P. 187 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	oduclatb 18601	Being a complete lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ 𝐷 = (ODual‘𝑂)    ⇒   ⊢ (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat)
Theorem	clatl 18602	A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5779 to shorten proof and eliminate joindmss 18471 and meetdmss 18485?
⊢ (𝐾 ∈ CLat → 𝐾 ∈ Lat)
Theorem	isglbd 18603*	Properties that determine the greatest lower bound of a complete lattice. (Contributed by Mario Carneiro, 19-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐻 ≤ 𝑦)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝐻)    &   ⊢ (𝜑 → 𝐾 ∈ CLat)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐻 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘𝑆) = 𝐻)
Theorem	lublem 18604*	Lemma for the least upper bound properties in a complete lattice. (Contributed by NM, 19-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)))
Theorem	lubub 18605	The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ≤ (𝑈‘𝑆))
Theorem	lubl 18606*	The LUB of a complete lattice subset is the least bound. (Contributed by NM, 19-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝐵) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 → (𝑈‘𝑆) ≤ 𝑋))
Theorem	lubss 18607	Subset law for least upper bounds. (chsupss 31831 analog.) (Contributed by NM, 20-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇) → (𝑈‘𝑆) ≤ (𝑈‘𝑇))
Theorem	lubel 18608	An element of a set is less than or equal to the least upper bound of the set. (Contributed by NM, 21-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ∈ 𝑆 ∧ 𝑆 ⊆ 𝐵) → 𝑋 ≤ (𝑈‘𝑆))
Theorem	lubun 18609	The LUB of a union. (Contributed by NM, 5-Mar-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑈‘(𝑆 ∪ 𝑇)) = ((𝑈‘𝑆) ∨ (𝑈‘𝑇)))
Theorem	clatglb 18610*	Properties of greatest lower bound of a complete lattice. (Contributed by NM, 5-Dec-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (∀𝑦 ∈ 𝑆 (𝐺‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝐺‘𝑆))))
Theorem	clatglble 18611	The greatest lower bound is the least element. (Contributed by NM, 5-Dec-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑆) ≤ 𝑋)
Theorem	clatleglb 18612*	Two ways of expressing "less than or equal to the greatest lower bound." (Contributed by NM, 5-Dec-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑋 ≤ (𝐺‘𝑆) ↔ ∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦))
Theorem	clatglbss 18613	Subset law for greatest lower bound. (Contributed by Mario Carneiro, 16-Apr-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝑇) → (𝐺‘𝑇) ≤ (𝐺‘𝑆))
9.5.3  Distributive lattices
Syntax	cdlat 18614	The class of distributive lattices.
class DLat
Definition	df-dlat 18615*	A distributive lattice is a lattice in which meets distribute over joins, or equivalently (latdisd 18591) joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))}
Theorem	isdlat 18616*	Property of being a distributive lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))
Theorem	dlatmjdi 18617	In a distributive lattice, meets distribute over joins. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍)))
Theorem	dlatl 18618	A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat)
Theorem	odudlatb 18619	The dual of a distributive lattice is a distributive lattice and conversely. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐷 = (ODual‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ DLat ↔ 𝐷 ∈ DLat))
Theorem	dlatjmdi 18620	In a distributive lattice, joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∧ 𝑍)) = ((𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑍)))
9.5.4  Subset order structures
Syntax	cipo 18621	Class function defining inclusion posets.
class toInc
Definition	df-ipo 18622*	For any family of sets, define the poset of that family ordered by inclusion. See ipobas 18625, ipolerval 18626, and ipole 18628 for its contract. EDITORIAL: I'm not thrilled with the name. Any suggestions? (Contributed by Stefan O'Rear, 30-Jan-2015.) (New usage is discouraged.)
⊢ toInc = (𝑓 ∈ V ↦ ⦋{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} / 𝑜⦌({⟨(Base‘ndx), 𝑓⟩, ⟨(TopSet‘ndx), (ordTop‘𝑜)⟩} ∪ {⟨(le‘ndx), 𝑜⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))
Theorem	ipostr 18623	The structure of df-ipo 18622 is a structure defining indices up to 11. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), ⊥ ⟩}) Struct ⟨1, ;11⟩
Theorem	ipoval 18624*	Value of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}    ⇒   ⊢ (𝐹 ∈ 𝑉 → 𝐼 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩} ∪ {⟨(le‘ndx), ≤ ⟩, ⟨(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})⟩}))
Theorem	ipobas 18625	Base set of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by Mario Carneiro, 25-Oct-2015.)
