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Theorem List for Metamath Proof Explorer - 13901-14000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremyoneda 13901* The Yoneda Lemma. There is a natural isomorphism between the functors and , where is the natural transformations from Yon to , and is the evaluation functor. Here we need two universes to state the claim: the smaller universe is used for forming the functor category op , which itself does not (necessarily) live in but instead is an element of the larger universe . (If is a Grothendieck universe, then it will be closed under this "presheaf" operation, and so we can set in this case.) (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f        Nat

Theoremyonffth 13902 The Yoneda Lemma. The Yoneda embedding, the curried Hom functor, is full and faithful, and hence is a representation of the category as a full subcategory of the category of presheaves on . (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon       oppCat              FuncCat                      f        Full Faith

Theoremyoniso 13903* If the codomain is recoverable from a hom-set, then the Yoneda embedding is injective on objects, and hence is an isomorphism from into a full subcategory of a presheaf category. (Contributed by Mario Carneiro, 30-Jan-2017.)
Yon       oppCat              CatCat                     FuncCat        s                             f

PART 9  BASIC ORDER THEORY

9.1  Presets and directed sets using extensible structures

Syntaxcpreset 13904 Extend class notation with the class of all presets.

Syntaxcdrs 13905 Extend class notation with the class of all directed sets.
Dirset

Definitiondf-preset 13906* Define the class of preordered sets (presets). A preset is a set equipped with a transitive and reflexive relation.

Preorders are a natural generalization of order for sets where there is a well-defined ordering, but it in some sense "fails to capture the whole story", in that there may be pairs of elements which are indistinguishable under the order. Two elements which are not equal but are less-or-equal to each other behave the same under all order operations and may be thought of as "tied".

A preorder can naturally be strengthened by requiring that there are no ties, resulting in a partial order, or by stating that all comparable pairs of elements are tied, resulting in an equivalence relation. Every preorder naturally factors into these two types; the tied relation on a preorder is an equivalence relation and the quotient under that relation is a partial order. (Contributed by FL, 17-Nov-2014.) (Revised by Stefan O'Rear, 31-Jan-2015.)

Definitiondf-drs 13907* Define the class of directed sets. A directed set is a nonempty preordered set where every pair of elements have some upper bound. Note that it is not required that there exist a least upper bound.

There is no consensus in the literature over whether directed sets are allowed to be empty. It is slightly more convenient for us if they are not. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Dirset

Theoremisprs 13908* Property of being a preordered set. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Theoremprslem 13909 Lemma for prsref 13910 and prstr 13911. (Contributed by Mario Carneiro, 1-Feb-2015.)

Theoremprsref 13910 Less-or-equal is reflexive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremprstr 13911 Less-or-equal is transitive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremisdrs 13912* Property of being a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsdir 13913* Direction of a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsprs 13914 A directed set is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsbn0 13915 The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsdirfi 13916* Any finite number of elements in a directed set have a common upper bound. Here is where the non-emptiness constraint in df-drs 13907 first comes into play; without it we would need an additional constraint that not be empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremisdrs2 13917* Directed sets may be defined in terms of finite subsets. Again, without nonemptiness we would need to restrict to nonempty subsets here. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

9.2  Posets and lattices using extensible structures

9.2.1  Posets

Syntaxcpo 13918 Extend class notation with the class of posets.

Syntaxcplt 13919 Extend class notation with less-than for posets.

Syntaxclub 13920 Extend class notation with poset least upper bound.

Syntaxcglb 13921 Extend class notation with poset greatest lower bound.

Syntaxcjn 13922 Extend class notation with poset join.

Syntaxcmee 13923 Extend class notation with poset meet.

Definitiondf-poset 13924* Define the class of posets. Definition of poset in [Crawley] p. 1. Note that Crawley-Dilworth require that a poset base set be nonempty, but we follow the convention of most authors who don't make this a requirement.

The quantifiers provide a notational shorthand to allow us to refer to the base and ordering relation as and the definition rather than having to repeat and throughout. These quantifiers can be eliminated with ceqsex2v 2763 and related theorems. (Contributed by NM, 18-Oct-2012.)

Theoremispos 13925* The predicate "is a poset." (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremispos2 13926* A poset is an antisymmetric preset.

EDITORIAL: could become the definition of poset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremposprs 13927 A poset is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremposi 13928 Lemma for poset properties. (Contributed by NM, 11-Sep-2011.)

