P. 173 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17201-17300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xpsaddlem 17201*	Lemma for xpsadd 17202 and xpsmul 17203. (Contributed by Mario Carneiro, 15-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &   ⊢ · = (𝐸‘𝑅)    &   ⊢ × = (𝐸‘𝑆)    &   ⊢ ∙ = (𝐸‘𝑇)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   ⊢ 𝑈 = ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})    &   ⊢ ((𝜑 ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran 𝐹 ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ ran 𝐹) → ((◡𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) ∙ (◡𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (◡𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})))    &   ⊢ (({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘𝑈) ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ (Base‘𝑈)) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸‘𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))))    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
Theorem	xpsadd 17202	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)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &   ⊢ · = (+g‘𝑅)    &   ⊢ × = (+g‘𝑆)    &   ⊢ ∙ = (+g‘𝑇)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
Theorem	xpsmul 17203	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)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)    &   ⊢ (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    &   ⊢ ∙ = (.r‘𝑇)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∙ ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
Theorem	xpssca 17204	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)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐺 = (Scalar‘𝑇))
Theorem	xpsvsca 17205	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)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ · = ( ·𝑠 ‘𝑅)    &   ⊢ × = ( ·𝑠 ‘𝑆)    &   ⊢ ∙ = ( ·𝑠 ‘𝑇)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑌)    &   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ 𝑋)    &   ⊢ (𝜑 → (𝐴 × 𝐶) ∈ 𝑌)    ⇒   ⊢ (𝜑 → (𝐴 ∙ ⟨𝐵, 𝐶⟩) = ⟨(𝐴 · 𝐵), (𝐴 × 𝐶)⟩)
Theorem	xpsless 17206	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)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ ≤ = (le‘𝑇)    ⇒   ⊢ (𝜑 → ≤ ⊆ ((𝑋 × 𝑌) × (𝑋 × 𝑌)))
Theorem	xpsle 17207	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)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ ≤ = (le‘𝑇)    &   ⊢ 𝑀 = (le‘𝑅)    &   ⊢ 𝑁 = (le‘𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ≤ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝑀𝐶 ∧ 𝐵𝑁𝐷)))
7.2  Moore spaces
Syntax	cmre 17208	The class of Moore systems.
class Moore
Syntax	cmrc 17209	The class function generating Moore closures.
class mrCls
Syntax	cmri 17210	mrInd is a class function which takes a Moore system to its set of independent sets.
class mrInd
Syntax	cacs 17211	The class of algebraic closure (Moore) systems.
class ACS
Definition	df-mre 17212*	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 22137) and vector spaces (lssmre 20143) 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 17216, mresspw 17218, mre1cl 17220 and mreintcl 17221 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 17226); as such the disjoint union of all Moore collections is sometimes considered as ∪ ran Moore, justified by mreunirn 17227. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.)
⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))})
Definition	df-mrc 17213*	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 22138) and linear span (mrclsp 20166). A Moore closure operation 𝑁 is (1) extensive, i.e., 𝑥 ⊆ (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcssid 17243), (2) isotone, i.e., 𝑥 ⊆ 𝑦 implies that (𝑁‘𝑥) ⊆ (𝑁‘𝑦) for all subsets 𝑥 and 𝑦 of the base set (mrcss 17242), and (3) idempotent, i.e., (𝑁‘(𝑁‘𝑥)) = (𝑁‘𝑥) for all subsets 𝑥 of the base set (mrcidm 17245.) 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 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}))
Definition	df-mri 17214*	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 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))})
Definition	df-acs 17215*	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 9331 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(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
Theorem	ismre 17216*	Property of being a Moore collection on some base set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝐶 ∈ (Moore‘𝑋) ↔ (𝐶 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝐶 ∧ ∀𝑠 ∈ 𝒫 𝐶(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝐶)))
Theorem	fnmre 17217	The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 21981 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ Moore Fn V
Theorem	mresspw 17218	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)
Theorem	mress 17219	A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
Theorem	mre1cl 17220	In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶)
Theorem	mreintcl 17221	A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶)
Theorem	mreiincl 17222*	A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐼 ≠ ∅ ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → ∩ 𝑦 ∈ 𝐼 𝑆 ∈ 𝐶)
Theorem	mrerintcl 17223	The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶)
Theorem	mreriincl 17224*	The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝐼 𝑆 ∈ 𝐶) → (𝑋 ∩ ∩ 𝑦 ∈ 𝐼 𝑆) ∈ 𝐶)
Theorem	mreincl 17225	Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶)
Theorem	mreuni 17226	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)
Theorem	mreunirn 17227	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))
Theorem	ismred 17228*	Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶 ∧ 𝑠 ≠ ∅) → ∩ 𝑠 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋))
Theorem	ismred2 17229*	Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (𝜑 → 𝐶 ⊆ 𝒫 𝑋)    &   ⊢ ((𝜑 ∧ 𝑠 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑠) ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (Moore‘𝑋))
Theorem	mremre 17230	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.)
⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))
Theorem	submre 17231	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) ∧ 𝐴 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))
7.2.1  Moore closures
Theorem	mrcflem 17232*	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 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}):𝒫 𝑋⟶𝐶)
Theorem	fnmrc 17233	Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ mrCls Fn ∪ ran Moore
Theorem	mrcfval 17234*	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 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}))
Theorem	mrcf 17235	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 𝑋⟶𝐶)
Theorem	mrcval 17236*	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) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) = ∩ {𝑠 ∈ 𝐶 ∣ 𝑈 ⊆ 𝑠})
Theorem	mrccl 17237	The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶)
Theorem	mrcsncl 17238	The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝑋) → (𝐹‘{𝑈}) ∈ 𝐶)
Theorem	mrcid 17239	The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝐶) → (𝐹‘𝑈) = 𝑈)
Theorem	mrcssv 17240	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) → (𝐹‘𝑈) ⊆ 𝑋)
Theorem	mrcidb 17241	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) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) = 𝑈))
Theorem	mrcss 17242	Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘𝑈) ⊆ (𝐹‘𝑉))
Theorem	mrcssid 17243	The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ (𝐹‘𝑈))
Theorem	mrcidb2 17244	A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) ⊆ 𝑈))
Theorem	mrcidm 17245	The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘(𝐹‘𝑈)) = (𝐹‘𝑈))
Theorem	mrcsscl 17246	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) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ∈ 𝐶) → (𝐹‘𝑈) ⊆ 𝑉)
Theorem	mrcuni 17247	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‘∪ (𝐹 “ 𝑈)))
Theorem	mrcun 17248	Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋 ∧ 𝑉 ⊆ 𝑋) → (𝐹‘(𝑈 ∪ 𝑉)) = (𝐹‘((𝐹‘𝑈) ∪ (𝐹‘𝑉))))
Theorem	mrcssvd 17249	The Moore closure of a set is a subset of the base. Deduction form of mrcssv 17240. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    ⇒   ⊢ (𝜑 → (𝑁‘𝐵) ⊆ 𝑋)
Theorem	mrcssd 17250	Moore closure preserves subset ordering. Deduction form of mrcss 17242. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑁‘𝑈) ⊆ (𝑁‘𝑉))
Theorem	mrcssidd 17251	A set is contained in its Moore closure. Deduction form of mrcssid 17243. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈))
Theorem	mrcidmd 17252	Moore closure is idempotent. Deduction form of mrcidm 17245. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑁‘(𝑁‘𝑈)) = (𝑁‘𝑈))
Theorem	mressmrcd 17253	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.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇))    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))
Theorem	submrc 17254	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) ∧ 𝐷 ∈ 𝐶 ∧ 𝑈 ⊆ 𝐷) → (𝐺‘𝑈) = (𝐹‘𝑈))
Theorem	mrieqvlemd 17255	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 17264 and mrieqv2d 17265. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)))
7.2.2  Independent sets in a Moore system
Theorem	mrisval 17256*	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 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ (𝑁‘(𝑠 ∖ {𝑥}))})
Theorem	ismri 17257*	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) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
Theorem	ismri2 17258*	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) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
Theorem	ismri2d 17259*	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)    &   ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
Theorem	ismri2dd 17260*	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)    &   ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))    ⇒   ⊢ (𝜑 → 𝑆 ∈ 𝐼)
Theorem	mriss 17261	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) ∧ 𝑆 ∈ 𝐼) → 𝑆 ⊆ 𝑋)
Theorem	mrissd 17262	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)    &   ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    ⇒   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
Theorem	ismri2dad 17263	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)    &   ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})))
Theorem	mrieqvd 17264*	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.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁‘𝑆)))
