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Mirrors > Home > ILE Home > Th. List > cos2bnd | Unicode version |
Description: Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
cos2bnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7cn 8932 | . . . . . 6 | |
2 | 9cn 8936 | . . . . . 6 | |
3 | 9re 8935 | . . . . . . 7 | |
4 | 9pos 8952 | . . . . . . 7 | |
5 | 3, 4 | gt0ap0ii 8517 | . . . . . 6 # |
6 | divnegap 8593 | . . . . . 6 # | |
7 | 1, 2, 5, 6 | mp3an 1326 | . . . . 5 |
8 | 2cn 8919 | . . . . . . 7 | |
9 | 2, 5 | pm3.2i 270 | . . . . . . 7 # |
10 | divsubdirap 8595 | . . . . . . 7 # | |
11 | 8, 2, 9, 10 | mp3an 1326 | . . . . . 6 |
12 | 2, 8 | negsubdi2i 8175 | . . . . . . . 8 |
13 | 7p2e9 8999 | . . . . . . . . . 10 | |
14 | 2, 8, 1 | subadd2i 8177 | . . . . . . . . . 10 |
15 | 13, 14 | mpbir 145 | . . . . . . . . 9 |
16 | 15 | negeqi 8083 | . . . . . . . 8 |
17 | 12, 16 | eqtr3i 2187 | . . . . . . 7 |
18 | 17 | oveq1i 5846 | . . . . . 6 |
19 | 11, 18 | eqtr3i 2187 | . . . . 5 |
20 | 2, 5 | dividapi 8632 | . . . . . 6 |
21 | 20 | oveq2i 5847 | . . . . 5 |
22 | 7, 19, 21 | 3eqtr2ri 2192 | . . . 4 |
23 | ax-1cn 7837 | . . . . . . . 8 | |
24 | 8, 23, 2, 5 | divassapi 8655 | . . . . . . 7 |
25 | 2t1e2 9001 | . . . . . . . 8 | |
26 | 25 | oveq1i 5846 | . . . . . . 7 |
27 | 24, 26 | eqtr3i 2187 | . . . . . 6 |
28 | 3cn 8923 | . . . . . . . . . 10 | |
29 | 3ap0 8944 | . . . . . . . . . 10 # | |
30 | 23, 28, 29 | sqdivapi 10528 | . . . . . . . . 9 |
31 | sq1 10538 | . . . . . . . . . 10 | |
32 | sq3 10541 | . . . . . . . . . 10 | |
33 | 31, 32 | oveq12i 5848 | . . . . . . . . 9 |
34 | 30, 33 | eqtri 2185 | . . . . . . . 8 |
35 | cos1bnd 11686 | . . . . . . . . . 10 | |
36 | 35 | simpli 110 | . . . . . . . . 9 |
37 | 0le1 8370 | . . . . . . . . . . 11 | |
38 | 3pos 8942 | . . . . . . . . . . 11 | |
39 | 1re 7889 | . . . . . . . . . . . 12 | |
40 | 3re 8922 | . . . . . . . . . . . 12 | |
41 | 39, 40 | divge0i 8797 | . . . . . . . . . . 11 |
42 | 37, 38, 41 | mp2an 423 | . . . . . . . . . 10 |
43 | 0re 7890 | . . . . . . . . . . 11 | |
44 | recoscl 11648 | . . . . . . . . . . . 12 | |
45 | 39, 44 | ax-mp 5 | . . . . . . . . . . 11 |
46 | 40, 29 | rerecclapi 8664 | . . . . . . . . . . . . 13 |
47 | 43, 46, 45 | lelttri 7995 | . . . . . . . . . . . 12 |
48 | 42, 36, 47 | mp2an 423 | . . . . . . . . . . 11 |
49 | 43, 45, 48 | ltleii 7992 | . . . . . . . . . 10 |
50 | 46, 45 | lt2sqi 10532 | . . . . . . . . . 10 |
51 | 42, 49, 50 | mp2an 423 | . . . . . . . . 9 |
52 | 36, 51 | mpbi 144 | . . . . . . . 8 |
53 | 34, 52 | eqbrtrri 3999 | . . . . . . 7 |
54 | 2pos 8939 | . . . . . . . 8 | |
55 | 3, 5 | rerecclapi 8664 | . . . . . . . . 9 |
56 | 45 | resqcli 10529 | . . . . . . . . 9 |
57 | 2re 8918 | . . . . . . . . 9 | |
58 | 55, 56, 57 | ltmul2i 8809 | . . . . . . . 8 |
59 | 54, 58 | ax-mp 5 | . . . . . . 7 |
60 | 53, 59 | mpbi 144 | . . . . . 6 |
61 | 27, 60 | eqbrtrri 3999 | . . . . 5 |
62 | 57, 3, 5 | redivclapi 8666 | . . . . . 6 |
63 | 57, 56 | remulcli 7904 | . . . . . 6 |
64 | ltsub1 8347 | . . . . . 6 | |
65 | 62, 63, 39, 64 | mp3an 1326 | . . . . 5 |
66 | 61, 65 | mpbi 144 | . . . 4 |
