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Theorem List for Metamath Proof Explorer - 37701-37800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremishlat2 37701* 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.)
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 ∈ AtLat) ∧ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯))) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
 
Theoremishlat3N 37702* The predicate "is a Hilbert lattice". Note that the superposition principle is expressed in the compact form βˆƒπ‘§ ∈ 𝐴(π‘₯ ∨ 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.)
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 βˆƒπ‘§ ∈ 𝐴 (π‘₯ ∨ 𝑧) = (𝑦 ∨ 𝑧) ∧ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))))
 
TheoremishlatiN 37703* Properties that determine a Hilbert lattice. (Contributed by NM, 13-Nov-2011.) (New usage is discouraged.)
𝐾 ∈ OML    &   πΎ ∈ CLat    &   πΎ ∈ AtLat    &   π΅ = (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)    &   βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((π‘₯ β‰  𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  π‘₯ ∧ 𝑧 β‰  𝑦 ∧ 𝑧 ≀ (π‘₯ ∨ 𝑦))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ π‘₯ ≀ 𝑧 ∧ π‘₯ ≀ (𝑧 ∨ 𝑦)) β†’ 𝑦 ≀ (𝑧 ∨ π‘₯)))    &   βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 ))    β‡’   πΎ ∈ HL
 
Theoremhlomcmcv 37704 A Hilbert lattice is orthomodular, complete, and has the covering (exchange) property. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL β†’ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat))
 
Theoremhloml 37705 A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL β†’ 𝐾 ∈ OML)
 
Theoremhlclat 37706 A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
 
Theoremhlcvl 37707 A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL β†’ 𝐾 ∈ CvLat)
 
Theoremhlatl 37708 A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
 
Theoremhlol 37709 A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL β†’ 𝐾 ∈ OL)
 
Theoremhlop 37710 A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL β†’ 𝐾 ∈ OP)
 
Theoremhllat 37711 A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
 
Theoremhllatd 37712 Deduction form of hllat 37711. A Hilbert lattice is a lattice. (Contributed by BJ, 14-Aug-2022.)
(πœ‘ β†’ 𝐾 ∈ HL)    β‡’   (πœ‘ β†’ 𝐾 ∈ Lat)
 
Theoremhlomcmat 37713 A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ HL β†’ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
 
Theoremhlpos 37714 A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)
(𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
 
Theoremhlatjcl 37715 Closure of join operation. Frequently-used special case of latjcl 18263 for atoms. (Contributed by NM, 15-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐴) β†’ (𝑋 ∨ π‘Œ) ∈ 𝐡)
 
Theoremhlatjcom 37716 Commutatitivity of join operation. Frequently-used special case of latjcom 18271 for atoms. (Contributed by NM, 15-Jun-2012.)
∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐴) β†’ (𝑋 ∨ π‘Œ) = (π‘Œ ∨ 𝑋))
 
Theoremhlatjidm 37717 Idempotence of join operation. Frequently-used special case of latjcom 18271 for atoms. (Contributed by NM, 15-Jul-2012.)
∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴) β†’ (𝑋 ∨ 𝑋) = 𝑋)
 
Theoremhlatjass 37718 Lattice join is associative. Frequently-used special case of latjass 18307 for atoms. (Contributed by NM, 27-Jul-2012.)
∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
 
Theoremhlatj12 37719 Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 18309 for atoms. (Contributed by NM, 4-Jun-2012.)
∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 ∨ (𝑄 ∨ 𝑅)) = (𝑄 ∨ (𝑃 ∨ 𝑅)))
 
Theoremhlatj32 37720 Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 18309 for atoms. (Contributed by NM, 21-Jul-2012.)
∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑃 ∨ 𝑅) ∨ 𝑄))
 
Theoremhlatjrot 37721 Rotate lattice join of 3 classes. Frequently-used special case of latjrot 18312 for atoms. (Contributed by NM, 2-Aug-2012.)
∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑅 ∨ 𝑃) ∨ 𝑄))
 
Theoremhlatj4 37722 Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 18313 for atoms. (Contributed by NM, 9-Aug-2012.)
∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∨ (𝑅 ∨ 𝑆)) = ((𝑃 ∨ 𝑅) ∨ (𝑄 ∨ 𝑆)))
 
