| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 11263 | . 2 ⊢ 0 ∈ ℝ | |
| 2 | 1 | leidi 11797 | 1 ⊢ 0 ≤ 0 |
| Colors of variables: wff setvar class |
| Syntax hints: class class class wbr 5143 0cc0 11155 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: nn0ledivnn 13148 xsubge0 13303 xmulge0 13326 0e0icopnf 13498 0e0iccpnf 13499 0elunit 13509 0mod 13942 sqlecan 14248 discr 14279 cnpart 15279 sqrt0 15280 resqrex 15289 sqrt00 15302 fsumabs 15837 rpnnen2lem4 16253 divalglem7 16436 pcmptdvds 16932 prmreclem4 16957 prmreclem5 16958 prmreclem6 16959 ramz2 17062 ramz 17063 isabvd 20813 prdsxmetlem 24378 metustto 24566 cfilucfil 24572 nmolb2d 24739 nmoi 24749 nmoix 24750 nmoleub 24752 nmo0 24756 pcoval1 25046 pco0 25047 minveclem7 25469 ovolfiniun 25536 ovolicc1 25551 ioorf 25608 itg1ge0a 25746 mbfi1fseqlem5 25754 itg2const 25775 itg2const2 25776 itg2splitlem 25783 itg2cnlem1 25796 itg2cnlem2 25797 iblss 25840 itgle 25845 ibladdlem 25855 iblabs 25864 iblabsr 25865 iblmulc2 25866 bddmulibl 25874 bddiblnc 25877 c1lip1 26036 dveq0 26039 dv11cn 26040 fta1g 26209 abelthlem2 26476 sinq12ge0 26550 cxpge0 26725 abscxp2 26735 log2ublem3 26991 chtwordi 27199 ppiwordi 27205 chpub 27264 bposlem1 27328 bposlem6 27333 dchrisum0flblem2 27553 qabvle 27669 ostth2lem2 27678 colinearalg 28925 eucrct2eupth 30264 ex-po 30454 nvz0 30687 nmlnoubi 30815 nmblolbii 30818 blocnilem 30823 siilem2 30871 minvecolem7 30902 pjneli 31742 nmbdoplbi 32043 nmcoplbi 32047 nmbdfnlbi 32068 nmcfnlbi 32071 nmopcoi 32114 unierri 32123 leoprf2 32146 leoprf 32147 stle0i 32258 fzo0opth 32807 m1pmeq 33608 xrge0iifcnv 33932 xrge0iifiso 33934 xrge0iifhom 33936 esumrnmpt2 34069 dstfrvclim1 34480 ballotlemrc 34533 signsply0 34566 chtvalz 34644 poimirlem23 37650 mblfinlem2 37665 itg2addnclem 37678 itg2gt0cn 37682 ibladdnclem 37683 itgaddnclem2 37686 iblabsnc 37691 iblmulc2nc 37692 ftc1anclem5 37704 ftc1anclem7 37706 ftc1anclem8 37707 ftc1anc 37708 areacirclem1 37715 areacirclem4 37718 mettrifi 37764 aks6d1c1 42117 bcled 42179 bcle2d 42180 readvrec2 42391 monotoddzzfi 42954 rmxypos 42959 rmygeid 42976 stoweidlem55 46070 fourierdlem14 46136 fourierdlem20 46142 fourierdlem92 46213 fourierdlem93 46214 fouriersw 46246 isomennd 46546 ovnssle 46576 hoidmvlelem3 46612 ovnhoilem1 46616 nnlog2ge0lt1 48487 dig1 48529 sepfsepc 48825 seppcld 48827 ex-gte 49248 |
| Copyright terms: Public domain | W3C validator |