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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 36495 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML) | |
2 | hlclat 36496 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat) | |
3 | hlatl 36498 | . 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 2114 CLatccla 17719 OMLcoml 36313 AtLatcal 36402 HLchlt 36488 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-cvlat 36460 df-hlat 36489 |
This theorem is referenced by: hlatmstcOLDN 36535 hlatle 36536 hlrelat1 36538 pmaple 36899 pol1N 37048 polpmapN 37050 pmaplubN 37062 |
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