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Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-join 18301* Define poset join. (Contributed by NM, 12-Sep-2011.) (Revised by Mario Carneiro, 3-Nov-2015.)
join = (𝑝 ∈ V ↦ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦} (lubβ€˜π‘)𝑧})
 
Definitiondf-meet 18302* Define poset meet. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 8-Sep-2018.)
meet = (𝑝 ∈ V ↦ {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦} (glbβ€˜π‘)𝑧})
 
Theoremlubfval 18303* Value of the least upper bound function of a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑠 𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑠 𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    β‡’   (πœ‘ β†’ π‘ˆ = ((𝑠 ∈ 𝒫 𝐡 ↦ (β„©π‘₯ ∈ 𝐡 πœ“)) β†Ύ {𝑠 ∣ βˆƒ!π‘₯ ∈ 𝐡 πœ“}))
 
Theoremlubdm 18304* Domain of the least upper bound function of a poset. (Contributed by NM, 6-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑠 𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑠 𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    β‡’   (πœ‘ β†’ dom π‘ˆ = {𝑠 ∈ 𝒫 𝐡 ∣ βˆƒ!π‘₯ ∈ 𝐡 πœ“})
 
Theoremlubfun 18305 The LUB is a function. (Contributed by NM, 9-Sep-2018.)
π‘ˆ = (lubβ€˜πΎ)    β‡’   Fun π‘ˆ
 
Theoremlubeldm 18306* Member of the domain of the least upper bound function of a poset. (Contributed by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝑆 ∈ dom π‘ˆ ↔ (𝑆 βŠ† 𝐡 ∧ βˆƒ!π‘₯ ∈ 𝐡 πœ“)))
 
Theoremlubelss 18307 A member of the domain of the least upper bound function is a subset of the base set. (Contributed by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ dom π‘ˆ)    β‡’   (πœ‘ β†’ 𝑆 βŠ† 𝐡)
 
Theoremlubeu 18308* Unique existence proper of a member of the domain of the least upper bound function of a poset. (Contributed by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ dom π‘ˆ)    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ 𝐡 πœ“)
 
Theoremlubval 18309* Value of the least upper bound function of a poset. Out-of-domain arguments (those not satisfying 𝑆 ∈ dom π‘ˆ) are allowed for convenience, evaluating to the empty set. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    β‡’   (πœ‘ β†’ (π‘ˆβ€˜π‘†) = (β„©π‘₯ ∈ 𝐡 πœ“))
 
Theoremlubcl 18310 The least upper bound function value belongs to the base set. (Contributed by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ dom π‘ˆ)    β‡’   (πœ‘ β†’ (π‘ˆβ€˜π‘†) ∈ 𝐡)
 
Theoremlubprop 18311* Properties of greatest lower bound of a poset. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ dom π‘ˆ)    β‡’   (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ (π‘ˆβ€˜π‘†) ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ 𝑧 β†’ (π‘ˆβ€˜π‘†) ≀ 𝑧)))
 
Theoremluble 18312 The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ dom π‘ˆ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑆)    β‡’   (πœ‘ β†’ 𝑋 ≀ (π‘ˆβ€˜π‘†))
 
Theoremlublecllem 18313* Lemma for lublecl 18314 and lubid 18315. (Contributed by NM, 8-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Poset)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ((βˆ€π‘§ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑋}𝑧 ≀ π‘₯ ∧ βˆ€π‘€ ∈ 𝐡 (βˆ€π‘§ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑋}𝑧 ≀ 𝑀 β†’ π‘₯ ≀ 𝑀)) ↔ π‘₯ = 𝑋))
 
Theoremlublecl 18314* The set of all elements less than a given element has an LUB. (Contributed by NM, 8-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Poset)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑋} ∈ dom π‘ˆ)
 
Theoremlubid 18315* The LUB of elements less than or equal to a fixed value equals that value. (Contributed by NM, 19-Oct-2011.) (Revised by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Poset)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (π‘ˆβ€˜{𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑋}) = 𝑋)
 
Theoremglbfval 18316* Value of the greatest lower function of a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &   (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑠 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑠 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝐺 = ((𝑠 ∈ 𝒫 𝐡 ↦ (β„©π‘₯ ∈ 𝐡 πœ“)) β†Ύ {𝑠 ∣ βˆƒ!π‘₯ ∈ 𝐡 πœ“}))
 
