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Mirrors > Home > ILE Home > Th. List > cos01bnd | Unicode version |
Description: Bounds on the cosine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
cos01bnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7924 | . . . . . . . . 9 | |
2 | 1re 7877 | . . . . . . . . 9 | |
3 | elioc2 9840 | . . . . . . . . 9 | |
4 | 1, 2, 3 | mp2an 423 | . . . . . . . 8 |
5 | 4 | simp1bi 997 | . . . . . . 7 |
6 | eqid 2157 | . . . . . . . 8 | |
7 | 6 | recos4p 11616 | . . . . . . 7 |
8 | 5, 7 | syl 14 | . . . . . 6 |
9 | 8 | eqcomd 2163 | . . . . 5 |
10 | 5 | recoscld 11621 | . . . . . . 7 |
11 | 10 | recnd 7906 | . . . . . 6 |
12 | 5 | resqcld 10577 | . . . . . . . . 9 |
13 | 12 | rehalfcld 9079 | . . . . . . . 8 |
14 | resubcl 8139 | . . . . . . . 8 | |
15 | 2, 13, 14 | sylancr 411 | . . . . . . 7 |
16 | 15 | recnd 7906 | . . . . . 6 |
17 | ax-icn 7827 | . . . . . . . . . 10 | |
18 | 5 | recnd 7906 | . . . . . . . . . 10 |
19 | mulcl 7859 | . . . . . . . . . 10 | |
20 | 17, 18, 19 | sylancr 411 | . . . . . . . . 9 |
21 | 4nn0 9109 | . . . . . . . . 9 | |
22 | 6 | eftlcl 11585 | . . . . . . . . 9 |
23 | 20, 21, 22 | sylancl 410 | . . . . . . . 8 |
24 | 23 | recld 10838 | . . . . . . 7 |
25 | 24 | recnd 7906 | . . . . . 6 |
26 | 11, 16, 25 | subaddd 8204 | . . . . 5 |
27 | 9, 26 | mpbird 166 | . . . 4 |
28 | 27 | fveq2d 5472 | . . 3 |
29 | 25 | abscld 11081 | . . . 4 |
30 | 23 | abscld 11081 | . . . 4 |
31 | 6nn 8998 | . . . . 5 | |
32 | nndivre 8869 | . . . . 5 | |
33 | 12, 31, 32 | sylancl 410 | . . . 4 |
34 | absrele 10983 | . . . . 5 | |
35 | 23, 34 | syl 14 | . . . 4 |
36 | reexpcl 10436 | . . . . . . 7 | |
37 | 5, 21, 36 | sylancl 410 | . . . . . 6 |
38 | nndivre 8869 | . . . . . 6 | |
39 | 37, 31, 38 | sylancl 410 | . . . . 5 |
40 | 6 | ef01bndlem 11653 | . . . . 5 |
41 | 2nn0 9107 | . . . . . . . 8 | |
42 | 41 | a1i 9 | . . . . . . 7 |
43 | 4z 9197 | . . . . . . . . 9 | |
44 | 2re 8903 | . . . . . . . . . 10 | |
45 | 4re 8910 | . . . . . . . . . 10 | |
46 | 2lt4 9006 | . . . . . . . . . 10 | |
47 | 44, 45, 46 | ltleii 7979 | . . . . . . . . 9 |
48 | 2z 9195 | . . . . . . . . . 10 | |
49 | 48 | eluz1i 9446 | . . . . . . . . 9 |
50 | 43, 47, 49 | mpbir2an 927 | . . . . . . . 8 |
51 | 50 | a1i 9 | . . . . . . 7 |
52 | 4 | simp2bi 998 | . . . . . . . 8 |
53 | 0re 7878 | . . . . . . . . 9 | |
54 | ltle 7964 | . . . . . . . . 9 | |
55 | 53, 5, 54 | sylancr 411 | . . . . . . . 8 |
56 | 52, 55 | mpd 13 | . . . . . . 7 |
57 | 4 | simp3bi 999 | . . . . . . 7 |
58 | 5, 42, 51, 56, 57 | leexp2rd 10581 | . . . . . 6 |
59 | 6re 8914 | . . . . . . . 8 | |
60 | 59 | a1i 9 | . . . . . . 7 |
61 | 6pos 8934 | . . . . . . . 8 | |
62 | 61 | a1i 9 | . . . . . . 7 |
63 | lediv1 8740 | . . . . . . 7 | |
64 | 37, 12, 60, 62, 63 | syl112anc 1224 | . . . . . 6 |
65 | 58, 64 | mpbid 146 | . . . . 5 |
66 | 30, 39, 33, 40, 65 | ltletrd 8298 | . . . 4 |
67 | 29, 30, 33, 35, 66 | lelttrd 8000 | . . 3 |
68 | 28, 67 | eqbrtrd 3986 | . 2 |
69 | 10, 15, 33 | absdifltd 11078 | . . 3 |
70 | 1cnd 7894 | . . . . . . 7 | |
71 | 13 | recnd 7906 | . . . . . . 7 |
72 | 33 | recnd 7906 | . . . . . . 7 |
73 | 70, 71, 72 | subsub4d 8217 | . . . . . 6 |
74 | halfpm6th 9053 | . . . . . . . . . . 11 | |
