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Mirrors > Home > ILE Home > Th. List > sin01gt0 | Unicode version |
Description: The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Wolf Lammen, 25-Sep-2020.) |
Ref | Expression |
---|---|
sin01gt0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7991 |
. . . . . . . 8
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2 | 1re 7944 |
. . . . . . . 8
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3 | elioc2 9920 |
. . . . . . . 8
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4 | 1, 2, 3 | mp2an 426 |
. . . . . . 7
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5 | 4 | simp1bi 1012 |
. . . . . 6
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6 | 3nn0 9180 |
. . . . . 6
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7 | reexpcl 10520 |
. . . . . 6
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8 | 5, 6, 7 | sylancl 413 |
. . . . 5
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9 | 3re 8979 |
. . . . . 6
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10 | 3ap0 9001 |
. . . . . 6
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11 | redivclap 8674 |
. . . . . 6
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12 | 9, 10, 11 | mp3an23 1329 |
. . . . 5
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13 | 8, 12 | syl 14 |
. . . 4
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14 | 3z 9268 |
. . . . . . . . 9
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15 | expgt0 10536 |
. . . . . . . . 9
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16 | 14, 15 | mp3an2 1325 |
. . . . . . . 8
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17 | 16 | 3adant3 1017 |
. . . . . . 7
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18 | 4, 17 | sylbi 121 |
. . . . . 6
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19 | 0lt1 8071 |
. . . . . . . 8
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20 | 2, 19 | pm3.2i 272 |
. . . . . . 7
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21 | 3pos 8999 |
. . . . . . . 8
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22 | 9, 21 | pm3.2i 272 |
. . . . . . 7
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23 | 1lt3 9076 |
. . . . . . . 8
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24 | ltdiv2 8830 |
. . . . . . . 8
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25 | 23, 24 | mpbii 148 |
. . . . . . 7
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26 | 20, 22, 25 | mp3an12 1327 |
. . . . . 6
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27 | 8, 18, 26 | syl2anc 411 |
. . . . 5
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28 | 8 | recnd 7973 |
. . . . . 6
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29 | 28 | div1d 8723 |
. . . . 5
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30 | 27, 29 | breqtrd 4026 |
. . . 4
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31 | 1nn0 9178 |
. . . . . . 7
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32 | 31 | a1i 9 |
. . . . . 6
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33 | 1le3 9116 |
. . . . . . . 8
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34 | 1z 9265 |
. . . . . . . . 9
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35 | 34 | eluz1i 9521 |
. . . . . . . 8
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36 | 14, 33, 35 | mpbir2an 942 |
. . . . . . 7
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37 | 36 | a1i 9 |
. . . . . 6
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38 | 4 | simp2bi 1013 |
. . . . . . 7
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39 | 0re 7945 |
. . . . . . . 8
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40 | ltle 8032 |
. . . . . . . 8
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41 | 39, 5, 40 | sylancr 414 |
. . . . . . 7
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42 | 38, 41 | mpd 13 |
. . . . . 6
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43 | 4 | simp3bi 1014 |
. . . . . 6
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44 | 5, 32, 37, 42, 43 | leexp2rd 10666 |
. . . . 5
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45 | 5 | recnd 7973 |
. . . . . 6
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46 | 45 | exp1d 10631 |
. . . . 5
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47 | 44, 46 | breqtrd 4026 |
. . . 4
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48 | 13, 8, 5, 30, 47 | ltletrd 8367 |
. . 3
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49 | 13, 5 | posdifd 8476 |
. . 3
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50 | 48, 49 | mpbid 147 |
. 2
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51 | sin01bnd 11746 |
. . 3
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52 | 51 | simpld 112 |
. 2
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53 | 5, 13 | resubcld 8325 |
. . 3
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54 | 5 | resincld 11712 |
. . 3
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55 | lttr 8018 |
. . 3
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56 | 39, 53, 54, 55 | mp3an2i 1342 |
. 2
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57 | 50, 52, 56 | mp2and 433 |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-mulrcl 7898 ax-addcom 7899 ax-mulcom 7900 ax-addass 7901 ax-mulass 7902 ax-distr 7903 ax-i2m1 7904 ax-0lt1 7905 ax-1rid 7906 ax-0id 7907 ax-rnegex 7908 ax-precex 7909 ax-cnre 7910 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 ax-pre-apti 7914 ax-pre-ltadd 7915 ax-pre-mulgt0 7916 ax-pre-mulext 7917 ax-arch 7918 ax-caucvg 7919 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-frec 6386 df-1o 6411 df-oadd 6415 df-er 6529 df-en 6735 df-dom 6736 df-fin 6737 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-sub 8117 df-neg 8118 df-reap 8519 df-ap 8526 df-div 8616 df-inn 8906 df-2 8964 df-3 8965 df-4 8966 df-5 8967 df-6 8968 df-7 8969 df-8 8970 df-n0 9163 df-z 9240 df-uz 9515 df-q 9606 df-rp 9638 df-ioc 9877 df-ico 9878 df-fz 9993 df-fzo 10126 df-seqfrec 10429 df-exp 10503 df-fac 10687 df-ihash 10737 df-shft 10805 df-cj 10832 df-re 10833 df-im 10834 df-rsqrt 10988 df-abs 10989 df-clim 11268 df-sumdc 11343 df-ef 11637 df-sin 11639 |
This theorem is referenced by: sin02gt0 11752 sincos1sgn 11753 sincos4thpi 13921 |
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