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Theorem List for Metamath Proof Explorer - 37601-37700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcmtbr2N 37601 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 30324 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    &   πΆ = (cmβ€˜πΎ)    β‡’   ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ 𝑋 = ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)))))
 
Theoremcmtbr3N 37602 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 30336 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    &   πΆ = (cmβ€˜πΎ)    β‡’   ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) = (𝑋 ∧ π‘Œ)))
 
Theoremcmtbr4N 37603 Alternate definition for the commutes relation. (cmbr4i 30329 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    &   πΆ = (cmβ€˜πΎ)    β‡’   ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ (𝑋 ∧ (( βŠ₯ β€˜π‘‹) ∨ π‘Œ)) ≀ π‘Œ))
 
TheoremlecmtN 37604 Ordered elements commute. (lecmi 30330 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΆ = (cmβ€˜πΎ)    β‡’   ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ π‘‹πΆπ‘Œ))
 
TheoremcmtidN 37605 Any element commutes with itself. (cmidi 30338 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   πΆ = (cmβ€˜πΎ)    β‡’   ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡) β†’ 𝑋𝐢𝑋)
 
Theoremomlfh1N 37606 Foulis-Holland Theorem, part 1. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Part of Theorem 5 in [Kalmbach] p. 25. (fh1 30346 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   πΆ = (cmβ€˜πΎ)    β‡’   ((𝐾 ∈ OML ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ (π‘‹πΆπ‘Œ ∧ 𝑋𝐢𝑍)) β†’ (𝑋 ∧ (π‘Œ ∨ 𝑍)) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑍)))
 
Theoremomlfh3N 37607 Foulis-Holland Theorem, part 3. Dual of omlfh1N 37606. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   πΆ = (cmβ€˜πΎ)    β‡’   ((𝐾 ∈ OML ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ (π‘‹πΆπ‘Œ ∧ 𝑋𝐢𝑍)) β†’ (𝑋 ∨ (π‘Œ ∧ 𝑍)) = ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ 𝑍)))
 
Theoremomlmod1i2N 37608 Analogue of modular law atmod1i2 38208 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   πΆ = (cmβ€˜πΎ)    β‡’   ((𝐾 ∈ OML ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ (𝑋 ≀ 𝑍 ∧ π‘ŒπΆπ‘)) β†’ (𝑋 ∨ (π‘Œ ∧ 𝑍)) = ((𝑋 ∨ π‘Œ) ∧ 𝑍))
 
TheoremomlspjN 37609 Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    β‡’   ((𝐾 ∈ OML ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ ((𝑋 ∨ ( βŠ₯ β€˜π‘Œ)) ∧ π‘Œ) = 𝑋)
 
21.24.10  Atomic lattices with covering property
 
Syntaxccvr 37610 Extend class notation with covers relation.
class β‹–
 
Syntaxcatm 37611 Extend class notation with atoms.
class Atoms
 
Syntaxcal 37612 Extend class notation with atomic lattices.
class AtLat
 
Syntaxclc 37613 Extend class notation with lattices with the covering property.
class CvLat
 
Definitiondf-covers 37614* Define the covers relation ("is covered by") for posets. "π‘Ž is covered by 𝑏 " means that π‘Ž is strictly less than 𝑏 and there is nothing in between. See cvrval 37617 for the relation form. (Contributed by NM, 18-Sep-2011.)
β‹– = (𝑝 ∈ V ↦ {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (Baseβ€˜π‘) ∧ 𝑏 ∈ (Baseβ€˜π‘)) ∧ π‘Ž(ltβ€˜π‘)𝑏 ∧ Β¬ βˆƒπ‘§ ∈ (Baseβ€˜π‘)(π‘Ž(ltβ€˜π‘)𝑧 ∧ 𝑧(ltβ€˜π‘)𝑏))})
 
Definitiondf-ats 37615* Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)
Atoms = (𝑝 ∈ V ↦ {π‘Ž ∈ (Baseβ€˜π‘) ∣ (0.β€˜π‘)( β‹– β€˜π‘)π‘Ž})
 
Theoremcvrfval 37616* Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   (𝐾 ∈ 𝐴 β†’ 𝐢 = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ π‘₯ < 𝑦 ∧ Β¬ βˆƒπ‘§ ∈ 𝐡 (π‘₯ < 𝑧 ∧ 𝑧 < 𝑦))})
 
