 Home Metamath Proof ExplorerTheorem List (p. 169 of 425) < 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 (1-26947) Hilbert Space Explorer (26948-28472) Users' Mathboxes (28473-42426)

Theorem List for Metamath Proof Explorer - 16801-16900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlatmidm 16801 Lattice join is idempotent. (inidm 3687 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)

Theoremlatabs1 16802 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 27547 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)

Theoremlatabs2 16803 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 27548 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)

Theoremlatledi 16804 An ortholattice is distributive in one ordering direction. (ledi 27571 analog.) (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) (𝑋 (𝑌 𝑍)))

Theoremlatmlej11 16805 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) (𝑋 𝑍))

Theoremlatmlej12 16806 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) (𝑍 𝑋))

Theoremlatmlej21 16807 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑋) (𝑋 𝑍))

Theoremlatmlej22 16808 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑋) (𝑍 𝑋))

Theoremlubsn 16809 The least upper bound of a singleton. (chsupsn 27444 analog.) (Contributed by NM, 20-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑈‘{𝑋}) = 𝑋)

Theoremlatjass 16810 Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 27564 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))

Theoremlatj12 16811 Swap 1st and 2nd members of lattice join. (chj12 27565 analog.) (Contributed by NM, 4-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑌 (𝑋 𝑍)))

Theoremlatj32 16812 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 2-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))

Theoremlatj13 16813 Swap 1st and 3rd members of lattice join. (Contributed by NM, 4-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑍 (𝑌 𝑋)))

Theoremlatj31 16814 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 23-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑍 𝑌) 𝑋))

Theoremlatjrot 16815 Rotate lattice join of 3 classes. (Contributed by NM, 23-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑍 𝑋) 𝑌))

Theoremlatj4 16816 Rearrangement of lattice join of 4 classes. (chj4 27566 analog.) (Contributed by NM, 14-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑌) (𝑍 𝑊)) = ((𝑋 𝑍) (𝑌 𝑊)))

Theoremlatj4rot 16817 Rotate lattice join of 4 classes. (Contributed by NM, 11-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑌) (𝑍 𝑊)) = ((𝑊 𝑋) (𝑌 𝑍)))

Theoremlatjjdi 16818 Lattice join distributes over itself. (Contributed by NM, 30-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))

Theoremlatjjdir 16819 Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) (𝑌 𝑍)))

Theoremmod1ile 16820 The weak direction of the modular law (e.g., pmod1i 34042, atmod1i1 34051) that holds in any lattice. (Contributed by NM, 11-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 → (𝑋 (𝑌 𝑍)) ((𝑋 𝑌) 𝑍)))

Theoremmod2ile 16821 The weak direction of the modular law (e.g., pmod2iN 34043) that holds in any lattice. (Contributed by NM, 11-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 𝑋 → ((𝑋 𝑌) 𝑍) (𝑋 (𝑌 𝑍))))

Syntaxccla 16822 Extend class notation with complete lattices.
class CLat

Definitiondf-clat 16823 Define the class of all complete lattices, where every subset of the base set has an LUB and a GLB. (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}

Theoremisclat 16824 The predicate "is a complete lattice." (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)       (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))

Theoremclatpos 16825 A complete lattice is a poset. (Contributed by NM, 8-Sep-2018.)
(𝐾 ∈ CLat → 𝐾 ∈ Poset)

Theoremclatlem 16826 Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → ((𝑈𝑆) ∈ 𝐵 ∧ (𝐺𝑆) ∈ 𝐵))

Theoremclatlubcl 16827 Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → (𝑈𝑆) ∈ 𝐵)

Theoremclatlubcl2 16828 Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝑈)

Theoremclatglbcl 16829 Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → (𝐺𝑆) ∈ 𝐵)

Theoremclatglbcl2 16830 Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → 𝑆 ∈ dom 𝐺)

Theoremclatl 16831 A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5035 to shorten proof and eliminate joindmss 16722 and meetdmss 16736?
(𝐾 ∈ CLat → 𝐾 ∈ Lat)

