Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7780 | . . . . . . . . 9 | |
2 | 1re 7733 | . . . . . . . . 9 | |
3 | elioc2 9687 | . . . . . . . . 9 | |
4 | 1, 2, 3 | mp2an 422 | . . . . . . . 8 |
5 | 4 | simp1bi 981 | . . . . . . 7 |
6 | eqid 2117 | . . . . . . . 8 | |
7 | 6 | recos4p 11353 | . . . . . . 7 |
8 | 5, 7 | syl 14 | . . . . . 6 |
9 | 8 | eqcomd 2123 | . . . . 5 |
10 | 5 | recoscld 11358 | . . . . . . 7 |
11 | 10 | recnd 7762 | . . . . . 6 |
12 | 5 | resqcld 10418 | . . . . . . . . 9 |
13 | 12 | rehalfcld 8934 | . . . . . . . 8 |
14 | resubcl 7994 | . . . . . . . 8 | |
15 | 2, 13, 14 | sylancr 410 | . . . . . . 7 |
16 | 15 | recnd 7762 | . . . . . 6 |
17 | ax-icn 7683 | . . . . . . . . . 10 | |
18 | 5 | recnd 7762 | . . . . . . . . . 10 |
19 | mulcl 7715 | . . . . . . . . . 10 | |
20 | 17, 18, 19 | sylancr 410 | . . . . . . . . 9 |
21 | 4nn0 8964 | . . . . . . . . 9 | |
22 | 6 | eftlcl 11321 | . . . . . . . . 9 |
23 | 20, 21, 22 | sylancl 409 | . . . . . . . 8 |
24 | 23 | recld 10678 | . . . . . . 7 |
25 | 24 | recnd 7762 | . . . . . 6 |
26 | 11, 16, 25 | subaddd 8059 | . . . . 5 |
27 | 9, 26 | mpbird 166 | . . . 4 |
28 | 27 | fveq2d 5393 | . . 3 |
29 | 25 | abscld 10921 | . . . 4 |
30 | 23 | abscld 10921 | . . . 4 |
31 | 6nn 8853 | . . . . 5 | |
32 | nndivre 8724 | . . . . 5 | |
33 | 12, 31, 32 | sylancl 409 | . . . 4 |
34 | absrele 10823 | . . . . 5 | |
35 | 23, 34 | syl 14 | . . . 4 |
36 | reexpcl 10278 | . . . . . . 7 | |
37 | 5, 21, 36 | sylancl 409 | . . . . . 6 |
38 | nndivre 8724 | . . . . . 6 | |
39 | 37, 31, 38 | sylancl 409 | . . . . 5 |
40 | 6 | ef01bndlem 11390 | . . . . 5 |
41 | 2nn0 8962 | . . . . . . . 8 | |
42 | 41 | a1i 9 | . . . . . . 7 |
43 | 4z 9052 | . . . . . . . . 9 | |
44 | 2re 8758 | . . . . . . . . . 10 | |
45 | 4re 8765 | . . . . . . . . . 10 | |
46 | 2lt4 8861 | . . . . . . . . . 10 | |
47 | 44, 45, 46 | ltleii 7834 | . . . . . . . . 9 |
48 | 2z 9050 | . . . . . . . . . 10 | |
49 | 48 | eluz1i 9301 | . . . . . . . . 9 |
50 | 43, 47, 49 | mpbir2an 911 | . . . . . . . 8 |
51 | 50 | a1i 9 | . . . . . . 7 |
52 | 4 | simp2bi 982 | . . . . . . . 8 |
53 | 0re 7734 | . . . . . . . . 9 | |
54 | ltle 7819 | . . . . . . . . 9 | |
55 | 53, 5, 54 | sylancr 410 | . . . . . . . 8 |
56 | 52, 55 | mpd 13 | . . . . . . 7 |
57 | 4 | simp3bi 983 | . . . . . . 7 |
58 | 5, 42, 51, 56, 57 | leexp2rd 10422 | . . . . . 6 |
59 | 6re 8769 | . . . . . . . 8 | |
60 | 59 | a1i 9 | . . . . . . 7 |
61 | 6pos 8789 | . . . . . . . 8 | |
62 | 61 | a1i 9 | . . . . . . 7 |
63 | lediv1 8595 | . . . . . . 7 | |
64 | 37, 12, 60, 62, 63 | syl112anc 1205 | . . . . . 6 |
65 | 58, 64 | mpbid 146 | . . . . 5 |
66 | 30, 39, 33, 40, 65 | ltletrd 8153 | . . . 4 |
67 | 29, 30, 33, 35, 66 | lelttrd 7855 | . . 3 |
68 | 28, 67 | eqbrtrd 3920 | . 2 |
69 | 10, 15, 33 | absdifltd 10918 | . . 3 |
70 | 1cnd 7750 | . . . . . . 7 | |
71 | 13 | recnd 7762 | . . . . . . 7 |
72 | 33 | recnd 7762 | . . . . . . 7 |
73 | 70, 71, 72 | subsub4d 8072 | . . . . . 6 |
