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Mirrors > Home > HSE Home > Th. List > 0leop | Structured version Unicode 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0hmop 25559 |
. 2
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2 | leoprf 25704 |
. 2
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3 | 1, 2 | ax-mp 5 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-inf2 7961 ax-cc 8718 ax-cnex 9452 ax-resscn 9453 ax-1cn 9454 ax-icn 9455 ax-addcl 9456 ax-addrcl 9457 ax-mulcl 9458 ax-mulrcl 9459 ax-mulcom 9460 ax-addass 9461 ax-mulass 9462 ax-distr 9463 ax-i2m1 9464 ax-1ne0 9465 ax-1rid 9466 ax-rnegex 9467 ax-rrecex 9468 ax-cnre 9469 ax-pre-lttri 9470 ax-pre-lttrn 9471 ax-pre-ltadd 9472 ax-pre-mulgt0 9473 ax-pre-sup 9474 ax-addf 9475 ax-mulf 9476 ax-hilex 24573 ax-hfvadd 24574 ax-hvcom 24575 ax-hvass 24576 ax-hv0cl 24577 ax-hvaddid 24578 ax-hfvmul 24579 ax-hvmulid 24580 ax-hvmulass 24581 ax-hvdistr1 24582 ax-hvdistr2 24583 ax-hvmul0 24584 ax-hfi 24653 ax-his1 24656 ax-his2 24657 ax-his3 24658 ax-his4 24659 ax-hcompl 24776 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-int 4240 df-iun 4284 df-iin 4285 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-se 4791 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-isom 5538 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-of 6433 df-om 6590 df-1st 6690 df-2nd 6691 df-supp 6804 df-recs 6945 df-rdg 6979 df-1o 7033 df-2o 7034 df-oadd 7037 df-omul 7038 df-er 7214 df-map 7329 df-pm 7330 df-ixp 7377 df-en 7424 df-dom 7425 df-sdom 7426 df-fin 7427 df-fsupp 7735 df-fi 7775 df-sup 7805 df-oi 7838 df-card 8223 df-acn 8226 df-cda 8451 df-pnf 9534 df-mnf 9535 df-xr 9536 df-ltxr 9537 df-le 9538 df-sub 9711 df-neg 9712 df-div 10108 df-nn 10437 df-2 10494 df-3 10495 df-4 10496 df-5 10497 df-6 10498 df-7 10499 df-8 10500 df-9 10501 df-10 10502 df-n0 10694 df-z 10761 df-dec 10870 df-uz 10976 df-q 11068 df-rp 11106 df-xneg 11203 df-xadd 11204 df-xmul 11205 df-ioo 11418 df-ico 11420 df-icc 11421 df-fz 11558 df-fzo 11669 df-fl 11762 df-seq 11927 df-exp 11986 df-hash 12224 df-cj 12709 df-re 12710 df-im 12711 df-sqr 12845 df-abs 12846 df-clim 13087 df-rlim 13088 df-sum 13285 df-struct 14297 df-ndx 14298 df-slot 14299 df-base 14300 df-sets 14301 df-ress 14302 df-plusg 14373 df-mulr 14374 df-starv 14375 df-sca 14376 df-vsca 14377 df-ip 14378 df-tset 14379 df-ple 14380 df-ds 14382 df-unif 14383 df-hom 14384 df-cco 14385 df-rest 14483 df-topn 14484 df-0g 14502 df-gsum 14503 df-topgen 14504 df-pt 14505 df-prds 14508 df-xrs 14562 df-qtop 14567 df-imas 14568 df-xps 14570 df-mre 14646 df-mrc 14647 df-acs 14649 df-mnd 15537 df-submnd 15587 df-mulg 15670 df-cntz 15957 df-cmn 16403 df-psmet 17937 df-xmet 17938 df-met 17939 df-bl 17940 df-mopn 17941 df-fbas 17942 df-fg 17943 df-cnfld 17947 df-top 18638 df-bases 18640 df-topon 18641 df-topsp 18642 df-cld 18758 df-ntr 18759 df-cls 18760 df-nei 18837 df-cn 18966 df-cnp 18967 df-lm 18968 df-haus 19054 df-tx 19270 df-hmeo 19463 df-fil 19554 df-fm 19646 df-flim 19647 df-flf 19648 df-xms 20030 df-ms 20031 df-tms 20032 df-cfil 20901 df-cau 20902 df-cmet 20903 df-grpo 23850 df-gid 23851 df-ginv 23852 df-gdiv 23853 df-ablo 23941 df-subgo 23961 df-vc 24096 df-nv 24142 df-va 24145 df-ba 24146 df-sm 24147 df-0v 24148 df-vs 24149 df-nmcv 24150 df-ims 24151 df-dip 24268 df-ssp 24292 df-ph 24385 df-cbn 24436 df-hnorm 24542 df-hba 24543 df-hvsub 24545 df-hlim 24546 df-hcau 24547 df-sh 24781 df-ch 24796 df-oc 24827 df-ch0 24828 df-shs 24883 df-pjh 24970 df-hosum 25306 df-homul 25307 df-hodif 25308 df-h0op 25324 df-hmop 25420 df-leop 25428 |
This theorem is referenced by: (None) |
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