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Theorem List for Metamath Proof Explorer - 14301-14400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcyon 14301 Extend class notation with the Yoneda embedding.
Yon

Definitiondf-hof 14302* Define the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from oppCat to , whose object part is the hom-function , and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
HomF f comp comp

Definitiondf-yon 14303 Define the Yoneda embedding, which is the currying of the (opposite) Hom functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
Yon oppCat curryF HomFoppCat

Theoremhofval 14304* Value of the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from oppCat to , whose object part is the hom-function , and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF                            comp       f

Theoremhof1fval 14305 The object part of the Hom functor is the f operation, which is just a functionalized version of . That is, it is a two argument function, which maps to the set of morphisms from to . (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF              f

Theoremhof1 14306 The object part of the Hom functor maps to the set of morphisms from to . (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF

Theoremhof2fval 14307* The morphism part of the Hom functor, for morphisms (which since the first argument is contravariant means morphisms and ), yields a function (a morphism of ) mapping to . (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF                                                        comp

Theoremhof2val 14308* The morphism part of the Hom functor, for morphisms (which since the first argument is contravariant means morphisms and ), yields a function (a morphism of ) mapping to . (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF                                                        comp

Theoremhof2 14309 The morphism part of the Hom functor, for morphisms (which since the first argument is contravariant means morphisms and ), yields a function (a morphism of ) mapping to . (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF                                                        comp

Theoremhofcllem 14310 Lemma for hofcl 14311. (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF       oppCat                            f                                                                                            comp comp comp

Theoremhofcl 14311 Closure of the Hom functor. Note that the codomain is the category for any universe which contains each Hom-set. This corresponds to the assertion that be locally small (with respect to ). (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF       oppCat                            f        c

Theoremoppchofcl 14312 Closure of the opposite Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
oppCat       HomF                            f        c

Theoremyonval 14313 Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon              oppCat       HomF       curryF

Theoremyoncl 14314 The Yoneda embedding is a functor from the category to the category of presheaves on . (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon              oppCat              FuncCat               f

Theoremyon1cl 14315 The Yoneda embedding at an object of is a presheaf on , also known as the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon                            oppCat                     f

Theoremyon11 14316 Value of the Yoneda embedding at an object. The partially evaluated Yoneds embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon

Theoremyon12 14317 Value of the Yoneda embedding at a morphism. The partially evaluated Yoneds embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon                                          comp

Theoremyon2 14318 Value of the Yoneda embedding at a morphism. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon                                          comp

Theoremhofpropd 14319 If two categories have the same set of objects, morphisms, and compositions, then they have the same Hom functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf                     HomF HomF

Theoremyonpropd 14320 If two categories have the same set of objects, morphisms, and compositions, then they have the same Yoneda functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf                     Yon Yon

Theoremoppcyon 14321 Value of the opposite Yoneda embedding. (Contributed by Mario Carneiro, 26-Jan-2017.)
oppCat       Yon       HomF              curryF

Theoremoyoncl 14322 The opposite Yoneda embedding is a functor from oppCat to the functor category . (Contributed by Mario Carneiro, 26-Jan-2017.)
oppCat       Yon                            f        FuncCat

Theoremoyon1cl 14323 The opposite Yoneda embedding at an object of is a functor from to Set, also known as the covariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
oppCat       Yon                            f

Theoremyonedalem1 14324 Lemma for yoneda 14335. (Contributed by Mario Carneiro, 28-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f        c c

Theoremyonedalem21 14325 Lemma for yoneda 14335. (Contributed by Mario Carneiro, 28-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                      Nat

Theoremyonedalem3a 14326* Lemma for yoneda 14335. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                      Nat        Nat

Theoremyonedalem4a 14327* Lemma for yoneda 14335. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f

Theoremyonedalem4b 14328* Lemma for yoneda 14335. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f

Theoremyonedalem4c 14329* Lemma for yoneda 14335. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                                    Nat

Theoremyonedalem22 14330 Lemma for yoneda 14335. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                                    Nat

Theoremyonedalem3b 14331* Lemma for yoneda 14335. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                                    Nat               Nat        comp comp

Theoremyonedalem3 14332* Lemma for yoneda 14335. (Contributed by Mario Carneiro, 28-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f        Nat        c Nat

Theoremyonedainv 14333* The Yoneda Lemma with explicit inverse. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f        Nat        Inv

Theoremyonffthlem 14334* Lemma for yonffth 14336. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f        Nat        Inv              Full Faith

Theoremyoneda 14335* 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 14336 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 14337* 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 14338 Extend class notation with the class of all presets.

Syntaxcdrs 14339 Extend class notation with the class of all directed sets.
Dirset

Definitiondf-preset 14340* 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 14341* 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 14342* Property of being a preordered set. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Theoremprslem 14343 Lemma for prsref 14344 and prstr 14345. (Contributed by Mario Carneiro, 1-Feb-2015.)