Theoremglbdm 18317* Domain of the greatest lower bound function of a poset. (Contributed by NM, 6-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &   (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑠 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑠 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    β‡’   (πœ‘ β†’ dom 𝐺 = {𝑠 ∈ 𝒫 𝐡 ∣ βˆƒ!π‘₯ ∈ 𝐡 πœ“})
 
Theoremglbfun 18318 The GLB is a function. (Contributed by NM, 9-Sep-2018.)
𝐺 = (glbβ€˜πΎ)    β‡’   Fun 𝐺
 
Theoremglbeldm 18319* Member of the domain of the greatest lower bound function of a poset. (Contributed by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &   (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝑆 ∈ dom 𝐺 ↔ (𝑆 βŠ† 𝐡 ∧ βˆƒ!π‘₯ ∈ 𝐡 πœ“)))
 
Theoremglbelss 18320 A member of the domain of the greatest lower bound function is a subset of the base set. (Contributed by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ dom 𝐺)    β‡’   (πœ‘ β†’ 𝑆 βŠ† 𝐡)
 
Theoremglbeu 18321* Unique existence proper of a member of the domain of the greatest lower bound function of a poset. (Contributed by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &   (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ dom 𝐺)    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ 𝐡 πœ“)
 
Theoremglbval 18322* Value of the greatest lower bound function of a poset. Out-of-domain arguments (those not satisfying 𝑆 ∈ dom π‘ˆ) are allowed for convenience, evaluating to the empty set on both sides of the equality. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &   (πœ“ ↔ (βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    β‡’   (πœ‘ β†’ (πΊβ€˜π‘†) = (β„©π‘₯ ∈ 𝐡 πœ“))
 
Theoremglbcl 18323 The least upper bound function value belongs to the base set. (Contributed by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ dom 𝐺)    β‡’   (πœ‘ β†’ (πΊβ€˜π‘†) ∈ 𝐡)
 
Theoremglbprop 18324* Properties of greatest lower bound of a poset. (Contributed by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (glbβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ dom π‘ˆ)    β‡’   (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝑆 (π‘ˆβ€˜π‘†) ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ (π‘ˆβ€˜π‘†))))
 
Theoremglble 18325 The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π‘ˆ = (glbβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ dom π‘ˆ)    &   (πœ‘ β†’ 𝑋 ∈ 𝑆)    β‡’   (πœ‘ β†’ (π‘ˆβ€˜π‘†) ≀ 𝑋)
 
Theoremjoinfval 18326* Value of join function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove joinfval2 18327 first to reduce net proof size (existence part)?
π‘ˆ = (lubβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ ∨ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}π‘ˆπ‘§})
 
Theoremjoinfval2 18327* Value of join function for a poset-type structure. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.)
π‘ˆ = (lubβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ ∨ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom π‘ˆ ∧ 𝑧 = (π‘ˆβ€˜{π‘₯, 𝑦}))})
 
Theoremjoindm 18328* Domain of join function for a poset-type structure. (Contributed by NM, 16-Sep-2018.)
π‘ˆ = (lubβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ dom ∨ = {⟨π‘₯, π‘¦βŸ© ∣ {π‘₯, 𝑦} ∈ dom π‘ˆ})
 
Theoremjoindef 18329 Two ways to say that a join is defined. (Contributed by NM, 9-Sep-2018.)
π‘ˆ = (lubβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ π‘Š)    &   (πœ‘ β†’ π‘Œ ∈ 𝑍)    β‡’   (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ↔ {𝑋, π‘Œ} ∈ dom π‘ˆ))
 
Theoremjoinval 18330 Join value. Since both sides evaluate to βˆ… when they don't exist, for convenience we drop the {𝑋, π‘Œ} ∈ dom π‘ˆ requirement. (Contributed by NM, 9-Sep-2018.)
π‘ˆ = (lubβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ π‘Š)    &   (πœ‘ β†’ π‘Œ ∈ 𝑍)    β‡’   (πœ‘ β†’ (𝑋 ∨ π‘Œ) = (π‘ˆβ€˜{𝑋, π‘Œ}))
 
Theoremjoincl 18331 Closure of join of elements in the domain. (Contributed by NM, 12-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ )    β‡’   (πœ‘ β†’ (𝑋 ∨ π‘Œ) ∈ 𝐡)
 
