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Mirrors > Home > MPE Home > Th. List > 0le0 | Structured version Visualization version GIF version |
Description: Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.) |
Ref | Expression |
---|---|
0le0 | ⊢ 0 ≤ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 9919 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1 | leidi 10441 | 1 ⊢ 0 ≤ 0 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4583 0cc0 9815 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: nn0ledivnn 11817 xsubge0 11963 xmulge0 11986 0e0icopnf 12153 0e0iccpnf 12154 0elunit 12161 0mod 12563 sqlecan 12833 discr 12863 cnpart 13828 sqr0lem 13829 resqrex 13839 sqrt00 13852 fsumabs 14374 rpnnen2lem4 14785 divalglem7 14960 pcmptdvds 15436 prmreclem4 15461 prmreclem5 15462 prmreclem6 15463 ramz2 15566 ramz 15567 isabvd 18643 prdsxmetlem 21983 metustto 22168 cfilucfil 22174 nmolb2d 22332 nmoi 22342 nmoix 22343 nmoleub 22345 nmo0 22349 pcoval1 22621 pco0 22622 minveclem7 23014 ovolfiniun 23076 ovolicc1 23091 ioorf 23147 itg1ge0a 23284 mbfi1fseqlem5 23292 itg2const 23313 itg2const2 23314 itg2splitlem 23321 itg2cnlem1 23334 itg2cnlem2 23335 iblss 23377 itgle 23382 ibladdlem 23392 iblabs 23401 iblabsr 23402 iblmulc2 23403 bddmulibl 23411 c1lip1 23564 dveq0 23567 dv11cn 23568 fta1g 23731 abelthlem2 23990 sinq12ge0 24064 cxpge0 24229 abscxp2 24239 log2ublem3 24475 chtwordi 24682 ppiwordi 24688 chpub 24745 bposlem1 24809 bposlem6 24814 dchrisum0flblem2 24998 qabvle 25114 ostth2lem2 25123 colinearalg 25590 ex-po 26684 nvz0 26907 nmlnoubi 27035 nmblolbii 27038 blocnilem 27043 siilem2 27091 minvecolem7 27123 pjneli 27966 nmbdoplbi 28267 nmcoplbi 28271 nmbdfnlbi 28292 nmcfnlbi 28295 nmopcoi 28338 unierri 28347 leoprf2 28370 leoprf 28371 stle0i 28482 xrge0iifcnv 29307 xrge0iifiso 29309 xrge0iifhom 29311 esumrnmpt2 29457 dstfrvclim1 29866 ballotlemrc 29919 signsply0 29954 poimirlem23 32602 mblfinlem2 32617 itg2addnclem 32631 itg2gt0cn 32635 ibladdnclem 32636 itgaddnclem2 32639 iblabsnc 32644 iblmulc2nc 32645 bddiblnc 32650 ftc1anclem5 32659 ftc1anclem7 32661 ftc1anclem8 32662 ftc1anc 32663 areacirclem1 32670 areacirclem4 32673 mettrifi 32723 monotoddzzfi 36525 rmxypos 36532 rmygeid 36549 stoweidlem55 38948 fourierdlem14 39014 fourierdlem20 39020 fourierdlem92 39091 fourierdlem93 39092 fouriersw 39124 isomennd 39421 ovnssle 39451 hoidmvlelem3 39487 ovnhoilem1 39491 eucrct2eupth 41413 nnlog2ge0lt1 42158 dig1 42200 ex-gte 42269 |
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