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Mirrors > Home > HSE Home > Th. List > 0leop | Structured version Visualization version GIF version |
Description: The zero operator is a positive operator. (The literature calls it "positive", even though in some sense it is really "nonnegative".) Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0leop | ⊢ 0hop ≤op 0hop |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0hmop 30061 | . 2 ⊢ 0hop ∈ HrmOp | |
2 | leoprf 30206 | . 2 ⊢ ( 0hop ∈ HrmOp → 0hop ≤op 0hop ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 0hop ≤op 0hop |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 class class class wbr 5050 0hop ch0o 29021 HrmOpcho 29028 ≤op cleo 29036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-inf2 9253 ax-cc 10046 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 ax-addf 10805 ax-mulf 10806 ax-hilex 29077 ax-hfvadd 29078 ax-hvcom 29079 ax-hvass 29080 ax-hv0cl 29081 ax-hvaddid 29082 ax-hfvmul 29083 ax-hvmulid 29084 ax-hvmulass 29085 ax-hvdistr1 29086 ax-hvdistr2 29087 ax-hvmul0 29088 ax-hfi 29157 ax-his1 29160 ax-his2 29161 ax-his3 29162 ax-his4 29163 ax-hcompl 29280 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-iin 4904 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-se 5507 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-isom 6386 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-of 7466 df-om 7642 df-1st 7758 df-2nd 7759 df-supp 7901 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-2o 8200 df-oadd 8203 df-omul 8204 df-er 8388 df-map 8507 df-pm 8508 df-ixp 8576 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-fsupp 8983 df-fi 9024 df-sup 9055 df-inf 9056 df-oi 9123 df-card 9552 df-acn 9555 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-7 11895 df-8 11896 df-9 11897 df-n0 12088 df-z 12174 df-dec 12291 df-uz 12436 df-q 12542 df-rp 12584 df-xneg 12701 df-xadd 12702 df-xmul 12703 df-ioo 12936 df-ico 12938 df-icc 12939 df-fz 13093 df-fzo 13236 df-fl 13364 df-seq 13572 df-exp 13633 df-hash 13894 df-cj 14659 df-re 14660 df-im 14661 df-sqrt 14795 df-abs 14796 df-clim 15046 df-rlim 15047 df-sum 15247 df-struct 16697 df-sets 16714 df-slot 16732 df-ndx 16742 df-base 16758 df-ress 16782 df-plusg 16812 df-mulr 16813 df-starv 16814 df-sca 16815 df-vsca 16816 df-ip 16817 df-tset 16818 df-ple 16819 df-ds 16821 df-unif 16822 df-hom 16823 df-cco 16824 df-rest 16924 df-topn 16925 df-0g 16943 df-gsum 16944 df-topgen 16945 df-pt 16946 df-prds 16949 df-xrs 17004 df-qtop 17009 df-imas 17010 df-xps 17012 df-mre 17086 df-mrc 17087 df-acs 17089 df-mgm 18111 df-sgrp 18160 df-mnd 18171 df-submnd 18216 df-mulg 18486 df-cntz 18708 df-cmn 19169 df-psmet 20352 df-xmet 20353 df-met 20354 df-bl 20355 df-mopn 20356 df-fbas 20357 df-fg 20358 df-cnfld 20361 df-top 21788 df-topon 21805 df-topsp 21827 df-bases 21840 df-cld 21913 df-ntr 21914 df-cls 21915 df-nei 21992 df-cn 22121 df-cnp 22122 df-lm 22123 df-haus 22209 df-tx 22456 df-hmeo 22649 df-fil 22740 df-fm 22832 df-flim 22833 df-flf 22834 df-xms 23215 df-ms 23216 df-tms 23217 df-cfil 24149 df-cau 24150 df-cmet 24151 df-grpo 28571 df-gid 28572 df-ginv 28573 df-gdiv 28574 df-ablo 28623 df-vc 28637 df-nv 28670 df-va 28673 df-ba 28674 df-sm 28675 df-0v 28676 df-vs 28677 df-nmcv 28678 df-ims 28679 df-dip 28779 df-ssp 28800 df-ph 28891 df-cbn 28941 df-hnorm 29046 df-hba 29047 df-hvsub 29049 df-hlim 29050 df-hcau 29051 df-sh 29285 df-ch 29299 df-oc 29330 df-ch0 29331 df-shs 29386 df-pjh 29473 df-hosum 29808 df-homul 29809 df-hodif 29810 df-h0op 29826 df-hmop 29922 df-leop 29930 |
This theorem is referenced by: (None) |
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