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Mirrors > Home > ILE Home > Th. List > cos01gt0 | Unicode version |
Description: The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
cos01gt0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7955 | . . . . . . . . . 10 | |
2 | 1re 7908 | . . . . . . . . . 10 | |
3 | elioc2 9882 | . . . . . . . . . 10 | |
4 | 1, 2, 3 | mp2an 424 | . . . . . . . . 9 |
5 | 4 | simp1bi 1007 | . . . . . . . 8 |
6 | 5 | resqcld 10624 | . . . . . . 7 |
7 | 6 | recnd 7937 | . . . . . 6 |
8 | 2cn 8938 | . . . . . . 7 | |
9 | 3cn 8942 | . . . . . . . 8 | |
10 | 3ap0 8963 | . . . . . . . 8 # | |
11 | 9, 10 | pm3.2i 270 | . . . . . . 7 # |
12 | div12ap 8600 | . . . . . . 7 # | |
13 | 8, 11, 12 | mp3an13 1323 | . . . . . 6 |
14 | 7, 13 | syl 14 | . . . . 5 |
15 | 2z 9229 | . . . . . . . . . 10 | |
16 | expgt0 10498 | . . . . . . . . . 10 | |
17 | 15, 16 | mp3an2 1320 | . . . . . . . . 9 |
18 | 17 | 3adant3 1012 | . . . . . . . 8 |
19 | 4, 18 | sylbi 120 | . . . . . . 7 |
20 | 2lt3 9037 | . . . . . . . . . 10 | |
21 | 2re 8937 | . . . . . . . . . . 11 | |
22 | 3re 8941 | . . . . . . . . . . 11 | |
23 | 3pos 8961 | . . . . . . . . . . 11 | |
24 | 21, 22, 22, 23 | ltdiv1ii 8834 | . . . . . . . . . 10 |
25 | 20, 24 | mpbi 144 | . . . . . . . . 9 |
26 | 9, 10 | dividapi 8651 | . . . . . . . . 9 |
27 | 25, 26 | breqtri 4012 | . . . . . . . 8 |
28 | 21, 22, 10 | redivclapi 8685 | . . . . . . . . 9 |
29 | ltmul2 8761 | . . . . . . . . 9 | |
30 | 28, 2, 29 | mp3an12 1322 | . . . . . . . 8 |
31 | 27, 30 | mpbii 147 | . . . . . . 7 |
32 | 6, 19, 31 | syl2anc 409 | . . . . . 6 |
33 | 7 | mulid1d 7926 | . . . . . 6 |
34 | 32, 33 | breqtrd 4013 | . . . . 5 |
35 | 14, 34 | eqbrtrd 4009 | . . . 4 |
36 | 0re 7909 | . . . . . . . . 9 | |
37 | ltle 7996 | . . . . . . . . 9 | |
38 | 36, 37 | mpan 422 | . . . . . . . 8 |
39 | 38 | imdistani 443 | . . . . . . 7 |
40 | le2sq2 10540 | . . . . . . . 8 | |
41 | 2, 40 | mpanr1 435 | . . . . . . 7 |
42 | 39, 41 | stoic3 1424 | . . . . . 6 |
43 | 4, 42 | sylbi 120 | . . . . 5 |
44 | sq1 10558 | . . . . 5 | |
45 | 43, 44 | breqtrdi 4028 | . . . 4 |
46 | redivclap 8637 | . . . . . . . 8 # | |
47 | 22, 10, 46 | mp3an23 1324 | . . . . . . 7 |
48 | 6, 47 | syl 14 | . . . . . 6 |
49 | remulcl 7891 | . . . . . 6 | |
50 | 21, 48, 49 | sylancr 412 | . . . . 5 |
51 | ltletr 7998 | . . . . . 6 | |
52 | 2, 51 | mp3an3 1321 | . . . . 5 |
53 | 50, 6, 52 | syl2anc 409 | . . . 4 |
54 | 35, 45, 53 | mp2and 431 | . . 3 |
55 | posdif 8363 | . . . 4 | |
56 | 50, 2, 55 | sylancl 411 | . . 3 |
57 | 54, 56 | mpbid 146 | . 2 |
58 | cos01bnd 11710 | . . 3 | |
59 | 58 | simpld 111 | . 2 |
60 | resubcl 8172 | . . . 4 | |
61 | 2, 50, 60 | sylancr 412 | . . 3 |
62 | 5 | recoscld 11676 | . . 3 |
63 | lttr 7982 | . . 3 | |
64 | 36, 61, 62, 63 | mp3an2i 1337 | . 2 |
65 | 57, 59, 64 | mp2and 431 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 class class class wbr 3987 cfv 5196 (class class class)co 5851 cc 7761 cr 7762 cc0 7763 c1 7764 cmul 7768 cxr 7942 clt 7943 cle 7944 cmin 8079 # cap 8489 cdiv 8578 c2 8918 c3 8919 cz 9201 cioc 9835 cexp 10464 ccos 11597 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 ax-pre-mulext 7881 ax-arch 7882 ax-caucvg 7883 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-frec 6368 df-1o 6393 df-oadd 6397 df-er 6510 df-en 6716 df-dom 6717 df-fin 6718 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 df-div 8579 df-inn 8868 df-2 8926 df-3 8927 df-4 8928 df-5 8929 df-6 8930 df-7 8931 df-8 8932 df-n0 9125 df-z 9202 df-uz 9477 df-q 9568 df-rp 9600 df-ioc 9839 df-ico 9840 df-fz 9955 df-fzo 10088 df-seqfrec 10391 df-exp 10465 df-fac 10649 df-ihash 10699 df-shft 10768 df-cj 10795 df-re 10796 df-im 10797 df-rsqrt 10951 df-abs 10952 df-clim 11231 df-sumdc 11306 df-ef 11600 df-cos 11603 |
This theorem is referenced by: sin02gt0 11715 sincos1sgn 11716 tangtx 13514 |
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