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Theorem List for Metamath Proof Explorer - 14601-14700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremacsficld 14601 In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 14597. (Contributed by David Moews, 1-May-2017.)
ACS       mrCls

Theoremacsficl2d 14602* In an algebraic closure system, an element is in the closure of a set if and only if it is in the closure of a finite subset. Alternate form of acsficl 14597. Deduction form. (Contributed by David Moews, 1-May-2017.)
ACS       mrCls

Theoremacsfiindd 14603 In an algebraic closure system, a set is independent if and only if all its finite subsets are independent. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
ACS       mrCls       mrInd

Theoremacsmapd 14604* In an algebraic closure system, if is contained in the closure of , there is a map from into the set of finite subsets of such that the closure of contains . This is proven by applying acsficl2d 14602 to each element of . See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
ACS       mrCls

Theoremacsmap2d 14605* In an algebraic closure system, if and have the same closure and is independent, then there is a map from into the set of finite subsets of such that equals the union of . This is proven by taking the map from acsmapd 14604 and observing that, since and have the same closure, the closure of must contain . Since is independent, by mrissmrcd 13865, must equal . See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
ACS       mrCls       mrInd

Theoremacsinfd 14606 In an algebraic closure system, if and have the same closure and is infinite independent, then is infinite. This follows from applying unirnffid 7398 to the map given in acsmap2d 14605. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
ACS       mrCls       mrInd

Theoremacsdomd 14607 In an algebraic closure system, if and have the same closure and is infinite independent, then dominates . This follows from applying acsinfd 14606 and then applying unirnfdomd 8442 to the map given in acsmap2d 14605. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
ACS       mrCls       mrInd

Theoremacsinfdimd 14608 In an algebraic closure system, if two independent sets have equal closure and one is infinite, then they are equinumerous. This is proven by using acsdomd 14607 twice with acsinfd 14606. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
ACS       mrCls       mrInd

Theoremacsexdimd 14609* In an algebraic closure system whose closure operator has the exchange property, if two independent sets have equal closure, they are equinumerous. See mreexfidimd 13875 for the finite case and acsinfdimd 14608 for the infinite case. This is a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
ACS       mrCls       mrInd

Theoremmrelatglb 14610 Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.)
toInc              Moore

Theoremmrelatglb0 14611 The empty intersection in a Moore space is realized by the base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
toInc              Moore

Theoremmrelatlub 14612 Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
toInc       mrCls              Moore

Theoremmreclat 14613 A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.)
toInc       Moore

9.2.5  Distributive lattices

Theoremlatmass 14614 Lattice meet is associative. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremlatdisdlem 14615* Lemma for latdisd 14616. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremlatdisd 14616* In a lattice, joins distribute over meets if and only if meets distribute over joins; the distributive property is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)

Syntaxcdlat 14617 The class of distributive lattices.
DLat

Definitiondf-dlat 14618* A distributive lattice is a lattice in which meets distribute over joins, or equivalently (latdisd 14616) joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
DLat

Theoremisdlat 14619* Property of being a distributive lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
DLat

Theoremdlatmjdi 14620 In a distributive lattice, meets distribute over joins. (Contributed by Stefan O'Rear, 30-Jan-2015.)
DLat

Theoremdlatl 14621 A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
DLat

Theoremodudlatb 14622 The dual of a distributive lattice is a distributive lattice and conversely. (Contributed by Stefan O'Rear, 30-Jan-2015.)
ODual       DLat DLat

Theoremdlatjmdi 14623 In a distributive lattice, joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
DLat

9.2.6  Posets and lattices as relations

Syntaxcps 14624 Extend class notation with the class of all posets.

Syntaxctsr 14625 Extend class notation with the class of all totally ordered sets.

Syntaxcspw 14626 Extend class notation with the supremum of an ordered set.

Syntaxcinf 14627 Extend class notation with the infimum of an ordered set.

Syntaxcla 14628 Extend class notation with the class of all lattices.

Definitiondf-ps 14629 Define the class of all posets (partially ordered sets) with weak ordering (e.g. "less than or equal to" instead of "less than"). A poset is a relation which is transitive, reflexive, and antisymmetric. (Contributed by NM, 11-May-2008.)

Definitiondf-tsr 14630 Define the class of all totally ordered sets. (Contributed by FL, 1-Nov-2009.)

Definitiondf-spw 14631* Define suprema under weak orderings. Unlike df-sup 7446 for strong orderings, is evaluates to a member of the field of iff the supremum exists. Read as the -supremum of set . (Contributed by NM, 13-May-2008.)

Definitiondf-nfw 14632* Define the class of all infima of a weak ordering relation. (Contributed by FL, 6-Sep-2009.)

Definitiondf-lar 14633* Define the class of all lattices, which are posets in which every two elements have upper and lower bounds. (Contributed by NM, 12-Jun-2008.)

Theoremisps 14634 The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.)

Theorempsrel 14635 A poset is a relation. (Contributed by NM, 12-May-2008.)

Theorempsref2 14636 A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)

Theorempstr2 14637 A poset is transitive. (Contributed by FL, 3-Aug-2009.)

Theorempslem 14638 Lemma for psref 14640 and others. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theorempsdmrn 14639 The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.)

Theorempsref 14640 A poset is reflexive. (Contributed by NM, 13-May-2008.)

