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Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxpsc0 16201 The pair function maps 0 to 𝐴. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐴𝑉 → (({𝐴} +𝑐 {𝐵})‘∅) = 𝐴)

Theoremxpsc1 16202 The pair function maps 1 to 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐵𝑉 → (({𝐴} +𝑐 {𝐵})‘1𝑜) = 𝐵)

Theoremxpscfv 16203 The value of the pair function at an element of 2𝑜. (Contributed by Mario Carneiro, 14-Aug-2015.)
((𝐴𝑉𝐵𝑊𝐶 ∈ 2𝑜) → (({𝐴} +𝑐 {𝐵})‘𝐶) = if(𝐶 = ∅, 𝐴, 𝐵))

Theoremxpsfrnel 16204* Elementhood in the target space of the function 𝐹 appearing in xpsval 16213. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐺X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2𝑜 ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1𝑜) ∈ 𝐵))

Theoremxpsfeq 16205 A function on 2𝑜 is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝐺 Fn 2𝑜({(𝐺‘∅)} +𝑐 {(𝐺‘1𝑜)}) = 𝐺)

Theoremxpsfrnel2 16206* Elementhood in the target space of the function 𝐹 appearing in xpsval 16213. (Contributed by Mario Carneiro, 15-Aug-2015.)
(({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋𝐴𝑌𝐵))

Theoremxpscf 16207 Equivalent condition for the pair function to be a proper function on 𝐴. (Contributed by Mario Carneiro, 20-Aug-2015.)
(({𝑋} +𝑐 {𝑌}):2𝑜𝐴 ↔ (𝑋𝐴𝑌𝐴))

Theoremxpsfval 16208* The value of the function appearing in xpsval 16213. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))       ((𝑋𝐴𝑌𝐵) → (𝑋𝐹𝑌) = ({𝑋} +𝑐 {𝑌}))

Theoremxpsff1o 16209* The function appearing in xpsval 16213 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2𝑜 = {∅, 1𝑜}. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))       𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)

Theoremxpsfrn 16210* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))       ran 𝐹 = X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)

Theoremxpsfrn2 16211* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))       ((𝐴𝑉𝐵𝑊) → ran 𝐹 = X𝑘 ∈ 2𝑜 (({𝐴} +𝑐 {𝐵})‘𝑘))

Theoremxpsff1o2 16212* The function appearing in xpsval 16213 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2𝑜 = {∅, 1𝑜}. (Contributed by Mario Carneiro, 24-Jan-2015.)
𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))       𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹

Theoremxpsval 16213* Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝐹 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))    &   𝐺 = (Scalar‘𝑅)    &   𝑈 = (𝐺Xs({𝑅} +𝑐 {𝑆}))       (𝜑𝑇 = (𝐹s 𝑈))

Theoremxpslem 16214* The indexed structure product that appears in xpsval 16213 has the same base as the target of the function 𝐹. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝐹 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))    &   𝐺 = (Scalar‘𝑅)    &   𝑈 = (𝐺Xs({𝑅} +𝑐 {𝑆}))       (𝜑 → ran 𝐹 = (Base‘𝑈))

Theoremxpsbas 16215 The base set of the binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)       (𝜑 → (𝑋 × 𝑌) = (Base‘𝑇))

Theoremxpsaddlem 16216* Lemma for xpsadd 16217 and xpsmul 16218. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &    · = (𝐸𝑅)    &    × = (𝐸𝑆)    &    = (𝐸𝑇)    &   𝐹 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))    &   𝑈 = ((Scalar‘𝑅)Xs({𝑅} +𝑐 {𝑆}))    &   ((𝜑({𝐴} +𝑐 {𝐵}) ∈ ran 𝐹({𝐶} +𝑐 {𝐷}) ∈ ran 𝐹) → ((𝐹({𝐴} +𝑐 {𝐵})) (𝐹({𝐶} +𝑐 {𝐷}))) = (𝐹‘(({𝐴} +𝑐 {𝐵})(𝐸𝑈)({𝐶} +𝑐 {𝐷}))))    &   ((({𝑅} +𝑐 {𝑆}) Fn 2𝑜({𝐴} +𝑐 {𝐵}) ∈ (Base‘𝑈) ∧ ({𝐶} +𝑐 {𝐷}) ∈ (Base‘𝑈)) → (({𝐴} +𝑐 {𝐵})(𝐸𝑈)({𝐶} +𝑐 {𝐷})) = (𝑘 ∈ 2𝑜 ↦ ((({𝐴} +𝑐 {𝐵})‘𝑘)(𝐸‘(({𝑅} +𝑐 {𝑆})‘𝑘))(({𝐶} +𝑐 {𝐷})‘𝑘))))       (𝜑 → (⟨𝐴, 𝐵𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