Theoremcvrval 37617* Binary relation expressing 𝐡 covers 𝐴, which means that 𝐡 is larger than 𝐴 and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (cvbr 31010 analog.) (Contributed by NM, 18-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ (𝑋 < π‘Œ ∧ Β¬ βˆƒπ‘§ ∈ 𝐡 (𝑋 < 𝑧 ∧ 𝑧 < π‘Œ))))
 
Theoremcvrlt 37618 The covers relation implies the less-than relation. (cvpss 31013 analog.) (Contributed by NM, 8-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ π‘‹πΆπ‘Œ) β†’ 𝑋 < π‘Œ)
 
Theoremcvrnbtwn 37619 There is no element between the two arguments of the covers relation. (cvnbtwn 31014 analog.) (Contributed by NM, 18-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ π‘‹πΆπ‘Œ) β†’ Β¬ (𝑋 < 𝑍 ∧ 𝑍 < π‘Œ))
 
Theoremncvr1 37620 No element covers the lattice unity. (Contributed by NM, 8-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    1 = (1.β€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ Β¬ 1 𝐢𝑋)
 
TheoremcvrletrN 37621 Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((π‘‹πΆπ‘Œ ∧ π‘Œ ≀ 𝑍) β†’ 𝑋 < 𝑍))
 
Theoremcvrval2 37622* Binary relation expressing π‘Œ covers 𝑋. Definition of covers in [Kalmbach] p. 15. (cvbr2 31011 analog.) (Contributed by NM, 16-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ (𝑋 < π‘Œ ∧ βˆ€π‘§ ∈ 𝐡 ((𝑋 < 𝑧 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 = π‘Œ))))
 
Theoremcvrnbtwn2 37623 The covers relation implies no in-betweenness. (cvnbtwn2 31015 analog.) (Contributed by NM, 17-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ π‘‹πΆπ‘Œ) β†’ ((𝑋 < 𝑍 ∧ 𝑍 ≀ π‘Œ) ↔ 𝑍 = π‘Œ))
 
Theoremcvrnbtwn3 37624 The covers relation implies no in-betweenness. (cvnbtwn3 31016 analog.) (Contributed by NM, 4-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ π‘‹πΆπ‘Œ) β†’ ((𝑋 ≀ 𝑍 ∧ 𝑍 < π‘Œ) ↔ 𝑋 = 𝑍))
 
Theoremcvrcon3b 37625 Contraposition law for the covers relation. (cvcon3 31012 analog.) (Contributed by NM, 4-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ ( βŠ₯ β€˜π‘Œ)𝐢( βŠ₯ β€˜π‘‹)))
 
Theoremcvrle 37626 The covers relation implies the "less than or equal to" relation. (Contributed by NM, 12-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ π‘‹πΆπ‘Œ) β†’ 𝑋 ≀ π‘Œ)
 
Theoremcvrnbtwn4 37627 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 31017 analog.) (Contributed by NM, 18-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ π‘‹πΆπ‘Œ) β†’ ((𝑋 ≀ 𝑍 ∧ 𝑍 ≀ π‘Œ) ↔ (𝑋 = 𝑍 ∨ 𝑍 = π‘Œ)))
 
Theoremcvrnle 37628 The covers relation implies the negation of the converse "less than or equal to" relation. (Contributed by NM, 18-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ π‘‹πΆπ‘Œ) β†’ Β¬ π‘Œ ≀ 𝑋)
 
Theoremcvrne 37629 The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ π‘‹πΆπ‘Œ) β†’ 𝑋 β‰  π‘Œ)
 
TheoremcvrnrefN 37630 The covers relation is not reflexive. (cvnref 31019 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ Β¬ 𝑋𝐢𝑋)
 
Theoremcvrcmp 37631 If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ (𝑍𝐢𝑋 ∧ π‘πΆπ‘Œ)) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 = π‘Œ))
 
Theoremcvrcmp2 37632 If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ OP ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ (𝑋𝐢𝑍 ∧ π‘ŒπΆπ‘)) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 = π‘Œ))
 
Theorempats 37633* The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (𝐾 ∈ 𝐷 β†’ 𝐴 = {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯})
 
Theoremisat 37634 The predicate "is an atom". (ela 31067 analog.) (Contributed by NM, 18-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (𝐾 ∈ 𝐷 β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ 0 𝐢𝑃)))
 
Theoremisat2 37635 The predicate "is an atom". (elatcv0 31069 analog.) (Contributed by NM, 18-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐡) β†’ (𝑃 ∈ 𝐴 ↔ 0 𝐢𝑃))
 
Theorematcvr0 37636 An atom covers zero. (atcv0 31070 analog.) (Contributed by NM, 4-Nov-2011.)
0 = (0.β€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐴) β†’ 0 𝐢𝑃)
 