⊢ 𝐼 = (toInc‘𝐹)    ⇒   ⊢ (𝐹 ∈ 𝑉 → 𝐹 = (Base‘𝐼))
Theorem	ipolerval 18626*	Relation of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐼 = (toInc‘𝐹)    ⇒   ⊢ (𝐹 ∈ 𝑉 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼))
Theorem	ipotset 18627	Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ ≤ = (le‘𝐼)    ⇒   ⊢ (𝐹 ∈ 𝑉 → (ordTop‘ ≤ ) = (TopSet‘𝐼))
Theorem	ipole 18628	Weak order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ ≤ = (le‘𝐼)    ⇒   ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑋 ≤ 𝑌 ↔ 𝑋 ⊆ 𝑌))
Theorem	ipolt 18629	Strict order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ < = (lt‘𝐼)    ⇒   ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑋 < 𝑌 ↔ 𝑋 ⊊ 𝑌))
Theorem	ipopos 18630	The inclusion poset on a family of sets is actually a poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐼 = (toInc‘𝐹)    ⇒   ⊢ 𝐼 ∈ Poset
Theorem	isipodrs 18631*	Condition for a family of sets to be directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ ((toInc‘𝐴) ∈ Dirset ↔ (𝐴 ∈ V ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ∪ 𝑦) ⊆ 𝑧))
Theorem	ipodrscl 18632	Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ ((toInc‘𝐴) ∈ Dirset → 𝐴 ∈ V)
Theorem	ipodrsfi 18633*	Finite upper bound property for directed collections of sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (((toInc‘𝐴) ∈ Dirset ∧ 𝑋 ⊆ 𝐴 ∧ 𝑋 ∈ Fin) → ∃𝑧 ∈ 𝐴 ∪ 𝑋 ⊆ 𝑧)
Theorem	fpwipodrs 18634	The finite subsets of any set are directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → (toInc‘(𝒫 𝐴 ∩ Fin)) ∈ Dirset)
Theorem	ipodrsima 18635*	The monotone image of a directed set. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (𝜑 → 𝐹 Fn 𝒫 𝐴)    &   ⊢ ((𝜑 ∧ (𝑢 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝐴)) → (𝐹‘𝑢) ⊆ (𝐹‘𝑣))    &   ⊢ (𝜑 → (toInc‘𝐵) ∈ Dirset)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝒫 𝐴)    &   ⊢ (𝜑 → (𝐹 “ 𝐵) ∈ 𝑉)    ⇒   ⊢ (𝜑 → (toInc‘(𝐹 “ 𝐵)) ∈ Dirset)
Theorem	isacs3lem 18636*	An algebraic closure system satisfies isacs3 18644. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
Theorem	acsdrsel 18637	An algebraic closure system contains all directed unions of closed sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑌 ⊆ 𝐶 ∧ (toInc‘𝑌) ∈ Dirset) → ∪ 𝑌 ∈ 𝐶)
Theorem	isacs4lem 18638*	In a closure system in which directed unions of closed sets are closed, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡))))
Theorem	isacs5lem 18639*	If closure commutes with directed unions, then the closure of a set is the closure of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡))) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin))))
Theorem	acsdrscl 18640	In an algebraic closure system, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑌 ⊆ 𝒫 𝑋 ∧ (toInc‘𝑌) ∈ Dirset) → (𝐹‘∪ 𝑌) = ∪ (𝐹 “ 𝑌))
Theorem	acsficl 18641	A closure in an algebraic closure system is the union of the closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝐹‘𝑆) = ∪ (𝐹 “ (𝒫 𝑆 ∩ Fin)))
Theorem	isacs5 18642*	A closure system is algebraic iff the closure of a generating set is the union of the closures of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin))))
Theorem	isacs4 18643*	A closure system is algebraic iff closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝒫 𝑋((toInc‘𝑠) ∈ Dirset → (𝐹‘∪ 𝑠) = ∪ (𝐹 “ 𝑠))))
Theorem	isacs3 18644*	A closure system is algebraic iff directed unions of closed sets are closed. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
Theorem	acsficld 18645	In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 18641. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑁‘𝑆) = ∪ (𝑁 “ (𝒫 𝑆 ∩ Fin)))
Theorem	acsficl2d 18646*	In an algebraic closure system, an element is in the closure of a set if and only if it is in the closure of a finite subset. Alternate form of acsficl 18641. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑌 ∈ (𝑁‘𝑆) ↔ ∃𝑥 ∈ (𝒫 𝑆 ∩ Fin)𝑌 ∈ (𝑁‘𝑥)))
Theorem	acsfiindd 18647	In an algebraic closure system, a set is independent if and only if all its finite subsets are independent. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼))
