![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > hlomcmat | Structured version Visualization version GIF version |
Description: A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
hlomcmat | ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hloml 38691 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML) | |
2 | hlclat 38692 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
3 | hlatl 38694 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
4 | 1, 2, 3 | 3jca 1127 | 1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2105 CLatccla 18461 OMLcoml 38509 AtLatcal 38598 HLchlt 38684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7415 df-cvlat 38656 df-hlat 38685 |
This theorem is referenced by: hlatmstcOLDN 38732 hlatle 38733 hlrelat1 38735 pmaple 39096 pol1N 39245 polpmapN 39247 pmaplubN 39259 |
Copyright terms: Public domain | W3C validator |