Theorem hlomcmat 39485

Description: A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)

Assertion

Ref	Expression
hlomcmat	⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))

Proof of Theorem hlomcmat

Step	Hyp	Ref	Expression
1		hloml 39477	. 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
2		hlclat 39478	. 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
3		hlatl 39480	. 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
4	1, 2, 3	3jca 1128	1 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2113  CLatccla 18406  OMLcoml 39295  AtLatcal 39384  HLchlt 39470

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-ov 7355  df-cvlat 39442  df-hlat 39471

This theorem is referenced by:  hlatmstcOLDN  39517  hlatle  39518  hlrelat1  39520  pmaple  39881  pol1N  40030  polpmapN  40032  pmaplubN  40044