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Mirrors > Home > ILE Home > Th. List > sin02gt0 | Unicode version |
Description: The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
sin02gt0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7836 |
. . . . . . 7
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2 | 2re 8814 |
. . . . . . 7
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3 | elioc2 9749 |
. . . . . . 7
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4 | 1, 2, 3 | mp2an 423 |
. . . . . 6
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5 | rehalfcl 8971 |
. . . . . . 7
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6 | 5 | 3ad2ant1 1003 |
. . . . . 6
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7 | 4, 6 | sylbi 120 |
. . . . 5
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8 | resincl 11463 |
. . . . . 6
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9 | recoscl 11464 |
. . . . . 6
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10 | 8, 9 | remulcld 7820 |
. . . . 5
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11 | 7, 10 | syl 14 |
. . . 4
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12 | 2pos 8835 |
. . . . . . . . . 10
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13 | divgt0 8654 |
. . . . . . . . . 10
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14 | 2, 12, 13 | mpanr12 436 |
. . . . . . . . 9
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15 | 14 | 3adant3 1002 |
. . . . . . . 8
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16 | 2, 12 | pm3.2i 270 |
. . . . . . . . . . . 12
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17 | lediv1 8651 |
. . . . . . . . . . . 12
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18 | 2, 16, 17 | mp3an23 1308 |
. . . . . . . . . . 11
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19 | 18 | biimpa 294 |
. . . . . . . . . 10
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20 | 2div2e1 8876 |
. . . . . . . . . 10
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21 | 19, 20 | breqtrdi 3977 |
. . . . . . . . 9
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22 | 21 | 3adant2 1001 |
. . . . . . . 8
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23 | 6, 15, 22 | 3jca 1162 |
. . . . . . 7
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24 | 1re 7789 |
. . . . . . . 8
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25 | elioc2 9749 |
. . . . . . . 8
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26 | 1, 24, 25 | mp2an 423 |
. . . . . . 7
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27 | 23, 4, 26 | 3imtr4i 200 |
. . . . . 6
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28 | sin01gt0 11504 |
. . . . . 6
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29 | 27, 28 | syl 14 |
. . . . 5
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30 | cos01gt0 11505 |
. . . . . 6
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31 | 27, 30 | syl 14 |
. . . . 5
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32 | axmulgt0 7860 |
. . . . . . 7
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33 | 8, 9, 32 | syl2anc 409 |
. . . . . 6
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34 | 7, 33 | syl 14 |
. . . . 5
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35 | 29, 31, 34 | mp2and 430 |
. . . 4
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36 | axmulgt0 7860 |
. . . . . 6
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37 | 2, 36 | mpan 421 |
. . . . 5
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38 | 12, 37 | mpani 427 |
. . . 4
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39 | 11, 35, 38 | sylc 62 |
. . 3
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40 | 7 | recnd 7818 |
. . . 4
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41 | sin2t 11492 |
. . . 4
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42 | 40, 41 | syl 14 |
. . 3
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43 | 39, 42 | breqtrrd 3964 |
. 2
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44 | 4 | simp1bi 997 |
. . . . 5
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45 | 44 | recnd 7818 |
. . . 4
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46 | 2cn 8815 |
. . . . 5
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47 | 2ap0 8837 |
. . . . 5
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48 | divcanap2 8464 |
. . . . 5
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49 | 46, 47, 48 | mp3an23 1308 |
. . . 4
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50 | 45, 49 | syl 14 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51 | 50 | fveq2d 5433 |
. 2
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52 | 43, 51 | breqtrd 3962 |
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 |
This theorem depends on definitions: df-bi 116 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-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-en 6643 df-dom 6644 df-fin 6645 df-sup 6879 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-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-ioc 9706 df-ico 9707 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 |
This theorem is referenced by: sincos2sgn 11508 cos12dec 11510 sin0pilem1 12910 sin0pilem2 12911 sinhalfpilem 12920 sincosq1lem 12954 |
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