Theoremjoindmss 18332 Subset property of domain of join. (Contributed by NM, 12-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    β‡’   (πœ‘ β†’ dom ∨ βŠ† (𝐡 Γ— 𝐡))
 
Theoremjoinval2lem 18333* Lemma for joinval2 18334 and joineu 18335. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu 18335 into joinlem 18336?
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)) ↔ ((𝑋 ≀ π‘₯ ∧ π‘Œ ≀ π‘₯) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
 
Theoremjoinval2 18334* Value of join for a poset with LUB expanded. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 11-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 ∨ π‘Œ) = (β„©π‘₯ ∈ 𝐡 ((𝑋 ≀ π‘₯ ∧ π‘Œ ≀ π‘₯) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
 
Theoremjoineu 18335* Uniqueness of join of elements in the domain. (Contributed by NM, 12-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ )    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ 𝐡 ((𝑋 ≀ π‘₯ ∧ π‘Œ ≀ π‘₯) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ π‘₯ ≀ 𝑧)))
 
Theoremjoinlem 18336* Lemma for join properties. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ )    β‡’   (πœ‘ β†’ ((𝑋 ≀ (𝑋 ∨ π‘Œ) ∧ π‘Œ ≀ (𝑋 ∨ π‘Œ)) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ (𝑋 ∨ π‘Œ) ≀ 𝑧)))
 
Theoremlejoin1 18337 A join's first argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ )    β‡’   (πœ‘ β†’ 𝑋 ≀ (𝑋 ∨ π‘Œ))
 
Theoremlejoin2 18338 A join's second argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ )    β‡’   (πœ‘ β†’ π‘Œ ≀ (𝑋 ∨ π‘Œ))
 
Theoremjoinle 18339 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Poset)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ )    β‡’   (πœ‘ β†’ ((𝑋 ≀ 𝑍 ∧ π‘Œ ≀ 𝑍) ↔ (𝑋 ∨ π‘Œ) ≀ 𝑍))
 
Theoremmeetfval 18340* Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove meetfval2 18341 first to reduce net proof size (existence part)?
𝐺 = (glbβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ ∧ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ {π‘₯, 𝑦}𝐺𝑧})
 
Theoremmeetfval2 18341* Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.)
𝐺 = (glbβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ ∧ = {⟨⟨π‘₯, π‘¦βŸ©, π‘§βŸ© ∣ ({π‘₯, 𝑦} ∈ dom 𝐺 ∧ 𝑧 = (πΊβ€˜{π‘₯, 𝑦}))})
 
Theoremmeetdm 18342* Domain of meet function for a poset-type structure. (Contributed by NM, 16-Sep-2018.)
𝐺 = (glbβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ dom ∧ = {⟨π‘₯, π‘¦βŸ© ∣ {π‘₯, 𝑦} ∈ dom 𝐺})
 
Theoremmeetdef 18343 Two ways to say that a meet is defined. (Contributed by NM, 9-Sep-2018.)
𝐺 = (glbβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ π‘Š)    &   (πœ‘ β†’ π‘Œ ∈ 𝑍)    β‡’   (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ ↔ {𝑋, π‘Œ} ∈ dom 𝐺))
 
Theoremmeetval 18344 Meet value. Since both sides evaluate to βˆ… when they don't exist, for convenience we drop the {𝑋, π‘Œ} ∈ dom 𝐺 requirement. (Contributed by NM, 9-Sep-2018.)
𝐺 = (glbβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ π‘Š)    &   (πœ‘ β†’ π‘Œ ∈ 𝑍)    β‡’   (πœ‘ β†’ (𝑋 ∧ π‘Œ) = (πΊβ€˜{𝑋, π‘Œ}))
 
Theoremmeetcl 18345 Closure of meet of elements in the domain. (Contributed by NM, 12-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ )    β‡’   (πœ‘ β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
 
Theoremmeetdmss 18346 Subset property of domain of meet. (Contributed by NM, 12-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    β‡’   (πœ‘ β†’ dom ∧ βŠ† (𝐡 Γ— 𝐡))
 
Theoremmeetval2lem 18347* Lemma for meetval2 18348 and meeteu 18349. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 18349 into meetlem 18350?
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((βˆ€π‘¦ ∈ {𝑋, π‘Œ}π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)) ↔ ((π‘₯ ≀ 𝑋 ∧ π‘₯ ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯))))
 