Theoremisglbd 16832* Properties that determine the greatest lower bound of a complete lattice. (Contributed by Mario Carneiro, 19-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   ((𝜑𝑦𝑆) → 𝐻 𝑦)    &   ((𝜑𝑥𝐵 ∧ ∀𝑦𝑆 𝑥 𝑦) → 𝑥 𝐻)    &   (𝜑𝐾 ∈ CLat)    &   (𝜑𝑆𝐵)    &   (𝜑𝐻𝐵)       (𝜑 → (𝐺𝑆) = 𝐻)

Theoremlublem 16833* Lemma for the least upper bound properties in a complete lattice. (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → (∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)))

Theoremlubub 16834 The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵𝑋𝑆) → 𝑋 (𝑈𝑆))

Theoremlubl 16835* The LUB of a complete lattice subset is the least bound. (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵𝑋𝐵) → (∀𝑦𝑆 𝑦 𝑋 → (𝑈𝑆) 𝑋))

Theoremlubss 16836 Subset law for least upper bounds. (chsupss 27373 analog.) (Contributed by NM, 20-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑇𝐵𝑆𝑇) → (𝑈𝑆) (𝑈𝑇))

Theoremlubel 16837 An element of a set is less than or equal to the least upper bound of the set. (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑋𝑆𝑆𝐵) → 𝑋 (𝑈𝑆))

Theoremlubun 16838 The LUB of a union. (Contributed by NM, 5-Mar-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵𝑇𝐵) → (𝑈‘(𝑆𝑇)) = ((𝑈𝑆) (𝑈𝑇)))

Theoremclatglb 16839* Properties of greatest lower bound of a complete lattice. (Contributed by NM, 5-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵) → (∀𝑦𝑆 (𝐺𝑆) 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 (𝐺𝑆))))

Theoremclatglble 16840 The greatest lower bound is the least element. (Contributed by NM, 5-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑆𝐵𝑋𝑆) → (𝐺𝑆) 𝑋)

Theoremclatleglb 16841* Two ways of expressing "less than or equal to the greatest lower bound." (Contributed by NM, 5-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑋𝐵𝑆𝐵) → (𝑋 (𝐺𝑆) ↔ ∀𝑦𝑆 𝑋 𝑦))

Theoremclatglbss 16842 Subset law for greatest lower bound. (Contributed by Mario Carneiro, 16-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)       ((𝐾 ∈ CLat ∧ 𝑇𝐵𝑆𝑇) → (𝐺𝑇) (𝐺𝑆))

9.2.3  The dual of an ordered set

Syntaxcodu 16843 Class function defining dual orders.
class ODual

Definitiondf-odu 16844 Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 16848, oduleval 16846, and oduleg 16847 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 16903. (Contributed by Stefan O'Rear, 29-Jan-2015.)

ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), (le‘𝑤)⟩))

Theoremoduval 16845 Value of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &    = (le‘𝑂)       𝐷 = (𝑂 sSet ⟨(le‘ndx), ⟩)

Theoremoduleval 16846 Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &    = (le‘𝑂)        = (le‘𝐷)

Theoremoduleg 16847 Truth of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &    = (le‘𝑂)    &   𝐺 = (le‘𝐷)       ((𝐴𝑉𝐵𝑊) → (𝐴𝐺𝐵𝐵 𝐴))

Theoremodubas 16848 Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &   𝐵 = (Base‘𝑂)       𝐵 = (Base‘𝐷)

Theorempospropd 16849* Posethood is determined only by structure components and only by the value of the relation within the base set. (Contributed by Stefan O'Rear, 29-Jan-2015.)
(𝜑𝐾𝑉)    &   (𝜑𝐿𝑊)    &   (𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(le‘𝐾)𝑦𝑥(le‘𝐿)𝑦))       (𝜑 → (𝐾 ∈ Poset ↔ 𝐿 ∈ Poset))

Theoremodupos 16850 Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂 ∈ Poset → 𝐷 ∈ Poset)

Theoremoduposb 16851 Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂𝑉 → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset))

Theoremmeet0 16852 Lemma for odujoin 16857. (Contributed by Stefan O'Rear, 29-Jan-2015.) TODO (df-riota 6388 update): This proof increased from 152 bytes to 547 bytes after the df-riota 6388 change. Any way to shorten it? join0 16853 also.
(meet‘∅) = ∅