Theorem	acsmapd 18648*	In an algebraic closure system, if 𝑇 is contained in the closure of 𝑆, there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that the closure of ∪ ran 𝑓 contains 𝑇. This is proven by applying acsficl2d 18646 to each element of 𝑇. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑆))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓)))
Theorem	acsmap2d 18649*	In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is independent, then there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that 𝑆 equals the union of ran 𝑓. This is proven by taking the map 𝑓 from acsmapd 18648 and observing that, since 𝑆 and 𝑇 have the same closure, the closure of ∪ ran 𝑓 must contain 𝑆. Since 𝑆 is independent, by mrissmrcd 17734, ∪ ran 𝑓 must equal 𝑆. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑋)    &   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓))
Theorem	acsinfd 18650	In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is infinite independent, then 𝑇 is infinite. This follows from applying unirnffid 9318 to the map given in acsmap2d 18649. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑋)    &   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))    &   ⊢ (𝜑 → ¬ 𝑆 ∈ Fin)    ⇒   ⊢ (𝜑 → ¬ 𝑇 ∈ Fin)
Theorem	acsdomd 18651	In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is infinite independent, then 𝑇 dominates 𝑆. This follows from applying acsinfd 18650 and then applying unirnfdomd 10580 to the map given in acsmap2d 18649. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑋)    &   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))    &   ⊢ (𝜑 → ¬ 𝑆 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝑆 ≼ 𝑇)
Theorem	acsinfdimd 18652	In an algebraic closure system, if two independent sets have equal closure and one is infinite, then they are equinumerous. This is proven by using acsdomd 18651 twice with acsinfd 18650. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐼)    &   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))    &   ⊢ (𝜑 → ¬ 𝑆 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝑆 ≈ 𝑇)
Theorem	acsexdimd 18653*	In an algebraic closure system whose closure operator has the exchange property, if two independent sets have equal closure, they are equinumerous. See mreexfidimd 17744 for the finite case and acsinfdimd 18652 for the infinite case. This is a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐼)    &   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))    ⇒   ⊢ (𝜑 → 𝑆 ≈ 𝑇)
Theorem	mrelatglb 18654	Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) See mrelatglbALT 49930 for an alternate proof.
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐺 = (glb‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈)
Theorem	mrelatglb0 18655	The empty intersection in a Moore space is realized by the base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐺 = (glb‘𝐼)    ⇒   ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐺‘∅) = 𝑋)
Theorem	mrelatlub 18656	Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) See mrelatlubALT 49929 for an alternate proof.
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐹 = (mrCls‘𝐶)    &   ⊢ 𝐿 = (lub‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))
Theorem	mreclatBAD 18657*	A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7374 update): Reprove using isclat 18594 instead of the isclatBAD. hypothesis. See commented-out mreclat above. See mreclat 49931 for a good version.
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ (𝐼 ∈ CLat ↔ (𝐼 ∈ Poset ∧ ∀𝑥(𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼)))))    ⇒   ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat)
9.6  Posets, directed sets, and lattices as relations
9.6.1  Posets and lattices as relations See commented-out notes for lattices as relations.
Syntax	cps 18658	Extend class notation with the class of all posets.
class PosetRel
Syntax	ctsr 18659	Extend class notation with the class of all totally ordered sets.
class TosetRel
Definition	df-ps 18660	Define the class of all posets (partially ordered sets) with weak ordering (e.g., "less than or equal to" instead of "less than"). A poset is a relation which is transitive, reflexive, and antisymmetric. (Contributed by NM, 11-May-2008.)
⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟))}
Definition	df-tsr 18661	Define the class of all totally ordered sets. (Contributed by FL, 1-Nov-2009.)
⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)}
Theorem	isps 18662	The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.)
⊢ (𝑅 ∈ 𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))
Theorem	psrel 18663	A poset is a relation. (Contributed by NM, 12-May-2008.)
⊢ (𝐴 ∈ PosetRel → Rel 𝐴)
Theorem	psref2 18664	A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))
Theorem	pstr2 18665	A poset is transitive. (Contributed by FL, 3-Aug-2009.)
⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
Theorem	pslem 18666	Lemma for psref 18668 and others. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)))
Theorem	psdmrn 18667	The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.)
⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
Theorem	psref 18668	A poset is reflexive. (Contributed by NM, 13-May-2008.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)
Theorem	psrn 18669	The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
Theorem	psasym 18670	A poset is antisymmetric. (Contributed by NM, 12-May-2008.)
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)
Theorem	pstr 18671	A poset is transitive. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Theorem	cnvps 18672	The converse of a poset is a poset. In the general case (◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 18673 for a special case where the property holds. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
Theorem	cnvpsb 18673	The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)
⊢ (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ ◡𝑅 ∈ PosetRel))
Theorem	psss 18674	Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.)
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
Theorem	psssdm2 18675	Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴))
Theorem	psssdm 18676	Field of a subposet. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ⊆ 𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
Theorem	istsr 18677	The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)))
Theorem	istsr2 18678*	The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
Theorem	tsrlin 18679	A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴))
Theorem	tsrlemax 18680	Two ways of saying a number is less than or equal to the maximum of two others. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))
Theorem	tsrps 18681	A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
Theorem	cnvtsr 18682	The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )
Theorem	tsrss 18683	Any subset of a totally ordered set is totally ordered. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Nov-2013.)
⊢ (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel )
Theorem	ledm 18684	The domain of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
⊢ ℝ* = dom ≤
Theorem	lern 18685	The range of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ ℝ* = ran ≤
Theorem	lefld 18686	The field of the 'less or equal to' relationship on the extended real. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
⊢ ℝ* = ∪ ∪ ≤
Theorem	letsr 18687	The "less than or equal to" relationship on the extended reals is a toset. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ ≤ ∈ TosetRel
9.6.2  Directed sets, nets
Syntax	cdir 18688	Extend class notation with the class of directed sets.
class DirRel
Syntax	ctail 18689	Extend class notation with the tail function for directed sets.
class tail
Definition	df-dir 18690	Define the class of directed sets (the order relation itself is sometimes called a direction, and a directed set is a set equipped with a direction). (Contributed by Jeff Hankins, 25-Nov-2009.)
⊢ DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ∧ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟 × ∪ ∪ 𝑟) ⊆ (◡𝑟 ∘ 𝑟)))}
Definition	df-tail 18691*	Define the tail function for directed sets. (Contributed by Jeff Hankins, 25-Nov-2009.)
⊢ tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟 ↦ (𝑟 “ {𝑥})))
Theorem	isdir 18692	A condition for a relation to be a direction. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ 𝐴 = ∪ ∪ 𝑅    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (◡𝑅 ∘ 𝑅)))))
Theorem	reldir 18693	A direction is a relation. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ (𝑅 ∈ DirRel → Rel 𝑅)
Theorem	dirdm 18694	A direction's domain is equal to its field. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)
Theorem	dirref 18695	A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)
Theorem	dirtr 18696	A direction is transitive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ (((𝑅 ∈ DirRel ∧ 𝐶 ∈ 𝑉) ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
Theorem	dirge 18697*	For any two elements of a directed set, there exists a third element greater than or equal to both. Note that this does not say that the two elements have a least upper bound. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))
Theorem	tsrdir 18698	A totally ordered set is a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)
9.7  Chains
Syntax	cchn 18699	Extend class notation with the class of (finite) chains.
class ( < Chain 𝐴)
Definition	df-chn 18700*	Define the class of (finite) chains. A chain is defined to be a sequence of objects, where each object is less than the next one in the sequence. The term "chain" is usually used in order theory. In the context of algebra, chains are often called "towers", for example for fields, or "series", for example for subgroup or subnormal series. (Contributed by Thierry Arnoux, 19-Jun-2025.)
⊢ ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}