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Mirrors > Home > ILE Home > Th. List > cosq34lt1 | Unicode version |
Description: Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 19-Mar-2024.) |
Ref | Expression |
---|---|
cosq34lt1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pire 12915 |
. . . . . . . 8
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2 | 2re 8814 |
. . . . . . . . . 10
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3 | 2, 1 | remulcli 7804 |
. . . . . . . . 9
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4 | 3 | rexri 7847 |
. . . . . . . 8
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5 | elico2 9750 |
. . . . . . . 8
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6 | 1, 4, 5 | mp2an 423 |
. . . . . . 7
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7 | 6 | simp1bi 997 |
. . . . . 6
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8 | 7 | recnd 7818 |
. . . . 5
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9 | 2cn 8815 |
. . . . . . 7
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10 | picn 12916 |
. . . . . . 7
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11 | 9, 10 | mulcli 7795 |
. . . . . 6
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12 | 11 | a1i 9 |
. . . . 5
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13 | 8, 12 | subcld 8097 |
. . . 4
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14 | cosneg 11470 |
. . . 4
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15 | 13, 14 | syl 14 |
. . 3
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16 | 12 | mulm1d 8196 |
. . . . . 6
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17 | 16 | oveq2d 5798 |
. . . . 5
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18 | 8, 12 | negsubd 8103 |
. . . . 5
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19 | 17, 18 | eqtrd 2173 |
. . . 4
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20 | 19 | fveq2d 5433 |
. . 3
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21 | neg1z 9110 |
. . . 4
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22 | cosper 12939 |
. . . 4
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23 | 8, 21, 22 | sylancl 410 |
. . 3
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24 | 15, 20, 23 | 3eqtr2d 2179 |
. 2
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25 | 0xr 7836 |
. . . . 5
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26 | 1 | rexri 7847 |
. . . . 5
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27 | 0re 7790 |
. . . . . . 7
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28 | pipos 12917 |
. . . . . . 7
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29 | 27, 1, 28 | ltleii 7890 |
. . . . . 6
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30 | 29 | a1i 9 |
. . . . 5
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31 | lbicc2 9797 |
. . . . 5
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32 | 25, 26, 30, 31 | mp3an12i 1320 |
. . . 4
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33 | 3 | a1i 9 |
. . . . . . 7
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34 | 7, 33 | resubcld 8167 |
. . . . . 6
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35 | 34 | renegcld 8166 |
. . . . 5
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36 | 27 | a1i 9 |
. . . . . 6
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37 | 6 | simp3bi 999 |
. . . . . . . 8
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38 | 7, 33 | posdifd 8318 |
. . . . . . . 8
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39 | 37, 38 | mpbid 146 |
. . . . . . 7
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40 | 8, 12 | negsubdi2d 8113 |
. . . . . . 7
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41 | 39, 40 | breqtrrd 3964 |
. . . . . 6
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42 | 36, 35, 41 | ltled 7905 |
. . . . 5
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43 | 1 | a1i 9 |
. . . . . . 7
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44 | ax-1cn 7737 |
. . . . . . . . . 10
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45 | 9, 44, 10 | subdiri 8194 |
. . . . . . . . 9
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46 | 2m1e1 8862 |
. . . . . . . . . . 11
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47 | 46 | oveq1i 5792 |
. . . . . . . . . 10
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48 | 10 | mulid2i 7793 |
. . . . . . . . . 10
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49 | 47, 48 | eqtri 2161 |
. . . . . . . . 9
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50 | 48 | oveq2i 5793 |
. . . . . . . . 9
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51 | 45, 49, 50 | 3eqtr3ri 2170 |
. . . . . . . 8
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52 | 6 | simp2bi 998 |
. . . . . . . 8
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53 | 51, 52 | eqbrtrid 3971 |
. . . . . . 7
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54 | 33, 43, 7, 53 | subled 8334 |
. . . . . 6
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55 | 40, 54 | eqbrtrd 3958 |
. . . . 5
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56 | 27, 1 | elicc2i 9752 |
. . . . 5
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57 | 35, 42, 55, 56 | syl3anbrc 1166 |
. . . 4
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58 | 32, 57, 41 | cosordlem 12978 |
. . 3
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59 | cos0 11473 |
. . 3
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60 | 58, 59 | breqtrdi 3977 |
. 2
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61 | 24, 60 | eqbrtrrd 3960 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 ax-pre-suploc 7765 ax-addf 7766 ax-mulf 7767 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-disj 3915 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-map 6552 df-pm 6553 df-en 6643 df-dom 6644 df-fin 6645 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-xneg 9589 df-xadd 9590 df-ioo 9705 df-ioc 9706 df-ico 9707 df-icc 9708 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-fac 10504 df-bc 10526 df-ihash 10554 df-shft 10619 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155 df-ef 11391 df-sin 11393 df-cos 11394 df-pi 11396 df-rest 12161 df-topgen 12180 df-psmet 12195 df-xmet 12196 df-met 12197 df-bl 12198 df-mopn 12199 df-top 12204 df-topon 12217 df-bases 12249 df-ntr 12304 df-cn 12396 df-cnp 12397 df-tx 12461 df-cncf 12766 df-limced 12833 df-dvap 12834 |
This theorem is referenced by: cos02pilt1 12980 |
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