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Theorem List for Metamath Proof Explorer - 13801-13900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxpsadd 13801 Value of the addition operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
s

Theoremxpsmul 13802 Value of the multiplication operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
s

Theoremxpssca 13803 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.)
s        Scalar                     Scalar

Theoremxpsvsca 13804 Value of the scalar multiplication function in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
s        Scalar

Theoremxpsless 13805 Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
s

Theoremxpsle 13806 Value of the ordering in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

7.2  Moore spaces

Syntaxcmre 13807 The class of Moore systems.
Moore

Syntaxcmrc 13808 The class function generating Moore closures.
mrCls

Syntaxcmri 13809 mrInd is a class function which takes a Moore system to its set of independent sets.
mrInd

Syntaxcacs 13810 The class of algebraic closure (Moore) systems.
ACS

Definitiondf-mre 13811* 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 17142) and vector spaces (lssmre 16042) 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 13815, mresspw 13817, mre1cl 13819 and mreintcl 13820 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13825); as such the disjoint union of all Moore collections is sometimes considered as Moore, justified by mreunirn 13826. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.)

Moore

Definitiondf-mrc 13812* 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 17143) and linear span (mrclsp 16065).

A Moore closure operation is (1) extensive, i.e., for all subsets of the base set (mrcssid 13842), (2) isotone, i.e., implies that for all subsets and of the base set (mrcss 13841), and (3) idempotent, i.e., for all subsets of the base set (mrcidm 13844.) 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 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 Moore

Definitiondf-mri 13813* 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 Moore mrCls

Definitiondf-acs 13814* 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 7598 is to be avoided), we consider a single function defined on finite sets instead. (Contributed by Stefan O'Rear, 2-Apr-2015.)
ACS Moore

Theoremismre 13815* Property of being a Moore collection on some base set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremfnmre 13816 The Moore collection generator is a well-behaved function. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremmresspw 13817 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.)
Moore

Theoremmress 13818 A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Moore

Theoremmre1cl 13819 In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremmreintcl 13820 A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremmreiincl 13821* A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Moore

Theoremmrerintcl 13822 The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Moore

Theoremmreriincl 13823* The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Moore

Theoremmreincl 13824 Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremmreuni 13825 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.)
Moore

Theoremmreunirn 13826 Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore Moore

Theoremismred 13827* Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremismred2 13828* Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Moore

Theoremmremre 13829 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.)
Moore Moore

Theoremsubmre 13830 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.)
Moore Moore

7.2.1  Moore closures

Theoremmrcflem 13831* The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Moore

Theoremfnmrc 13832 Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
mrCls Moore

Theoremmrcfval 13833* Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcf 13834 The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcval 13835* Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
mrCls       Moore

Theoremmrccl 13836 The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcsncl 13837 The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcid 13838 The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcssv 13839 The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcidb 13840 A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcss 13841 Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcssid 13842 The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcidb2 13843 A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
mrCls       Moore

Theoremmrcidm 13844 The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcsscl 13845 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.)
mrCls       Moore

Theoremmrcuni 13846 Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcun 13847 Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcssvd 13848 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 13839. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmrcssd 13849 Moore closure preserves subset ordering. Deduction form of mrcss 13841. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmrcssidd 13850 A set is contained in its Moore closure. Deduction form of mrcssid 13842. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmrcidmd 13851 Moore closure is idempotent. Deduction form of mrcidm 13844. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmressmrcd 13852 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.)
Moore       mrCls

Theoremsubmrc 13853 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.)
mrCls       mrCls        Moore

Theoremmrieqvlemd 13854 In a Moore system, if is a member of , and have the same closure if and only if is in the closure of . Used in the proof of mrieqvd 13863 and mrieqv2d 13864. Deduction form. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

7.2.2  Independent sets in a Moore system

Theoremmrisval 13855* Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
mrCls       mrInd       Moore

Theoremismri 13856* Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
mrCls       mrInd       Moore

Theoremismri2 13857* Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)
mrCls       mrInd       Moore

Theoremismri2d 13858* Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
mrCls       mrInd       Moore

Theoremismri2dd 13859* 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.)
mrCls       mrInd       Moore

Theoremmriss 13860 An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.)
mrInd       Moore

Theoremmrissd 13861 An independent set of a Moore system is a subset of the base set. Deduction form. (Contributed by David Moews, 1-May-2017.)
mrInd       Moore

Theoremismri2dad 13862 Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
mrCls       mrInd       Moore

Theoremmrieqvd 13863* 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.)
Moore       mrCls       mrInd

Theoremmrieqv2d 13864* 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.)
Moore       mrCls       mrInd

Theoremmrissmrcd 13865 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 13852, and so are equal by mrieqv2d 13864.) (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmrissmrid 13866 In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexd 13867* In a Moore system, the closure operator is said to have the exchange property if, for all elements and of the base set and subsets of the base set such that is in the closure of but not in the closure of , is in the closure of (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.)

Theoremmreexmrid 13868* 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.)
Moore       mrCls       mrInd

Theoremmreexexlemd 13869* This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 13873. (Contributed by David Moews, 1-May-2017.)

Theoremmreexexlem2d 13870* Used in mreexexlem4d 13872 to prove the induction step in mreexexd 13873. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexexlem3d 13871* Base case of the induction in mreexexd 13873. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexexlem4d 13872* Induction step of the induction in mreexexd 13873. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexexd 13873* Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if and are disjoint from , is independent, is contained in the closure of , and either or is finite, then there is a subset of equinumerous to such that is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either or is finite. The theorem is proven by induction using mreexexlem3d 13871 for the base case and mreexexlem4d 13872 for the induction step. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexdomd 13874* In a Moore system whose closure operator has the exchange property, if is independent and contained in the closure of , and either or is finite, then dominates . This is an immediate consequence of mreexexd 13873. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexfidimd 13875* 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 13874 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

7.2.3  Algebraic closure systems

Theoremisacs 13876* 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.)
ACS Moore

Theoremacsmre 13877 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
ACS Moore

Theoremisacs2 13878* 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.)
mrCls       ACS Moore

Theoremacsfiel 13879* 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.)
mrCls       ACS

Theoremacsfiel2 13880* 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.)
mrCls       ACS

Theoremacsmred 13881 An algebraic closure system is also a Moore system. Deduction form of acsmre 13877. (Contributed by David Moews, 1-May-2017.)
ACS       Moore

Theoremisacs1i 13882* A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremmreacs 13883 Algebraicity is a composible property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS Moore

Theoremacsfn 13884* Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn0 13885* Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn1 13886* Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn1c 13887* Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn2 13888* Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

PART 8  BASIC CATEGORY THEORY

8.1  Categories

8.1.1  Categories

Syntaxccat 13889 Extend class notation with the class of categories.

Syntaxccid 13890 Extend class notation with the identity arrow of a category.

Syntaxchomf 13891 Extend class notation to include functionalized Hom-set extractor.
f

Syntaxccomf 13892 Extend class notation to include functionalized composition operation.
compf

Definitiondf-cat 13893* 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 to 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. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
comp

Definitiondf-cid 13894* 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.)
comp

Definitiondf-homf 13895* Define the functionalized Hom-set operator, which is exactly like but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Definitiondf-comf 13896* 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 comp

Theoremiscat 13897* The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremiscatd 13898* Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatidex 13899* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatideu 13900* Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

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