Theorempsrn 14641 The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)

Theorempsasym 14642 A poset is antisymmetric. (Contributed by NM, 12-May-2008.)

Theorempstr 14643 A poset is transitive. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremcnvps 14644 The converse of a poset is a poset. In the general case is not true. See cnvpsb 14645 for a special case where the property holds. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)

Theoremcnvpsb 14645 The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)

Theorempsss 14646 Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.)

Theorempsssdm2 14647 Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theorempsssdm 14648 Field of a subposet. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 9-Sep-2015.)

Theoremistsr 14649 The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremistsr2 14650* The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremtsrlin 14651 A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremtsrlemax 14652 Two ways of saying a number is less than or equal to the maximum of two others. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremtsrps 14653 A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremcnvtsr 14654 The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremtsrss 14655 Any subset of a totally ordered set is totally ordered. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Nov-2013.)

Theoremspwval2 14656* Value of supremum under a weak ordering. Read as "the -supremum of ." is the field of a relation by relfld 5395. Unlike df-sup 7446 for strong orderings, the supremum exists iff belongs to the field. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)

Theoremspwval 14657* Value of supremum under a weak ordering. Read as "the -supremum of ." is the field of a relation by relfld 5395. Unlike df-sup 7446 for strong orderings, the supremum exists iff belongs to the field. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)

Theoremspwmo 14658* A poset has at most one supremum. (Contributed by NM, 13-May-2008.) (Revised by NM, 16-Jun-2017.)

Theoremspweu 14659* A supremum is unique. (Contributed by NM, 15-May-2008.)

Theoremspwpr2 14660* Property of supremum defining condition for an unordered pair. (Contributed by NM, 24-Jun-2008.)

Theoremspwex 14661* A supremum exists iff belongs to the domain of . (Contributed by NM, 15-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)

Theoremspwcl 14662* Closure of a supremum. (Contributed by NM, 15-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)

Theoremspwpr4 14663* Supremum of an unordered pair. (Contributed by NM, 7-Jul-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)

Theoremspwpr4c 14664 Supremum of an unordered pair of comparable elements. (Contributed by NM, 7-Jul-2008.)

Theoremisla 14665* The predicate "is a lattice" i.e. a poset in which any two elements have upper and lower bounds. (Contributed by NM, 12-Jun-2008.)

Theoremlaspwcl 14666 Closure of the supremum (join) of two lattice elements. (Contributed by NM, 12-Jun-2008.)

Theoremlanfwcl 14667 Closure of the infimum (meet) of two lattice elements. (Contributed by NM, 20-Jun-2008.)

Theoremlaps 14668 A lattice is a poset. (Contributed by NM, 12-Jun-2008.)

Theoremledm 14669 domain of is . (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)

Theoremlern 14670 The range of is . (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremlefld 14671 The field of the 'less or equal to' relationship on the extended real. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)

Theoremletsr 14672 The "less than or equal to" relationship on the extended reals is a toset. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)

9.2.7  Directed sets, nets

Syntaxcdir 14673 Extend class notation with the class of all directed sets.

Syntaxctail 14674 Extend class notation with the tail function.

Definitiondf-dir 14675 Define the class of all directed sets/directions. (Contributed by Jeff Hankins, 25-Nov-2009.)

Definitiondf-tail 14676* Define the tail function for directed sets. (Contributed by Jeff Hankins, 25-Nov-2009.)

Theoremisdir 14677 A condition for a relation to be a direction. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremreldir 14678 A direction is a relation. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremdirdm 14679 A direction's domain is equal to its field. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremdirref 14680 A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremdirtr 14681 A direction is transitive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremdirge 14682* For any two elements of a directed set, there exists a third element greater than or equal to both. (Note that this does not say that the two elements have a least upper bound.) (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Theoremtsrdir 14683 A totally ordered set is a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

PART 10  BASIC ALGEBRAIC STRUCTURES

10.1  Monoids

10.1.1  Definition and basic properties

Syntaxcmnd 14684 Extend class notation with class of all monoids.

Syntaxcgrp 14685 Extend class notation with class of all groups.

Syntaxcminusg 14686 Extend class notation with inverse of group element.

Syntaxcplusf 14687 Extend class notation with group addition as a function.

Syntaxcsg 14688 Extend class notation with group subtraction (or division) operation.

Syntaxcmg 14689 Extend class notation with a function mapping a group operation to the power operation for the group.
.g

Definitiondf-mnd 14690* Definition of a monoid. A monoid is a set equipped with an everywhere defined internal operation (so, a magma, see mndcl 14695), whose operation is associative (so, a semigroup, see mndass 14696) and has a two-sided neutral element (see mndid 14697). (Contributed by Mario Carneiro, 6-Jan-2015.)

Definitiondf-plusf 14691* Define group addition function. Usually we will use directly instead of , and they have the same behavior in most cases. The main advantage of is that it is a guaranteed function (mndplusf 14706), while only has closure (mndcl 14695). (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremismnd 14692* The predicate "is a monoid." (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremmgmidmo 14693* A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.)

Theoremmndlem1 14694 Lemma for monoid properties. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremmndcl 14695 Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremmndass 14696 A monoid operation is associative. (Contributed by NM, 14-Aug-2011.)

Theoremmndid 14697* A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)

Theoremmndideu 14698* The two-sided identity element of a monoid is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by Mario Carneiro, 8-Dec-2014.)

Theoremmnd32g 14699 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremmnd12g 14700 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

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