Theoremxpsadd 16217 Value of the addition operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &    · = (+g𝑅)    &    × = (+g𝑆)    &    = (+g𝑇)       (𝜑 → (⟨𝐴, 𝐵𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

Theoremxpsmul 16218 Value of the multiplication operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &    · = (.r𝑅)    &    × = (.r𝑆)    &    = (.r𝑇)       (𝜑 → (⟨𝐴, 𝐵𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)

Theoremxpssca 16219 Value of the scalar field of a binary structure product. For concreteness, we choose the scalar field to match the left argument, but in most cases where this slot is meaningful both factors will have the same scalar field, so that it doesn't matter which factor is chosen. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝐺 = (Scalar‘𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)       (𝜑𝐺 = (Scalar‘𝑇))

Theoremxpsvsca 16220 Value of the scalar multiplication function in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝐺 = (Scalar‘𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   𝐾 = (Base‘𝐺)    &    · = ( ·𝑠𝑅)    &    × = ( ·𝑠𝑆)    &    = ( ·𝑠𝑇)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑌)    &   (𝜑 → (𝐴 · 𝐵) ∈ 𝑋)    &   (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)       (𝜑 → (𝐴 𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)

Theoremxpsless 16221 Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &    = (le‘𝑇)       (𝜑 ⊆ ((𝑋 × 𝑌) × (𝑋 × 𝑌)))

Theoremxpsle 16222 Value of the ordering in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &    = (le‘𝑇)    &   𝑀 = (le‘𝑅)    &   𝑁 = (le‘𝑆)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)       (𝜑 → (⟨𝐴, 𝐵𝐶, 𝐷⟩ ↔ (𝐴𝑀𝐶𝐵𝑁𝐷)))

7.2  Moore spaces

Syntaxcmre 16223 The class of Moore systems.
class Moore

Syntaxcmrc 16224 The class function generating Moore closures.
class mrCls

Syntaxcmri 16225 mrInd is a class function which takes a Moore system to its set of independent sets.
class mrInd

Syntaxcacs 16226 The class of algebraic closure (Moore) systems.
class ACS

Definitiondf-mre 16227* Define a Moore collection, which is a family of subsets of a base set which preserve arbitrary intersection. Elements of a Moore collection are termed closed; Moore collections generalize the notion of closedness from topologies (cldmre 20863) and vector spaces (lssmre 18947) to the most general setting in which such concepts make sense. Definition of Moore collection of sets in [Schechter] p. 78. A Moore collection may also be called a closure system (Section 0.6 in [Gratzer] p. 23.) The name Moore collection is after Eliakim Hastings Moore, who discussed these systems in Part I of [Moore] p. 53 to 76.

See ismre 16231, mresspw 16233, mre1cl 16235 and mreintcl 16236 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 16241); as such the disjoint union of all Moore collections is sometimes considered as ran Moore, justified by mreunirn 16242. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.)

Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → 𝑠𝑐))})

Definitiondf-mrc 16228* Define the Moore closure of a generating set, which is the smallest closed set containing all generating elements. Definition of Moore closure in [Schechter] p. 79. This generalizes topological closure (mrccls 20864) and linear span (mrclsp 18970).

A Moore closure operation 𝑁 is (1) extensive, i.e., 𝑥 ⊆ (𝑁𝑥) for all subsets 𝑥 of the base set (mrcssid 16258), (2) isotone, i.e., 𝑥𝑦 implies that (𝑁𝑥) ⊆ (𝑁𝑦) for all subsets 𝑥 and 𝑦 of the base set (mrcss 16257), and (3) idempotent, i.e., (𝑁‘(𝑁𝑥)) = (𝑁𝑥) for all subsets 𝑥 of the base set (mrcidm 16260.) Operators satisfying these three properties are in bijective correspondence with Moore collections, so these properties may be used to give an alternate characterization of a Moore collection by providing a closure operation 𝑁 on the set of subsets of a given base set which satisfies (1), (2), and (3); the closed sets can be recovered as those sets which equal their closures (Section 4.5 in [Schechter] p. 82.) (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by David Moews, 1-May-2017.)

mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))

Definitiondf-mri 16229* In a Moore system, a set is independent if no element of the set is in the closure of the set with the element removed (Section 0.6 in [Gratzer] p. 27; Definition 4.1.1 in [FaureFrolicher] p. 83.) mrInd is a class function which takes a Moore system to its set of independent sets. (Contributed by David Moews, 1-May-2017.)
mrInd = (𝑐 ran Moore ↦ {𝑠 ∈ 𝒫 𝑐 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})