Theoremhlatlej1 37723 A join's first argument is less than or equal to the join. Special case of latlej1 18272 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
 
Theoremhlatlej2 37724 A join's second argument is less than or equal to the join. Special case of latlej2 18273 to show an atom is on a line. (Contributed by NM, 15-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑄 ≀ (𝑃 ∨ 𝑄))
 
TheoremglbconN 37725* 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 df-riota 7306. (Revised by SN, 3-Jan-2025.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐡) β†’ (πΊβ€˜π‘†) = ( βŠ₯ β€˜(π‘ˆβ€˜{π‘₯ ∈ 𝐡 ∣ ( βŠ₯ β€˜π‘₯) ∈ 𝑆})))
 
TheoremglbconNOLD 37726* Obsolete version of glbconN 37725 as of 3-Jan-2025. (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐡) β†’ (πΊβ€˜π‘†) = ( βŠ₯ β€˜(π‘ˆβ€˜{π‘₯ ∈ 𝐡 ∣ ( βŠ₯ β€˜π‘₯) ∈ 𝑆})))
 
TheoremglbconxN 37727* De Morgan's law for GLB and LUB. Index-set version of glbconN 37725, where we read 𝑆 as 𝑆(𝑖). (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ βˆ€π‘– ∈ 𝐼 𝑆 ∈ 𝐡) β†’ (πΊβ€˜{π‘₯ ∣ βˆƒπ‘– ∈ 𝐼 π‘₯ = 𝑆}) = ( βŠ₯ β€˜(π‘ˆβ€˜{π‘₯ ∣ βˆƒπ‘– ∈ 𝐼 π‘₯ = ( βŠ₯ β€˜π‘†)})))
 
Theorematnlej1 37728 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.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑃 β‰  𝑄)
 
Theorematnlej2 37729 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.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑃 β‰  𝑅)
 
Theoremhlsuprexch 37730* A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ ((𝑃 β‰  𝑄 β†’ βˆƒπ‘§ ∈ 𝐴 (𝑧 β‰  𝑃 ∧ 𝑧 β‰  𝑄 ∧ 𝑧 ≀ (𝑃 ∨ 𝑄))) ∧ βˆ€π‘§ ∈ 𝐡 ((Β¬ 𝑃 ≀ 𝑧 ∧ 𝑃 ≀ (𝑧 ∨ 𝑄)) β†’ 𝑄 ≀ (𝑧 ∨ 𝑃))))
 
Theoremhlexch1 37731 A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))
 
Theoremhlexch2 37732 A Hilbert 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 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑄 ∨ 𝑋) β†’ 𝑄 ≀ (𝑃 ∨ 𝑋)))
 
Theoremhlexchb1 37733 A Hilbert 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 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
 
Theoremhlexchb2 37734 A Hilbert 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 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ Β¬ 𝑃 ≀ 𝑋) β†’ (𝑃 ≀ (𝑄 ∨ 𝑋) ↔ (𝑃 ∨ 𝑋) = (𝑄 ∨ 𝑋)))
 
Theoremhlsupr 37735* A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
 
Theoremhlsupr2 37736* A Hilbert lattice has the superposition property. (Contributed by NM, 25-Nov-2012.)
∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ))
 
Theoremhlhgt4 37737* A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &    1 = (1.β€˜πΎ)    β‡’   (𝐾 ∈ HL β†’ βˆƒπ‘₯ ∈ 𝐡 βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐡 (( 0 < π‘₯ ∧ π‘₯ < 𝑦) ∧ (𝑦 < 𝑧 ∧ 𝑧 < 1 )))
 
Theoremhlhgt2 37738* A Hilbert lattice has a height of at least 2. (Contributed by NM, 4-Dec-2011.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &    1 = (1.β€˜πΎ)    β‡’   (𝐾 ∈ HL β†’ βˆƒπ‘₯ ∈ 𝐡 ( 0 < π‘₯ ∧ π‘₯ < 1 ))
 
Theoremhl0lt1N 37739 Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)
< = (ltβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &    1 = (1.β€˜πΎ)    β‡’   (𝐾 ∈ HL β†’ 0 < 1 )
 
