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Mathbox for Norm Megill |
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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 35432 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML) | |
2 | hlclat 35433 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
3 | hlatl 35435 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
4 | 1, 2, 3 | 3jca 1164 | 1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 ∈ wcel 2166 CLatccla 17460 OMLcoml 35250 AtLatcal 35339 HLchlt 35425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-iota 6086 df-fv 6131 df-ov 6908 df-cvlat 35397 df-hlat 35426 |
This theorem is referenced by: hlatmstcOLDN 35472 hlatle 35473 hlrelat1 35475 pmaple 35836 pol1N 35985 polpmapN 35987 pmaplubN 35999 |
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