Home Metamath Proof ExplorerTheorem List (p. 355 of 437) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28364) Hilbert Space Explorer (28365-29889) Users' Mathboxes (29890-43671)

Theorem List for Metamath Proof Explorer - 35401-35500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisoml 35401* The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))

TheoremisomliN 35402* Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
𝐾 ∈ OL    &   𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))))       𝐾 ∈ OML

Theoremomlol 35403 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OML → 𝐾 ∈ OL)

Theoremomlop 35404 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
(𝐾 ∈ OML → 𝐾 ∈ OP)

Theoremomllat 35405 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
(𝐾 ∈ OML → 𝐾 ∈ Lat)

Theoremomllaw 35406 The orthomodular law. (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))

Theoremomllaw2N 35407 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 29033 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 (( 𝑋) 𝑌)) = 𝑌))

Theoremomllaw3 35408 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 28884 analog.) (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = 𝑌))

Theoremomllaw4 35409 Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋))

Theoremomllaw5N 35410 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 29061 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) (𝑋 𝑌))) = (𝑋 𝑌))

TheoremcmtcomlemN 35411 Lemma for cmtcomN 35412. (cmcmlem 29039 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌𝐶𝑋))

TheoremcmtcomN 35412 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 29040 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌𝐶𝑋))

Theoremcmt2N 35413 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 29041 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋𝐶( 𝑌)))

Theoremcmt3N 35414 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 29043 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ( 𝑋)𝐶𝑌))

Theoremcmt4N 35415 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 29043 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ( 𝑋)𝐶( 𝑌)))

Theoremcmtbr2N 35416 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 29044 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))

Theoremcmtbr3N 35417 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 29056 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))

Theoremcmtbr4N 35418 Alternate definition for the commutes relation. (cmbr4i 29049 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) 𝑌))

TheoremlecmtN 35419 Ordered elements commute. (lecmi 29050 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑋𝐶𝑌))

TheoremcmtidN 35420 Any element commutes with itself. (cmidi 29058 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵) → 𝑋𝐶𝑋)

Theoremomlfh1N 35421 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 29066 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))

Theoremomlfh3N 35422 Foulis-Holland Theorem, part 3. Dual of omlfh1N 35421. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))

Theoremomlmod1i2N 35423 Analogue of modular law atmod1i2 36022 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) 𝑍))

TheoremomlspjN 35424 Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → ((𝑋 ( 𝑌)) 𝑌) = 𝑋)

20.23.10  Atomic lattices with covering property

Syntaxccvr 35425 Extend class notation with covers relation.
class

Syntaxcatm 35426 Extend class notation with atoms.
class Atoms

Syntaxcal 35427 Extend class notation with atomic lattices.
class AtLat

Syntaxclc 35428 Extend class notation with lattices with the covering property.
class CvLat

Definitiondf-covers 35429* Define the covers relation ("is covered by") for posets. "𝑎 is covered by 𝑏 " means that 𝑎 is strictly less than 𝑏 and there is nothing in between. See cvrval 35432 for the relation form. (Contributed by NM, 18-Sep-2011.)
⋖ = (𝑝 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑝) ∧ 𝑏 ∈ (Base‘𝑝)) ∧ 𝑎(lt‘𝑝)𝑏 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑎(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑏))})

Definitiondf-ats 35430* Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)
Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎})

Theoremcvrfval 35431* Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})

Theoremcvrval 35432* Binary relation expressing 𝐵 covers 𝐴, which means that 𝐵 is larger than 𝐴 and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (cvbr 29730 analog.) (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))

Theoremcvrlt 35433 The covers relation implies the less-than relation. (cvpss 29733 analog.) (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)

Theoremcvrnbtwn 35434 There is no element between the two arguments of the covers relation. (cvnbtwn 29734 analog.) (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))

Theoremncvr1 35435 No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.)
𝐵 = (Base‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ¬ 1 𝐶𝑋)

TheoremcvrletrN 35436 Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌 𝑍) → 𝑋 < 𝑍))

Theoremcvrval2 35437* Binary relation expressing 𝑌 covers 𝑋. Definition of covers in [Kalmbach] p. 15. (cvbr2 29731 analog.) (Contributed by NM, 16-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∀𝑧𝐵 ((𝑋 < 𝑧𝑧 𝑌) → 𝑧 = 𝑌))))