Theoremhlexch3 37740 A Hilbert lattice has the exchange property. (atexch 31109 analog.) (Contributed by NM, 15-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ 𝑄 ≀ (𝑋 ∨ 𝑃)))
 
Theoremhlexch4N 37741 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
 
Theoremhlatexchb1 37742 A version of hlexchb1 37733 for atoms. (Contributed by NM, 15-Nov-2011.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑅) β†’ (𝑃 ≀ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄)))
 
Theoremhlatexchb2 37743 A version of hlexchb2 37734 for atoms. (Contributed by NM, 7-Feb-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑅) β†’ (𝑃 ≀ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
 
Theoremhlatexch1 37744 Atom exchange property. (Contributed by NM, 7-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑅) β†’ (𝑃 ≀ (𝑅 ∨ 𝑄) β†’ 𝑄 ≀ (𝑅 ∨ 𝑃)))
 
Theoremhlatexch2 37745 Atom exchange property. (Contributed by NM, 8-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑅) β†’ (𝑃 ≀ (𝑄 ∨ 𝑅) β†’ 𝑄 ≀ (𝑃 ∨ 𝑅)))
 
TheoremhlatmstcOLDN 37746* 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, 21-Oct-2011.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (π‘ˆβ€˜{𝑦 ∈ 𝐴 ∣ 𝑦 ≀ 𝑋}) = 𝑋)
 
Theoremhlatle 37747* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 31099 analog.) (Contributed by NM, 4-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ βˆ€π‘ ∈ 𝐴 (𝑝 ≀ 𝑋 β†’ 𝑝 ≀ π‘Œ)))
 
Theoremhlateq 37748* The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 31101 analog.) (Contributed by NM, 24-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘ ∈ 𝐴 (𝑝 ≀ 𝑋 ↔ 𝑝 ≀ π‘Œ) ↔ 𝑋 = π‘Œ))
 
Theoremhlrelat1 37749* 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 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 < π‘Œ β†’ βˆƒπ‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑋 ∧ 𝑝 ≀ π‘Œ)))
 
Theoremhlrelat5N 37750* 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ βˆƒπ‘ ∈ 𝐴 (𝑋 < (𝑋 ∨ 𝑝) ∧ 𝑝 ≀ π‘Œ))
 
Theoremhlrelat 37751* A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 31092 analog.) (Contributed by NM, 4-Feb-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ βˆƒπ‘ ∈ 𝐴 (𝑋 < (𝑋 ∨ 𝑝) ∧ (𝑋 ∨ 𝑝) ≀ π‘Œ))
 
Theoremhlrelat2 37752* A consequence of relative atomicity. (chrelat2i 31093 analog.) (Contributed by NM, 5-Feb-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (Β¬ 𝑋 ≀ π‘Œ ↔ βˆƒπ‘ ∈ 𝐴 (𝑝 ≀ 𝑋 ∧ Β¬ 𝑝 ≀ π‘Œ)))
 
TheoremexatleN 37753 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑅 ≀ 𝑋 ↔ 𝑅 = 𝑃))
 
Theoremhl2at 37754* A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)
𝐴 = (Atomsβ€˜πΎ)    β‡’   (𝐾 ∈ HL β†’ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 𝑝 β‰  π‘ž)
 
Theorematex 37755 At least one atom exists. (Contributed by NM, 15-Jul-2012.)
𝐴 = (Atomsβ€˜πΎ)    β‡’   (𝐾 ∈ HL β†’ 𝐴 β‰  βˆ…)
 
TheoremintnatN 37756 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (Β¬ π‘Œ ≀ 𝑋 ∧ (𝑋 ∧ π‘Œ) ∈ 𝐴)) β†’ Β¬ π‘Œ ∈ 𝐴)
 
Theorem2llnne2N 37757 Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑃 ≀ (𝑅 ∨ 𝑄)) β†’ (𝑅 ∨ 𝑃) β‰  (𝑅 ∨ 𝑄))
 
Theorem2llnneN 37758 Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑅 ∨ 𝑃) β‰  (𝑅 ∨ 𝑄))
 