Theorem	mrieqv2d 17265*	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.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑠(𝑠 ⊊ 𝑆 → (𝑁‘𝑠) ⊊ (𝑁‘𝑆))))
Theorem	mrissmrcd 17266	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 17253, and so are equal by mrieqv2d 17265.) (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇))    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    ⇒   ⊢ (𝜑 → 𝑆 = 𝑇)
Theorem	mrissmrid 17267	In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → 𝑇 ∈ 𝐼)
Theorem	mreexd 17268*	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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑁‘(𝑆 ∪ {𝑌})))    &   ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘𝑆))    ⇒   ⊢ (𝜑 → 𝑌 ∈ (𝑁‘(𝑆 ∪ {𝑍})))
Theorem	mreexmrid 17269*	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.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘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 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘𝑆))    ⇒   ⊢ (𝜑 → (𝑆 ∪ {𝑌}) ∈ 𝐼)
Theorem	mreexexlemd 17270*	This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 17274. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝑋 ∈ 𝐽)    &   ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))    &   ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))    &   ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))    &   ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)    &   ⊢ (𝜑 → (𝐹 ≈ 𝐾 ∨ 𝐺 ≈ 𝐾))    &   ⊢ (𝜑 → ∀𝑡∀𝑢 ∈ 𝒫 (𝑋 ∖ 𝑡)∀𝑣 ∈ 𝒫 (𝑋 ∖ 𝑡)(((𝑢 ≈ 𝐾 ∨ 𝑣 ≈ 𝐾) ∧ 𝑢 ⊆ (𝑁‘(𝑣 ∪ 𝑡)) ∧ (𝑢 ∪ 𝑡) ∈ 𝐼) → ∃𝑖 ∈ 𝒫 𝑣(𝑢 ≈ 𝑖 ∧ (𝑖 ∪ 𝑡) ∈ 𝐼)))    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
Theorem	mreexexlem2d 17271*	Used in mreexexlem4d 17273 to prove the induction step in mreexexd 17274. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))    &   ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))    &   ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))    &   ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (¬ 𝑔 ∈ (𝐹 ∖ {𝑌}) ∧ ((𝐹 ∖ {𝑌}) ∪ (𝐻 ∪ {𝑔})) ∈ 𝐼))
Theorem	mreexexlem3d 17272*	Base case of the induction in mreexexd 17274. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))    &   ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))    &   ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))    &   ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)    &   ⊢ (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅))    ⇒   ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼))
Theorem	mreexexlem4d 17273*	Induction step of the induction in mreexexd 17274. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))    &   ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))    &   ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))    &   ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)    &   ⊢ (𝜑 → 𝐿 ∈ ω)    &   ⊢ (𝜑 → ∀ℎ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐿 ∨ 𝑔 ≈ 𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)))    &   ⊢ (𝜑 → (𝐹 ≈ suc 𝐿 ∨ 𝐺 ≈ suc 𝐿))    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
Theorem	mreexexd 17274*	Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if 𝐹 and 𝐺 are disjoint from 𝐻, (𝐹 ∪ 𝐻) is independent, 𝐹 is contained in the closure of (𝐺 ∪ 𝐻), and either 𝐹 or 𝐺 is finite, then there is a subset 𝑞 of 𝐺 equinumerous to 𝐹 such that (𝑞 ∪ 𝐻) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either (𝐴 ∖ 𝐵) or (𝐵 ∖ 𝐴) is finite. The theorem is proven by induction using mreexexlem3d 17272 for the base case and mreexexlem4d 17273 for the induction step. (Contributed by David Moews, 1-May-2017.) Remove dependencies on ax-rep 5205 and ax-ac2 10150. (Revised by Brendan Leahy, 2-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))    &   ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))    &   ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))    &   ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)    &   ⊢ (𝜑 → (𝐹 ∈ Fin ∨ 𝐺 ∈ Fin))    ⇒   ⊢ (𝜑 → ∃𝑞 ∈ 𝒫 𝐺(𝐹 ≈ 𝑞 ∧ (𝑞 ∪ 𝐻) ∈ 𝐼))
Theorem	mreexdomd 17275*	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 17274. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘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 ⊆ (𝑁‘𝑇))    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑋)    &   ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin))    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    ⇒   ⊢ (𝜑 → 𝑆 ≼ 𝑇)