Theoremjoin0 16853 Lemma for odumeet 16855. (Contributed by Stefan O'Rear, 29-Jan-2015.)
(join‘∅) = ∅

Theoremoduglb 16854 Greatest lower bounds in a dual order are least upper bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &   𝑈 = (lub‘𝑂)       (𝑂𝑉𝑈 = (glb‘𝐷))

Theoremodumeet 16855 Meets in a dual order are joins in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &    = (join‘𝑂)        = (meet‘𝐷)

Theoremodulub 16856 Least upper bounds in a dual order are greatest lower bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &   𝐿 = (glb‘𝑂)       (𝑂𝑉𝐿 = (lub‘𝐷))

Theoremodujoin 16857 Joins in a dual order are meets in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)    &    = (meet‘𝑂)        = (join‘𝐷)

Theoremodulatb 16858 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂𝑉 → (𝑂 ∈ Lat ↔ 𝐷 ∈ Lat))

Theoremoduclatb 16859 Being a complete lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat)

Theoremodulat 16860 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂 ∈ Lat → 𝐷 ∈ Lat)

Theoremposlubmo 16861* Least upper bounds in a poset are unique if they exist. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by NM, 16-Jun-2017.)
= (le‘𝐾)    &   𝐵 = (Base‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))

Theoremposglbmo 16862* Greatest lower bounds in a poset are unique if they exist. (Contributed by NM, 20-Sep-2018.)
= (le‘𝐾)    &   𝐵 = (Base‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))

Theoremposlubd 16863* Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
= (le‘𝐾)    &   𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑆𝐵)    &   (𝜑𝑇𝐵)    &   ((𝜑𝑥𝑆) → 𝑥 𝑇)    &   ((𝜑𝑦𝐵 ∧ ∀𝑥𝑆 𝑥 𝑦) → 𝑇 𝑦)       (𝜑 → (𝑈𝑆) = 𝑇)

Theoremposlubdg 16864* Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
= (le‘𝐾)    &   (𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝑈 = (lub‘𝐾))    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑆𝐵)    &   (𝜑𝑇𝐵)    &   ((𝜑𝑥𝑆) → 𝑥 𝑇)    &   ((𝜑𝑦𝐵 ∧ ∀𝑥𝑆 𝑥 𝑦) → 𝑇 𝑦)       (𝜑 → (𝑈𝑆) = 𝑇)

Theoremposglbd 16865* Properties which determine the greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
= (le‘𝐾)    &   (𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐺 = (glb‘𝐾))    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑆𝐵)    &   (𝜑𝑇𝐵)    &   ((𝜑𝑥𝑆) → 𝑇 𝑥)    &   ((𝜑𝑦𝐵 ∧ ∀𝑥𝑆 𝑦 𝑥) → 𝑦 𝑇)       (𝜑 → (𝐺𝑆) = 𝑇)

9.2.4  Subset order structures

Syntaxcipo 16866 Class function defining inclusion posets.
class toInc

Definitiondf-ipo 16867* For any family of sets, define the poset of that family ordered by inclusion. See ipobas 16870, ipolerval 16871, and ipole 16873 for its contract.

EDITORIAL: I'm not thrilled with the name. Any suggestions? (Contributed by Stefan O'Rear, 30-Jan-2015.) (New usage is discouraged.)

toInc = (𝑓 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓𝑥𝑦)} / 𝑜({⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} ∪ {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥𝑓 {𝑦𝑓 ∣ (𝑦𝑥) = ∅})⟩}))

Theoremipostr 16868 The structure of df-ipo 16867 is a structure defining indexes up to 11. (Contributed by Mario Carneiro, 25-Oct-2015.)
({⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), ⟩}) Struct ⟨1, 11⟩

Theoremipoval 16869* Value of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐼 = (toInc‘𝐹)    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)}       (𝐹𝑉𝐼 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ )⟩} ∪ {⟨(le‘ndx), ⟩, ⟨(oc‘ndx), (𝑥𝐹 {𝑦𝐹 ∣ (𝑦𝑥) = ∅})⟩}))

