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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 35377 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) | |
2 | 1 | simp2d 1174 | 1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 CLatccla 17422 OMLcoml 35196 CvLatclc 35286 HLchlt 35371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-iota 6064 df-fv 6109 df-ov 6881 df-hlat 35372 |
This theorem is referenced by: hlomcmat 35386 glbconN 35398 pmaple 35782 pmapglbx 35790 polsubN 35928 2polvalN 35935 2polssN 35936 3polN 35937 2pmaplubN 35947 paddunN 35948 poldmj1N 35949 pnonsingN 35954 ispsubcl2N 35968 psubclinN 35969 paddatclN 35970 polsubclN 35973 poml4N 35974 diaglbN 37076 diaintclN 37079 dibglbN 37187 dibintclN 37188 dihglblem2N 37315 dihglblem3N 37316 dihglblem4 37318 dihglbcpreN 37321 dihglblem6 37361 dihintcl 37365 dochval2 37373 dochcl 37374 dochvalr 37378 dochss 37386 |
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