| 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 39358 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML) | |
| 2 | hlclat 39359 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
| 3 | hlatl 39361 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
| 4 | 1, 2, 3 | 3jca 1129 | 1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2108 CLatccla 18543 OMLcoml 39176 AtLatcal 39265 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-cvlat 39323 df-hlat 39352 |
| This theorem is referenced by: hlatmstcOLDN 39399 hlatle 39400 hlrelat1 39402 pmaple 39763 pol1N 39912 polpmapN 39914 pmaplubN 39926 |
| Copyright terms: Public domain | W3C validator |