| Mathbox for Norm Megill |
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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 39357 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) | |
| 2 | 1 | simp2d 1144 | 1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 CLatccla 18543 OMLcoml 39176 CvLatclc 39266 HLchlt 39351 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-hlat 39352 |
| This theorem is referenced by: hlomcmat 39366 glbconN 39378 glbconNOLD 39379 pmaple 39763 pmapglbx 39771 polsubN 39909 2polvalN 39916 2polssN 39917 3polN 39918 2pmaplubN 39928 paddunN 39929 poldmj1N 39930 pnonsingN 39935 ispsubcl2N 39949 psubclinN 39950 paddatclN 39951 polsubclN 39954 poml4N 39955 diaglbN 41057 diaintclN 41060 dibglbN 41168 dibintclN 41169 dihglblem2N 41296 dihglblem3N 41297 dihglblem4 41299 dihglbcpreN 41302 dihglblem6 41342 dihintcl 41346 dochval2 41354 dochcl 41355 dochvalr 41359 dochss 41367 |
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