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Theorem List for Metamath Proof Explorer - 36701-36800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcvlatexch3 36701 Atom exchange property. (Contributed by NM, 29-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄𝑃𝑅)) → (𝑃 (𝑄 𝑅) → (𝑃 𝑄) = (𝑃 𝑅)))

Theoremcvlcvr1 36702 The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 30179 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋𝐵𝑃𝐴) → (¬ 𝑃 𝑋𝑋𝐶(𝑋 𝑃)))

Theoremcvlcvrp 36703 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 30199 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋𝐵𝑃𝐴) → ((𝑋 𝑃) = 0𝑋𝐶(𝑋 𝑃)))

Theoremcvlatcvr1 36704 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄𝑃𝐶(𝑃 𝑄)))

Theoremcvlatcvr2 36705 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄𝑃𝐶(𝑄 𝑃)))

Theoremcvlsupr2 36706 Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄. (Contributed by NM, 5-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄))))

Theoremcvlsupr3 36707 Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄, which can replace the superposition part of ishlat1 36715, (𝑥𝑦 → ∃𝑧𝐴(𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦)) ), with the simpler 𝑧𝐴(𝑥 𝑧) = (𝑦 𝑧) as shown in ishlat3N 36717. (Contributed by NM, 5-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑅) = (𝑄 𝑅) ↔ (𝑃𝑄 → (𝑅𝑃𝑅𝑄𝑅 (𝑃 𝑄)))))

Theoremcvlsupr4 36708 Consequence of superposition condition (𝑃 𝑅) = (𝑄 𝑅). (Contributed by NM, 9-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅 (𝑃 𝑄))

Theoremcvlsupr5 36709 Consequence of superposition condition (𝑃 𝑅) = (𝑄 𝑅). (Contributed by NM, 9-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑃)

Theoremcvlsupr6 36710 Consequence of superposition condition (𝑃 𝑅) = (𝑄 𝑅). (Contributed by NM, 9-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → 𝑅𝑄)

Theoremcvlsupr7 36711 Consequence of superposition condition (𝑃 𝑅) = (𝑄 𝑅). (Contributed by NM, 24-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑅 𝑄))

Theoremcvlsupr8 36712 Consequence of superposition condition (𝑃 𝑅) = (𝑄 𝑅). (Contributed by NM, 24-Nov-2012.)
𝐴 = (Atoms‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ (𝑃 𝑅) = (𝑄 𝑅))) → (𝑃 𝑄) = (𝑃 𝑅))

20.24.11  Hilbert lattices

Syntaxchlt 36713 Extend class notation with Hilbert lattices.
class HL

Definitiondf-hlat 36714* Define the class of Hilbert lattices, which are complete, atomic lattices satisfying the superposition principle and minimum height. (Contributed by NM, 5-Nov-2012.)
HL = {𝑙 ∈ ((OML ∩ CLat) ∩ CvLat) ∣ (∀𝑎 ∈ (Atoms‘𝑙)∀𝑏 ∈ (Atoms‘𝑙)(𝑎𝑏 → ∃𝑐 ∈ (Atoms‘𝑙)(𝑐𝑎𝑐𝑏𝑐(le‘𝑙)(𝑎(join‘𝑙)𝑏))) ∧ ∃𝑎 ∈ (Base‘𝑙)∃𝑏 ∈ (Base‘𝑙)∃𝑐 ∈ (Base‘𝑙)(((0.‘𝑙)(lt‘𝑙)𝑎𝑎(lt‘𝑙)𝑏) ∧ (𝑏(lt‘𝑙)𝑐𝑐(lt‘𝑙)(1.‘𝑙))))}

Theoremishlat1 36715* The predicate "is a Hilbert lattice", which is: is orthomodular (𝐾 ∈ OML), complete (𝐾 ∈ CLat), atomic and satisfies the exchange (or covering) property (𝐾 ∈ CvLat), satisfies the superposition principle, and has a minimum height of 4 (witnessed here by 0, x, y, z, 1). (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))

Theoremishlat2 36716* The predicate "is a Hilbert lattice". Here we replace 𝐾 ∈ CvLat with the weaker 𝐾 ∈ AtLat and show the exchange property explicitly. (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥))) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))

Theoremishlat3N 36717* The predicate "is a Hilbert lattice". Note that the superposition principle is expressed in the compact form 𝑧𝐴(𝑥 𝑧) = (𝑦 𝑧). The exchange property and atomicity are provided by 𝐾 ∈ CvLat, and "minimum height 4" is shown explicitly. (Contributed by NM, 8-Nov-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥 𝑧) = (𝑦 𝑧) ∧ ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))))

