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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlol | Structured version Visualization version GIF version |
Description: A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.) |
Ref | Expression |
---|---|
hlol | ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hloml 39338 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML) | |
2 | omlol 39221 | . 2 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 OLcol 39155 OMLcoml 39156 HLchlt 39331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-ov 7433 df-oml 39160 df-hlat 39332 |
This theorem is referenced by: hlop 39343 cvrexch 39402 atle 39418 athgt 39438 2at0mat0 39507 dalem24 39679 pmapjat1 39835 atmod1i1m 39840 llnexchb2lem 39850 dalawlem2 39854 dalawlem6 39858 dalawlem7 39859 dalawlem11 39863 dalawlem12 39864 poldmj1N 39910 pmapj2N 39911 2polatN 39914 lhpmcvr3 40007 lhp2at0 40014 lhp2at0nle 40017 lhpelim 40019 lhpmod2i2 40020 lhpmod6i1 40021 lhprelat3N 40022 lhple 40024 4atex2-0aOLDN 40060 trljat1 40148 trljat2 40149 cdlemc1 40173 cdlemc6 40178 cdleme0cp 40196 cdleme0cq 40197 cdleme0e 40199 cdleme1 40209 cdleme2 40210 cdleme3c 40212 cdleme4 40220 cdleme5 40222 cdleme7c 40227 cdleme7e 40229 cdleme8 40232 cdleme9 40235 cdleme10 40236 cdleme15b 40257 cdlemednpq 40281 cdleme20c 40293 cdleme20d 40294 cdleme20j 40300 cdleme22cN 40324 cdleme22d 40325 cdleme22e 40326 cdleme22eALTN 40327 cdleme23b 40332 cdleme30a 40360 cdlemefrs29pre00 40377 cdlemefrs29bpre0 40378 cdlemefrs29cpre1 40380 cdleme32fva 40419 cdleme35b 40432 cdleme35d 40434 cdleme35e 40435 cdleme42a 40453 cdleme42ke 40467 cdlemeg46frv 40507 cdlemg2fv2 40582 cdlemg2m 40586 cdlemg10bALTN 40618 cdlemg12e 40629 cdlemg31d 40682 trlcoabs2N 40704 trlcolem 40708 trljco 40722 cdlemh2 40798 cdlemh 40799 cdlemi1 40800 cdlemk4 40816 cdlemk9 40821 cdlemk9bN 40822 cdlemkid2 40906 dia2dimlem1 41046 dia2dimlem2 41047 dia2dimlem3 41048 doca2N 41108 djajN 41119 cdlemn10 41188 dihvalcqat 41221 dih1 41268 dihglbcpreN 41282 dihmeetbclemN 41286 dihmeetlem7N 41292 dihjatc1 41293 djhlj 41383 djh01 41394 dihjatc 41399 |
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