![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > hlop | Structured version Visualization version GIF version |
Description: A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.) |
Ref | Expression |
---|---|
hlop | ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlol 35169 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) | |
2 | olop 35022 | . 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2139 OPcops 34980 OLcol 34982 HLchlt 35158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-iota 6012 df-fv 6057 df-ov 6817 df-ol 34986 df-oml 34987 df-hlat 35159 |
This theorem is referenced by: glbconN 35184 glbconxN 35185 hlhgt2 35196 hl0lt1N 35197 hl2at 35212 cvrexch 35227 atcvr0eq 35233 lnnat 35234 atle 35243 cvrat4 35250 athgt 35263 1cvrco 35279 1cvratex 35280 1cvrjat 35282 1cvrat 35283 ps-2 35285 llnn0 35323 lplnn0N 35354 llncvrlpln 35365 lvoln0N 35398 lplncvrlvol 35423 dalemkeop 35432 pmapeq0 35573 pmapglb2N 35578 pmapglb2xN 35579 2atm2atN 35592 polval2N 35713 polsubN 35714 pol1N 35717 2polpmapN 35720 2polvalN 35721 poldmj1N 35735 pmapj2N 35736 2polatN 35739 pnonsingN 35740 ispsubcl2N 35754 polsubclN 35759 poml4N 35760 pmapojoinN 35775 pl42lem1N 35786 lhp2lt 35808 lhp0lt 35810 lhpn0 35811 lhpexnle 35813 lhpoc2N 35822 lhpocnle 35823 lhpj1 35829 lhpmod2i2 35845 lhpmod6i1 35846 lhprelat3N 35847 ltrnatb 35944 ltrnmwOLD 35959 trlcl 35972 trlle 35992 cdleme3c 36038 cdleme7e 36055 cdleme22b 36149 cdlemg12e 36455 cdlemg12g 36457 tendoid 36581 tendo0tp 36597 cdlemk39s-id 36748 tendoex 36783 dia0eldmN 36849 dia2dimlem2 36874 dia2dimlem3 36875 docaclN 36933 doca2N 36935 djajN 36946 dib0 36973 dih0 37089 dih0bN 37090 dih0rn 37093 dih1 37095 dih1rn 37096 dih1cnv 37097 dihmeetlem18N 37133 dih1dimatlem 37138 dihlspsnssN 37141 dihlspsnat 37142 dihatexv 37147 dihglb2 37151 dochcl 37162 doch0 37167 doch1 37168 dochvalr3 37172 doch2val2 37173 dochss 37174 dochocss 37175 dochoc 37176 dochnoncon 37200 djhlj 37210 dihjatc 37226 |
Copyright terms: Public domain | W3C validator |