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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlclat | Structured version Visualization version GIF version |
Description: A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.) |
Ref | Expression |
---|---|
hlclat | ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlomcmcv 39338 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) | |
2 | 1 | simp2d 1142 | 1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 CLatccla 18556 OMLcoml 39157 CvLatclc 39247 HLchlt 39332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-hlat 39333 |
This theorem is referenced by: hlomcmat 39347 glbconN 39359 glbconNOLD 39360 pmaple 39744 pmapglbx 39752 polsubN 39890 2polvalN 39897 2polssN 39898 3polN 39899 2pmaplubN 39909 paddunN 39910 poldmj1N 39911 pnonsingN 39916 ispsubcl2N 39930 psubclinN 39931 paddatclN 39932 polsubclN 39935 poml4N 39936 diaglbN 41038 diaintclN 41041 dibglbN 41149 dibintclN 41150 dihglblem2N 41277 dihglblem3N 41278 dihglblem4 41280 dihglbcpreN 41283 dihglblem6 41323 dihintcl 41327 dochval2 41335 dochcl 41336 dochvalr 41340 dochss 41348 |
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