Theoremcvr1 37759 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 31083 analog.) (Contributed by NM, 17-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) β†’ (Β¬ 𝑃 ≀ 𝑋 ↔ 𝑋𝐢(𝑋 ∨ 𝑃)))
 
Theoremcvr2N 37760 Less-than and covers equivalence in a Hilbert lattice. (chcv2 31084 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) β†’ (𝑋 < (𝑋 ∨ 𝑃) ↔ 𝑋𝐢(𝑋 ∨ 𝑃)))
 
Theoremhlrelat3 37761* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 37751. (Contributed by NM, 2-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ βˆƒπ‘ ∈ 𝐴 (𝑋𝐢(𝑋 ∨ 𝑝) ∧ (𝑋 ∨ 𝑝) ≀ π‘Œ))
 
Theoremcvrval3 37762* Binary relation expressing π‘Œ covers 𝑋. (Contributed by NM, 16-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ βˆƒπ‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑋 ∧ (𝑋 ∨ 𝑝) = π‘Œ)))
 
Theoremcvrval4N 37763* Binary relation expressing π‘Œ covers 𝑋. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ (𝑋 < π‘Œ ∧ βˆƒπ‘ ∈ 𝐴 (𝑋 ∨ 𝑝) = π‘Œ)))
 
Theoremcvrval5 37764* Binary relation expressing 𝑋 covers 𝑋 ∧ π‘Œ. (Contributed by NM, 7-Dec-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ π‘Œ)𝐢𝑋 ↔ βˆƒπ‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Œ ∧ (𝑝 ∨ (𝑋 ∧ π‘Œ)) = 𝑋)))
 
Theoremcvrp 37765 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 31103 analog.) (Contributed by NM, 18-Nov-2011.)
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 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴) β†’ ((𝑋 ∧ 𝑃) = 0 ↔ 𝑋𝐢(𝑋 ∨ 𝑃)))
 
Theorematcvr1 37766 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 β‰  𝑄 ↔ 𝑃𝐢(𝑃 ∨ 𝑄)))
 
Theorematcvr2 37767 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 β‰  𝑄 ↔ 𝑃𝐢(𝑄 ∨ 𝑃)))
 
Theoremcvrexchlem 37768 Lemma for cvrexch 37769. (cvexchlem 31096 analog.) (Contributed by NM, 18-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ π‘Œ)πΆπ‘Œ β†’ 𝑋𝐢(𝑋 ∨ π‘Œ)))
 
Theoremcvrexch 37769 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 31097 analog.) (Contributed by NM, 18-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∧ π‘Œ)πΆπ‘Œ ↔ 𝑋𝐢(𝑋 ∨ π‘Œ)))
 
Theoremcvratlem 37770 Lemma for cvrat 37771. (atcvatlem 31113 analog.) (Contributed by NM, 22-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) β†’ (Β¬ 𝑃(leβ€˜πΎ)𝑋 β†’ 𝑋 ∈ 𝐴))
 
Theoremcvrat 37771 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 31114 analog.) (Contributed by NM, 22-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) β†’ 𝑋 ∈ 𝐴))
 
Theoremltltncvr 37772 A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 < π‘Œ ∧ π‘Œ < 𝑍) β†’ Β¬ 𝑋𝐢𝑍))
 
Theoremltcvrntr 37773 Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 < π‘Œ ∧ π‘ŒπΆπ‘) β†’ Β¬ 𝑋𝐢𝑍))
 
Theoremcvrntr 37774 The covers relation is not transitive. (cvntr 31020 analog.) (Contributed by NM, 18-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((π‘‹πΆπ‘Œ ∧ π‘ŒπΆπ‘) β†’ Β¬ 𝑋𝐢𝑍))
 
Theorematcvr0eq 37775 The covers relation is not transitive. (atcv0eq 31107 analog.) (Contributed by NM, 29-Nov-2011.)
∨ = (joinβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ ( 0 𝐢(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
 
Theoremlnnat 37776 A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)
∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 β‰  𝑄 ↔ Β¬ (𝑃 ∨ 𝑄) ∈ 𝐴))
 
Theorematcvrj0 37777 Two atoms covering the zero subspace are equal. (atcv1 31108 analog.) (Contributed by NM, 29-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐢(𝑃 ∨ 𝑄)) β†’ (𝑋 = 0 ↔ 𝑃 = 𝑄))
 
