2e9 10128 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
154, 9, 2subadd2i 9393 . . . . . . . . . 10  |-  ( ( 9  -  2 )  =  7  <->  ( 7  +  2 )  =  9 )
1614, 15mpbir 202 . . . . . . . . 9  |-  ( 9  -  2 )  =  7
1716negeqi 9304 . . . . . . . 8  |-  -u (
9  -  2 )  =  -u 7
1813, 17eqtr3i 2460 . . . . . . 7  |-  ( 2  -  9 )  = 
-u 7
1918oveq1i 6094 . . . . . 6  |-  ( ( 2  -  9 )  /  9 )  =  ( -u 7  / 
9 )
2012, 19eqtr3i 2460 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( -u 7  / 
9 )
214, 6dividi 9752 . . . . . 6  |-  ( 9  /  9 )  =  1
2221oveq2i 6095 . . . . 5  |-  ( ( 2  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 2  / 
9 )  -  1 )
238, 20, 223eqtr2ri 2465 . . . 4  |-  ( ( 2  /  9 )  -  1 )  = 
-u ( 7  / 
9 )
24 ax-1cn 9053 . . . . . . . 8  |-  1  e.  CC
259, 24, 4, 6divassi 9775 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  x.  (
1  /  9 ) )
269mulid1i 9097 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
2726oveq1i 6094 . . . . . . 7  |-  ( ( 2  x.  1 )  /  9 )  =  ( 2  /  9
)
2825, 27eqtr3i 2460 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  =  ( 2  /  9
)
29 3cn 10077 . . . . . . . . . 10  |-  3  e.  CC
30 3ne0 10090 . . . . . . . . . 10  |-  3  =/=  0
3124, 29, 30sqdivi 11471 . . . . . . . . 9  |-  ( ( 1  /  3 ) ^ 2 )  =  ( ( 1 ^ 2 )  /  (
3 ^ 2 ) )
32 sq1 11481 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
33 sq3 11483 . . . . . . . . . 10  |-  ( 3 ^ 2 )  =  9
3432, 33oveq12i 6096 . . . . . . . . 9  |-  ( ( 1 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 1  /  9
)
3531, 34eqtri 2458 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  =  ( 1  /  9
)
36 cos1bnd 12793 . . . . . . . . . 10  |-  ( ( 1  /  3 )  <  ( cos `  1
)  /\  ( cos `  1 )  <  (
2  /  3 ) )
3736simpli 446 . . . . . . . . 9  |-  ( 1  /  3 )  < 
( cos `  1
)
38 0le1 9556 . . . . . . . . . . 11  |-  0  <_  1
39 3pos 10089 . . . . . . . . . . 11  |-  0  <  3
40 1re 9095 . . . . . . . . . . . 12  |-  1  e.  RR
41 3re 10076 . . . . . . . . . . . 12  |-  3  e.  RR
4240, 41divge0i 9925 . . . . . . . . . . 11  |-  ( ( 0  <_  1  /\  0  <  3 )  -> 
0  <_  ( 1  /  3 ) )
4338, 39, 42mp2an 655 . . . . . . . . . 10  |-  0  <_  ( 1  /  3
)
44 0re 9096 . . . . . . . . . . 11  |-  0  e.  RR
45 recoscl 12747 . . . . . . . . . . . 12  |-  ( 1  e.  RR  ->  ( cos `  1 )  e.  RR )
4640, 45ax-mp 5 . . . . . . . . . . 11  |-  ( cos `  1 )  e.  RR
4741, 30rereccli 9784 . . . . . . . . . . . . 13  |-  ( 1  /  3 )  e.  RR
4844, 47, 46lelttri 9205 . . . . . . . . . . . 12  |-  ( ( 0  <_  ( 1  /  3 )  /\  ( 1  /  3
)  <  ( cos `  1 ) )  -> 
0  <  ( cos `  1 ) )
4943, 37, 48mp2an 655 . . . . . . . . . . 11  |-  0  <  ( cos `  1
)
5044, 46, 49ltleii 9201 . . . . . . . . . 10  |-  0  <_  ( cos `  1
)
5147, 46lt2sqi 11475 . . . . . . . . . 10  |-  ( ( 0  <_  ( 1  /  3 )  /\  0  <_  ( cos `  1
) )  ->  (
( 1  /  3
)  <  ( cos `  1 )  <->  ( (
1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 ) ) )
5243, 50, 51mp2an 655 . . . . . . . . 9  |-  ( ( 1  /  3 )  <  ( cos `  1
)  <->  ( ( 1  /  3 ) ^
2 )  <  (
( cos `  1
) ^ 2 ) )
5337, 52mpbi 201 . . . . . . . 8  |-  ( ( 1  /  3 ) ^ 2 )  < 
( ( cos `  1
) ^ 2 )
5435, 53eqbrtrri 4236 . . . . . . 7  |-  ( 1  /  9 )  < 
( ( cos `  1
