Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 34643 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat)) | |
2 | 1 | simp2d 1074 | 1 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 CLatccla 17107 OMLcoml 34462 CvLatclc 34552 HLchlt 34637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-hlat 34638 |
This theorem is referenced by: hlomcmat 34651 glbconN 34663 pmaple 35047 pmapglbx 35055 polsubN 35193 2polvalN 35200 2polssN 35201 3polN 35202 2pmaplubN 35212 paddunN 35213 poldmj1N 35214 pnonsingN 35219 ispsubcl2N 35233 psubclinN 35234 paddatclN 35235 polsubclN 35238 poml4N 35239 diaglbN 36344 diaintclN 36347 dibglbN 36455 dibintclN 36456 dihglblem2N 36583 dihglblem3N 36584 dihglblem4 36586 dihglbcpreN 36589 dihglblem6 36629 dihintcl 36633 dochval2 36641 dochcl 36642 dochvalr 36646 dochss 36654 |
Copyright terms: Public domain | W3C validator |