Theoremcvrat2 37778 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 31115 analog.) (Contributed by NM, 30-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢(𝑃 ∨ 𝑄))) β†’ 𝑋 ∈ 𝐴)
 
TheorematcvrneN 37779 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑄 β‰  𝑅)
 
Theorematcvrj1 37780 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) β†’ 𝑃𝐢(𝑄 ∨ 𝑅))
 
Theorematcvrj2b 37781 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅)) ↔ 𝑃𝐢(𝑄 ∨ 𝑅)))
 
Theorematcvrj2 37782 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) β†’ 𝑃𝐢(𝑄 ∨ 𝑅))
 
TheorematleneN 37783 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) β†’ 𝑄 β‰  𝑅)
 
Theorematltcvr 37784 An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)
< = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 < (𝑄 ∨ 𝑅) ↔ 𝑃𝐢(𝑄 ∨ 𝑅)))
 
Theorematle 37785* Any nonzero element has an atom under it. (Contributed by NM, 28-Jun-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ βˆƒπ‘ ∈ 𝐴 𝑝 ≀ 𝑋)
 
Theorematlt 37786 Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)
< = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 < (𝑃 ∨ 𝑄) ↔ 𝑃 β‰  𝑄))
 
Theorematlelt 37787 Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 < 𝑋)) β†’ 𝑃 < 𝑋)
 
Theorem2atlt 37788* Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    < = (ltβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 < 𝑋) β†’ βˆƒπ‘ž ∈ 𝐴 (π‘ž β‰  𝑃 ∧ π‘ž < 𝑋))
 
TheorematexchcvrN 37789 Atom exchange property. Version of hlatexch2 37745 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   πΆ = ( β‹– β€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑅) β†’ (𝑃𝐢(𝑄 ∨ 𝑅) β†’ 𝑄𝐢(𝑃 ∨ 𝑅)))
 
TheorematexchltN 37790 Atom exchange property. Version of hlatexch2 37745 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
< = (ltβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 β‰  𝑅) β†’ (𝑃 < (𝑄 ∨ 𝑅) β†’ 𝑄 < (𝑃 ∨ 𝑅)))
 
Theoremcvrat3 37791 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 31124 analog.) (Contributed by NM, 30-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
 
Theoremcvrat4 37792* A condition implying existence of an atom with the properties shown. Lemma 3.2.20 in [PtakPulmannova] p. 68. Also Lemma 9.2(delta) in [MaedaMaeda] p. 41. (atcvat4i 31125 analog.) (Contributed by NM, 30-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))))
 
Theoremcvrat42 37793* Commuted version of cvrat4 37792. (Contributed by NM, 28-Jan-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (π‘Ÿ ∨ 𝑄))))
 
Theorem2atjm 37794 The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) = 𝑃)
 
Theorematbtwn 37795 Property of a 3rd atom 𝑅 on a line 𝑃 ∨ 𝑄 intersecting element 𝑋 at 𝑃. (Contributed by NM, 30-Jul-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑅 β‰  𝑃 ↔ Β¬ 𝑅 ≀ 𝑋))
 
TheorematbtwnexOLDN 37796* There exists a 3rd atom π‘Ÿ on a line 𝑃 ∨ 𝑄 intersecting element 𝑋 at 𝑃, such that π‘Ÿ is different from 𝑄 and not in 𝑋. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑄 ∧ Β¬ π‘Ÿ ≀ 𝑋 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
 
Theorematbtwnex 37797* Given atoms 𝑃 in 𝑋 and 𝑄 not in 𝑋, there exists an atom π‘Ÿ not in 𝑋 such that the line 𝑄 ∨ π‘Ÿ intersects 𝑋 at 𝑃. (Contributed by NM, 1-Aug-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑄 ∧ Β¬ π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))
 
Theorem3noncolr2 37798 Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑄 β‰  𝑅 ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)))
 
Theorem3noncolr1N 37799 Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑅 β‰  𝑃 ∧ Β¬ 𝑄 ≀ (𝑅 ∨ 𝑃)))
 
Theoremhlatcon3 37800 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46966
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