Theoremmeetval2 18348* Value of meet for a poset with LUB expanded. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 11-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 ∧ π‘Œ) = (β„©π‘₯ ∈ 𝐡 ((π‘₯ ≀ 𝑋 ∧ π‘₯ ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯))))
 
Theoremmeeteu 18349* Uniqueness of meet of elements in the domain. (Contributed by NM, 12-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ )    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ 𝐡 ((π‘₯ ≀ 𝑋 ∧ π‘₯ ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯)))
 
Theoremmeetlem 18350* Lemma for meet properties. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ )    β‡’   (πœ‘ β†’ (((𝑋 ∧ π‘Œ) ≀ 𝑋 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ (𝑋 ∧ π‘Œ))))
 
Theoremlemeet1 18351 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ )    β‡’   (πœ‘ β†’ (𝑋 ∧ π‘Œ) ≀ 𝑋)
 
Theoremlemeet2 18352 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ )    β‡’   (πœ‘ β†’ (𝑋 ∧ π‘Œ) ≀ π‘Œ)
 
Theoremmeetle 18353 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Poset)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ )    β‡’   (πœ‘ β†’ ((𝑍 ≀ 𝑋 ∧ 𝑍 ≀ π‘Œ) ↔ 𝑍 ≀ (𝑋 ∧ π‘Œ)))
 
TheoremjoincomALT 18354 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.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) = (π‘Œ ∨ 𝑋))
 
Theoremjoincom 18355 The join of a poset is commutative. (The antecedent βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ∧ βŸ¨π‘Œ, π‘‹βŸ© ∈ 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.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ∧ βŸ¨π‘Œ, π‘‹βŸ© ∈ dom ∨ )) β†’ (𝑋 ∨ π‘Œ) = (π‘Œ ∨ 𝑋))
 
TheoremmeetcomALT 18356 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.)
𝐡 = (Baseβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) = (π‘Œ ∧ 𝑋))
 
Theoremmeetcom 18357 The meet of a poset is commutative. (The antecedent βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ ∧ βŸ¨π‘Œ, π‘‹βŸ© ∈ 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.)
𝐡 = (Baseβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ ∧ βŸ¨π‘Œ, π‘‹βŸ© ∈ dom ∧ )) β†’ (𝑋 ∧ π‘Œ) = (π‘Œ ∧ 𝑋))
 
Theoremjoin0 18358 Lemma for odumeet 18363. (Contributed by Stefan O'Rear, 29-Jan-2015.)
(joinβ€˜βˆ…) = βˆ…
 
Theoremmeet0 18359 Lemma for odujoin 18361. (Contributed by Stefan O'Rear, 29-Jan-2015.) TODO (df-riota 7365 update): This proof increased from 152 bytes to 547 bytes after the df-riota 7365 change. Any way to shorten it? join0 18358 also.
(meetβ€˜βˆ…) = βˆ…
 
Theoremodulub 18360 Least upper bounds in a dual order are greatest lower bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODualβ€˜π‘‚)    &   πΏ = (glbβ€˜π‘‚)    β‡’   (𝑂 ∈ 𝑉 β†’ 𝐿 = (lubβ€˜π·))
 
Theoremodujoin 18361 Joins in a dual order are meets in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODualβ€˜π‘‚)    &    ∧ = (meetβ€˜π‘‚)    β‡’    ∧ = (joinβ€˜π·)
 
Theoremoduglb 18362 Greatest lower bounds in a dual order are least upper bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODualβ€˜π‘‚)    &   π‘ˆ = (lubβ€˜π‘‚)    β‡’   (𝑂 ∈ 𝑉 β†’ π‘ˆ = (glbβ€˜π·))
 
Theoremodumeet 18363 Meets in a dual order are joins in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODualβ€˜π‘‚)    &    ∨ = (joinβ€˜π‘‚)    β‡’    ∨ = (meetβ€˜π·)
 
Theoremposlubmo 18364* 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.)
≀ = (leβ€˜πΎ)    &   π΅ = (Baseβ€˜πΎ)    β‡’   ((𝐾 ∈ Poset ∧ 𝑆 βŠ† 𝐡) β†’ βˆƒ*π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)))
 