Theorematbase 37637 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 31072 analog.) (Contributed by NM, 10-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
 
Theorematssbase 37638 The set of atoms is a subset of the base set. (atssch 31071 analog.) (Contributed by NM, 21-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   π΄ βŠ† 𝐡
 
Theorem0ltat 37639 An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)
0 = (0.β€˜πΎ)    &    < = (ltβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) β†’ 0 < 𝑃)
 
Theoremleatb 37640 A poset element less than or equal to an atom equals either zero or the atom. (atss 31074 analog.) (Contributed by NM, 17-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) β†’ (𝑋 ≀ 𝑃 ↔ (𝑋 = 𝑃 ∨ 𝑋 = 0 )))
 
Theoremleat 37641 A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≀ 𝑃) β†’ (𝑋 = 𝑃 ∨ 𝑋 = 0 ))
 
Theoremleat2 37642 A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 β‰  0 ∧ 𝑋 ≀ 𝑃)) β†’ 𝑋 = 𝑃)
 
Theoremleat3 37643 A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≀ 𝑃) β†’ (𝑋 ∈ 𝐴 ∨ 𝑋 = 0 ))
 
Theoremmeetat 37644 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) β†’ ((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 ))
 
Theoremmeetat2 37645 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) β†’ ((𝑋 ∧ 𝑃) ∈ 𝐴 ∨ (𝑋 ∧ 𝑃) = 0 ))
 
Definitiondf-atl 37646* Define the class of atomic lattices, in which every nonzero element is greater than or equal to an atom. We also ensure the existence of a lattice zero, since a lattice by itself may not have a zero. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
AtLat = {π‘˜ ∈ Lat ∣ ((Baseβ€˜π‘˜) ∈ dom (glbβ€˜π‘˜) ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)(π‘₯ β‰  (0.β€˜π‘˜) β†’ βˆƒπ‘ ∈ (Atomsβ€˜π‘˜)𝑝(leβ€˜π‘˜)π‘₯))}
 
Theoremisatl 37647* The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐡 ∈ dom 𝐺 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)))
 
Theorematllat 37648 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
(𝐾 ∈ AtLat β†’ 𝐾 ∈ Lat)
 
Theorematlpos 37649 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ AtLat β†’ 𝐾 ∈ Poset)
 
Theorematl0dm 37650 Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    β‡’   (𝐾 ∈ AtLat β†’ 𝐡 ∈ dom 𝐺)
 
Theorematl0cl 37651 An atomic lattice has a zero element. We can use this in place of op0cl 37532 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
𝐡 = (Baseβ€˜πΎ)    &    0 = (0.β€˜πΎ)    β‡’   (𝐾 ∈ AtLat β†’ 0 ∈ 𝐡)
 
Theorematl0le 37652 Orthoposet zero is less than or equal to any element. (ch0le 30169 analog.) (Contributed by NM, 12-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐡) β†’ 0 ≀ 𝑋)
 
Theorematlle0 37653 An element less than or equal to zero equals zero. (chle0 30171 analog.) (Contributed by NM, 21-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ≀ 0 ↔ 𝑋 = 0 ))
 
Theorematlltn0 37654 A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    0 = (0.β€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐡) β†’ ( 0 < 𝑋 ↔ 𝑋 β‰  0 ))
 
Theoremisat3 37655* The predicate "is an atom". (elat2 31068 analog.) (Contributed by NM, 27-Apr-2014.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (𝐾 ∈ AtLat β†’ (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐡 ∧ 𝑃 β‰  0 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ ≀ 𝑃 β†’ (π‘₯ = 𝑃 ∨ π‘₯ = 0 )))))
 
Theorematn0 37656 An atom is not zero. (atne0 31073 analog.) (Contributed by NM, 5-Nov-2012.)
0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 𝑃 β‰  0 )
 
Theorematnle0 37657 An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)
≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ Β¬ 𝑃 ≀ 0 )
 
Theorematlen0 37658 A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≀ 𝑋) β†’ 𝑋 β‰  0 )
 
Theorematcmp 37659 If two atoms are comparable, they are equal. (atsseq 31075 analog.) (Contributed by NM, 13-Oct-2011.)
≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ≀ 𝑄 ↔ 𝑃 = 𝑄))
 
Theorematncmp 37660 Frequently-used variation of atcmp 37659. (Contributed by NM, 29-Jun-2012.)
≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (Β¬ 𝑃 ≀ 𝑄 ↔ 𝑃 β‰  𝑄))
 