Definitiondf-acs 16230* An important subclass of Moore systems are those which can be interpreted as closure under some collection of operators of finite arity (the collection itself is not required to be finite). These are termed algebraic closure systems; similar to definition (A) of an algebraic closure system in [Schechter] p. 84, but to avoid the complexity of an arbitrary mixed collection of functions of various arities (especially if the axiom of infinity omex 8525 is to be avoided), we consider a single function defined on finite sets instead. (Contributed by Stefan O'Rear, 2-Apr-2015.)
ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})

Theoremismre 16231* Property of being a Moore collection on some base set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋𝑋𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → 𝑠𝐶)))

Theoremfnmre 16232 The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 20709 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore Fn V

Theoremmresspw 16233 A Moore collection is a subset of the power of the base set; each closed subset of the system is actually a subset of the base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)

Theoremmress 16234 A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)

Theoremmre1cl 16235 In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)

Theoremmreintcl 16236 A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶𝑆 ≠ ∅) → 𝑆𝐶)

Theoremmreiincl 16237* A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦𝐼 𝑆𝐶) → 𝑦𝐼 𝑆𝐶)

Theoremmrerintcl 16238 The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (𝑋 𝑆) ∈ 𝐶)

Theoremmreriincl 16239* The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦𝐼 𝑆𝐶) → (𝑋 𝑦𝐼 𝑆) ∈ 𝐶)

Theoremmreincl 16240 Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶)

Theoremmreuni 16241 Since the entire base set of a Moore collection is the greatest element of it, the base set can be recovered from a Moore collection by set union. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)

Theoremmreunirn 16242 Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐶 ran Moore ↔ 𝐶 ∈ (Moore‘ 𝐶))

Theoremismred 16243* Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝜑𝐶 ⊆ 𝒫 𝑋)    &   (𝜑𝑋𝐶)    &   ((𝜑𝑠𝐶𝑠 ≠ ∅) → 𝑠𝐶)       (𝜑𝐶 ∈ (Moore‘𝑋))

Theoremismred2 16244* Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝜑𝐶 ⊆ 𝒫 𝑋)    &   ((𝜑𝑠𝐶) → (𝑋 𝑠) ∈ 𝐶)       (𝜑𝐶 ∈ (Moore‘𝑋))

Theoremmremre 16245 The Moore collections of subsets of a space, viewed as a kind of subset of the power set, form a Moore collection in their own right on the power set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝑋𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))

Theoremsubmre 16246 The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))

7.2.1  Moore closures

Theoremmrcflem 16247* The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)
(𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}):𝒫 𝑋𝐶)

Theoremfnmrc 16248 Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
mrCls Fn ran Moore

Theoremmrcfval 16249* Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))

Theoremmrcf 16250 The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)

Theoremmrcval 16251* Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})

Theoremmrccl 16252 The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝐶)

Theoremmrcsncl 16253 The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹‘{𝑈}) ∈ 𝐶)

Theoremmrcid 16254 The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝐶) → (𝐹𝑈) = 𝑈)

Theoremmrcssv 16255 The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑈) ⊆ 𝑋)

Theoremmrcidb 16256 A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (Moore‘𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) = 𝑈))

Theoremmrcss 16257 Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑈) ⊆ (𝐹𝑉))

Theoremmrcssid 16258 The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ⊆ (𝐹𝑈))

Theoremmrcidb2 16259 A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝑈𝐶 ↔ (𝐹𝑈) ⊆ 𝑈))

Theoremmrcidm 16260 The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹‘(𝐹𝑈)) = (𝐹𝑈))

Theoremmrcsscl 16261 The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝐶) → (𝐹𝑈) ⊆ 𝑉)

Theoremmrcuni 16262 Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) = (𝐹 (𝐹𝑈)))

Theoremmrcun 16263 Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹‘(𝑈𝑉)) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))

Theoremmrcssvd 16264 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 16255. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)       (𝜑 → (𝑁𝐵) ⊆ 𝑋)

Theoremmrcssd 16265 Moore closure preserves subset ordering. Deduction form of mrcss 16257. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑉)    &   (𝜑𝑉𝑋)       (𝜑 → (𝑁𝑈) ⊆ (𝑁𝑉))

Theoremmrcssidd 16266 A set is contained in its Moore closure. Deduction form of mrcssid 16258. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑋)       (𝜑𝑈 ⊆ (𝑁𝑈))

