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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatl | Structured version Visualization version GIF version |
Description: A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.) |
Ref | Expression |
---|---|
hlatl | ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcvl 33664 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat) | |
2 | cvlatl 33630 | . 2 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 AtLatcal 33569 CvLatclc 33570 HLchlt 33655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-cvlat 33627 df-hlat 33656 |
This theorem is referenced by: hllat 33668 hlomcmat 33669 intnatN 33711 cvratlem 33725 atcvrj0 33732 atcvrneN 33734 atcvrj1 33735 atcvrj2b 33736 atltcvr 33739 cvrat4 33747 2atjm 33749 atbtwn 33750 3dim2 33772 2dim 33774 1cvrjat 33779 ps-2 33782 ps-2b 33786 islln3 33814 llnnleat 33817 llnexatN 33825 2llnmat 33828 2atm 33831 2llnm3N 33873 2llnm4 33874 2llnmeqat 33875 dalem21 33998 dalem24 34001 dalem25 34002 dalem54 34030 dalem55 34031 dalem57 34033 pmapat 34067 pmapeq0 34070 isline4N 34081 2lnat 34088 2llnma1b 34090 cdlema2N 34096 cdlemblem 34097 pmapjat1 34157 llnexchb2lem 34172 pol1N 34214 pnonsingN 34237 pclfinclN 34254 lhpocnle 34320 lhpmat 34334 lhpmatb 34335 lhp2at0 34336 lhp2atnle 34337 lhp2at0nle 34339 lhpat3 34350 4atexlemcnd 34376 ltrnmwOLD 34456 trlatn0 34477 ltrnnidn 34479 trlnidatb 34482 trlnle 34491 trlval3 34492 trlval4 34493 cdlemc5 34500 cdleme0e 34522 cdleme3 34542 cdleme7c 34550 cdleme7ga 34553 cdleme7 34554 cdleme11k 34573 cdleme15b 34580 cdleme16b 34584 cdleme16e 34587 cdleme16f 34588 cdlemednpq 34604 cdleme20zN 34606 cdleme20yOLD 34608 cdleme20j 34624 cdleme22aa 34645 cdleme22cN 34648 cdleme22d 34649 cdlemf2 34868 cdlemb3 34912 cdlemg12e 34953 cdlemg17dALTN 34970 cdlemg19a 34989 cdlemg27b 35002 cdlemg31d 35006 cdlemg33c 35014 cdlemg33e 35016 trlcone 35034 cdlemi 35126 tendotr 35136 cdlemk17 35164 cdlemk52 35260 cdleml1N 35282 dian0 35346 dia0 35359 dia2dimlem1 35371 dia2dimlem2 35372 dia2dimlem3 35373 dih0cnv 35590 dihmeetlem4preN 35613 dihmeetlem7N 35617 dihmeetlem17N 35630 dihlspsnat 35640 dihatexv 35645 |
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