74 | halfpm6th 8908 | . . . . . . . . . . 11 | |
75 | 74 | simpri 112 | . . . . . . . . . 10 |
76 | 75 | oveq2i 5753 | . . . . . . . . 9 |
77 | 12 | recnd 7762 | . . . . . . . . . 10 |
78 | 2cn 8759 | . . . . . . . . . . . 12 | |
79 | 2ap0 8781 | . . . . . . . . . . . 12 # | |
80 | 78, 79 | recclapi 8470 | . . . . . . . . . . 11 |
81 | 6cn 8770 | . . . . . . . . . . . 12 | |
82 | 31 | nnap0i 8719 | . . . . . . . . . . . 12 # |
83 | 81, 82 | recclapi 8470 | . . . . . . . . . . 11 |
84 | adddi 7720 | . . . . . . . . . . 11 | |
85 | 80, 83, 84 | mp3an23 1292 | . . . . . . . . . 10 |
86 | 77, 85 | syl 14 | . . . . . . . . 9 |
87 | 76, 86 | syl5eqr 2164 | . . . . . . . 8 |
88 | 3cn 8763 | . . . . . . . . . . 11 | |
89 | 3ap0 8784 | . . . . . . . . . . 11 # | |
90 | 88, 89 | pm3.2i 270 | . . . . . . . . . 10 # |
91 | div12ap 8422 | . . . . . . . . . 10 # | |
92 | 78, 90, 91 | mp3an13 1291 | . . . . . . . . 9 |
93 | 77, 92 | syl 14 | . . . . . . . 8 |
94 | divrecap 8416 | . . . . . . . . . . 11 # | |
95 | 78, 79, 94 | mp3an23 1292 | . . . . . . . . . 10 |
96 | 77, 95 | syl 14 | . . . . . . . . 9 |
97 | divrecap 8416 | . . . . . . . . . . 11 # | |
98 | 81, 82, 97 | mp3an23 1292 | . . . . . . . . . 10 |
99 | 77, 98 | syl 14 | . . . . . . . . 9 |
100 | 96, 99 | oveq12d 5760 | . . . . . . . 8 |
101 | 87, 93, 100 | 3eqtr4rd 2161 | . . . . . . 7 |
102 | 101 | oveq2d 5758 | . . . . . 6 |
103 | 73, 102 | eqtrd 2150 | . . . . 5 |
104 | 103 | breq1d 3909 | . . . 4 |
105 | 70, 71, 72 | subsubd 8069 | . . . . . 6 |
106 | 74 | simpli 110 | . . . . . . . . . 10 |
107 | 106 | oveq2i 5753 | . . . . . . . . 9 |
108 | subdi 8115 | . . . . . . . . . . 11 | |
109 | 80, 83, 108 | mp3an23 1292 | . . . . . . . . . 10 |
110 | 77, 109 | syl 14 | . . . . . . . . 9 |
111 | 107, 110 | syl5eqr 2164 | . . . . . . . 8 |
112 | divrecap 8416 | . . . . . . . . . 10 # | |
113 | 88, 89, 112 | mp3an23 1292 | . . . . . . . . 9 |
114 | 77, 113 | syl 14 | . . . . . . . 8 |
115 | 96, 99 | oveq12d 5760 | . . . . . . . 8 |
116 | 111, 114, 115 | 3eqtr4rd 2161 | . . . . . . 7 |
117 | 116 | oveq2d 5758 | . . . . . 6 |
118 | 105, 117 | eqtr3d 2152 | . . . . 5 |
119 | 118 | breq2d 3911 | . . . 4 |
120 | 104, 119 | anbi12d 464 | . . 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 947 wceq 1316 wcel 1465 class class class wbr 3899 cmpt 3959 cfv 5093 (class class class)co 5742 cc 7586 cr 7587 cc0 7588 c1 7589 ci 7590 caddc 7591 cmul 7593 cxr 7767 clt 7768 cle 7769 cmin 7901 # cap 8311 cdiv 8400 cn 8688 c2 8739 c3 8740 c4 8741 c6 8743 cn0 8945 cz 9022 cuz 9294 cioc 9640 cexp 10260 cfa 10439 cre 10580 cabs 10737 csu 11090 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-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-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-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-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-cos 11284 |
This theorem is referenced by: cos1bnd 11393 cos01gt0 11396 tangtx 12846 |
Copyright terms: Public domain | W3C validator |