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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempreotr2 24401 A preset is transitive. (Contributed by FL, 23-May-2011.) (Revised by Mario Carneiro, 3-May-2015.)
PresetRel

Theoremaltprs2 24402 The composite of a preset with itself. (Contributed by FL, 13-May-2011.)
PresetRel

Theoremint2pre 24403 The intersection of two presets is a preset. (Contributed by FL, 28-Dec-2011.)
PresetRel PresetRel PresetRel

Theoremsqpre 24404 A square product is a preset. (Contributed by FL, 28-Dec-2011.)
PresetRel

Theoremindpre 24405 The relation induced by a preset on a part of its field is a preset. (Contributed by FL, 28-Dec-2011.)
PresetRel PresetRel

Theoremposprsr 24406 A partial order is a preset. (Contributed by FL, 1-May-2011.)
PresetRel

Theoremposispre 24407 A poset is a preset. (Contributed by FL, 19-Sep-2011.)
PresetRel

Theoremempos 24408 The empty set is a poset. (Contributed by FL, 6-Oct-2008.)

Theoremdupre1 24409 The converse of a preset is a preset. The case PresetRel PresetRel is true only if is a relation. See dupre2 24410. (Contributed by FL, 5-Jan-2009.)
PresetRel PresetRel

Theoremdupre2 24410 The converse of a preset is a preset. (Contributed by FL, 19-Sep-2011.)
PresetRel PresetRel

Theoremnfwval 24411 An infimum is the supremum of the converse relation. (Contributed by FL, 6-Sep-2009.) (Revised by Mario Carneiro, 10-Sep-2015.)

Definitiondf-mxl 24412* Define the maximal elements of a set. I.e. the elements of the set that are not smaller than the other elements. Meaningful if is at least a preset. Read as the maximal elements of the set preordered by . Bourbaki E III 8. Experimental. (Contributed by FL, 16-May-2011.)

Definitiondf-mnl 24413* Define the minimal elements of a set. I.e. the elements of the set that are not greater than the other elements. Meaningful is is at least a preset. Read as the minimal elements of the set preordered by . Bourbaki E III 8. Experimental. (Contributed by FL, 19-Sep-2011.)

Definitiondf-ge 24414 Define the greatest element of a poset. I.e. the element of the poset that is larger than the other elements. Meaningful is is at least a poset (otherwise there could be more than one supremum due to cycles). Bourbaki E III 10. Experimental. (Contributed by FL, 19-Sep-2011.)

Definitiondf-ler 24415 Define the least element of a poset. I.e. the element of the poset that is smaller than the other elements. Meaningful is is at least a poset. Experimental. (Contributed by FL, 19-Sep-2011.)
leR

Theoremgepsup 24416 The greatest element of a poset is the supremum of the poset. (Contributed by FL, 19-Sep-2011.)

Theoremseinf 24417 The least element of a poset is the infimum of the poset. (Contributed by FL, 19-Sep-2011.)
leR

Theoremsege 24418 The least element of a poset is the greatest element of the converse poset. (Contributed by FL, 30-Dec-2011.)
leR

Definitiondf-ub 24419* Define the upper bounds of a set . Meaningful if is at least a preset, and a subset of the field of . Bourbaki E.III.9 def. 5. Experimental. (Contributed by FL, 16-May-2011.)

Definitiondf-lb 24420* Define the lower bounds of a set . Meaningful if is at least a preset, and a subset of the field of . Experimental. (Contributed by FL, 16-May-2011.)

Definitiondf-antidir 24421* An antidirected set (also called a set filtering on the left by Bourbaki) is a preset whose every pair of elements has a lower bound. (Contributed by FL, 17-Oct-2011.)
PresetRel

Theoremubos 24422* The upper bounds of . (Contributed by FL, 16-May-2011.)

Theoremubos2 24423* The upper bounds of . (Contributed by FL, 18-Sep-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
PresetRel

Theorempuub2 24424* The predicate " is an upper bound of ." (Contributed by FL, 16-May-2011.)
PresetRel

Theorempuub 24425* The predicate " is an upper bound of ." (Contributed by FL, 16-May-2011.)

Theoremprltub 24426 If is a preset, and is an upper bound of then is an upper bound of . Bourbaki E.III.9 nb 8. (Contributed by FL, 23-May-2011.) (Proof shortened by Mario Carneiro, 3-May-2015.)
PresetRel

Theoremubpar 24427 If is an upper bound of and then is an upper bound of . Bourbaki E.III.9 n 8. (Contributed by FL, 23-May-2011.)
PresetRel

Theoremsupdef 24428* If it exists, a supremum of is greater or equal to every element of and is the least upper bound of . Here the existence of the supremum is expressed by the idiom . (Contributed by FL, 23-May-2011.)

Theoremsupdefa 24429 The greatest element of a poset is greater than the other elements of the poset. (Contributed by FL, 19-Sep-2011.)