Theoremposref 13929 A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.)

Theoremposasymb 13930 A poset ordering is asymetric. (Contributed by NM, 21-Oct-2011.)

Theorempostr 13931 A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.)

Theorem0pos 13932 Technical lemma to simplify the statement of ipopos 14107. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 13058) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremisposd 13933* Properties that determine a poset (implicit structure version). (Contributed by Mario Carneiro, 29-Apr-2014.)

Theoremisposi 13934* Properties that determine a poset (implicit structure version). (Contributed by NM, 11-Sep-2011.)

Theoremisposix 13935* Properties that determine a poset (explicit structure version). Note that the numeric indices of the structure components are not mentioned explicitly in either the theorem or its proof. (Contributed by NM, 9-Nov-2012.)

Definitiondf-plt 13936 Define less-than ordering for posets and related structures. Unlike df-base 13027 and df-ple 13102, this is a derived component extractor and not an extensible structure component extractor that defines the poset. (Contributed by NM, 12-Oct-2011.) (Revised by Mario 8-Feb-2015.)

Theorempltfval 13937 Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theorempltval 13938 Less-than relation. (df-pss 3091 analog.) (Contributed by NM, 12-Oct-2011.)

Theorempltle 13939 Less-than implies less-than-or-equal. (pssss 3192 analog.) (Contributed by NM, 4-Dec-2011.)

Theorempltne 13940 Less-than relation. (df-pss 3091 analog.) (Contributed by NM, 2-Dec-2011.)

Theorempltirr 13941 The less-than relation is not reflexive. (pssirr 3196 analog.) (Contributed by NM, 7-Feb-2012.)

Theorempleval2i 13942 One direction of pleval2 13943. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theorempleval2 13943 Less-than-or-equal in terms of less-than. (sspss 3195 analog.) (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)

Theorempltnle 13944 Less-than implies not inverse less-than-or-equal. (Contributed by NM, 18-Oct-2011.)

Theorempltval3 13945 Alternate expression for less-than relation. (dfpss3 3183 analog.) (Contributed by NM, 4-Nov-2011.)

Theorempltnlt 13946 The less-than relation implies the negation of its inverse. (Contributed by NM, 18-Oct-2011.)

Theorempltn2lp 13947 The less-than relation has no 2-cycle loops. (pssn2lp 3197 analog.) (Contributed by NM, 2-Dec-2011.)

Theoremplttr 13948 The less-than relation is transitive. (psstr 3200 analog.) (Contributed by NM, 2-Dec-2011.)

Theorempltletr 13949 Transitive law for chained less-than and less-than-or-equal. (psssstr 3202 analog.) (Contributed by NM, 2-Dec-2011.)

Theoremplelttr 13950 Transitive law for chained less-than-or-equal and less-than. (sspsstr 3201 analog.) (Contributed by NM, 2-May-2012.)

Theorempospo 13951 Write a poset structure in terms of the proper-class poset predicate (strict less than version). (Contributed by Mario Carneiro, 8-Feb-2015.)

Definitiondf-lub 13952* Define poset least upper bound. If it doesn't exist, an undefined value not in the base set is returned. (Contributed by NM, 12-Sep-2011.)

Definitiondf-glb 13953* Define poset greatest lower bound. (Contributed by NM, 19-Jul-2012.)

Definitiondf-join 13954* Define poset join. (Contributed by NM, 12-Sep-2011.)

Definitiondf-meet 13955* Define poset meet. (Contributed by NM, 12-Sep-2011.)

Theoremlubfval 13956* Value of least upper bound function of a poset. (Contributed by NM, 12-Sep-2011.)

Theoremlubval 13957* Value of least upper bound of a poset. (Contributed by NM, 12-Sep-2011.)

Theoremlubprop 13958* Properties of greatest lower bound of a poset. (Contributed by NM, 22-Oct-2011.)

Theoremluble 13959 A greatest lower bound is a least element. (Contributed by NM, 22-Oct-2011.)

Theoremlubid 13960* The LUB of elements less than or equal to a fixed value equals that value. (Contributed by NM, 19-Oct-2011.)

Theoremglbfval 13961* Value of least upper bound function of a poset. (Contributed by NM, 19-Jul-2012.)

