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Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremjoindmss 17001 Subset property of domain of join. (Contributed by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)       (𝜑 → dom ⊆ (𝐵 × 𝐵))

Theoremjoinval2lem 17002* Lemma for joinval2 17003 and joineu 17004. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu into joinlem?
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))

Theoremjoinval2 17003* Value of join for a poset with LUB expanded. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 11-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 𝑌) = (𝑥𝐵 ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))

Theoremjoineu 17004* Uniqueness of join of elements in the domain. (Contributed by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑 → ∃!𝑥𝐵 ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧)))

Theoremjoinlem 17005* Lemma for join properties. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑 → ((𝑋 (𝑋 𝑌) ∧ 𝑌 (𝑋 𝑌)) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧)))

Theoremlejoin1 17006 A join's first argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑𝑋 (𝑋 𝑌))

Theoremlejoin2 17007 A join's second argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑𝑌 (𝑋 𝑌))

Theoremjoinle 17008 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (Contributed by NM, 16-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑 → ((𝑋 𝑍𝑌 𝑍) ↔ (𝑋 𝑌) 𝑍))

Theoremmeetfval 17009* Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove meetfval2 17010 first to reduce net proof size (existence part)?
𝐺 = (glb‘𝐾)    &    = (meet‘𝐾)       (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})

Theoremmeetfval2 17010* Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.)
𝐺 = (glb‘𝐾)    &    = (meet‘𝐾)       (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝐺𝑧 = (𝐺‘{𝑥, 𝑦}))})

Theoremmeetdm 17011* Domain of meet function for a poset-type structure. (Contributed by NM, 16-Sep-2018.)
𝐺 = (glb‘𝐾)    &    = (meet‘𝐾)       (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝐺})

Theoremmeetdef 17012 Two ways to say that a meet is defined. (Contributed by NM, 9-Sep-2018.)
𝐺 = (glb‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝑊)    &   (𝜑𝑌𝑍)       (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝐺))

Theoremmeetval 17013 Meet value. Since both sides evaluate to when they don't exist, for convenience we drop the {𝑋, 𝑌} ∈ dom 𝐺 requirement. (Contributed by NM, 9-Sep-2018.)
𝐺 = (glb‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝑊)    &   (𝜑𝑌𝑍)       (𝜑 → (𝑋 𝑌) = (𝐺‘{𝑋, 𝑌}))

Theoremmeetcl 17014 Closure of meet of elements in the domain. (Contributed by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑 → (𝑋 𝑌) ∈ 𝐵)

Theoremmeetdmss 17015 Subset property of domain of meet. (Contributed by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)       (𝜑 → dom ⊆ (𝐵 × 𝐵))

Theoremmeetval2lem 17016* Lemma for meetval2 17017 and meeteu 17018. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu into meetlem?
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)) ↔ ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))

Theoremmeetval2 17017* Value of meet for a poset with LUB expanded. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 11-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 𝑌) = (𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))

Theoremmeeteu 17018* Uniqueness of meet of elements in the domain. (Contributed by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑 → ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))

Theoremmeetlem 17019* Lemma for meet properties. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑 → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))

Theoremlemeet1 17020 A meet's first argument is less than or equal to the meet. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑 → (𝑋 𝑌) 𝑋)

Theoremlemeet2 17021 A meet's second argument is less than or equal to the meet. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑 → (𝑋 𝑌) 𝑌)

Theoremmeetle 17022 A meet is less than or equal to a third value iff each argument is less than or equal to the third value. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑 → ((𝑍 𝑋𝑍 𝑌) ↔ 𝑍 (𝑋 𝑌)))

TheoremjoincomALT 17023 The join of a poset commutes. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 16-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))

Theoremjoincom 17024 The join of a poset commutes. (The antecedent 𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑌, 𝑋⟩ ∈ dom i.e. "the joins exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑌, 𝑋⟩ ∈ dom )) → (𝑋 𝑌) = (𝑌 𝑋))

TheoremmeetcomALT 17025 The meet of a poset commutes. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 17-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))

Theoremmeetcom 17026 The meet of a poset commutes. (The antecedent 𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑌, 𝑋⟩ ∈ dom i.e. "the meets exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 17-Sep-2011.) (Revised by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑌, 𝑋⟩ ∈ dom )) → (𝑋 𝑌) = (𝑌 𝑋))