TheoremishlatiN 36718* Properties that determine a Hilbert lattice. (Contributed by NM, 13-Nov-2011.) (New usage is discouraged.)
𝐾 ∈ OML    &   𝐾 ∈ CLat    &   𝐾 ∈ AtLat    &   𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑥𝐴𝑦𝐴 ((𝑥𝑦 → ∃𝑧𝐴 (𝑧𝑥𝑧𝑦𝑧 (𝑥 𝑦))) ∧ ∀𝑧𝐵 ((¬ 𝑥 𝑧𝑥 (𝑧 𝑦)) → 𝑦 (𝑧 𝑥)))    &   𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 ))       𝐾 ∈ HL

Theoremhlomcmcv 36719 A Hilbert lattice is orthomodular, complete, and has the covering (exchange) property. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))

Theoremhloml 36720 A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ OML)

Theoremhlclat 36721 A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ CLat)

Theoremhlcvl 36722 A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL → 𝐾 ∈ CvLat)

Theoremhlatl 36723 A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ AtLat)

Theoremhlol 36724 A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ OL)

Theoremhlop 36725 A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ OP)

Theoremhllat 36726 A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ Lat)

Theoremhllatd 36727 Deduction form of hllat 36726. A Hilbert lattice is a lattice. (Contributed by BJ, 14-Aug-2022.)
(𝜑𝐾 ∈ HL)       (𝜑𝐾 ∈ Lat)

Theoremhlomcmat 36728 A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))

Theoremhlpos 36729 A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL → 𝐾 ∈ Poset)

Theoremhlatjcl 36730 Closure of join operation. Frequently-used special case of latjcl 17670 for atoms. (Contributed by NM, 15-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 𝑌) ∈ 𝐵)

Theoremhlatjcom 36731 Commutatitivity of join operation. Frequently-used special case of latjcom 17678 for atoms. (Contributed by NM, 15-Jun-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 𝑌) = (𝑌 𝑋))

Theoremhlatjidm 36732 Idempotence of join operation. Frequently-used special case of latjcom 17678 for atoms. (Contributed by NM, 15-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑋 𝑋) = 𝑋)

Theoremhlatjass 36733 Lattice join is associative. Frequently-used special case of latjass 17714 for atoms. (Contributed by NM, 27-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = (𝑃 (𝑄 𝑅)))

Theoremhlatj12 36734 Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 17716 for atoms. (Contributed by NM, 4-Jun-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 (𝑄 𝑅)) = (𝑄 (𝑃 𝑅)))

Theoremhlatj32 36735 Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 17716 for atoms. (Contributed by NM, 21-Jul-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑃 𝑅) 𝑄))

Theoremhlatjrot 36736 Rotate lattice join of 3 classes. Frequently-used special case of latjrot 17719 for atoms. (Contributed by NM, 2-Aug-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑅 𝑃) 𝑄))

Theoremhlatj4 36737 Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 17720 for atoms. (Contributed by NM, 9-Aug-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆)) = ((𝑃 𝑅) (𝑄 𝑆)))

Theoremhlatlej1 36738 A join's first argument is less than or equal to the join. Special case of latlej1 17679 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))

Theoremhlatlej2 36739 A join's second argument is less than or equal to the join. Special case of latlej2 17680 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))

TheoremglbconN 36740* De Morgan's law for GLB and LUB. This holds in any complete ortholattice, although we assume HL for convenience. (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐵) → (𝐺𝑆) = ( ‘(𝑈‘{𝑥𝐵 ∣ ( 𝑥) ∈ 𝑆})))

TheoremglbconxN 36741* De Morgan's law for GLB and LUB. Index-set version of glbconN 36740, where we read 𝑆 as 𝑆(𝑖). (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ HL ∧ ∀𝑖𝐼 𝑆𝐵) → (𝐺‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = 𝑆}) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑖𝐼 𝑥 = ( 𝑆)})))

Theorematnlej1 36742 If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑄)

Theorematnlej2 36743 If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑄 𝑅)) → 𝑃𝑅)

Theoremhlsuprexch 36744* A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ((𝑃𝑄 → ∃𝑧𝐴 (𝑧𝑃𝑧𝑄𝑧 (𝑃 𝑄))) ∧ ∀𝑧𝐵 ((¬ 𝑃 𝑧𝑃 (𝑧 𝑄)) → 𝑄 (𝑧 𝑃))))