) ^ 2 )
55 2pos 10087 . . . . . . . 8  |-  0  <  2
563, 6rereccli 9784 . . . . . . . . 9  |-  ( 1  /  9 )  e.  RR
5746resqcli 11472 . . . . . . . . 9  |-  ( ( cos `  1 ) ^ 2 )  e.  RR
58 2re 10074 . . . . . . . . 9  |-  2  e.  RR
5956, 57, 58ltmul2i 9937 . . . . . . . 8  |-  ( 0  <  2  ->  (
( 1  /  9
)  <  ( ( cos `  1 ) ^
2 )  <->  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) ) ) )
6055, 59ax-mp 5 . . . . . . 7  |-  ( ( 1  /  9 )  <  ( ( cos `  1 ) ^
2 )  <->  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) ) )
6154, 60mpbi 201 . . . . . 6  |-  ( 2  x.  ( 1  / 
9 ) )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6228, 61eqbrtrri 4236 . . . . 5  |-  ( 2  /  9 )  < 
( 2  x.  (
( cos `  1
) ^ 2 ) )
6358, 3, 6redivcli 9786 . . . . . 6  |-  ( 2  /  9 )  e.  RR
6458, 57remulcli 9109 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  e.  RR
65 ltsub1 9529 . . . . . 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 1280 . . . . 5  |-  ( ( 2  /  9 )  <  ( 2  x.  ( ( cos `  1
) ^ 2 ) )  <->  ( ( 2  /  9 )  - 
1 )  <  (
( 2  x.  (
( cos `  1
) ^ 2 ) )  -  1 ) )
6762, 66mpbi 201 . . . 4  |-  ( ( 2  /  9 )  -  1 )  < 
( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
6823, 67eqbrtrri 4236 . . 3  |-  -u (
7  /  9 )  <  ( ( 2  x.  ( ( cos `  1 ) ^
2 ) )  - 
1 )
6926fveq2i 5734 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( cos `  2 )
70 cos2t 12784 . . . . 5  |-  ( 1  e.  CC  ->  ( cos `  ( 2  x.  1 ) )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 ) )
7124, 70ax-mp 5 . . . 4  |-  ( cos `  ( 2  x.  1 ) )  =  ( ( 2  x.  (
( cos `  1
) ^ 2 ) )  -  1 )
7269, 71eqtr3i 2460 . . 3  |-  ( cos `  2 )  =  ( ( 2  x.  ( ( cos `  1
) ^ 2 ) )  -  1 )
7368, 72breqtrri 4240 . 2  |-  -u (
7  /  9 )  <  ( cos `  2
)
7436simpri 450 . . . . . . . . 9  |-  ( cos `  1 )  < 
( 2  /  3
)
75 2nn0 10243 . . . . . . . . . . . 12  |-  2  e.  NN0
7675nn0ge0i 10254 . . . . . . . . . . 11  |-  0  <_  2
7758, 41divge0i 9925 . . . . . . . . . . 11  |-  ( ( 0  <_  2  /\  0  <  3 )  -> 
0  <_  ( 2  /  3 ) )
7876, 39, 77mp2an 655 . . . . . . . . . 10  |-  0  <_  ( 2  /  3
)
7958, 41, 30redivcli 9786 . . . . . . . . . . 11  |-  ( 2  /  3 )  e.  RR
8046, 79lt2sqi 11475 . . . . . . . . . 10  |-  ( ( 0  <_  ( cos `  1 )  /\  0  <_  ( 2  /  3
) )  ->  (
( cos `  1
)  <  ( 2  /  3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) ) )
8150, 78, 80mp2an 655 . . . . . . . . 9  |-  ( ( cos `  1 )  <  ( 2  / 
3 )  <->  ( ( cos `  1 ) ^
2 )  <  (
( 2  /  3
) ^ 2 ) )
8274, 81mpbi 201 . . . . . . . 8  |-  ( ( cos `  1 ) ^ 2 )  < 
( ( 2  / 
3 ) ^ 2 )
839, 29, 30sqdivi 11471 . . . . . . . . 9  |-  ( ( 2  /  3 ) ^ 2 )  =  ( ( 2 ^ 2 )  /  (
3 ^ 2 ) )
84 sq2 11482 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
8584, 33oveq12i 6096 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  /  ( 3 ^ 2 ) )  =  ( 4  /  9
)
8683, 85eqtri 2458 . . . . . . . 8  |-  ( ( 2  /  3 ) ^ 2 )  =  ( 4  /  9
)
8782, 86breqtri 4238 . . . . . . 7  |-  ( ( cos `  1 ) ^ 2 )  < 
( 4  /  9
)
88 4re 10078 . . . . . . . . . 10  |-  4  e.  RR
8988, 3, 6redivcli 9786 . . . . . . . . 9  |-  ( 4  /  9 )  e.  RR
9057, 89, 58ltmul2i 9937 . . . . . . . 8  |-  ( 0  <  2  ->  (
( ( cos `  1
) ^ 2 )  <  ( 4  / 
