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Theorem List for Metamath Proof Explorer - 16701-16800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlubprop 16701* Properties of greatest lower bound of a poset. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝑈)       (𝜑 → (∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)))
 
Theoremluble 16702 The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝑈)    &   (𝜑𝑋𝑆)       (𝜑𝑋 (𝑈𝑆))
 
Theoremlublecllem 16703* Lemma for lublecl 16704 and lubid 16705. (Contributed by NM, 8-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑋𝐵)       ((𝜑𝑥𝐵) → ((∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤)) ↔ 𝑥 = 𝑋))
 
Theoremlublecl 16704* The set of all elements less than a given element has an LUB. (Contributed by NM, 8-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑋𝐵)       (𝜑 → {𝑦𝐵𝑦 𝑋} ∈ dom 𝑈)
 
Theoremlubid 16705* 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.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜑𝐾 ∈ Poset)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑈‘{𝑦𝐵𝑦 𝑋}) = 𝑋)
 
Theoremglbfval 16706* Value of the greatest lower function of a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑠 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑠 𝑧 𝑦𝑧 𝑥)))    &   (𝜑𝐾𝑉)       (𝜑𝐺 = ((𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥𝐵 𝜓}))
 
Theoremglbdm 16707* Domain of the greatest lower bound function of a poset. (Contributed by NM, 6-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑠 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑠 𝑧 𝑦𝑧 𝑥)))    &   (𝜑𝐾𝑉)       (𝜑 → dom 𝐺 = {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥𝐵 𝜓})
 
Theoremglbfun 16708 The GLB is a function. (Contributed by NM, 9-Sep-2018.)
𝐺 = (glb‘𝐾)       Fun 𝐺
 
Theoremglbeldm 16709* Member of the domain of the greatest lower bound function of a poset. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))    &   (𝜑𝐾𝑉)       (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆𝐵 ∧ ∃!𝑥𝐵 𝜓)))
 
Theoremglbelss 16710 A member of the domain of the greatest lower bound function is a subset of the base set. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝐺)       (𝜑𝑆𝐵)
 
Theoremglbeu 16711* Unique existence proper of a member of the domain of the greatest lower bound function of a poset. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝐺)       (𝜑 → ∃!𝑥𝐵 𝜓)
 
Theoremglbval 16712* Value of the greatest lower bound function of a poset. Out-of-domain arguments (those not satisfying 𝑆 ∈ dom 𝑈) 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.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))    &   (𝜑𝐾𝑉)    &   (𝜑𝑆𝐵)       (𝜑 → (𝐺𝑆) = (𝑥𝐵 𝜓))
 
Theoremglbcl 16713 The least upper bound function value belongs to the base set. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝐺)       (𝜑 → (𝐺𝑆) ∈ 𝐵)
 
Theoremglbprop 16714* Properties of greatest lower bound of a poset. (Contributed by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (glb‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝑈)       (𝜑 → (∀𝑦𝑆 (𝑈𝑆) 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 (𝑈𝑆))))
 
Theoremglble 16715 The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑈 = (glb‘𝐾)    &   (𝜑𝐾𝑉)    &   (𝜑𝑆 ∈ dom 𝑈)    &   (𝜑𝑋𝑆)       (𝜑 → (𝑈𝑆) 𝑋)
 
Theoremjoinfval 16716* Value of join function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove joinfval2 16717 first to reduce net proof size (existence part)?
𝑈 = (lub‘𝐾)    &    = (join‘𝐾)       (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
 
Theoremjoinfval2 16717* 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 16718* 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 16719 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 16720 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 16721 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 16722 Subset property of domain of join. (Contributed by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   (𝜑𝐾𝑉)       (𝜑 → dom ⊆ (𝐵 × 𝐵))
 
Theoremjoinval2lem 16723* Lemma for joinval2 16724 and joineu 16725. (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 16724* 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 16725* 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 16726* 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 16727 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 16728 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 16729 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 16730* Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove meetfval2 16731 first to reduce net proof size (existence part)?
𝐺 = (glb‘𝐾)    &    = (meet‘𝐾)       (𝐾𝑉 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝐺𝑧})
 
Theoremmeetfval2 16731* 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 16732* 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 16733 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 16734 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 16735 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 16736 Subset property of domain of meet. (Contributed by NM, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &   (𝜑𝐾𝑉)       (𝜑 → dom ⊆ (𝐵 × 𝐵))
 
Theoremmeetval2lem 16737* Lemma for meetval2 16738 and meeteu 16739. (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 16738* 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 16739* 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 16740* 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 16741 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 16742 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 16743 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 16744 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 16745 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 16746 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 16747 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 16748 Extend class notation with the class of all tosets.
class Toset
 
Definitiondf-toset 16749* 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 16750* 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 16751 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 16752 Extend class notation with poset zero.
class 0.
 
