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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 36497 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML) | |
2 | hlclat 36498 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
3 | hlatl 36500 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
4 | 1, 2, 3 | 3jca 1124 | 1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 ∈ wcel 2113 CLatccla 17720 OMLcoml 36315 AtLatcal 36404 HLchlt 36490 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-cvlat 36462 df-hlat 36491 |
This theorem is referenced by: hlatmstcOLDN 36537 hlatle 36538 hlrelat1 36540 pmaple 36901 pol1N 37050 polpmapN 37052 pmaplubN 37064 |
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