67 | 22, 66 | eqbrtrri 3999 | . . 3 |
68 | 25 | fveq2i 5483 | . . . 4 |
69 | cos2t 11677 | . . . . 5 | |
70 | 23, 69 | ax-mp 5 | . . . 4 |
71 | 68, 70 | eqtr3i 2187 | . . 3 |
72 | 67, 71 | breqtrri 4003 | . 2 |
73 | 35 | simpri 112 | . . . . . . . . 9 |
74 | 0le2 8938 | . . . . . . . . . . 11 | |
75 | 57, 40 | divge0i 8797 | . . . . . . . . . . 11 |
76 | 74, 38, 75 | mp2an 423 | . . . . . . . . . 10 |
77 | 57, 40, 29 | redivclapi 8666 | . . . . . . . . . . 11 |
78 | 45, 77 | lt2sqi 10532 | . . . . . . . . . 10 |
79 | 49, 76, 78 | mp2an 423 | . . . . . . . . 9 |
80 | 73, 79 | mpbi 144 | . . . . . . . 8 |
81 | 8, 28, 29 | sqdivapi 10528 | . . . . . . . . 9 |
82 | sq2 10540 | . . . . . . . . . 10 | |
83 | 82, 32 | oveq12i 5848 | . . . . . . . . 9 |
84 | 81, 83 | eqtri 2185 | . . . . . . . 8 |
85 | 80, 84 | breqtri 4001 | . . . . . . 7 |
86 | 4re 8925 | . . . . . . . . . 10 | |
87 | 86, 3, 5 | redivclapi 8666 | . . . . . . . . 9 |
88 | 56, 87, 57 | ltmul2i 8809 | . . . . . . . 8 |
89 | 54, 88 | ax-mp 5 | . . . . . . 7 |
90 | 85, 89 | mpbi 144 | . . . . . 6 |
91 | 4cn 8926 | . . . . . . . 8 | |
92 | 8, 91, 2, 5 | divassapi 8655 | . . . . . . 7 |
93 | 4t2e8 9006 | . . . . . . . . 9 | |
94 | 91, 8, 93 | mulcomli 7897 | . . . . . . . 8 |
95 | 94 | oveq1i 5846 | . . . . . . 7 |
96 | 92, 95 | eqtr3i 2187 | . . . . . 6 |
97 | 90, 96 | breqtri 4001 | . . . . 5 |
98 | 8re 8933 | . . . . . . 7 | |
99 | 98, 3, 5 | redivclapi 8666 | . . . . . 6 |
100 | ltsub1 8347 | . . . . . 6 | |
101 | 63, 99, 39, 100 | mp3an 1326 | . . . . 5 |
102 | 97, 101 | mpbi 144 | . . . 4 |
103 | 20 | oveq2i 5847 | . . . . 5 |
104 | divnegap 8593 | . . . . . . 7 # | |
105 | 23, 2, 5, 104 | mp3an 1326 | . . . . . 6 |
106 | 8cn 8934 | . . . . . . . . 9 | |
107 | 2, 106 | negsubdi2i 8175 | . . . . . . . 8 |
108 | 8p1e9 8988 | . . . . . . . . . 10 | |
109 | 2, 106, 23, 108 | subaddrii 8178 | . . . . . . . . 9 |
110 | 109 | negeqi 8083 | . . . . . . . 8 |
111 | 107, 110 | eqtr3i 2187 | . . . . . . 7 |
112 | 111 | oveq1i 5846 | . . . . . 6 |
113 | divsubdirap 8595 | . . . . . . 7 # | |
114 | 106, 2, 9, 113 | mp3an 1326 | . . . . . 6 |
115 | 105, 112, 114 | 3eqtr2ri 2192 | . . . . 5 |
116 | 103, 115 | eqtr3i 2187 | . . . 4 |
117 | 102, 116 | breqtri 4001 | . . 3 |
118 | 71, 117 | eqbrtri 3997 | . 2 |
119 | 72, 118 | pm3.2i 270 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 wcel 2135 class class class wbr 3976 cfv 5182 (class class class)co 5836 cc 7742 cr 7743 cc0 7744 c1 7745 caddc 7747 cmul 7749 clt 7924 cle 7925 cmin 8060 cneg 8061 # cap 8470 cdiv 8559 c2 8899 c3 8900 c4 8901 c7 8904 c8 8905 c9 8906 cexp 10444 ccos 11572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-disj 3954 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-sup 6940 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-ioc 9820 df-ico 9821 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-fac 10628 df-bc 10650 df-ihash 10678 df-shft 10743 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 df-ef 11575 df-sin 11577 df-cos 11578 |
This theorem is referenced by: sincos2sgn 11692 |
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