Theoremhlexch1 36745 A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Theoremhlexch2 36746 A Hilbert lattice has the exchange property. (Contributed by NM, 6-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) → 𝑄 (𝑃 𝑋)))

Theoremhlexchb1 36747 A Hilbert lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Theoremhlexchb2 36748 A Hilbert lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) ↔ (𝑃 𝑋) = (𝑄 𝑋)))

Theoremhlsupr 36749* A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄)))

Theoremhlsupr2 36750* A Hilbert lattice has the superposition property. (Contributed by NM, 25-Nov-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ∃𝑟𝐴 (𝑃 𝑟) = (𝑄 𝑟))

Theoremhlhgt4 36751* A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ HL → ∃𝑥𝐵𝑦𝐵𝑧𝐵 (( 0 < 𝑥𝑥 < 𝑦) ∧ (𝑦 < 𝑧𝑧 < 1 )))

Theoremhlhgt2 36752* A Hilbert lattice has a height of at least 2. (Contributed by NM, 4-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ HL → ∃𝑥𝐵 ( 0 < 𝑥𝑥 < 1 ))

Theoremhl0lt1N 36753 Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)
< = (lt‘𝐾)    &    0 = (0.‘𝐾)    &    1 = (1.‘𝐾)       (𝐾 ∈ HL → 0 < 1 )

Theoremhlexch3 36754 A Hilbert lattice has the exchange property. (atexch 30205 analog.) (Contributed by NM, 15-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Theoremhlexch4N 36755 A Hilbert lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 15-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Theoremhlatexchb1 36756 A version of hlexchb1 36747 for atoms. (Contributed by NM, 15-Nov-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑄) ↔ (𝑅 𝑃) = (𝑅 𝑄)))

Theoremhlatexchb2 36757 A version of hlexchb2 36748 for atoms. (Contributed by NM, 7-Feb-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))

Theoremhlatexch1 36758 Atom exchange property. (Contributed by NM, 7-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑄) → 𝑄 (𝑅 𝑃)))

Theoremhlatexch2 36759 Atom exchange property. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) → 𝑄 (𝑃 𝑅)))

TheoremhlatmstcOLDN 36760* An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 30186 analog.) (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑈‘{𝑦𝐴𝑦 𝑋}) = 𝑋)

Theoremhlatle 36761* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 30195 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌)))

Theoremhlateq 36762* The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 30197 analog.) (Contributed by NM, 24-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌) ↔ 𝑋 = 𝑌))

Theoremhlrelat1 36763* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 30187, with swapped, analog.) (Contributed by NM, 4-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ∃𝑝𝐴𝑝 𝑋𝑝 𝑌)))

Theoremhlrelat5N 36764* An atomistic lattice with 0 is relatively atomic, using the definition in Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝𝐴 (𝑋 < (𝑋 𝑝) ∧ 𝑝 𝑌))

Theoremhlrelat 36765* A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 30188 analog.) (Contributed by NM, 4-Feb-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝𝐴 (𝑋 < (𝑋 𝑝) ∧ (𝑋 𝑝) 𝑌))

Theoremhlrelat2 36766* A consequence of relative atomicity. (chrelat2i 30189 analog.) (Contributed by NM, 5-Feb-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ ∃𝑝𝐴 (𝑝 𝑋 ∧ ¬ 𝑝 𝑌)))

TheoremexatleN 36767 A condition for an atom to be less than or equal to a lattice element. Part of proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑅 (𝑃 𝑄))) → (𝑅 𝑋𝑅 = 𝑃))

Theoremhl2at 36768* A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)
𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL → ∃𝑝𝐴𝑞𝐴 𝑝𝑞)

Theorematex 36769 At least one atom exists. (Contributed by NM, 15-Jul-2012.)
𝐴 = (Atoms‘𝐾)       (𝐾 ∈ HL → 𝐴 ≠ ∅)

TheoremintnatN 36770 If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (¬ 𝑌 𝑋 ∧ (𝑋 𝑌) ∈ 𝐴)) → ¬ 𝑌𝐴)

Theorem2llnne2N 36771 Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑅𝐴) ∧ ¬ 𝑃 (𝑅 𝑄)) → (𝑅 𝑃) ≠ (𝑅 𝑄))