Theoremglbval 13962* Value of greatest lower bound of a poset. (Contributed by NM, 19-Jul-2012.)

Theoremglbprop 13963* Properties of greatest lower bound of a poset. (Contributed by NM, 19-Jul-2012.)

Theoremglble 13964 A greatest lower bound is a least element. (Contributed by NM, 12-Oct-2011.)

Theoremjoinfval 13965* Value of join function for a poset. (Contributed by NM, 12-Sep-2011.)

Theoremjoinval 13966 Value of join for a poset. (Contributed by NM, 12-Sep-2011.)

Theoremjoinval2 13967* Value of join for a poset with GLB expanded. (Contributed by NM, 16-Sep-2011.)

Theoremjoinlem 13968* Lemma for join properties. (Contributed by NM, 16-Sep-2011.)

Theoremlejoin1 13969 A join's first argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)

Theoremlejoin2 13970 A join's second argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)

Theoremjoinle 13971 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (Contributed by NM, 16-Sep-2011.)

Theoremmeetfval 13972* Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.)

Theoremmeetval 13973 Value of meet for a poset. (Contributed by NM, 12-Sep-2011.)

Theoremmeetval2 13974* Value of meet for a poset with GLB expanded. (Contributed by NM, 19-Jul-2012.)

Theoremmeetlem 13975* Lemma for meet properties. (Contributed by NM, 19-Jul-2012.)

Theoremlemeet1 13976 A meet's first argument is greater than or equal to the meet. (Contributed by NM, 19-Jul-2012.)

Theoremlemeet2 13977 A meet's second argument is greater than or equal to the meet. (Contributed by NM, 19-Jul-2012.)

Theoremmeetle 13978 A meet is greater than or equal to a third value iff each argument is greater than or equal to the third value. (Contributed by NM, 19-Jul-2012.)

TheoremjoincomALT 13979 The join of a poset commutes. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 16-Sep-2011.) (New usage is discouraged.)

Theoremjoincom 13980 The join of a poset commutes. (The antecedent i.e. "the joins exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 16-Sep-2011.)

TheoremmeetcomALT 13981 The meet of a poset commutes. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 17-Sep-2011.) (New usage is discouraged.)

Theoremmeetcom 13982 The meet of a poset commutes. (The antecedent i.e. "the meets exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 17-Sep-2011.)

Syntaxctos 13983 Extend class notation with the class of all tosets.
Toset

Definitiondf-toset 13984* Define the class of totally ordered sets (tosets). (Contributed by FL, 17-Nov-2014.)
Toset

Theoremistos 13985* The predicate "is a toset." (Contributed by FL, 17-Nov-2014.)
Toset

Theoremtosso 13986 Write the totally ordered set structure predicate in terms of the proper class strict order predicate. (Contributed by Mario Carneiro, 8-Feb-2015.)
Toset

Syntaxcp0 13987 Extend class notation with poset zero.

Syntaxcp1 13988 Extend class notation with poset unit.

Definitiondf-p0 13989 Define poset zero. (Contributed by NM, 12-Oct-2011.)

Definitiondf-p1 13990 Define poset unit. (Contributed by NM, 22-Oct-2011.)

Theoremp0val 13991 Value of poset zero. (Contributed by NM, 12-Oct-2011.)

Theoremp1val 13992 Value of poset zero. (Contributed by NM, 22-Oct-2011.)

Theoremp0le 13993 Poset zero (if defined) is less than any element. (Contributed by NM, 22-Oct-2011.)

Theoremple1 13994 Any element is less than or equal to poset one (if defined). (Contributed by NM, 22-Oct-2011.)

9.2.2  Lattices

Syntaxclat 13995 Extend class notation with the class of all lattices.

Definitiondf-lat 13996* Define the class of all lattices. A lattice is a poset in which the join and meet of any two elements always exists. (Contributed by NM, 18-Oct-2012.)

Theoremislat 13997* The predicate "is a lattice." (Contributed by NM, 18-Oct-2012.)

Theoremlatlem 13998 Lemma for lattice properties. (Contributed by NM, 14-Sep-2011.)

Theoremlatpos 13999 A lattice is a poset. (Contributed by NM, 17-Sep-2011.)

Theoremlatjcl 14000 Closure of join operation in a lattice. (chjcom 21915 analog.) (Contributed by NM, 14-Sep-2011.)

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