Theoremcvrnbtwn2 35438 The covers relation implies no in-betweenness. (cvnbtwn2 29735 analog.) (Contributed by NM, 17-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) ↔ 𝑍 = 𝑌))

Theoremcvrnbtwn3 35439 The covers relation implies no in-betweenness. (cvnbtwn3 29736 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 < 𝑌) ↔ 𝑋 = 𝑍))

Theoremcvrcon3b 35440 Contraposition law for the covers relation. (cvcon3 29732 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ( 𝑌)𝐶( 𝑋)))

Theoremcvrle 35441 The covers relation implies the "less than or equal to" relation. (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)

Theoremcvrnbtwn4 35442 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 29737 analog.) (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) ↔ (𝑋 = 𝑍𝑍 = 𝑌)))

Theoremcvrnle 35443 The covers relation implies the negation of the converse "less than or equal to" relation. (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → ¬ 𝑌 𝑋)

Theoremcvrne 35444 The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝑌)

TheoremcvrnrefN 35445 The covers relation is not reflexive. (cvnref 29739 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴𝑋𝐵) → ¬ 𝑋𝐶𝑋)

Theoremcvrcmp 35446 If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))

Theoremcvrcmp2 35447 If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌𝑋 = 𝑌))

Theorempats 35448* The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})

Theoremisat 35449 The predicate "is an atom". (ela 29787 analog.) (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃𝐵0 𝐶𝑃)))

Theoremisat2 35450 The predicate "is an atom". (elatcv0 29789 analog.) (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾𝐷𝑃𝐵) → (𝑃𝐴0 𝐶𝑃))

Theorematcvr0 35451 An atom covers zero. (atcv0 29790 analog.) (Contributed by NM, 4-Nov-2011.)
0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾𝐷𝑃𝐴) → 0 𝐶𝑃)

Theorematbase 35452 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 29792 analog.) (Contributed by NM, 10-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝑃𝐴𝑃𝐵)

Theorematssbase 35453 The set of atoms is a subset of the base set. (atssch 29791 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)       𝐴𝐵

Theorem0ltat 35454 An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)
0 = (0.‘𝐾)    &    < = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OP ∧ 𝑃𝐴) → 0 < 𝑃)

Theoremleatb 35455 A poset element less than or equal to an atom equals either zero or the atom. (atss 29794 analog.) (Contributed by NM, 17-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) → (𝑋 𝑃 ↔ (𝑋 = 𝑃𝑋 = 0 )))

Theoremleat 35456 A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) ∧ 𝑋 𝑃) → (𝑋 = 𝑃𝑋 = 0 ))

Theoremleat2 35457 A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) ∧ (𝑋0𝑋 𝑃)) → 𝑋 = 𝑃)

Theoremleat3 35458 A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) ∧ 𝑋 𝑃) → (𝑋𝐴𝑋 = 0 ))

Theoremmeetat 35459 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑃𝐴) → ((𝑋 𝑃) = 𝑃 ∨ (𝑋 𝑃) = 0 ))

Theoremmeetat2 35460 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑃𝐴) → ((𝑋 𝑃) ∈ 𝐴 ∨ (𝑋 𝑃) = 0 ))

Definitiondf-atl 35461* 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 = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑝 ∈ (Atoms‘𝑘)𝑝(le‘𝑘)𝑥))}

Theoremisatl 35462* 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.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))

Theorematllat 35463 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
(𝐾 ∈ AtLat → 𝐾 ∈ Lat)

Theorematlpos 35464 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ AtLat → 𝐾 ∈ Poset)

Theorematl0dm 35465 Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)       (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)

Theorematl0cl 35466 An atomic lattice has a zero element. We can use this in place of op0cl 35347 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)       (𝐾 ∈ AtLat → 0𝐵)

Theorematl0le 35467 Orthoposet zero is less than or equal to any element. (ch0le 28889 analog.) (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵) → 0 𝑋)

Theorematlle0 35468 An element less than or equal to zero equals zero. (chle0 28891 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵) → (𝑋 0𝑋 = 0 ))

Theorematlltn0 35469 A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵) → ( 0 < 𝑋𝑋0 ))