Theoremmrcidmd 16267 Moore closure is idempotent. Deduction form of mrcidm 16260. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑈𝑋)       (𝜑 → (𝑁‘(𝑁𝑈)) = (𝑁𝑈))

Theoremmressmrcd 16268 In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑆 ⊆ (𝑁𝑇))    &   (𝜑𝑇𝑆)       (𝜑 → (𝑁𝑆) = (𝑁𝑇))

Theoremsubmrc 16269 In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝐹 = (mrCls‘𝐶)    &   𝐺 = (mrCls‘(𝐶 ∩ 𝒫 𝐷))       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐷𝐶𝑈𝐷) → (𝐺𝑈) = (𝐹𝑈))

Theoremmrieqvlemd 16270 In a Moore system, if 𝑌 is a member of 𝑆, (𝑆 ∖ {𝑌}) and 𝑆 have the same closure if and only if 𝑌 is in the closure of (𝑆 ∖ {𝑌}). Used in the proof of mrieqvd 16279 and mrieqv2d 16280. Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑𝑆𝑋)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁𝑆)))

7.2.2  Independent sets in a Moore system

Theoremmrisval 16271* Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)       (𝐴 ∈ (Moore‘𝑋) → 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})

Theoremismri 16272* Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)       (𝐴 ∈ (Moore‘𝑋) → (𝑆𝐼 ↔ (𝑆𝑋 ∧ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))

Theoremismri2 16273* Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)       ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))

Theoremismri2d 16274* Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝑋)       (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))

Theoremismri2dd 16275* Definition of independence of a subset of the base set in a Moore system. One-way deduction form. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝑋)    &   (𝜑 → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))       (𝜑𝑆𝐼)

Theoremmriss 16276 An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.)
𝐼 = (mrInd‘𝐴)       ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝐼) → 𝑆𝑋)

Theoremmrissd 16277 An independent set of a Moore system is a subset of the base set. Deduction form. (Contributed by David Moews, 1-May-2017.)
𝐼 = (mrInd‘𝐴)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝐼)       (𝜑𝑆𝑋)

Theoremismri2dad 16278 Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝐴 ∈ (Moore‘𝑋))    &   (𝜑𝑆𝐼)    &   (𝜑𝑌𝑆)       (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))

Theoremmrieqvd 16279* In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝑋)       (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆)))

Theoremmrieqv2d 16280* In a Moore system, a set is independent if and only if all its proper subsets have closure properly contained in the closure of the set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝑋)       (𝜑 → (𝑆𝐼 ↔ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))

Theoremmrissmrcd 16281 In a Moore system, if an independent set is between a set and its closure, the two sets are equal (since the two sets must have equal closures by mressmrcd 16268, and so are equal by mrieqv2d 16280.) (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆 ⊆ (𝑁𝑇))    &   (𝜑𝑇𝑆)    &   (𝜑𝑆𝐼)       (𝜑𝑆 = 𝑇)

Theoremmrissmrid 16282 In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑𝑆𝐼)    &   (𝜑𝑇𝑆)       (𝜑𝑇𝐼)

Theoremmreexd 16283* In a Moore system, the closure operator is said to have the exchange property if, for all elements 𝑦 and 𝑧 of the base set and subsets 𝑆 of the base set such that 𝑧 is in the closure of (𝑆 ∪ {𝑦}) but not in the closure of 𝑆, 𝑦 is in the closure of (𝑆 ∪ {𝑧}) (Definition 3.1.9 in [FaureFrolicher] p. 57 to 58.) This theorem allows us to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.)
(𝜑𝑋𝑉)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆𝑋)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌})))    &   (𝜑 → ¬ 𝑍 ∈ (𝑁𝑆))       (𝜑𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍})))

Theoremmreexmrid 16284* In a Moore system whose closure operator has the exchange property, if a set is independent and an element is not in its closure, then adding the element to the set gives another independent set. Lemma 4.1.5 in [FaureFrolicher] p. 84. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆𝐼)    &   (𝜑𝑌𝑋)    &   (𝜑 → ¬ 𝑌 ∈ (𝑁𝑆))       (𝜑 → (𝑆 ∪ {𝑌}) ∈ 𝐼)

Theoremmreexexlemd 16285* This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 16289. (Contributed by David Moews, 1-May-2017.)
(𝜑𝑋𝐽)    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹𝐾𝐺𝐾))    &   (𝜑 → ∀𝑡𝑢 ∈ 𝒫 (𝑋𝑡)∀𝑣 ∈ 𝒫 (𝑋𝑡)(((𝑢𝐾𝑣𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣𝑡)) ∧ (𝑢𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢𝑖 ∧ (𝑖𝑡) ∈ 𝐼)))       (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))

