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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 8772 | . . . . . 6 | |
2 | 9cn 8776 | . . . . . 6 | |
3 | 9re 8775 | . . . . . . 7 | |
4 | 9pos 8792 | . . . . . . 7 | |
5 | 3, 4 | gt0ap0ii 8358 | . . . . . 6 # |
6 | divnegap 8434 | . . . . . 6 # | |
7 | 1, 2, 5, 6 | mp3an 1300 | . . . . 5 |
8 | 2cn 8759 | . . . . . . 7 | |
9 | 2, 5 | pm3.2i 270 | . . . . . . 7 # |
10 | divsubdirap 8436 | . . . . . . 7 # | |
11 | 8, 2, 9, 10 | mp3an 1300 | . . . . . 6 |
12 | 2, 8 | negsubdi2i 8016 | . . . . . . . 8 |
13 | 7p2e9 8839 | . . . . . . . . . 10 | |
14 | 2, 8, 1 | subadd2i 8018 | . . . . . . . . . 10 |
15 | 13, 14 | mpbir 145 | . . . . . . . . 9 |
16 | 15 | negeqi 7924 | . . . . . . . 8 |
17 | 12, 16 | eqtr3i 2140 | . . . . . . 7 |
18 | 17 | oveq1i 5752 | . . . . . 6 |
19 | 11, 18 | eqtr3i 2140 | . . . . 5 |
20 | 2, 5 | dividapi 8473 | . . . . . 6 |
21 | 20 | oveq2i 5753 | . . . . 5 |
22 | 7, 19, 21 | 3eqtr2ri 2145 | . . . 4 |
23 | ax-1cn 7681 | . . . . . . . 8 | |
24 | 8, 23, 2, 5 | divassapi 8496 | . . . . . . 7 |
25 | 2t1e2 8841 | . . . . . . . 8 | |
26 | 25 | oveq1i 5752 | . . . . . . 7 |
27 | 24, 26 | eqtr3i 2140 | . . . . . 6 |
28 | 3cn 8763 | . . . . . . . . . 10 | |
29 | 3ap0 8784 | . . . . . . . . . 10 # | |
30 | 23, 28, 29 | sqdivapi 10344 | . . . . . . . . 9 |
31 | sq1 10354 | . . . . . . . . . 10 | |
32 | sq3 10357 | . . . . . . . . . 10 | |
33 | 31, 32 | oveq12i 5754 | . . . . . . . . 9 |
34 | 30, 33 | eqtri 2138 | . . . . . . . 8 |
35 | cos1bnd 11393 | . . . . . . . . . 10 | |
36 | 35 | simpli 110 | . . . . . . . . 9 |
37 | 0le1 8211 | . . . . . . . . . . 11 | |
38 | 3pos 8782 | . . . . . . . . . . 11 | |
39 | 1re 7733 | . . . . . . . . . . . 12 | |
40 | 3re 8762 | . . . . . . . . . . . 12 | |
41 | 39, 40 | divge0i 8637 | . . . . . . . . . . 11 |
42 | 37, 38, 41 | mp2an 422 | . . . . . . . . . 10 |
43 | 0re 7734 | . . . . . . . . . . 11 | |
44 | recoscl 11355 | . . . . . . . . . . . 12 | |
45 | 39, 44 | ax-mp 5 | . . . . . . . . . . 11 |
46 | 40, 29 | rerecclapi 8505 | . . . . . . . . . . . . 13 |
47 | 43, 46, 45 | lelttri 7837 | . . . . . . . . . . . 12 |
48 | 42, 36, 47 | mp2an 422 | . . . . . . . . . . 11 |
49 | 43, 45, 48 | ltleii 7834 | . . . . . . . . . 10 |
50 | 46, 45 | lt2sqi 10348 | . . . . . . . . . 10 |
51 | 42, 49, 50 | mp2an 422 | . . . . . . . . 9 |
52 | 36, 51 | mpbi 144 | . . . . . . . 8 |
53 | 34, 52 | eqbrtrri 3921 | . . . . . . 7 |
54 | 2pos 8779 | . . . . . . . 8 | |
55 | 3, 5 | rerecclapi 8505 | . . . . . . . . 9 |
56 | 45 | resqcli 10345 | . . . . . . . . 9 |
57 | 2re 8758 | . . . . . . . . 9 | |
58 | 55, 56, 57 | ltmul2i 8649 | . . . . . . . 8 |
59 | 54, 58 | ax-mp 5 | . . . . . . 7 |
60 | 53, 59 | mpbi 144 | . . . . . 6 |
61 | 27, 60 | eqbrtrri 3921 | . . . . 5 |
62 | 57, 3, 5 | redivclapi 8507 | . . . . . 6 |
63 | 57, 56 | remulcli 7748 | . . . . . 6 |
64 | ltsub1 8188 | . . . . . 6 | |
65 | 62, 63, 39, 64 | mp3an 1300 | . . . . 5 |
66 | 61, 65 | mpbi 144 | . . . 4 |
67 | 22, 66 | eqbrtrri 3921 | . . 3 |
68 | 25 | fveq2i 5392 | . . . 4 |
69 | cos2t 11384 | . . . . 5 | |
70 | 23, 69 | ax-mp 5 | . . . 4 |
71 | 68, 70 | eqtr3i 2140 | . . 3 |
72 | 67, 71 | breqtrri 3925 | . 2 |
