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

Theoremmrcuni 13801 Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcun 13802 Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls       Moore

Theoremmrcssvd 13803 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 13794. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmrcssd 13804 Moore closure preserves subset ordering. Deduction form of mrcss 13796. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmrcssidd 13805 A set is contained in its Moore closure. Deduction form of mrcssid 13797. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmrcidmd 13806 Moore closure is idempotent. Deduction form of mrcidm 13799. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

Theoremmressmrcd 13807 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 13808 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 13809 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 13818 and mrieqv2d 13819. Deduction form. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls

7.2.2  Independent sets in a Moore system

Theoremmrisval 13810* Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
mrCls       mrInd       Moore

Theoremismri 13811* Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
mrCls       mrInd       Moore

Theoremismri2 13812* 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 13813* 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 13814* 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 13815 An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.)
mrInd       Moore

Theoremmrissd 13816 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 13817 Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
mrCls       mrInd       Moore

Theoremmrieqvd 13818* 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 13819* 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 13820 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 13807, and so are equal by mrieqv2d 13819.) (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmrissmrid 13821 In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexd 13822* 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 13823* 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 13824* This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 13828. (Contributed by David Moews, 1-May-2017.)

Theoremmreexexlem2d 13825* Used in mreexexlem4d 13827 to prove the induction step in mreexexd 13828. 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 13826* Base case of the induction in mreexexd 13828. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexexlem4d 13827* Induction step of the induction in mreexexd 13828. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexexd 13828* 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 13826 for the base case and mreexexlem4d 13827 for the induction step. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexdomd 13829* 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 13828. (Contributed by David Moews, 1-May-2017.)
Moore       mrCls       mrInd

Theoremmreexfidimd 13830* 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 13829 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 13831* 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 13832 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
ACS Moore

Theoremisacs2 13833* 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 13834* 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 13835* 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 13836 An algebraic closure system is also a Moore system. Deduction form of acsmre 13832. (Contributed by David Moews, 1-May-2017.)
ACS       Moore

Theoremisacs1i 13837* A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremmreacs 13838 Algebraicity is a composible property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS Moore

Theoremacsfn 13839* Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn0 13840* Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn1 13841* Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn1c 13842* Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn2 13843* 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 13844 Extend class notation with the class of categories.

Syntaxccid 13845 Extend class notation with the identity arrow of a category.

Syntaxchomf 13846 Extend class notation to include functionalized Hom-set extractor.
f

Syntaxccomf 13847 Extend class notation to include functionalized composition operation.
compf

Definitiondf-cat 13848* 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 13849* 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 13850* 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 13851* 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 13852* The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremiscatd 13853* Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatidex 13854* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatideu 13855* Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcidfval 13856* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremcidval 13857* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcidffn 13858 The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)

Theoremcidfn 13859 The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremcatidd 13860* Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremiscatd2 13861* Version of iscatd2 13861 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp

Theoremcatidcl 13862 Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcatlid 13863 Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatrid 13864 Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatcocl 13865 Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatass 13866 Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theorem0catg 13867 Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theorem0cat 13868 The empty set is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremproplem2 13869* Lemma for mndpropd 14676. (Contributed by Mario Carneiro, 6-Dec-2014.)

Theoremproplem 13870* Lemma for mndpropd 14676. (Contributed by Mario Carneiro, 6-Dec-2014.)

Theoremproplem3 13871 Lemma for property theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)

Theoremhomffval 13872* Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomfval 13873 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomffn 13874 The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomfeq 13875* Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
f f

Theoremhomfeqd 13876 If two structures have the same slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremhomfeqbas 13877 Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremhomfeqval 13878 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremcomfffval 13879* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomffval 13880* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomfval 13881 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomfffval2 13882* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomffval2 13883* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomfval2 13884 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomfffn 13885 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf

Theoremcomffn 13886 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf

Theoremcomfeq 13887* Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp       comp                            f f        compf compf

Theoremcomfeqd 13888 Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp comp       f f        compf compf

Theoremcomfeqval 13889 Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp       comp       f f        compf compf

Theoremcatpropd 13890 Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.)
f f        compf compf

Theoremcidpropd 13891 Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)
f f        compf compf

8.1.2  Opposite category

Syntaxcoppc 13892 The opposite category operation.
oppCat

Definitiondf-oppc 13893* Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat sSet tpos sSet comp tpos comp

Theoremoppcval 13894* Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat       sSet tpos sSet comp tpos

Theoremoppchomfval 13895 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat       tpos

Theoremoppchom 13896 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccofval 13897 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat                            comp tpos

Theoremoppcco 13898 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat                            comp

Theoremoppcbas 13899 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccatid 13900 Lemma for oppccat 13903. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat

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