75 | 74 | simpri 112 | . . . . . . . . . 10 |
76 | 75 | oveq2i 5835 | . . . . . . . . 9 |
77 | 12 | recnd 7906 | . . . . . . . . . 10 |
78 | 2cn 8904 | . . . . . . . . . . . 12 | |
79 | 2ap0 8926 | . . . . . . . . . . . 12 # | |
80 | 78, 79 | recclapi 8615 | . . . . . . . . . . 11 |
81 | 6cn 8915 | . . . . . . . . . . . 12 | |
82 | 31 | nnap0i 8864 | . . . . . . . . . . . 12 # |
83 | 81, 82 | recclapi 8615 | . . . . . . . . . . 11 |
84 | adddi 7864 | . . . . . . . . . . 11 | |
85 | 80, 83, 84 | mp3an23 1311 | . . . . . . . . . 10 |
86 | 77, 85 | syl 14 | . . . . . . . . 9 |
87 | 76, 86 | eqtr3id 2204 | . . . . . . . 8 |
88 | 3cn 8908 | . . . . . . . . . . 11 | |
89 | 3ap0 8929 | . . . . . . . . . . 11 # | |
90 | 88, 89 | pm3.2i 270 | . . . . . . . . . 10 # |
91 | div12ap 8567 | . . . . . . . . . 10 # | |
92 | 78, 90, 91 | mp3an13 1310 | . . . . . . . . 9 |
93 | 77, 92 | syl 14 | . . . . . . . 8 |
94 | divrecap 8561 | . . . . . . . . . . 11 # | |
95 | 78, 79, 94 | mp3an23 1311 | . . . . . . . . . 10 |
96 | 77, 95 | syl 14 | . . . . . . . . 9 |
97 | divrecap 8561 | . . . . . . . . . . 11 # | |
98 | 81, 82, 97 | mp3an23 1311 | . . . . . . . . . 10 |
99 | 77, 98 | syl 14 | . . . . . . . . 9 |
100 | 96, 99 | oveq12d 5842 | . . . . . . . 8 |
101 | 87, 93, 100 | 3eqtr4rd 2201 | . . . . . . 7 |
102 | 101 | oveq2d 5840 | . . . . . 6 |
103 | 73, 102 | eqtrd 2190 | . . . . 5 |
104 | 103 | breq1d 3975 | . . . 4 |
105 | 70, 71, 72 | subsubd 8214 | . . . . . 6 |
106 | 74 | simpli 110 | . . . . . . . . . 10 |
107 | 106 | oveq2i 5835 | . . . . . . . . 9 |
108 | subdi 8260 | . . . . . . . . . . 11 | |
109 | 80, 83, 108 | mp3an23 1311 | . . . . . . . . . 10 |
110 | 77, 109 | syl 14 | . . . . . . . . 9 |
111 | 107, 110 | eqtr3id 2204 | . . . . . . . 8 |
112 | divrecap 8561 | . . . . . . . . . 10 # | |
113 | 88, 89, 112 | mp3an23 1311 | . . . . . . . . 9 |
114 | 77, 113 | syl 14 | . . . . . . . 8 |
115 | 96, 99 | oveq12d 5842 | . . . . . . . 8 |
116 | 111, 114, 115 | 3eqtr4rd 2201 | . . . . . . 7 |
117 | 116 | oveq2d 5840 | . . . . . 6 |
118 | 105, 117 | eqtr3d 2192 | . . . . 5 |
119 | 118 | breq2d 3977 | . . . 4 |
120 | 104, 119 | anbi12d 465 | . . 3 |
121 | 69, 120 | bitrd 187 | . 2 |
122 | 68, 121 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 class class class wbr 3965 cmpt 4025 cfv 5170 (class class class)co 5824 cc 7730 cr 7731 cc0 7732 c1 7733 ci 7734 caddc 7735 cmul 7737 cxr 7911 clt 7912 cle 7913 cmin 8046 # cap 8456 cdiv 8545 cn 8833 c2 8884 c3 8885 c4 8886 c6 8888 cn0 9090 cz 9167 cuz 9439 cioc 9793 cexp 10418 cfa 10599 cre 10740 cabs 10897 csu 11250 ccos 11542 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-isom 5179 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-frec 6338 df-1o 6363 df-oadd 6367 df-er 6480 df-en 6686 df-dom 6687 df-fin 6688 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-5 8895 df-6 8896 df-7 8897 df-8 8898 df-n0 9091 df-z 9168 df-uz 9440 df-q 9529 df-rp 9561 df-ioc 9797 df-ico 9798 df-fz 9913 df-fzo 10042 df-seqfrec 10345 df-exp 10419 df-fac 10600 df-ihash 10650 df-shft 10715 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899 df-clim 11176 df-sumdc 11251 df-ef 11545 df-cos 11548 |
This theorem is referenced by: cos1bnd 11656 cos01gt0 11659 tangtx 13170 |
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