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Theorem List for Metamath Proof Explorer - 13501-13600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmreincl 13501 Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremmreuni 13502 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 13503 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 13504* Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Moore

Theoremismred2 13505* Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Moore

Theoremmremre 13506 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 13507 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 13508* The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Moore

Theoremfnmrc 13509 Moore-closure is a well behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
mrCls Moore

Theoremmrcfval 13510* Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcf 13511 The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcval 13512* 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 13513 The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcsncl 13514 The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcid 13515 The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcssv 13516 The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcidb 13517 A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcss 13518 Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcssid 13519 The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcidb2 13520 A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
mrCls       Moore

Theoremmrcidm 13521 The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcsscl 13522 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 13523 Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcun 13524 Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcssvd 13525 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 13516. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmrcssd 13526 Moore closure preserves subset ordering. Deduction form of mrcss 13518. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmrcssidd 13527 A set is contained in its Moore closure. Deduction form of mrcssid 13519. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmrcidmd 13528 Moore closure is idempotent. Deduction form of mrcidm 13521. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmressmrcd 13529 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 13530 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 13531 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 13540 and mrieqv2d 13541. Deduction form. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

7.2.2  Independent sets in a Moore system

Theoremmrisval 13532* Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
mrCls       mrInd       Moore

Theoremismri 13533* Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
mrCls       mrInd       Moore

Theoremismri2 13534* 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 13535* 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 13536* 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 13537 An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.)
mrInd       Moore

Theoremmrissd 13538 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 13539 Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
mrCls       mrInd       Moore

Theoremmrieqvd 13540* 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 13541* 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 13542 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 13529, and so are equal by mrieqv2d 13541.) (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmrissmrid 13543 In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexd 13544* 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 13545* 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 13546* This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 13550. (Contributed by David Moews, 1-May-2017.)

Theoremmreexexlem2d 13547* Used in mreexexlem4d 13549 to prove the induction step in mreexexd 13550. 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 13548* Base case of the induction in mreexexd 13550. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexexlem4d 13549* Induction step of the induction in mreexexd 13550. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexexd 13550* 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 13548 for the base case and mreexexlem4d 13549 for the induction step. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexdomd 13551* 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 13550. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexfidimd 13552* 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 13551 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 13553* 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 13554 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
ACS Moore

Theoremisacs2 13555* 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 13556* 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 13557* 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 13558 An algebraic closure system is also a Moore system. Deduction form of acsmre 13554. (Contributed by David Moews, 1-May-2017.)
ACS       Moore

Theoremisacs1i 13559* A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremmreacs 13560 Algebraicity is a composible property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS Moore

Theoremacsfn 13561* Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn0 13562* Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn1 13563* Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn1c 13564* Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn2 13565* 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 13566 Extend class notation with the class of categories.

Syntaxccid 13567 Extend class notation with the identity arrow of a category.

Syntaxchomf 13568 Extend class notation to include functionalized Hom-set extractor.
f

Syntaxccomf 13569 Extend class notation to include functionalized composition operation.
compf

Definitiondf-cat 13570* 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 13571* 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 13572* 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 13573* 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 13574* The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremiscatd 13575* Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatidex 13576* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatideu 13577* Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcidfval 13578* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremcidval 13579* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcidffn 13580 The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)

Theoremcidfn 13581 The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremcatidd 13582* Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremiscatd2 13583* Version of iscatd2 13583 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp

Theoremcatidcl 13584 Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcatlid 13585 Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatrid 13586 Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatcocl 13587 Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatass 13588 Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theorem0catg 13589 Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theorem0cat 13590 The empty set is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremproplem2 13591* Lemma for mndpropd 14398. (Contributed by Mario Carneiro, 6-Dec-2014.)

Theoremproplem 13592* Lemma for mndpropd 14398. (Contributed by Mario Carneiro, 6-Dec-2014.)

Theoremproplem3 13593 Lemma for property theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)

Theoremhomffval 13594* Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomfval 13595 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomffn 13596 The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomfeq 13597* Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
f f

Theoremhomfeqd 13598 If two structures have the same slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremhomfeqbas 13599 Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremhomfeqval 13600 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

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