HomeHome Metamath Proof Explorer
Theorem List (p. 182 of 466)
< 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  Metamath Proof Explorer
(1-29289)
  Hilbert Space Explorer  Hilbert Space Explorer
(29290-30812)
  Users' Mathboxes  Users' Mathboxes
(30813-46532)
 

Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremjoinfval2 18101* Value of join function for a poset-type structure. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.)
𝑈 = (lub‘𝐾)    &    = (join‘𝐾)       (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ({𝑥, 𝑦} ∈ dom 𝑈𝑧 = (𝑈‘{𝑥, 𝑦}))})
 
Theoremjoindm 18102* Domain of join function for a poset-type structure. (Contributed by NM, 16-Sep-2018.)
𝑈 = (lub‘𝐾)    &    = (join‘𝐾)       (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈})
 
Theoremjoindef 18103 Two ways to say that a join is defined. (Contributed by NM, 9-Sep-2018.)
𝑈 = (lub‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝑊)    &   (𝜑𝑌𝑍)       (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝑈))
 
Theoremjoinval 18104 Join value. Since both sides evaluate to when they don't exist, for convenience we drop the {𝑋, 𝑌} ∈ dom 𝑈 requirement. (Contributed by NM, 9-Sep-2018.)
𝑈 = (lub‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝑊)    &   (𝜑𝑌𝑍)       (𝜑 → (𝑋 𝑌) = (𝑈‘{𝑋, 𝑌}))
 
Theoremjoincl 18105 Closure of join of elements in the domain. (Contributed by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )       (𝜑 → (𝑋 𝑌) ∈ 𝐵)
 
Theoremjoindmss 18106 Subset property of domain of join. (Contributed by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)       (𝜑 → dom ⊆ (𝐵 × 𝐵))
 
Theoremjoinval2lem 18107* Lemma for joinval2 18108 and joineu 18109. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu 18109 into joinlem 18110?
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 18108* 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 18109* 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 18110* 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 18111 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 18112 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 18113 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 18114* Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove meetfval2 18115 first to reduce net proof size (existence part)?
𝐺 = (glb‘𝐾)    &    = (meet‘𝐾)       (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
 
Theoremmeetfval2 18115* 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 18116* 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 18117 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 18118 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 18119 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 18120 Subset property of domain of meet. (Contributed by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)       (𝜑 → dom ⊆ (𝐵 × 𝐵))
 
Theoremmeetval2lem 18121* Lemma for meetval2 18122 and meeteu 18123. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 18123 into meetlem 18124?
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 18122* 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 18123* 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 18124* 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 18125 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 18126 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 18127 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 18128 The join of a poset is commutative. (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 18129 The join of a poset is commutative. (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 18130 The meet of a poset is commutative. (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 18131 The meet of a poset is commutative. (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))
 
Theoremjoin0 18132 Lemma for odumeet 18137. (Contributed by Stefan O'Rear, 29-Jan-2015.)
(join‘∅) = ∅
 
Theoremmeet0 18133 Lemma for odujoin 18135. (Contributed by Stefan O'Rear, 29-Jan-2015.) TODO (df-riota 7241 update): This proof increased from 152 bytes to 547 bytes after the df-riota 7241 change. Any way to shorten it? join0 18132 also.
(meet‘∅) = ∅
 
Theoremodulub 18134 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 18135 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)
 
Theoremoduglb 18136 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 18137 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)
 
Theoremposlubmo 18138* 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 18139* 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 18140* 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 18141* 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)
 
Theoremposglbdg 18142* 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)
 
9.4  Totally ordered sets (tosets)
 
Syntaxctos 18143 Extend class notation with the class of all tosets.
class Toset
 
Definitiondf-toset 18144* 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 18145* 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 18146 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) ⊆ )))
 
Theoremtospos 18147 A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ Toset → 𝐹 ∈ Poset)
 
Theoremtleile 18148 In a Toset, any two elements are comparable. (Contributed by Thierry Arnoux, 11-Feb-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
 
Theoremtltnle 18149 In a Toset, "less than" is equivalent to the negation of the converse of "less than or equal to", see pltnle 18065. (Contributed by Thierry Arnoux, 11-Feb-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 𝑋))
 
Syntaxcp0 18150 Extend class notation with poset zero.
class 0.
 
Syntaxcp1 18151 Extend class notation with poset unit.
class 1.
 