Theoremprsref 14344 Less-or-equal is reflexive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremprstr 14345 Less-or-equal is transitive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremisdrs 14346* Property of being a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsdir 14347* Direction of a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsprs 14348 A directed set is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsbn0 14349 The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsdirfi 14350* Any finite number of elements in a directed set have a common upper bound. Here is where the non-emptiness constraint in df-drs 14341 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 14351* 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 14352 Extend class notation with the class of posets.

Syntaxcplt 14353 Extend class notation with less-than for posets.

Syntaxclub 14354 Extend class notation with poset least upper bound.

Syntaxcglb 14355 Extend class notation with poset greatest lower bound.

Syntaxcjn 14356 Extend class notation with poset join.

Syntaxcmee 14357 Extend class notation with poset meet.

Definitiondf-poset 14358* 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 2953 and related theorems. (Contributed by NM, 18-Oct-2012.)

Theoremispos 14359* The predicate "is a poset." (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremispos2 14360* A poset is an antisymmetric preset.

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

Theoremposprs 14361 A poset is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremposi 14362 Lemma for poset properties. (Contributed by NM, 11-Sep-2011.)

Theoremposref 14363 A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.)

Theoremposasymb 14364 A poset ordering is asymetric. (Contributed by NM, 21-Oct-2011.)

Theorempostr 14365 A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.)

Theorem0pos 14366 Technical lemma to simplify the statement of ipopos 14541. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 13460) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremisposd 14367* Properties that determine a poset (implicit structure version). (Contributed by Mario Carneiro, 29-Apr-2014.)

Theoremisposi 14368* Properties that determine a poset (implicit structure version). (Contributed by NM, 11-Sep-2011.)

Theoremisposix 14369* 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 14370 Define less-than ordering for posets and related structures. Unlike df-base 13429 and df-ple 13504, 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 Carneiro, 8-Feb-2015.)

Theorempltfval 14371 Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theorempltval 14372 Less-than relation. (df-pss 3296 analog.) (Contributed by NM, 12-Oct-2011.)

Theorempltle 14373 Less-than implies less-than-or-equal. (pssss 3402 analog.) (Contributed by NM, 4-Dec-2011.)

Theorempltne 14374 Less-than relation. (df-pss 3296 analog.) (Contributed by NM, 2-Dec-2011.)

Theorempltirr 14375 The less-than relation is not reflexive. (pssirr 3407 analog.) (Contributed by NM, 7-Feb-2012.)

Theorempleval2i 14376 One direction of pleval2 14377. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theorempleval2 14377 Less-than-or-equal in terms of less-than. (sspss 3406 analog.) (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)

Theorempltnle 14378 Less-than implies not inverse less-than-or-equal. (Contributed by NM, 18-Oct-2011.)

Theorempltval3 14379 Alternate expression for less-than relation. (dfpss3 3393 analog.) (Contributed by NM, 4-Nov-2011.)

Theorempltnlt 14380 The less-than relation implies the negation of its inverse. (Contributed by NM, 18-Oct-2011.)

Theorempltn2lp 14381 The less-than relation has no 2-cycle loops. (pssn2lp 3408 analog.) (Contributed by NM, 2-Dec-2011.)

Theoremplttr 14382 The less-than relation is transitive. (psstr 3411 analog.) (Contributed by NM, 2-Dec-2011.)

Theorempltletr 14383 Transitive law for chained less-than and less-than-or-equal. (psssstr 3413 analog.) (Contributed by NM, 2-Dec-2011.)

Theoremplelttr 14384 Transitive law for chained less-than-or-equal and less-than. (sspsstr 3412 analog.) (Contributed by NM, 2-May-2012.)

Theorempospo 14385 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 14386* 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 14387* Define poset greatest lower bound. (Contributed by NM, 19-Jul-2012.)

Definitiondf-join 14388* Define poset join. (Contributed by NM, 12-Sep-2011.)

Definitiondf-meet 14389* Define poset meet. (Contributed by NM, 12-Sep-2011.)

Theoremlubfval 14390* Value of the least upper bound function of a poset. (Contributed by NM, 12-Sep-2011.)

Theoremlubval 14391* Value of the least upper bound of a poset. (Contributed by NM, 12-Sep-2011.)

Theoremlubprop 14392* Properties of greatest lower bound of a poset. (Contributed by NM, 22-Oct-2011.)

Theoremluble 14393 The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.)

Theoremlubid 14394* The LUB of elements less than or equal to a fixed value equals that value. (Contributed by NM, 19-Oct-2011.)

Theoremglbfval 14395* Value of the least upper bound function of a poset. (Contributed by NM, 19-Jul-2012.)

Theoremglbval 14396* Value of greatest lower bound of a poset. (Contributed by NM, 19-Jul-2012.)

Theoremglbprop 14397* Properties of greatest lower bound of a poset. (Contributed by NM, 19-Jul-2012.)

Theoremglble 14398 The greatest lower bound is the least element. (Contributed by NM, 12-Oct-2011.)

Theoremjoinfval 14399* Value of join function for a poset. (Contributed by NM, 12-Sep-2011.)

Theoremjoinval 14400 Value of join for a poset. (Contributed by NM, 12-Sep-2011.)

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