Theoremipobas 16870 Base set of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by Mario Carneiro, 25-Oct-2015.)
𝐼 = (toInc‘𝐹)       (𝐹𝑉𝐹 = (Base‘𝐼))

Theoremipolerval 16871* Relation of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐼 = (toInc‘𝐹)       (𝐹𝑉 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹𝑥𝑦)} = (le‘𝐼))

Theoremipotset 16872 Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015.)
𝐼 = (toInc‘𝐹)    &    = (le‘𝐼)       (𝐹𝑉 → (ordTop‘ ) = (TopSet‘𝐼))

Theoremipole 16873 Weak order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐼 = (toInc‘𝐹)    &    = (le‘𝐼)       ((𝐹𝑉𝑋𝐹𝑌𝐹) → (𝑋 𝑌𝑋𝑌))

Theoremipolt 16874 Strict order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐼 = (toInc‘𝐹)    &    < = (lt‘𝐼)       ((𝐹𝑉𝑋𝐹𝑌𝐹) → (𝑋 < 𝑌𝑋𝑌))

Theoremipopos 16875 The inclusion poset on a family of sets is actually a poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
𝐼 = (toInc‘𝐹)       𝐼 ∈ Poset

Theoremisipodrs 16876* Condition for a family of sets to be directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
((toInc‘𝐴) ∈ Dirset ↔ (𝐴 ∈ V ∧ 𝐴 ≠ ∅ ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑦) ⊆ 𝑧))

Theoremipodrscl 16877 Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.)
((toInc‘𝐴) ∈ Dirset → 𝐴 ∈ V)

Theoremipodrsfi 16878* Finite upper bound property for directed collections of sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(((toInc‘𝐴) ∈ Dirset ∧ 𝑋𝐴𝑋 ∈ Fin) → ∃𝑧𝐴 𝑋𝑧)

Theoremfpwipodrs 16879 The finite subsets of any set are directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝐴𝑉 → (toInc‘(𝒫 𝐴 ∩ Fin)) ∈ Dirset)

Theoremipodrsima 16880* The monotone image of a directed set. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝜑𝐹 Fn 𝒫 𝐴)    &   ((𝜑 ∧ (𝑢𝑣𝑣𝐴)) → (𝐹𝑢) ⊆ (𝐹𝑣))    &   (𝜑 → (toInc‘𝐵) ∈ Dirset)    &   (𝜑𝐵 ⊆ 𝒫 𝐴)    &   (𝜑 → (𝐹𝐵) ∈ 𝑉)       (𝜑 → (toInc‘(𝐹𝐵)) ∈ Dirset)

Theoremisacs3lem 16881* An algebraic closure system satisfies isacs3 16889. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝐶 ∈ (ACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶)))

Theoremacsdrsel 16882 An algebraic closure system contains all directed unions of closed sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
((𝐶 ∈ (ACS‘𝑋) ∧ 𝑌𝐶 ∧ (toInc‘𝑌) ∈ Dirset) → 𝑌𝐶)

Theoremisacs4lem 16883* In a closure system in which directed unions of closed sets are closed, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶)) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹 𝑡) = (𝐹𝑡))))

Theoremisacs5lem 16884* If closure commutes with directed unions, then the closure of a set is the closure of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹 𝑡) = (𝐹𝑡))) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝐹𝑠) = (𝐹 “ (𝒫 𝑠 ∩ Fin))))

Theoremacsdrscl 16885 In an algebraic closure system, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑌 ⊆ 𝒫 𝑋 ∧ (toInc‘𝑌) ∈ Dirset) → (𝐹 𝑌) = (𝐹𝑌))

Theoremacsficl 16886 A closure in an algebraic closure system is the union of the closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → (𝐹𝑆) = (𝐹 “ (𝒫 𝑆 ∩ Fin)))

Theoremisacs5 16887* A closure system is algebraic iff the closure of a generating set is the union of the closures of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝐹𝑠) = (𝐹 “ (𝒫 𝑠 ∩ Fin))))

Theoremisacs4 16888* A closure system is algebraic iff closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝒫 𝑋((toInc‘𝑠) ∈ Dirset → (𝐹 𝑠) = (𝐹𝑠))))