Syntaxctos 17027 Extend class notation with the class of all tosets.
class Toset

Definitiondf-toset 17028* Define the class of totally ordered sets (tosets). (Contributed by FL, 17-Nov-2014.)
Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥)}

Theoremistos 17029* The predicate "is a toset." (Contributed by FL, 17-Nov-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))

Theoremtosso 17030 Write the totally ordered set structure predicate in terms of the proper class strict order predicate. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       (𝐾𝑉 → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))

Syntaxcp0 17031 Extend class notation with poset zero.
class 0.

Syntaxcp1 17032 Extend class notation with poset unit.
class 1.

Definitiondf-p0 17033 Define poset zero. (Contributed by NM, 12-Oct-2011.)
0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))

Definitiondf-p1 17034 Define poset unit. (Contributed by NM, 22-Oct-2011.)
1. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝)))

Theoremp0val 17035 Value of poset zero. (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    0 = (0.‘𝐾)       (𝐾𝑉0 = (𝐺𝐵))

Theoremp1val 17036 Value of poset zero. (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &    1 = (1.‘𝐾)       (𝐾𝑉1 = (𝑈𝐵))

Theoremp0le 17037 Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝐵 ∈ dom 𝐺)       (𝜑0 𝑋)

Theoremple1 17038 Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &    = (le‘𝐾)    &    1 = (1.‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝐵 ∈ dom 𝑈)       (𝜑𝑋 1 )

9.2.2  Lattices

Syntaxclat 17039 Extend class notation with the class of all lattices.
class Lat

Definitiondf-lat 17040 Define the class of all lattices. A lattice is a poset in which the join and meet of any two elements always exists. (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))}

Theoremislat 17041 The predicate "is a lattice." (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))

Theoremlatcl2 17042 The join and meet of any two elements exist. (Contributed by NM, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾 ∈ Lat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑋, 𝑌⟩ ∈ dom ))

Theoremlatlem 17043 Lemma for lattice properties. (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑌) ∈ 𝐵))

Theoremlatpos 17044 A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
(𝐾 ∈ Lat → 𝐾 ∈ Poset)

Theoremlatjcl 17045 Closure of join operation in a lattice. (chjcom 28349 analog.) (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)

Theoremlatmcl 17046 Closure of meet operation in a lattice. (incom 3803 analog.) (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)

Theoremlatref 17047 A lattice ordering is reflexive. (ssid 3622 analog.) (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → 𝑋 𝑋)

Theoremlatasymb 17048 A lattice ordering is asymmetric. (eqss 3616 analog.) (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))

Theoremlatasym 17049 A lattice ordering is asymmetric. (eqss 3616 analog.) (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))

Theoremlattr 17050 A lattice ordering is transitive. (sstr 3609 analog.) (Contributed by NM, 17-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))

Theoremlatasymd 17051 Deduce equality from lattice ordering. (eqssd 3618 analog.) (Contributed by NM, 18-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   (𝜑𝐾 ∈ Lat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑋 𝑌)    &   (𝜑𝑌 𝑋)       (𝜑𝑋 = 𝑌)

Theoremlattrd 17052 A lattice ordering is transitive. Deduction version of lattr 17050. (Contributed by NM, 3-Sep-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   (𝜑𝐾 ∈ Lat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑌)    &   (𝜑𝑌 𝑍)       (𝜑𝑋 𝑍)

Theoremlatjcom 17053 The join of a lattice commutes. (chjcom 28349 analog.) (Contributed by NM, 16-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))

Theoremlatlej1 17054 A join's first argument is less than or equal to the join. (chub1 28350 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋 (𝑋 𝑌))

Theoremlatlej2 17055 A join's second argument is less than or equal to the join. (chub2 28351 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌 (𝑋 𝑌))

Theoremlatjle12 17056 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 28352 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍𝑌 𝑍) ↔ (𝑋 𝑌) 𝑍))

Theoremlatleeqj1 17057 Less-than-or-equal-to in terms of join. (chlejb1 28355 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 𝑌) = 𝑌))