Theoremisat3 35470* The predicate "is an atom". (elat2 29788 analog.) (Contributed by NM, 27-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ AtLat → (𝑃𝐴 ↔ (𝑃𝐵𝑃0 ∧ ∀𝑥𝐵 (𝑥 𝑃 → (𝑥 = 𝑃𝑥 = 0 )))))

Theorematn0 35471 An atom is not zero. (atne0 29793 analog.) (Contributed by NM, 5-Nov-2012.)
0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃0 )

Theorematnle0 35472 An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)
= (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → ¬ 𝑃 0 )

Theorematlen0 35473 A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑃𝐴) ∧ 𝑃 𝑋) → 𝑋0 )

Theorematcmp 35474 If two atoms are comparable, they are equal. (atsseq 29795 analog.) (Contributed by NM, 13-Oct-2011.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄𝑃 = 𝑄))

Theorematncmp 35475 Frequently-used variation of atcmp 35474. (Contributed by NM, 29-Jun-2012.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (¬ 𝑃 𝑄𝑃𝑄))

Theorematnlt 35476 Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.)
< = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → ¬ 𝑃 < 𝑄)

Theorematcvreq0 35477 An element covered by an atom must be zero. (atcveq0 29796 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑃𝐴) → (𝑋𝐶𝑃𝑋 = 0 ))

TheorematncvrN 35478 Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → ¬ 𝑃𝐶𝑄)

Theorematlex 35479* Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 29808 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑦𝐴 𝑦 𝑋)

Theorematnle 35480 Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 29824 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = 0 ))

Theorematnem0 35481 The meet of distinct atoms is zero. (atnemeq0 29825 analog.) (Contributed by NM, 5-Nov-2012.)
= (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑄𝐴) → (𝑃𝑄 ↔ (𝑃 𝑄) = 0 ))

Theorematlatmstc 35482* 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 29810 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    1 = (lub‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋𝐵) → ( 1 ‘{𝑦𝐴𝑦 𝑋}) = 𝑋)

Theorematlatle 35483* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 29819 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ∀𝑝𝐴 (𝑝 𝑋𝑝 𝑌)))

Theorematlrelat1 35484* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 29811, with swapped, analog.) (Contributed by NM, 4-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ∃𝑝𝐴𝑝 𝑋𝑝 𝑌)))

Definitiondf-cvlat 35485* 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 = {𝑘 ∈ AtLat ∣ ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))}

Theoremiscvlat 35486* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))

Theoremiscvlat2N 35487* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝𝐴𝑞𝐴𝑥𝐵 (((𝑝 𝑥) = 0𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))

Theoremcvlatl 35488 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ CvLat → 𝐾 ∈ AtLat)

Theoremcvllat 35489 An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ CvLat → 𝐾 ∈ Lat)

TheoremcvlposN 35490 An atomic lattice with the covering property is a poset. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
(𝐾 ∈ CvLat → 𝐾 ∈ Poset)

Theoremcvlexch1 35491 An atomic covering lattice has the exchange property. (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Theoremcvlexch2 35492 An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) → 𝑄 (𝑃 𝑋)))

Theoremcvlexchb1 35493 An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Theoremcvlexchb2 35494 An atomic covering lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ ¬ 𝑃 𝑋) → (𝑃 (𝑄 𝑋) ↔ (𝑃 𝑋) = (𝑄 𝑋)))

Theoremcvlexch3 35495 An atomic covering lattice has the exchange property. (atexch 29829 analog.) (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) → 𝑄 (𝑋 𝑃)))

Theoremcvlexch4N 35496 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.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 (𝑋 𝑄) ↔ (𝑋 𝑃) = (𝑋 𝑄)))

Theoremcvlatexchb1 35497 A version of cvlexchb1 35493 for atoms. (Contributed by NM, 5-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑄) ↔ (𝑅 𝑃) = (𝑅 𝑄)))

Theoremcvlatexchb2 35498 A version of cvlexchb2 35494 for atoms. (Contributed by NM, 5-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) ↔ (𝑃 𝑅) = (𝑄 𝑅)))

Theoremcvlatexch1 35499 Atom exchange property. (Contributed by NM, 5-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑄) → 𝑄 (𝑅 𝑃)))

Theoremcvlatexch2 35500 Atom exchange property. (Contributed by NM, 5-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑄 𝑅) → 𝑄 (𝑃 𝑅)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43671
 Copyright terms: Public domain < Previous  Next >