Theorem	mreexfidimd 17276*	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 17275 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (Moore‘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 → 𝑆 ∈ Fin)    &   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))    ⇒   ⊢ (𝜑 → 𝑆 ≈ 𝑇)
7.2.3  Algebraic closure systems
Theorem	isacs 17277*	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(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))
Theorem	acsmre 17278	Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
Theorem	isacs2 17279*	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(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
Theorem	acsfiel 17280*	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) → (𝑆 ∈ 𝐶 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆)))
Theorem	acsfiel2 17281*	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) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆))
Theorem	acsmred 17282	An algebraic closure system is also a Moore system. Deduction form of acsmre 17278. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    ⇒   ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
Theorem	isacs1i 17283*	A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 ∣ ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋))
Theorem	mreacs 17284	Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (𝑋 ∈ 𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))
Theorem	acsfn 17285*	Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)} ∈ (ACS‘𝑋))
Theorem	acsfn0 17286*	Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ 𝐾 ∈ 𝑎} ∈ (ACS‘𝑋))
Theorem	acsfn1 17287*	Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋))
Theorem	acsfn1c 17288*	Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝐾 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝐾 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋))
Theorem	acsfn2 17289*	Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋))
PART 8  BASIC CATEGORY THEORY
8.1  Categories
8.1.1  Categories
Syntax	ccat 17290	Extend class notation with the class of categories.
class Cat
Syntax	ccid 17291	Extend class notation with the identity arrow of a category.
class Id
Syntax	chomf 17292	Extend class notation to include functionalized Hom-set extractor.
class Homf
Syntax	ccomf 17293	Extend class notation to include functionalized composition operation.
class compf
Definition	df-cat 17294*	A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated with those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. Definition in [Lang] p. 53, without the axiom CAT 1, i.e., pairwise disjointness of hom-sets (cat1 17728). See setc2obas 17725 and setc2ohom 17726 for a counterexample. In contrast to definition 3.1 of [Adamek] p. 21, where "A category is a quadruple A = (O, hom, id, o)", a category is defined as an extensible structure consisting of three slots: the objects "O" ((Base‘𝑐)), the morphisms "hom" ((Hom ‘𝑐)) and the composition law "o" ((comp‘𝑐)). The identities "id" are defined by their properties related to morphisms and their composition, see condition 3.1(b) in [Adamek] p. 21 and df-cid 17295. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
⊢ Cat = {𝑐 ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))))}
Definition	df-cid 17295*	Define the category identity arrow. Since it is uniquely defined when it exists, we do not need to add it to the data of the category, and instead extract it by uniqueness. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ Id = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓))))
Definition	df-homf 17296*	Define the functionalized Hom-set operator, which is exactly like Hom but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
Definition	df-comf 17297*	Define the functionalized composition operator, which is exactly like comp but is guaranteed to be a function of the proper type. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓))))
Theorem	iscat 17298*	The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ Cat ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))))
Theorem	iscatd 17299*	Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → · = (comp‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (𝑥𝐻𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(⟨𝑥, 𝑥⟩ · 𝑦) 1 ) = 𝑓)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    &   ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))    ⇒   ⊢ (𝜑 → 𝐶 ∈ Cat)
Theorem	catidex 17300*	Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))