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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7812 | . . . . . . . 8 | |
2 | 1re 7765 | . . . . . . . 8 | |
3 | elioc2 9719 | . . . . . . . 8 | |
4 | 1, 2, 3 | mp2an 422 | . . . . . . 7 |
5 | 4 | simp1bi 996 | . . . . . 6 |
6 | 3nn0 8995 | . . . . . 6 | |
7 | reexpcl 10310 | . . . . . 6 | |
8 | 5, 6, 7 | sylancl 409 | . . . . 5 |
9 | 3re 8794 | . . . . . 6 | |
10 | 3ap0 8816 | . . . . . 6 # | |
11 | redivclap 8491 | . . . . . 6 # | |
12 | 9, 10, 11 | mp3an23 1307 | . . . . 5 |
13 | 8, 12 | syl 14 | . . . 4 |
14 | 3z 9083 | . . . . . . . . 9 | |
15 | expgt0 10326 | . . . . . . . . 9 | |
16 | 14, 15 | mp3an2 1303 | . . . . . . . 8 |
17 | 16 | 3adant3 1001 | . . . . . . 7 |
18 | 4, 17 | sylbi 120 | . . . . . 6 |
19 | 0lt1 7889 | . . . . . . . 8 | |
20 | 2, 19 | pm3.2i 270 | . . . . . . 7 |
21 | 3pos 8814 | . . . . . . . 8 | |
22 | 9, 21 | pm3.2i 270 | . . . . . . 7 |
23 | 1lt3 8891 | . . . . . . . 8 | |
24 | ltdiv2 8645 | . . . . . . . 8 | |
25 | 23, 24 | mpbii 147 | . . . . . . 7 |
26 | 20, 22, 25 | mp3an12 1305 | . . . . . 6 |
27 | 8, 18, 26 | syl2anc 408 | . . . . 5 |
28 | 8 | recnd 7794 | . . . . . 6 |
29 | 28 | div1d 8540 | . . . . 5 |
30 | 27, 29 | breqtrd 3954 | . . . 4 |
31 | 1nn0 8993 | . . . . . . 7 | |
32 | 31 | a1i 9 | . . . . . 6 |
33 | 1le3 8931 | . . . . . . . 8 | |
34 | 1z 9080 | . . . . . . . . 9 | |
35 | 34 | eluz1i 9333 | . . . . . . . 8 |
36 | 14, 33, 35 | mpbir2an 926 | . . . . . . 7 |
37 | 36 | a1i 9 | . . . . . 6 |
38 | 4 | simp2bi 997 | . . . . . . 7 |
39 | 0re 7766 | . . . . . . . 8 | |
40 | ltle 7851 | . . . . . . . 8 | |
41 | 39, 5, 40 | sylancr 410 | . . . . . . 7 |
42 | 38, 41 | mpd 13 | . . . . . 6 |
43 | 4 | simp3bi 998 | . . . . . 6 |
44 | 5, 32, 37, 42, 43 | leexp2rd 10454 | . . . . 5 |
45 | 5 | recnd 7794 | . . . . . 6 |
46 | 45 | exp1d 10419 | . . . . 5 |
47 | 44, 46 | breqtrd 3954 | . . . 4 |
48 | 13, 8, 5, 30, 47 | ltletrd 8185 | . . 3 |
49 | 13, 5 | posdifd 8294 | . . 3 |
50 | 48, 49 | mpbid 146 | . 2 |
51 | sin01bnd 11464 | . . 3 | |
52 | 51 | simpld 111 | . 2 |
53 | 5, 13 | resubcld 8143 | . . 3 |
54 | 5 | resincld 11430 | . . 3 |
55 | lttr 7838 | . . 3 | |
56 | 39, 53, 54, 55 | mp3an2i 1320 | . 2 |
57 | 50, 52, 56 | mp2and 429 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wcel 1480 class class class wbr 3929 cfv 5123 (class class class)co 5774 cr 7619 cc0 7620 c1 7621 cxr 7799 clt 7800 cle 7801 cmin 7933 # cap 8343 cdiv 8432 c3 8772 cn0 8977 cz 9054 cuz 9326 cioc 9672 cexp 10292 csin 11350 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-ioc 9676 df-ico 9677 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-fac 10472 df-ihash 10522 df-shft 10587 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 df-ef 11354 df-sin 11356 |
This theorem is referenced by: sin02gt0 11470 sincos1sgn 11471 sincos4thpi 12921 |
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