Theorematnlt 37661 Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.)
< = (ltβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ Β¬ 𝑃 < 𝑄)
 
Theorematcvreq0 37662 An element covered by an atom must be zero. (atcveq0 31076 analog.) (Contributed by NM, 4-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) β†’ (𝑋𝐢𝑃 ↔ 𝑋 = 0 ))
 
TheorematncvrN 37663 Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
𝐢 = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ Β¬ 𝑃𝐢𝑄)
 
Theorematlex 37664* Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 31088 analog.) (Contributed by NM, 21-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ 𝑋)
 
Theorematnle 37665 Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 31104 analog.) (Contributed by NM, 5-Nov-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (Β¬ 𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))
 
Theorematnem0 37666 The meet of distinct atoms is zero. (atnemeq0 31105 analog.) (Contributed by NM, 5-Nov-2012.)
∧ = (meetβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 β‰  𝑄 ↔ (𝑃 ∧ 𝑄) = 0 ))
 
Theorematlatmstc 37667* 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 31090 analog.) (Contributed by NM, 5-Nov-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    1 = (lubβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐡) β†’ ( 1 β€˜{𝑦 ∈ 𝐴 ∣ 𝑦 ≀ 𝑋}) = 𝑋)
 
Theorematlatle 37668* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 31099 analog.) (Contributed by NM, 5-Nov-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ βˆ€π‘ ∈ 𝐴 (𝑝 ≀ 𝑋 β†’ 𝑝 ≀ π‘Œ)))
 
Theorematlrelat1 37669* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 31091, with ∧ swapped, analog.) (Contributed by NM, 4-Dec-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 < π‘Œ β†’ βˆƒπ‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑋 ∧ 𝑝 ≀ π‘Œ)))
 
Definitiondf-cvlat 37670* Define the class of atomic lattices with the covering property. (This is actually the exchange property, but they are equivalent. The literature usually uses the covering property terminology.) (Contributed by NM, 5-Nov-2012.)
CvLat = {π‘˜ ∈ AtLat ∣ βˆ€π‘Ž ∈ (Atomsβ€˜π‘˜)βˆ€π‘ ∈ (Atomsβ€˜π‘˜)βˆ€π‘ ∈ (Baseβ€˜π‘˜)((Β¬ π‘Ž(leβ€˜π‘˜)𝑐 ∧ π‘Ž(leβ€˜π‘˜)(𝑐(joinβ€˜π‘˜)𝑏)) β†’ 𝑏(leβ€˜π‘˜)(𝑐(joinβ€˜π‘˜)π‘Ž))}
 
Theoremiscvlat 37671* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ βˆ€π‘ ∈ 𝐴 βˆ€π‘ž ∈ 𝐴 βˆ€π‘₯ ∈ 𝐡 ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝))))
 
Theoremiscvlat2N 37672* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ βˆ€π‘ ∈ 𝐴 βˆ€π‘ž ∈ 𝐴 βˆ€π‘₯ ∈ 𝐡 (((𝑝 ∧ π‘₯) = 0 ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝))))
 
Theoremcvlatl 37673 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ CvLat β†’ 𝐾 ∈ AtLat)
 
Theoremcvllat 37674 An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ CvLat β†’ 𝐾 ∈ Lat)
 
TheoremcvlposN 37675 An atomic lattice with the covering property is a poset. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
(𝐾 ∈ CvLat β†’ 𝐾 ∈ Poset)
 
Theoremcvlexch1 37676 An atomic covering lattice has the exchange property. (Contributed by NM, 6-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))
 
Theoremcvlexch2 37677 An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑄 ∨ 𝑋) β†’ 𝑄 ≀ (𝑃 ∨ 𝑋)))
 
Theoremcvlexchb1 37678 An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
 
Theoremcvlexchb2 37679 An atomic covering lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑄 ∨ 𝑋) ↔ (𝑃 ∨ 𝑋) = (𝑄 ∨ 𝑋)))
 
Theoremcvlexch3 37680 An atomic covering lattice has the exchange property. (atexch 31109 analog.) (Contributed by NM, 5-Nov-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))
 
Theoremcvlexch4N 37681 An atomic covering lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
 
Theoremcvlatexchb1 37682 A version of cvlexchb1 37678 for atoms. (Contributed by NM, 5-Nov-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑅) β†’ (𝑃 ≀ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄)))
 
Theoremcvlatexchb2 37683 A version of cvlexchb2 37679 for atoms. (Contributed by NM, 5-Nov-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑅) β†’ (𝑃 ≀ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
 
