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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 36653 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML) | |
2 | hlclat 36654 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
3 | hlatl 36656 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
4 | 1, 2, 3 | 3jca 1125 | 1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2111 CLatccla 17709 OMLcoml 36471 AtLatcal 36560 HLchlt 36646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-cvlat 36618 df-hlat 36647 |
This theorem is referenced by: hlatmstcOLDN 36693 hlatle 36694 hlrelat1 36696 pmaple 37057 pol1N 37206 polpmapN 37208 pmaplubN 37220 |
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