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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 33661 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) | |
2 | 1 | simp2d 1067 | 1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 CLatccla 16930 OMLcoml 33480 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-hlat 33656 |
This theorem is referenced by: hlomcmat 33669 glbconN 33681 pmaple 34065 pmapglbx 34073 polsubN 34211 2polvalN 34218 2polssN 34219 3polN 34220 2pmaplubN 34230 paddunN 34231 poldmj1N 34232 pnonsingN 34237 ispsubcl2N 34251 psubclinN 34252 paddatclN 34253 polsubclN 34256 poml4N 34257 diaglbN 35362 diaintclN 35365 dibglbN 35473 dibintclN 35474 dihglblem2N 35601 dihglblem3N 35602 dihglblem4 35604 dihglbcpreN 35607 dihglblem6 35647 dihintcl 35651 dochval2 35659 dochcl 35660 dochvalr 35664 dochss 35672 |
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