Theoremmxlelt 24430* The maximal elements of the preset . (Contributed by FL, 16-May-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremmxlelt2 24431* The maximal elements of the preset . (Contributed by FL, 16-May-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
PresetRel

Theoremmnlelt2 24432* The minimal elements of the preset . (Contributed by FL, 19-Sep-2011.)
PresetRel

Theoremismxl2 24433* The predicate " is a maximal element of the preset " . (Contributed by FL, 22-May-2011.)
PresetRel

Theoremismnl2 24434* The predicate " is a minimal element of the preset " . (Contributed by FL, 19-Sep-2011.)
PresetRel

Theoremmnlmxl2 24435 The minimal elements of a preset are the maximal elements of the converse preset. (Contributed by FL, 19-Sep-2011.)
PresetRel

Theoremmxlmnl2 24436 The maximal elements of a preset are the minimal elements of the converse preset. (Contributed by FL, 19-Sep-2011.)
PresetRel

Theoremdefge3 24437* The greatest element of a poset is an element, when it exists, that is greater than the other elements of the poset. Use the idiom when you mean the greatest element of exists. (Contributed by FL, 30-Dec-2011.)

Theoremdefse3 24438* The least element of a poset is an element, when it exists, that is less than the other elements of the poset. Use the idiom leR when you mean the least element of exists. (Contributed by FL, 30-Dec-2011.)
leR leR

Theoremsupaub 24439 If it exists, a supremum of is an upper bound for . (Contributed by FL, 19-Sep-2011.)

Theoremsupwlub 24440* If it exists, a supremum of is the least upper bound for . (Contributed by FL, 19-Sep-2011.)

Theoremgeme2 24441 The greatest element of is a maximal element. (Contributed by FL, 19-Sep-2011.)

Theoreminposetlem 24442* Lemma for inposet 24444. Definition of inclusion of sets using a class. (Contributed by FL, 22-Sep-2008.)

Theoreminpc 24443* Inclusion is a proper class. (Contributed by FL, 22-Sep-2008.)

Theoreminposet 24444* Inclusion partially orders any set. (Contributed by FL, 22-Sep-2008.)

Theoremdefinc 24445* Definition of the inclusion. (Contributed by FL, 6-Sep-2009.)

Theoremdominc 24446* The domain of the inclusion relation is . (Contributed by FL, 6-Sep-2009.)

Theoremrninc 24447* The range of the inclusion relation is . (Contributed by FL, 6-Sep-2009.)

Theoremdomncnt 24448* Domain of the intersection of the inclusion with a square cross product. (Contributed by FL, 6-Sep-2009.)

Theoremranncnt 24449* Range of the intersection of the inclusion with a square cross product. (Contributed by FL, 6-Sep-2009.)

Theoremsupwval 24450 Value of an infimum under a weak ordering. (Contributed by FL, 19-Sep-2011.)

Theoremnfwpr4c 24451 Infimum of an unordered pair of comparable elements. (Contributed by FL, 19-Sep-2011.)

Theoremtolat 24452 A totally ordered set is a lattice. (Contributed by FL, 19-Sep-2011.)

Theoremdispos 24453 A restriction of the identity is a poset. (Contributed by FL, 2-Aug-2009.)

Theoremderef 24454 An idiom to "dereflexivate" a relation. (Contributed by FL, 30-Dec-2010.)

Theoremdfps2 24455 Alternate definition of a poset. Bourbaki E.III.2 prop. 1. (Contributed by FL, 30-Dec-2010.)

Theoremtoplat 24456* A topology when ordered by the inclusion is a lattice. This fact leads to the idea of pointless topology, that is a lattice looked at with the eyes of a topologist. (Contributed by FL, 6-Sep-2009.)

Theoremdfdir2 24457* A directed set (also called a set filtering on the right by Bourbaki) is a preordered set whose every pair of elements has an upper bound. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 21-Nov-2013.)
PresetRel

Theoremisdir2 24458* Alternate definition of a direction. (Contributed by FL, 19-Sep-2011.)
PresetRel

Theoremdirpre 24459 A direction is a preset. (Contributed by FL, 19-Sep-2011.)
PresetRel

Theoremdirub 24460* In a direction, every pair of elements has an upper bound. (Contributed by FL, 19-Sep-2011.)

Theoremlatdir 24461 A lattice is a direction. (Contributed by FL, 19-Sep-2011.)

Syntaxclbl 24462 Extend class notation to include bound lattice.

Definitiondf-bnlat 24463 A bound lattice is a lattice that has a greatest and a least element. (Contributed by FL, 21-May-2012.)
leR

16.11.14  Finite composites ( i. e. finite sums, products ... )

Syntaxcprd 24464 Extend class notation to include finite products/sums.