Theoremmreexexlem2d 16286* Used in mreexexlem4d 16288 to prove the induction step in mreexexd 16289. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑𝑌𝐹)       (𝜑 → ∃𝑔𝐺𝑔 ∈ (𝐹 ∖ {𝑌}) ∧ ((𝐹 ∖ {𝑌}) ∪ (𝐻 ∪ {𝑔})) ∈ 𝐼))

Theoremmreexexlem3d 16287* Base case of the induction in mreexexd 16289. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅))       (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹𝑖 ∧ (𝑖𝐻) ∈ 𝐼))

Theoremmreexexlem4d 16288* Induction step of the induction in mreexexd 16289. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑𝐿 ∈ ω)    &   (𝜑 → ∀𝑓 ∈ 𝒫 (𝑋)∀𝑔 ∈ 𝒫 (𝑋)(((𝑓𝐿𝑔𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔)) ∧ (𝑓) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓𝑗 ∧ (𝑗) ∈ 𝐼)))    &   (𝜑 → (𝐹 ≈ suc 𝐿𝐺 ≈ suc 𝐿))       (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))

Theoremmreexexd 16289* Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if 𝐹 and 𝐺 are disjoint from 𝐻, (𝐹𝐻) is independent, 𝐹 is contained in the closure of (𝐺𝐻), and either 𝐹 or 𝐺 is finite, then there is a subset 𝑞 of 𝐺 equinumerous to 𝐹 such that (𝑞𝐻) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either (𝐴𝐵) or (𝐵𝐴) is finite. The theorem is proven by induction using mreexexlem3d 16287 for the base case and mreexexlem4d 16288 for the induction step. (Contributed by David Moews, 1-May-2017.) Removed dependencies on ax-rep 4762 and ax-ac2 9270. (Revised by Brendan Leahy, 2-Jun-2021.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹 ∈ Fin ∨ 𝐺 ∈ Fin))       (𝜑 → ∃𝑞 ∈ 𝒫 𝐺(𝐹𝑞 ∧ (𝑞𝐻) ∈ 𝐼))

TheoremmreexexdOLD 16290* Obsolete proof of mreexexd 16289 as of 2-Jun-2021. Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if 𝐹 and 𝐺 are disjoint from 𝐻, (𝐹𝐻) is independent, 𝐹 is contained in the closure of (𝐺𝐻), and either 𝐹 or 𝐺 is finite, then there is a subset 𝑞 of 𝐺 equinumerous to 𝐹 such that (𝑞𝐻) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either (𝐴𝐵) or (𝐵𝐴) is finite. The theorem is proven by induction using mreexexlem3d 16287 for the base case and mreexexlem4d 16288 for the induction step. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝐹 ⊆ (𝑋𝐻))    &   (𝜑𝐺 ⊆ (𝑋𝐻))    &   (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))    &   (𝜑 → (𝐹𝐻) ∈ 𝐼)    &   (𝜑 → (𝐹 ∈ Fin ∨ 𝐺 ∈ Fin))       (𝜑 → ∃𝑞 ∈ 𝒫 𝐺(𝐹𝑞 ∧ (𝑞𝐻) ∈ 𝐼))

Theoremmreexdomd 16291* In a Moore system whose closure operator has the exchange property, if 𝑆 is independent and contained in the closure of 𝑇, and either 𝑆 or 𝑇 is finite, then 𝑇 dominates 𝑆. This is an immediate consequence of mreexexd 16289. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆 ⊆ (𝑁𝑇))    &   (𝜑𝑇𝑋)    &   (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin))    &   (𝜑𝑆𝐼)       (𝜑𝑆𝑇)

Theoremmreexfidimd 16292* In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd 16291 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (Moore‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   𝐼 = (mrInd‘𝐴)    &   (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   (𝜑𝑆𝐼)    &   (𝜑𝑇𝐼)    &   (𝜑𝑆 ∈ Fin)    &   (𝜑 → (𝑁𝑆) = (𝑁𝑇))       (𝜑𝑆𝑇)

7.2.3  Algebraic closure systems

Theoremisacs 16293* A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))

Theoremacsmre 16294 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
(𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))

Theoremisacs2 16295* In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))

Theoremacsfiel 16296* A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
𝐹 = (mrCls‘𝐶)       (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶 ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))

Theoremacsfiel2 16297* A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 3-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))

Theoremacsmred 16298 An algebraic closure system is also a Moore system. Deduction form of acsmre 16294. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (ACS‘𝑋))       (𝜑𝐴 ∈ (Moore‘𝑋))

Theoremisacs1i 16299* A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋))

Theoremmreacs 16300 Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))

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