Theorem2llnneN 36772 Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (𝑅 𝑃) ≠ (𝑅 𝑄))

Theoremcvr1 36773 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 30179 analog.) (Contributed by NM, 17-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) → (¬ 𝑃 𝑋𝑋𝐶(𝑋 𝑃)))

Theoremcvr2N 36774 Less-than and covers equivalence in a Hilbert lattice. (chcv2 30180 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) → (𝑋 < (𝑋 𝑃) ↔ 𝑋𝐶(𝑋 𝑃)))

Theoremhlrelat3 36775* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 36765. (Contributed by NM, 2-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝𝐴 (𝑋𝐶(𝑋 𝑝) ∧ (𝑋 𝑝) 𝑌))

Theoremcvrval3 36776* Binary relation expressing 𝑌 covers 𝑋. (Contributed by NM, 16-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ∃𝑝𝐴𝑝 𝑋 ∧ (𝑋 𝑝) = 𝑌)))

Theoremcvrval4N 36777* Binary relation expressing 𝑌 covers 𝑋. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∃𝑝𝐴 (𝑋 𝑝) = 𝑌)))

Theoremcvrval5 36778* Binary relation expressing 𝑋 covers 𝑋 𝑌. (Contributed by NM, 7-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑋 ↔ ∃𝑝𝐴𝑝 𝑌 ∧ (𝑝 (𝑋 𝑌)) = 𝑋)))

Theoremcvrp 36779 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 30199 analog.) (Contributed by NM, 18-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑃𝐴) → ((𝑋 𝑃) = 0𝑋𝐶(𝑋 𝑃)))

Theorematcvr1 36780 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄𝑃𝐶(𝑃 𝑄)))

Theorematcvr2 36781 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄𝑃𝐶(𝑄 𝑃)))

Theoremcvrexchlem 36782 Lemma for cvrexch 36783. (cvexchlem 30192 analog.) (Contributed by NM, 18-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌𝑋𝐶(𝑋 𝑌)))

Theoremcvrexch 36783 A Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (cvexchi 30193 analog.) (Contributed by NM, 18-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌𝑋𝐶(𝑋 𝑌)))

Theoremcvratlem 36784 Lemma for cvrat 36785. (atcvatlem 30209 analog.) (Contributed by NM, 22-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑋0𝑋 < (𝑃 𝑄))) → (¬ 𝑃(le‘𝐾)𝑋𝑋𝐴))

Theoremcvrat 36785 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 30210 analog.) (Contributed by NM, 22-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋0𝑋 < (𝑃 𝑄)) → 𝑋𝐴))

Theoremltltncvr 36786 A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → ¬ 𝑋𝐶𝑍))

Theoremltcvrntr 36787 Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))

Theoremcvrntr 36788 The covers relation is not transitive. (cvntr 30116 analog.) (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))

Theorematcvr0eq 36789 The covers relation is not transitive. (atcv0eq 30203 analog.) (Contributed by NM, 29-Nov-2011.)
= (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → ( 0 𝐶(𝑃 𝑄) ↔ 𝑃 = 𝑄))

Theoremlnnat 36790 A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄 ↔ ¬ (𝑃 𝑄) ∈ 𝐴))

Theorematcvrj0 36791 Two atoms covering the zero subspace are equal. (atcv1 30204 analog.) (Contributed by NM, 29-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ 𝑋𝐶(𝑃 𝑄)) → (𝑋 = 0𝑃 = 𝑄))

Theoremcvrat2 36792 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 30211 analog.) (Contributed by NM, 30-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ (𝑃𝑄𝑋𝐶(𝑃 𝑄))) → 𝑋𝐴)

TheorematcvrneN 36793 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
= (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄𝑅)

Theorematcvrj1 36794 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))

Theorematcvrj2b 36795 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄𝑅𝑃 (𝑄 𝑅)) ↔ 𝑃𝐶(𝑄 𝑅)))

Theorematcvrj2 36796 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))

TheorematleneN 36797 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑅𝑃 (𝑄 𝑅))) → 𝑄𝑅)

Theorematltcvr 36798 An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)
< = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 < (𝑄 𝑅) ↔ 𝑃𝐶(𝑄 𝑅)))

Theorematle 36799* Any nonzero element has an atom under it. (Contributed by NM, 28-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑋0 ) → ∃𝑝𝐴 𝑝 𝑋)

Theorematlt 36800 Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)
< = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 < (𝑃 𝑄) ↔ 𝑃𝑄))

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