Definitiondf-p0 18152 Define poset zero. (Contributed by NM, 12-Oct-2011.)
0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))
 
Definitiondf-p1 18153 Define poset unit. (Contributed by NM, 22-Oct-2011.)
1. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝)))
 
Theoremp0val 18154 Value of poset zero. (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    0 = (0.‘𝐾)       (𝐾𝑉0 = (𝐺𝐵))
 
Theoremp1val 18155 Value of poset zero. (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &    1 = (1.‘𝐾)       (𝐾𝑉1 = (𝑈𝐵))
 
Theoremp0le 18156 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 18157 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 )
 
9.5  Lattices
 
9.5.1  Lattices
 
Syntaxclat 18158 Extend class notation with the class of all lattices.
class Lat
 
Definitiondf-lat 18159 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 18160 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))))
 
Theoremodulatb 18161 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂𝑉 → (𝑂 ∈ Lat ↔ 𝐷 ∈ Lat))
 
Theoremodulat 18162 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODual‘𝑂)       (𝑂 ∈ Lat → 𝐷 ∈ Lat)
 
Theoremlatcl2 18163 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 18164 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 18165 A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
(𝐾 ∈ Lat → 𝐾 ∈ Poset)
 
Theoremlatjcl 18166 Closure of join operation in a lattice. (chjcom 29877 analog.) (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
 
Theoremlatmcl 18167 Closure of meet operation in a lattice. (incom 4136 analog.) (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
 
Theoremlatref 18168 A lattice ordering is reflexive. (ssid 3944 analog.) (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → 𝑋 𝑋)
 
Theoremlatasymb 18169 A lattice ordering is asymmetric. (eqss 3937 analog.) (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
 
Theoremlatasym 18170 A lattice ordering is asymmetric. (eqss 3937 analog.) (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))
 
Theoremlattr 18171 A lattice ordering is transitive. (sstr 3930 analog.) (Contributed by NM, 17-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))
 
Theoremlatasymd 18172 Deduce equality from lattice ordering. (eqssd 3939 analog.) (Contributed by NM, 18-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   (𝜑𝐾 ∈ Lat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑋 𝑌)    &   (𝜑𝑌 𝑋)       (𝜑𝑋 = 𝑌)
 
Theoremlattrd 18173 A lattice ordering is transitive. Deduction version of lattr 18171. (Contributed by NM, 3-Sep-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   (𝜑𝐾 ∈ Lat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑌)    &   (𝜑𝑌 𝑍)       (𝜑𝑋 𝑍)
 
Theoremlatjcom 18174 The join of a lattice commutes. (chjcom 29877 analog.) (Contributed by NM, 16-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
 
Theoremlatlej1 18175 A join's first argument is less than or equal to the join. (chub1 29878 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋 (𝑋 𝑌))
 
Theoremlatlej2 18176 A join's second argument is less than or equal to the join. (chub2 29879 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌 (𝑋 𝑌))
 
Theoremlatjle12 18177 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 29880 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍𝑌 𝑍) ↔ (𝑋 𝑌) 𝑍))
 
Theoremlatleeqj1 18178 "Less than or equal to" in terms of join. (chlejb1 29883 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 𝑌) = 𝑌))
 
Theoremlatleeqj2 18179 "Less than or equal to" in terms of join. (chlejb2 29884 analog.) (Contributed by NM, 14-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑌 𝑋) = 𝑌))
 
Theoremlatjlej1 18180 Add join to both sides of a lattice ordering. (chlej1i 29844 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑋 𝑍) (𝑌 𝑍)))
 
Theoremlatjlej2 18181 Add join to both sides of a lattice ordering. (chlej2i 29845 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑍 𝑋) (𝑍 𝑌)))
 
Theoremlatjlej12 18182 Add join to both sides of a lattice ordering. (chlej12i 29846 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 18183 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 18184 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 18185 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 18186 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 18187 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 18188 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 18189 Lattice join is idempotent. Analogue of unidm 4087. (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)
 
Theoremlatmcom 18190 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 18191 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 18192 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 18193 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 18194 "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 18195 "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 18196 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 18197 Add meet to both sides of a lattice ordering. (sslin 4169 analog.) (Contributed by NM, 10-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑍 𝑋) (𝑍 𝑌)))
 
Theoremlatmlem12 18198 Add join to both sides of a lattice ordering. (ss2in 4171 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 18199 Negation of "less than or equal to" expressed in terms of meet and less-than. (nssinpss 4191 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 18200 Equivalent expressions for "not less than" in a lattice. (chnle 29885 analog.) (Contributed by NM, 16-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑌 𝑋𝑋 < (𝑋 𝑌)))
    < Previous  Next >

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 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 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 46501-46532
  Copyright terms: Public domain < Previous  Next >