Theoremisacs3 16889* A closure system is algebraic iff directed unions of closed sets are closed. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → 𝑠𝐶)))

Theoremacsficld 16890 In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 16886. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑆𝑋)       (𝜑 → (𝑁𝑆) = (𝑁 “ (𝒫 𝑆 ∩ Fin)))

Theoremacsficl2d 16891* In an algebraic closure system, an element is in the closure of a set if and only if it is in the closure of a finite subset. Alternate form of acsficl 16886. Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑆𝑋)       (𝜑 → (𝑌 ∈ (𝑁𝑆) ↔ ∃𝑥 ∈ (𝒫 𝑆 ∩ Fin)𝑌 ∈ (𝑁𝑥)))

Theoremacsfiindd 16892 In an algebraic closure system, a set is independent if and only if all its finite subsets are independent. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝑋)       (𝜑 → (𝑆𝐼 ↔ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼))

Theoremacsmapd 16893* In an algebraic closure system, if 𝑇 is contained in the closure of 𝑆, there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that the closure of ran 𝑓 contains 𝑇. This is proven by applying acsficl2d 16891 to each element of 𝑇. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑆𝑋)    &   (𝜑𝑇 ⊆ (𝑁𝑆))       (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁 ran 𝑓)))

Theoremacsmap2d 16894* In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is independent, then there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that 𝑆 equals the union of ran 𝑓. This is proven by taking the map 𝑓 from acsmapd 16893 and observing that, since 𝑆 and 𝑇 have the same closure, the closure of ran 𝑓 must contain 𝑆. Since 𝑆 is independent, by mrissmrcd 16015, ran 𝑓 must equal 𝑆. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝐼)    &   (𝜑𝑇𝑋)    &   (𝜑 → (𝑁𝑆) = (𝑁𝑇))       (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ran 𝑓))

Theoremacsinfd 16895 In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is infinite independent, then 𝑇 is infinite. This follows from applying unirnffid 8017 to the map given in acsmap2d 16894. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝐼)    &   (𝜑𝑇𝑋)    &   (𝜑 → (𝑁𝑆) = (𝑁𝑇))    &   (𝜑 → ¬ 𝑆 ∈ Fin)       (𝜑 → ¬ 𝑇 ∈ Fin)

Theoremacsdomd 16896 In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is infinite independent, then 𝑇 dominates 𝑆. This follows from applying acsinfd 16895 and then applying unirnfdomd 9144 to the map given in acsmap2d 16894. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝐼)    &   (𝜑𝑇𝑋)    &   (𝜑 → (𝑁𝑆) = (𝑁𝑇))    &   (𝜑 → ¬ 𝑆 ∈ Fin)       (𝜑𝑆𝑇)

Theoremacsinfdimd 16897 In an algebraic closure system, if two independent sets have equal closure and one is infinite, then they are equinumerous. This is proven by using acsdomd 16896 twice with acsinfd 16895. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝐼)    &   (𝜑𝑇𝐼)    &   (𝜑 → (𝑁𝑆) = (𝑁𝑇))    &   (𝜑 → ¬ 𝑆 ∈ Fin)       (𝜑𝑆𝑇)

Theoremacsexdimd 16898* In an algebraic closure system whose closure operator has the exchange property, if two independent sets have equal closure, they are equinumerous. See mreexfidimd 16026 for the finite case and acsinfdimd 16897 for the infinite case. This is a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆𝐼)    &   (𝜑𝑇𝐼)    &   (𝜑 → (𝑁𝑆) = (𝑁𝑇))       (𝜑𝑆𝑇)

Theoremmrelatglb 16899 Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐼 = (toInc‘𝐶)    &   𝐺 = (glb‘𝐼)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶𝑈 ≠ ∅) → (𝐺𝑈) = 𝑈)

Theoremmrelatglb0 16900 The empty intersection in a Moore space is realized by the base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐼 = (toInc‘𝐶)    &   𝐺 = (glb‘𝐼)       (𝐶 ∈ (Moore‘𝑋) → (𝐺‘∅) = 𝑋)

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 42401-42426
 Copyright terms: Public domain < Previous  Next >