73 | 35 | simpri 112 | . . . . . . . . 9 |
74 | 0le2 8778 | . . . . . . . . . . 11 | |
75 | 57, 40 | divge0i 8637 | . . . . . . . . . . 11 |
76 | 74, 38, 75 | mp2an 422 | . . . . . . . . . 10 |
77 | 57, 40, 29 | redivclapi 8507 | . . . . . . . . . . 11 |
78 | 45, 77 | lt2sqi 10348 | . . . . . . . . . 10 |
79 | 49, 76, 78 | mp2an 422 | . . . . . . . . 9 |
80 | 73, 79 | mpbi 144 | . . . . . . . 8 |
81 | 8, 28, 29 | sqdivapi 10344 | . . . . . . . . 9 |
82 | sq2 10356 | . . . . . . . . . 10 | |
83 | 82, 32 | oveq12i 5754 | . . . . . . . . 9 |
84 | 81, 83 | eqtri 2138 | . . . . . . . 8 |
85 | 80, 84 | breqtri 3923 | . . . . . . 7 |
86 | 4re 8765 | . . . . . . . . . 10 | |
87 | 86, 3, 5 | redivclapi 8507 | . . . . . . . . 9 |
88 | 56, 87, 57 | ltmul2i 8649 | . . . . . . . 8 |
89 | 54, 88 | ax-mp 5 | . . . . . . 7 |
90 | 85, 89 | mpbi 144 | . . . . . 6 |
91 | 4cn 8766 | . . . . . . . 8 | |
92 | 8, 91, 2, 5 | divassapi 8496 | . . . . . . 7 |
93 | 4t2e8 8846 | . . . . . . . . 9 | |
94 | 91, 8, 93 | mulcomli 7741 | . . . . . . . 8 |
95 | 94 | oveq1i 5752 | . . . . . . 7 |
96 | 92, 95 | eqtr3i 2140 | . . . . . 6 |
97 | 90, 96 | breqtri 3923 | . . . . 5 |
98 | 8re 8773 | . . . . . . 7 | |
99 | 98, 3, 5 | redivclapi 8507 | . . . . . 6 |
100 | ltsub1 8188 | . . . . . 6 | |
101 | 63, 99, 39, 100 | mp3an 1300 | . . . . 5 |
102 | 97, 101 | mpbi 144 | . . . 4 |
103 | 20 | oveq2i 5753 | . . . . 5 |
104 | divnegap 8434 | . . . . . . 7 # | |
105 | 23, 2, 5, 104 | mp3an 1300 | . . . . . 6 |
106 | 8cn 8774 | . . . . . . . . 9 | |
107 | 2, 106 | negsubdi2i 8016 | . . . . . . . 8 |
108 | 8p1e9 8828 | . . . . . . . . . 10 | |
109 | 2, 106, 23, 108 | subaddrii 8019 | . . . . . . . . 9 |
110 | 109 | negeqi 7924 | . . . . . . . 8 |
111 | 107, 110 | eqtr3i 2140 | . . . . . . 7 |
112 | 111 | oveq1i 5752 | . . . . . 6 |
113 | divsubdirap 8436 | . . . . . . 7 # | |
114 | 106, 2, 9, 113 | mp3an 1300 | . . . . . 6 |
115 | 105, 112, 114 | 3eqtr2ri 2145 | . . . . 5 |
116 | 103, 115 | eqtr3i 2140 | . . . 4 |
117 | 102, 116 | breqtri 3923 | . . 3 |
118 | 71, 117 | eqbrtri 3919 | . 2 |
119 | 72, 118 | pm3.2i 270 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 wcel 1465 class class class wbr 3899 cfv 5093 (class class class)co 5742 cc 7586 cr 7587 cc0 7588 c1 7589 caddc 7591 cmul 7593 clt 7768 cle 7769 cmin 7901 cneg 7902 # cap 8311 cdiv 8400 c2 8739 c3 8740 c4 8741 c7 8744 c8 8745 c9 8746 cexp 10260 ccos 11278 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-disj 3877 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-isom 5102 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-frec 6256 df-1o 6281 df-oadd 6285 df-er 6397 df-en 6603 df-dom 6604 df-fin 6605 df-sup 6839 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 df-8 8753 df-9 8754 df-n0 8946 df-z 9023 df-uz 9295 df-q 9380 df-rp 9410 df-ioc 9644 df-ico 9645 df-fz 9759 df-fzo 9888 df-seqfrec 10187 df-exp 10261 df-fac 10440 df-bc 10462 df-ihash 10490 df-shft 10555 df-cj 10582 df-re 10583 df-im 10584 df-rsqrt 10738 df-abs 10739 df-clim 11016 df-sumdc 11091 df-ef 11281 df-sin 11283 df-cos 11284 |
This theorem is referenced by: sincos2sgn 11399 |
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