9 )  <->  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) ) ) )
9155, 90ax-mp 5 . . . . . . 7  |-  ( ( ( cos `  1
) ^ 2 )  <  ( 4  / 
9 )  <->  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) ) )
9287, 91mpbi 201 . . . . . 6  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 2  x.  (
4  /  9 ) )
93 4cn 10079 . . . . . . . 8  |-  4  e.  CC
949, 93, 4, 6divassi 9775 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 2  x.  (
4  /  9 ) )
95 4t2e8 10135 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
9693, 9, 95mulcomli 9102 . . . . . . . 8  |-  ( 2  x.  4 )  =  8
9796oveq1i 6094 . . . . . . 7  |-  ( ( 2  x.  4 )  /  9 )  =  ( 8  /  9
)
9894, 97eqtr3i 2460 . . . . . 6  |-  ( 2  x.  ( 4  / 
9 ) )  =  ( 8  /  9
)
9992, 98breqtri 4238 . . . . 5  |-  ( 2  x.  ( ( cos `  1 ) ^
2 ) )  < 
( 8  /  9
)
100 8re 10083 . . . . . . 7  |-  8  e.  RR
101100, 3, 6redivcli 9786 . . . . . 6  |-  ( 8  /  9 )  e.  RR
102 ltsub1 9529 . . . . . 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 1280 . . . . 5  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  <  ( 8  / 
9 )  <->  ( (
2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 ) )
10499, 103mpbi 201 . . . 4  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  < 
( ( 8  / 
9 )  -  1 )
10521oveq2i 6095 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  =  ( ( 8  / 
9 )  -  1 )
106 divneg 9714 . . . . . . 7  |-  ( ( 1  e.  CC  /\  9  e.  CC  /\  9  =/=  0 )  ->  -u (
1  /  9 )  =  ( -u 1  /  9 ) )
10724, 4, 6, 106mp3an 1280 . . . . . 6  |-  -u (
1  /  9 )  =  ( -u 1  /  9 )
108100recni 9107 . . . . . . . . 9  |-  8  e.  CC
1094, 108negsubdi2i 9391 . . . . . . . 8  |-  -u (
9  -  8 )  =  ( 8  -  9 )
110 8p1e9 10114 . . . . . . . . . 10  |-  ( 8  +  1 )  =  9
1114, 108, 24, 110subaddrii 9394 . . . . . . . . 9  |-  ( 9  -  8 )  =  1
112111negeqi 9304 . . . . . . . 8  |-  -u (
9  -  8 )  =  -u 1
113109, 112eqtr3i 2460 . . . . . . 7  |-  ( 8  -  9 )  = 
-u 1
114113oveq1i 6094 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( -u 1  / 
9 )
115 divsubdir 9715 . . . . . . 7  |-  ( ( 8  e.  CC  /\  9  e.  CC  /\  (
9  e.  CC  /\  9  =/=  0 ) )  ->  ( ( 8  -  9 )  / 
9 )  =  ( ( 8  /  9
)  -  ( 9  /  9 ) ) )
116108, 4, 10, 115mp3an 1280 . . . . . 6  |-  ( ( 8  -  9 )  /  9 )  =  ( ( 8  / 
9 )  -  (
9  /  9 ) )
117107, 114, 1163eqtr2ri 2465 . . . . 5  |-  ( ( 8  /  9 )  -  ( 9  / 
9 ) )  = 
-u ( 1  / 
9 )
118105, 117eqtr3i 2460 . . . 4  |-  ( ( 8  /  9 )  -  1 )  = 
-u ( 1  / 
9 )
119104, 118breqtri 4238 . . 3  |-  ( ( 2  x.  ( ( cos `  1 ) ^ 2 ) )  -  1 )  <  -u ( 1  /  9
)
12072, 119eqbrtri 4234 . 2  |-  ( cos `  2 )  <  -u ( 1  /  9
)
12173, 120pm3.2i 443 1  |-  ( -u ( 7  /  9
)  <  ( cos `  2 )  /\  ( cos `  2 )  <  -u ( 1  /  9
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998    x. cmul 9000    < clt 9125    <_ cle 9126    - cmin 9296   -ucneg 9297    / cdiv 9682   2c2 10054   3c3 10055   4c4 10056   7c7 10059   8c8 10060   9c9 10061   ^cexp 11387   cosccos 12672
This theorem is referenced by:  sincos2sgn  12800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-ioc 10926  df-ico 10927  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675  df-sin 12677  df-cos 12678
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