Theoremlatleeqj2 17058 Less-than-or-equal-to in terms of join. (chlejb2 28356 analog.) (Contributed by NM, 14-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑌 𝑋) = 𝑌))

Theoremlatjlej1 17059 Add join to both sides of a lattice ordering. (chlej1i 28316 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑋 𝑍) (𝑌 𝑍)))

Theoremlatjlej2 17060 Add join to both sides of a lattice ordering. (chlej2i 28317 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑍 𝑋) (𝑍 𝑌)))

Theoremlatjlej12 17061 Add join to both sides of a lattice ordering. (chlej12i 28318 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑌𝑍 𝑊) → (𝑋 𝑍) (𝑌 𝑊)))

Theoremlatnlej 17062 An idiom to express that a lattice element differs from two others. (Contributed by NM, 28-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → (𝑋𝑌𝑋𝑍))

Theoremlatnlej1l 17063 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → 𝑋𝑌)

Theoremlatnlej1r 17064 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → 𝑋𝑍)

Theoremlatnlej2 17065 An idiom to express that a lattice element differs from two others. (Contributed by NM, 10-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → (¬ 𝑋 𝑌 ∧ ¬ 𝑋 𝑍))

Theoremlatnlej2l 17066 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → ¬ 𝑋 𝑌)

Theoremlatnlej2r 17067 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ¬ 𝑋 (𝑌 𝑍)) → ¬ 𝑋 𝑍)

Theoremlatjidm 17068 Lattice join is idempotent. (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)

Theoremlatmcom 17069 The join of a lattice commutes. (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))

Theoremlatmle1 17070 A meet is less than or equal to its first argument. (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑋)

Theoremlatmle2 17071 A meet is less than or equal to its second argument. (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)

Theoremlatlem12 17072 An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑋 𝑍) ↔ 𝑋 (𝑌 𝑍)))

Theoremlatleeqm1 17073 Less-than-or-equal-to in terms of meet. (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 𝑌) = 𝑋))

Theoremlatleeqm2 17074 Less-than-or-equal-to in terms of meet. (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑌 𝑋) = 𝑋))

Theoremlatmlem1 17075 Add meet to both sides of a lattice ordering. (Contributed by NM, 10-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑋 𝑍) (𝑌 𝑍)))

Theoremlatmlem2 17076 Add meet to both sides of a lattice ordering. (sslin 3837 analog.) (Contributed by NM, 10-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑍 𝑋) (𝑍 𝑌)))

Theoremlatmlem12 17077 Add join to both sides of a lattice ordering. (ss2in 3838 analog.) (Contributed by NM, 10-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑌𝑍 𝑊) → (𝑋 𝑍) (𝑌 𝑊)))

Theoremlatnlemlt 17078 Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3854 analog.) (Contributed by NM, 5-Feb-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑋 𝑌 ↔ (𝑋 𝑌) < 𝑋))

Theoremlatnle 17079 Equivalent expressions for "not less than" in a lattice. (chnle 28357 analog.) (Contributed by NM, 16-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑌 𝑋𝑋 < (𝑋 𝑌)))

Theoremlatmidm 17080 Lattice join is idempotent. (inidm 3820 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)

Theoremlatabs1 17081 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 28359 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)

Theoremlatabs2 17082 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 28360 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)

Theoremlatledi 17083 An ortholattice is distributive in one ordering direction. (ledi 28383 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 17084 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 17085 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 17086 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 17087 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 17088 The least upper bound of a singleton. (chsupsn 28256 analog.) (Contributed by NM, 20-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑈‘{𝑋}) = 𝑋)

Theoremlatjass 17089 Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 28376 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))

Theoremlatj12 17090 Swap 1st and 2nd members of lattice join. (chj12 28377 analog.) (Contributed by NM, 4-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑌 (𝑋 𝑍)))

Theoremlatj32 17091 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 17092 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 17093 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 17094 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 17095 Rearrangement of lattice join of 4 classes. (chj4 28378 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 17096 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 17097 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 17098 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 17099 The weak direction of the modular law (e.g., pmod1i 34960, atmod1i1 34969) 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 17100 The weak direction of the modular law (e.g., pmod2iN 34961) 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))))

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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 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 42301-42322
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