Theoremposglbmo 18365* Greatest lower bounds in a poset are unique if they exist. (Contributed by NM, 20-Sep-2018.)
≀ = (leβ€˜πΎ)    &   π΅ = (Baseβ€˜πΎ)    β‡’   ((𝐾 ∈ Poset ∧ 𝑆 βŠ† 𝐡) β†’ βˆƒ*π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ 𝑆 𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)))
 
Theoremposlubd 18366* Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
≀ = (leβ€˜πΎ)    &   π΅ = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Poset)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    &   (πœ‘ β†’ 𝑇 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ≀ 𝑇)    &   ((πœ‘ ∧ 𝑦 ∈ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑆 π‘₯ ≀ 𝑦) β†’ 𝑇 ≀ 𝑦)    β‡’   (πœ‘ β†’ (π‘ˆβ€˜π‘†) = 𝑇)
 
Theoremposlubdg 18367* Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
≀ = (leβ€˜πΎ)    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ π‘ˆ = (lubβ€˜πΎ))    &   (πœ‘ β†’ 𝐾 ∈ Poset)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    &   (πœ‘ β†’ 𝑇 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ≀ 𝑇)    &   ((πœ‘ ∧ 𝑦 ∈ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑆 π‘₯ ≀ 𝑦) β†’ 𝑇 ≀ 𝑦)    β‡’   (πœ‘ β†’ (π‘ˆβ€˜π‘†) = 𝑇)
 
Theoremposglbdg 18368* Properties which determine the greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
≀ = (leβ€˜πΎ)    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐺 = (glbβ€˜πΎ))    &   (πœ‘ β†’ 𝐾 ∈ Poset)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐡)    &   (πœ‘ β†’ 𝑇 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑆) β†’ 𝑇 ≀ π‘₯)    &   ((πœ‘ ∧ 𝑦 ∈ 𝐡 ∧ βˆ€π‘₯ ∈ 𝑆 𝑦 ≀ π‘₯) β†’ 𝑦 ≀ 𝑇)    β‡’   (πœ‘ β†’ (πΊβ€˜π‘†) = 𝑇)
 
9.4  Totally ordered sets (tosets)
 
Syntaxctos 18369 Extend class notation with the class of all tosets.
class Toset
 
Definitiondf-toset 18370* Define the class of totally ordered sets (tosets). (Contributed by FL, 17-Nov-2014.)
Toset = {𝑓 ∈ Poset ∣ [(Baseβ€˜π‘“) / 𝑏][(leβ€˜π‘“) / π‘Ÿ]βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯π‘Ÿπ‘¦ ∨ π‘¦π‘Ÿπ‘₯)}
 
Theoremistos 18371* The predicate "is a toset". (Contributed by FL, 17-Nov-2014.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    β‡’   (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ∨ 𝑦 ≀ π‘₯)))
 
Theoremtosso 18372 Write the totally ordered set structure predicate in terms of the proper class strict order predicate. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ (𝐾 ∈ Toset ↔ ( < Or 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† ≀ )))
 
Theoremtospos 18373 A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ Toset β†’ 𝐹 ∈ Poset)
 
Theoremtleile 18374 In a Toset, any two elements are comparable. (Contributed by Thierry Arnoux, 11-Feb-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    β‡’   ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ∨ π‘Œ ≀ 𝑋))
 
Theoremtltnle 18375 In a Toset, "less than" is equivalent to the negation of the converse of "less than or equal to", see pltnle 18291. (Contributed by Thierry Arnoux, 11-Feb-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    < = (ltβ€˜πΎ)    β‡’   ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 < π‘Œ ↔ Β¬ π‘Œ ≀ 𝑋))
 
Syntaxcp0 18376 Extend class notation with poset zero.
class 0.
 
Syntaxcp1 18377 Extend class notation with poset unit.
class 1.
 
