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Mirrors > Home > ILE Home > Th. List > cos02pilt1 | Unicode version |
Description: Cosine is less than one between zero and . (Contributed by Jim Kingdon, 19-Mar-2024.) |
Ref | Expression |
---|---|
cos02pilt1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioore 9862 | . . . . . . 7 | |
2 | 1 | adantr 274 | . . . . . 6 |
3 | 2 | adantr 274 | . . . . 5 |
4 | pire 13466 | . . . . . . 7 | |
5 | 4 | a1i 9 | . . . . . 6 |
6 | simpr 109 | . . . . . 6 | |
7 | 5, 3, 6 | ltled 8031 | . . . . 5 |
8 | 0xr 7959 | . . . . . . . 8 | |
9 | 2re 8941 | . . . . . . . . . 10 | |
10 | 9, 4 | remulcli 7927 | . . . . . . . . 9 |
11 | 10 | rexri 7970 | . . . . . . . 8 |
12 | elioo2 9871 | . . . . . . . 8 | |
13 | 8, 11, 12 | mp2an 424 | . . . . . . 7 |
14 | 13 | simp3bi 1009 | . . . . . 6 |
15 | 14 | ad2antrr 485 | . . . . 5 |
16 | elico2 9887 | . . . . . 6 | |
17 | 4, 11, 16 | mp2an 424 | . . . . 5 |
18 | 3, 7, 15, 17 | syl3anbrc 1176 | . . . 4 |
19 | cosq34lt1 13530 | . . . 4 | |
20 | 18, 19 | syl 14 | . . 3 |
21 | 2 | adantr 274 | . . . . 5 |
22 | simplr 525 | . . . . 5 | |
23 | simpr 109 | . . . . 5 | |
24 | halfpire 13472 | . . . . . . 7 | |
25 | 24 | rexri 7970 | . . . . . 6 |
26 | 3re 8945 | . . . . . . . 8 | |
27 | 26, 24 | remulcli 7927 | . . . . . . 7 |
28 | 27 | rexri 7970 | . . . . . 6 |
29 | elioo2 9871 | . . . . . 6 | |
30 | 25, 28, 29 | mp2an 424 | . . . . 5 |
31 | 21, 22, 23, 30 | syl3anbrc 1176 | . . . 4 |
32 | elioore 9862 | . . . . . 6 | |
33 | 32 | recoscld 11680 | . . . . 5 |
34 | 0red 7914 | . . . . 5 | |
35 | 1red 7928 | . . . . 5 | |
36 | cosq23lt0 13513 | . . . . 5 | |
37 | 0lt1 8039 | . . . . . 6 | |
38 | 37 | a1i 9 | . . . . 5 |
39 | 33, 34, 35, 36, 38 | lttrd 8038 | . . . 4 |
40 | 31, 39 | syl 14 | . . 3 |
41 | 2lt3 9041 | . . . . . 6 | |
42 | 2pos 8962 | . . . . . . . 8 | |
43 | 9, 42 | pm3.2i 270 | . . . . . . 7 |
44 | 3pos 8965 | . . . . . . . 8 | |
45 | 26, 44 | pm3.2i 270 | . . . . . . 7 |
46 | pipos 13468 | . . . . . . . 8 | |
47 | 4, 46 | pm3.2i 270 | . . . . . . 7 |
48 | ltdiv2 8796 | . . . . . . 7 | |
49 | 43, 45, 47, 48 | mp3an 1332 | . . . . . 6 |
50 | 41, 49 | mpbi 144 | . . . . 5 |
51 | ltdivmul 8785 | . . . . . 6 | |
52 | 4, 24, 45, 51 | mp3an 1332 | . . . . 5 |
53 | 50, 52 | mpbi 144 | . . . 4 |
54 | axltwlin 7980 | . . . . 5 | |
55 | 4, 27, 2, 54 | mp3an12i 1336 | . . . 4 |
56 | 53, 55 | mpi 15 | . . 3 |
57 | 20, 40, 56 | mpjaodan 793 | . 2 |
58 | 4 | rexri 7970 | . . . . . 6 |
59 | 0re 7913 | . . . . . . 7 | |
60 | 59, 4, 46 | ltleii 8015 | . . . . . 6 |
61 | lbicc2 9934 | . . . . . 6 | |
62 | 8, 58, 60, 61 | mp3an 1332 | . . . . 5 |
63 | 62 | a1i 9 | . . . 4 |
64 | 1 | adantr 274 | . . . . 5 |
65 | 0red 7914 | . . . . . 6 | |
66 | 13 | simp2bi 1008 | . . . . . . 7 |
67 | 66 | adantr 274 | . . . . . 6 |
68 | 65, 64, 67 | ltled 8031 | . . . . 5 |
69 | 4 | a1i 9 | . . . . . 6 |
70 | simpr 109 | . . . . . 6 | |
71 | 64, 69, 70 | ltled 8031 | . . . . 5 |
72 | 59, 4 | elicc2i 9889 | . . . . 5 |
73 | 64, 68, 71, 72 | syl3anbrc 1176 | . . . 4 |
74 | 63, 73, 67 | cosordlem 13529 | . . 3 |
75 | cos0 11686 | . . 3 | |
76 | 74, 75 | breqtrdi 4028 | . 2 |
77 | pirp 13469 | . . . 4 | |
78 | rphalflt 9633 | . . . 4 | |
79 | 77, 78 | ax-mp 5 | . . 3 |
80 | axltwlin 7980 | . . . 4 | |
81 | 24, 4, 1, 80 | mp3an12i 1336 | . . 3 |
82 | 79, 81 | mpi 15 | . 2 |
83 | 57, 76, 82 | mpjaodan 793 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 703 w3a 973 wcel 2141 class class class wbr 3987 cfv 5196 (class class class)co 5851 cr 7766 cc0 7767 c1 7768 cmul 7772 cxr 7946 clt 7947 cle 7948 cdiv 8582 c2 8922 c3 8923 crp 9603 cioo 9838 cico 9840 cicc 9841 ccos 11601 cpi 11603 |
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 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 ax-arch 7886 ax-caucvg 7887 ax-pre-suploc 7888 ax-addf 7889 ax-mulf 7890 |
This theorem depends on definitions: df-bi 116 df-stab 826 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-disj 3965 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-of 6059 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-frec 6368 df-1o 6393 df-oadd 6397 df-er 6511 df-map 6626 df-pm 6627 df-en 6717 df-dom 6718 df-fin 6719 df-sup 6959 df-inf 6960 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-5 8933 df-6 8934 df-7 8935 df-8 8936 df-9 8937 df-n0 9129 df-z 9206 df-uz 9481 df-q 9572 df-rp 9604 df-xneg 9722 df-xadd 9723 df-ioo 9842 df-ioc 9843 df-ico 9844 df-icc 9845 df-fz 9959 df-fzo 10092 df-seqfrec 10395 df-exp 10469 df-fac 10653 df-bc 10675 df-ihash 10703 df-shft 10772 df-cj 10799 df-re 10800 df-im 10801 df-rsqrt 10955 df-abs 10956 df-clim 11235 df-sumdc 11310 df-ef 11604 df-sin 11606 df-cos 11607 df-pi 11609 df-rest 12574 df-topgen 12593 df-psmet 12746 df-xmet 12747 df-met 12748 df-bl 12749 df-mopn 12750 df-top 12755 df-topon 12768 df-bases 12800 df-ntr 12855 df-cn 12947 df-cnp 12948 df-tx 13012 df-cncf 13317 df-limced 13384 df-dvap 13385 |
This theorem is referenced by: cos0pilt1 13532 taupi 14067 |
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