Definitiondf-prod 24465* Define the composite for the law of a finite sequence of elements whose values are defined by the expression and whose set of indices is . may be empty. It may be thougt as a product (if is a multiplication), a sum (if is an addition) or whatever. The variable is normally a free variable in ( i.e. can be thought of as ). The definition is meaningful when is a finite set of sequential integers and is an internal operation. Our definition corresponds to the first part of the definition of df-sum 12036. The operation has been replaced by the generic operation . The reference to the concept of limit has been removed because one wants to use the product in contexts where limits are irrelevant. I could be still more generic and replace by a finite totally ordered set. I would then get the definition given by Bourbaki in the first chapter of the algebra book of his treatise ( A I.3 def.4 ). I don't because the present definition is easier to deal with and because there exists an order isomorphism between any finite totally ordered set and any finite sets of integers. I don't specify anything about because nothing is required of in the definition of . I hope it will be ok. Otherwise one could add . (Contributed by FL, 5-Sep-2010.)
GId

Syntaxcprd2 24466 Extend class notation to include finite supports products/sums.
prod2

Syntaxcprd3 24467 Extend class notation to include finite supports products/sums.
prod3

Definitiondf-prod2 24468* Definition of a sum or product operator to be used with generic structures defined by extensible structures. is the set of indices, the operation, an expression, is normally a free variable in . may be any extensible structure with a base set. Its base set may be infinite provided that the "support" is finite. The support is the set: . The base set of may be empty. must be an extensible structure with a law commutative, associative with a neutral element. (Contributed by FL, 17-Oct-2011.)
prod2

Definitiondf-prod3 24469* Definition of a sum or product operator to be used with generic structures defined by extensible structures. is the set of indices, the operation, an expression, is normally a free variable in . must be a total order. Its base set may be infinite provided that the "support" is finite. The support is the set: . The base set of may be empty. must be an associative law with a neutral element. (Contributed by FL, 17-Oct-2011.)
prod3

Theoremprodex 24470 A finite composite is a set. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprod0 24471 The value of . (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
GId

Theoremprodeq1 24472 Equality theorem for a composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeq2 24473 Equality theorem for a composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeq3ii 24474* Equality theorem for a composite. (Contributed by Mario Carneiro, 26-May-2014.)

Theoremprodeq3 24475* Equality theorem for a composite. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2014.)

Theoremnfprod1 24476* Bound-variable hypothesis builder for . (Contributed by FL, 14-Sep-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremnfprod 24477* Bound-variable hypothesis builder for . If is (effectively) not free in , and , it is not free in . (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremcbvprodi 24478 Change bound variable in a finite composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeq1d 24479 Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.)

Theoremprodeq2d 24480 Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.)

Theoremprodeq3d 24481* Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeq123d 24482* Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeq123i 24483* Conditions for two composites to be equal. (Contributed by FL, 6-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeqfv 24484* Convert a composite of function values to a composite of classes . (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremdffprod 24485 Special case of composite over a finite index set. (Contributed by FL, 5-Sep-2010.) (Proof shortened by Mario Carneiro, 26-May-2014.)

Theoremfprodser 24486* A finite composite expressed in terms of a partial composite of an infinite series. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprodserf 24487 Version of fprodser 24486 with a bound-variable hypothesis instead of a distinct variable condition. (Contributed by FL, 5-Sep-2010.)

Theoremfprod1i 24488* The finite composite of from to (i.e. a composite with only one term) is i.e. . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprodp1 24489* The composite of the next term in a finite composite of is the previous term composed with . (Contributed by Mario Carneiro, 26-May-2014.)

Theoremfprodp1i 24490* The composite of the next term in a finite composite of is the previous term composed with . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprod1s 24491 The finite composite of a sequence from to (i.e. a composite with only one term) is . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprod1fi 24492* The finite composite of a term from to (i.e. a composite with only one term) is , where is effectively not free in . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprodp1s 24493 The composite of the next term in a finite sum of is the previous term composed with . (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprodp1fi 24494* The composite of the next term in a finite composite of is the previous term composed with . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfsumprd 24495* Relation between and . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprod2 24496* The finite composite of from to (i.e. a composite with two terms) is i.e. . (Contributed by FL, 22-Nov-2010.) (Revised by Mario Carneiro, 26-May-2014.)

16.11.15  Operation properties

Syntaxccm1 24497 Extend class notation with a class that adds commutativity to semi-groups, monoids and so on.

Definitiondf-com1 24498* A device to add commutativity to magmas, semi-groups, monoids and so on. A commutative group is composed of 5 properties (internal operation, commutativity, associativity, existence of a neutral element and an inverse). If we switch on or off those four properties we get 32 structures. Instead of giving a name to those 32 structures, I suggest we use intersected classes and speak of or . (Contributed by FL, 5-Sep-2010.)

Theoremiscom 24499* The predicate "is a commutative operation". (Contributed by FL, 5-Sep-2010.)

Theoremiscomb 24500 The predicate "is a commutative operation". (Contributed by FL, 14-Sep-2010.)

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