Definitiondf-p0 18378 Define poset zero. (Contributed by NM, 12-Oct-2011.)
0. = (𝑝 ∈ V ↦ ((glbβ€˜π‘)β€˜(Baseβ€˜π‘)))
 
Definitiondf-p1 18379 Define poset unit. (Contributed by NM, 22-Oct-2011.)
1. = (𝑝 ∈ V ↦ ((lubβ€˜π‘)β€˜(Baseβ€˜π‘)))
 
Theoremp0val 18380 Value of poset zero. (Contributed by NM, 12-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &    0 = (0.β€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ 0 = (πΊβ€˜π΅))
 
Theoremp1val 18381 Value of poset zero. (Contributed by NM, 22-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &    1 = (1.β€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ 1 = (π‘ˆβ€˜π΅))
 
Theoremp0le 18382 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.)
𝐡 = (Baseβ€˜πΎ)    &   πΊ = (glbβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 𝐡 ∈ dom 𝐺)    β‡’   (πœ‘ β†’ 0 ≀ 𝑋)
 
Theoremple1 18383 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.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    1 = (1.β€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 𝐡 ∈ dom π‘ˆ)    β‡’   (πœ‘ β†’ 𝑋 ≀ 1 )
 
9.5  Lattices
 
9.5.1  Lattices
 
Syntaxclat 18384 Extend class notation with the class of all lattices.
class Lat
 
Definitiondf-lat 18385 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 (joinβ€˜π‘) = ((Baseβ€˜π‘) Γ— (Baseβ€˜π‘)) ∧ dom (meetβ€˜π‘) = ((Baseβ€˜π‘) Γ— (Baseβ€˜π‘)))}
 
Theoremislat 18386 The predicate "is a lattice". (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐡 Γ— 𝐡) ∧ dom ∧ = (𝐡 Γ— 𝐡))))
 
Theoremodulatb 18387 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODualβ€˜π‘‚)    β‡’   (𝑂 ∈ 𝑉 β†’ (𝑂 ∈ Lat ↔ 𝐷 ∈ Lat))
 
Theoremodulat 18388 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
𝐷 = (ODualβ€˜π‘‚)    β‡’   (𝑂 ∈ Lat β†’ 𝐷 ∈ Lat)
 
Theoremlatcl2 18389 The join and meet of any two elements exist. (Contributed by NM, 14-Sep-2018.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Lat)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ∧ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ ))
 
Theoremlatlem 18390 Lemma for lattice properties. (Contributed by NM, 14-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∨ π‘Œ) ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ∈ 𝐡))
 
Theoremlatpos 18391 A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
(𝐾 ∈ Lat β†’ 𝐾 ∈ Poset)
 
Theoremlatjcl 18392 Closure of join operation in a lattice. (chjcom 30759 analog.) (Contributed by NM, 14-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) ∈ 𝐡)
 
Theoremlatmcl 18393 Closure of meet operation in a lattice. (incom 4202 analog.) (Contributed by NM, 14-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
 
Theoremlatref 18394 A lattice ordering is reflexive. (ssid 4005 analog.) (Contributed by NM, 8-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ≀ 𝑋)
 
Theoremlatasymb 18395 A lattice ordering is asymmetric. (eqss 3998 analog.) (Contributed by NM, 22-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ≀ π‘Œ ∧ π‘Œ ≀ 𝑋) ↔ 𝑋 = π‘Œ))
 
Theoremlatasym 18396 A lattice ordering is asymmetric. (eqss 3998 analog.) (Contributed by NM, 8-Oct-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ≀ π‘Œ ∧ π‘Œ ≀ 𝑋) β†’ 𝑋 = π‘Œ))
 
Theoremlattr 18397 A lattice ordering is transitive. (sstr 3991 analog.) (Contributed by NM, 17-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ≀ π‘Œ ∧ π‘Œ ≀ 𝑍) β†’ 𝑋 ≀ 𝑍))
 
Theoremlatasymd 18398 Deduce equality from lattice ordering. (eqssd 4000 analog.) (Contributed by NM, 18-Nov-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Lat)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 ≀ π‘Œ)    &   (πœ‘ β†’ π‘Œ ≀ 𝑋)    β‡’   (πœ‘ β†’ 𝑋 = π‘Œ)
 
Theoremlattrd 18399 A lattice ordering is transitive. Deduction version of lattr 18397. (Contributed by NM, 3-Sep-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   (πœ‘ β†’ 𝐾 ∈ Lat)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 ≀ π‘Œ)    &   (πœ‘ β†’ π‘Œ ≀ 𝑍)    β‡’   (πœ‘ β†’ 𝑋 ≀ 𝑍)
 
Theoremlatjcom 18400 The join of a lattice commutes. (chjcom 30759 analog.) (Contributed by NM, 16-Sep-2011.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) = (π‘Œ ∨ 𝑋))
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