Theoremcvlatexch1 37684 Atom exchange property. (Contributed by NM, 5-Nov-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑅) β†’ (𝑃 ≀ (𝑅 ∨ 𝑄) β†’ 𝑄 ≀ (𝑅 ∨ 𝑃)))
 
Theoremcvlatexch2 37685 Atom exchange property. (Contributed by NM, 5-Nov-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑅) β†’ (𝑃 ≀ (𝑄 ∨ 𝑅) β†’ 𝑄 ≀ (𝑃 ∨ 𝑅)))
 
Theoremcvlatexch3 37686 Atom exchange property. (Contributed by NM, 29-Nov-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ (𝑃 ≀ (𝑄 ∨ 𝑅) β†’ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)))
 
Theoremcvlcvr1 37687 The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 31083 analog.) (Contributed by NM, 5-Nov-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) β†’ (Β¬ 𝑃 ≀ 𝑋 ↔ 𝑋𝐢(𝑋 ∨ 𝑃)))
 
Theoremcvlcvrp 37688 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 31103 analog.) (Contributed by NM, 5-Nov-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) β†’ ((𝑋 ∧ 𝑃) = 0 ↔ 𝑋𝐢(𝑋 ∨ 𝑃)))
 
Theoremcvlatcvr1 37689 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)
∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 β‰  𝑄 ↔ 𝑃𝐢(𝑃 ∨ 𝑄)))
 
Theoremcvlatcvr2 37690 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)
∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 β‰  𝑄 ↔ 𝑃𝐢(𝑄 ∨ 𝑃)))
 
Theoremcvlsupr2 37691 Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄. (Contributed by NM, 5-Nov-2012.)
𝐴 = (Atomsβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))))
 
Theoremcvlsupr3 37692 Two equivalent ways of expressing that 𝑅 is a superposition of 𝑃 and 𝑄, which can replace the superposition part of ishlat1 37700, (π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴(𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦)) ), with the simpler βˆƒπ‘§ ∈ 𝐴(π‘₯ ∨ 𝑧) = (𝑦 ∨ 𝑧) as shown in ishlat3N 37702. (Contributed by NM, 5-Nov-2012.)
𝐴 = (Atomsβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑃 β‰  𝑄 β†’ (𝑅 β‰  𝑃 ∧ 𝑅 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄)))))
 
Theoremcvlsupr4 37693 Consequence of superposition condition (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅). (Contributed by NM, 9-Nov-2012.)
𝐴 = (Atomsβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) β†’ 𝑅 ≀ (𝑃 ∨ 𝑄))
 
Theoremcvlsupr5 37694 Consequence of superposition condition (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅). (Contributed by NM, 9-Nov-2012.)
𝐴 = (Atomsβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) β†’ 𝑅 β‰  𝑃)
 
Theoremcvlsupr6 37695 Consequence of superposition condition (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅). (Contributed by NM, 9-Nov-2012.)
𝐴 = (Atomsβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) β†’ 𝑅 β‰  𝑄)
 
Theoremcvlsupr7 37696 Consequence of superposition condition (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅). (Contributed by NM, 24-Nov-2012.)
𝐴 = (Atomsβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) β†’ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄))
 
Theoremcvlsupr8 37697 Consequence of superposition condition (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅). (Contributed by NM, 24-Nov-2012.)
𝐴 = (Atomsβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) β†’ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))
 
21.24.11  Hilbert lattices
 
Syntaxchlt 37698 Extend class notation with Hilbert lattices.
class HL
 
Definitiondf-hlat 37699* 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) ∣ (βˆ€π‘Ž ∈ (Atomsβ€˜π‘™)βˆ€π‘ ∈ (Atomsβ€˜π‘™)(π‘Ž β‰  𝑏 β†’ βˆƒπ‘ ∈ (Atomsβ€˜π‘™)(𝑐 β‰  π‘Ž ∧ 𝑐 β‰  𝑏 ∧ 𝑐(leβ€˜π‘™)(π‘Ž(joinβ€˜π‘™)𝑏))) ∧ βˆƒπ‘Ž ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)βˆƒπ‘ ∈ (Baseβ€˜π‘™)(((0.β€˜π‘™)(ltβ€˜π‘™)π‘Ž ∧ π‘Ž(ltβ€˜π‘™)𝑏) ∧ (𝑏(ltβ€˜π‘™)𝑐 ∧ 𝑐(ltβ€˜π‘™)(1.β€˜π‘™))))}
 
Theoremishlat1 37700* 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &    1 = (1.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
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