Syntaxcp1 16753 Extend class notation with poset unit.
class 1.
 
Definitiondf-p0 16754 Define poset zero. (Contributed by NM, 12-Oct-2011.)
0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))
 
Definitiondf-p1 16755 Define poset unit. (Contributed by NM, 22-Oct-2011.)
1. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝)))
 
Theoremp0val 16756 Value of poset zero. (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    0 = (0.‘𝐾)       (𝐾𝑉0 = (𝐺𝐵))
 
Theoremp1val 16757 Value of poset zero. (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &    1 = (1.‘𝐾)       (𝐾𝑉1 = (𝑈𝐵))
 
Theoremp0le 16758 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 16759 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 16760 Extend class notation with the class of all lattices.
class Lat
 
Definitiondf-lat 16761 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 16762 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 16763 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 16764 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 16765 A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
(𝐾 ∈ Lat → 𝐾 ∈ Poset)
 
Theoremlatjcl 16766 Closure of join operation in a lattice. (chjcom 27537 analog.) (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
 
Theoremlatmcl 16767 Closure of meet operation in a lattice. (incom 3670 analog.) (Contributed by NM, 14-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
 
Theoremlatref 16768 A lattice ordering is reflexive. (ssid 3491 analog.) (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → 𝑋 𝑋)
 
Theoremlatasymb 16769 A lattice ordering is asymmetric. (eqss 3487 analog.) (Contributed by NM, 22-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))
 
Theoremlatasym 16770 A lattice ordering is asymmetric. (eqss 3487 analog.) (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌))
 
Theoremlattr 16771 A lattice ordering is transitive. (sstr 3480 analog.) (Contributed by NM, 17-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))
 
Theoremlatasymd 16772 Deduce equality from lattice ordering. (eqssd 3489 analog.) (Contributed by NM, 18-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   (𝜑𝐾 ∈ Lat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑋 𝑌)    &   (𝜑𝑌 𝑋)       (𝜑𝑋 = 𝑌)
 
Theoremlattrd 16773 A lattice ordering is transitive. Deduction version of lattr 16771. (Contributed by NM, 3-Sep-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   (𝜑𝐾 ∈ Lat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑌)    &   (𝜑𝑌 𝑍)       (𝜑𝑋 𝑍)
 
Theoremlatjcom 16774 The join of a lattice commutes. (chjcom 27537 analog.) (Contributed by NM, 16-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
 
Theoremlatlej1 16775 A join's first argument is less than or equal to the join. (chub1 27538 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋 (𝑋 𝑌))
 
Theoremlatlej2 16776 A join's second argument is less than or equal to the join. (chub2 27539 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌 (𝑋 𝑌))
 
Theoremlatjle12 16777 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 27540 analog.) (Contributed by NM, 17-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍𝑌 𝑍) ↔ (𝑋 𝑌) 𝑍))
 
Theoremlatleeqj1 16778 Less-than-or-equal-to in terms of join. (chlejb1 27543 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 𝑌) = 𝑌))
 
Theoremlatleeqj2 16779 Less-than-or-equal-to in terms of join. (chlejb2 27544 analog.) (Contributed by NM, 14-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑌 𝑋) = 𝑌))
 
Theoremlatjlej1 16780 Add join to both sides of a lattice ordering. (chlej1i 27504 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑋 𝑍) (𝑌 𝑍)))
 
Theoremlatjlej2 16781 Add join to both sides of a lattice ordering. (chlej2i 27505 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑍 𝑋) (𝑍 𝑌)))
 
Theoremlatjlej12 16782 Add join to both sides of a lattice ordering. (chlej12i 27506 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 16783 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 16784 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 16785 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 16786 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 16787 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 16788 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 16789 Lattice join is idempotent. (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)
 
Theoremlatmcom 16790 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 16791 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 16792 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 16793 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 16794 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 16795 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 16796 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 16797 Add meet to both sides of a lattice ordering. (sslin 3704 analog.) (Contributed by NM, 10-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑍 𝑋) (𝑍 𝑌)))
 
Theoremlatmlem12 16798 Add join to both sides of a lattice ordering. (ss2in 3705 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 16799 Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3721 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 16800 Equivalent expressions for "not less than" in a lattice. (chnle 27545 analog.) (Contributed by NM, 16-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (¬ 𝑌 𝑋𝑋 < (𝑋 𝑌)))
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