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Theorem List for Metamath Proof Explorer - 21901-22000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-ass 21901* A device to add associativity to various sorts of internal operations. The definition is meaningful when is a magma at least. (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)

Syntaxcexid 21902 Extend class notation with the class of all the internal operations with an identity element.

Definitiondf-exid 21903* A device to add an identity element to various sorts of internal operations. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremisass 21904* The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)

Theoremisexid 21905* The predicate has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

16.1.5  Group-like structures

Syntaxcmagm 21906 Extend class notation with the class of all magmas.

Definitiondf-mgm 21907* A magma is a binary internal operation. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremismgm 21908 The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremclmgm 21909 Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.)

Theoremopidon 21910 An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Theoremrngopid 21911 Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremopidon2 21912 An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremisexid2 21913* If , then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Theoremexidu1 21914* Unicity of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Theoremidrval 21915* The value of the identity element. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
GId

Theoremiorlid 21916 A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
GId

Theoremcmpidelt 21917 A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
GId

Syntaxcsem 21918 Extend class notation with the class of all semi-groups.

Definitiondf-sgr 21919 A semi-group is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpismgm 21920 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpisass 21921 A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremissmgrp 21922* The predicate "is a semi-group". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpmgm 21923 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpass 21924* A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Syntaxcmndo 21925 Extend class notation with the class of all monoids.
MndOp

Definitiondf-mndo 21926 A monoid is a semi-group with an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoissmgrp 21927 A monoid is a semi-group. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoisexid 21928 A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoismgm 21929 A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndomgmid 21930 A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
MndOp

Theoremismndo 21931* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremismndo1 21932* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremismndo2 21933* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremgrpomndo 21934 A group is a monoid. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

16.1.6  Examples of Abelian groups

Theoremablosn 21935 The Abelian group operation for the singleton group. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremgidsn 21936 The identity element of the trivial group. (Contributed by FL, 21-Jun-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Theoremginvsn 21937 The inverse function of the trivial group. (Contributed by FL, 21-Jun-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremcnaddablo 21938 Complex number addition is an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremcnid 21939 The group identity element of complex number addition is zero. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
GId

Theoremaddinv 21940 Value of the group inverse of complex number addition. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremreaddsubgo 21941 The real numbers under addition comprise a subgroup of the complex numbers under addition. (Contributed by Paul Chapman, 25-Apr-2008.) (New usage is discouraged.)

Theoremzaddsubgo 21942 The integers under addition comprise a subgroup of the complex numbers under addition. (Contributed by Paul Chapman, 25-Apr-2008.) (New usage is discouraged.)

Theoremablomul 21943 Nonzero complex number multiplication is an Abelian group operation. (Contributed by Steve Rodriguez, 12-Feb-2007.) (New usage is discouraged.)

Theoremmulid 21944 The group identity element of nonzero complex number multiplication is one. (Contributed by Steve Rodriguez, 23-Feb-2007.) (Revised by Mario Carneiro, 17-Dec-2013.) (New usage is discouraged.)
GId

16.1.7  Group homomorphism and isomorphism

Syntaxcghom 21945 Extend class notation to include the class of group homomorphisms.
GrpOpHom

Definitiondf-ghom 21946* Define the set of group homomorphisms from to . (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Syntaxcgiso 21947 Extend class notation to include the class of group isomorphisms.

Definitiondf-giso 21948* Define the set of group isomorphisms from to . (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Theoremelghomlem1 21949* Lemma for elghom 21951. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Theoremelghomlem2 21950* Lemma for elghom 21951. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Theoremelghom 21951* Membership in the set of group homomorphisms from to . (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
GrpOpHom

Theoremghomlin 21952 Linearity of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
GrpOpHom

Theoremghomid 21953 A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
GId       GId       GrpOpHom

Theoremghgrplem1 21954* Lemma for ghgrp 21956. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremghgrplem2 21955* Lemma for ghgrp 21956. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremghgrp 21956* The image of a group under a group homomorphism is a group. This is a stronger result than that usually found in the literature, since the target of the homomorphism (operator in our model) need not have any of the properties of a group as a prerequisite. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremghablo 21957* The image of an Abelian group under a group homomorphism is an Abelian group (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremghsubgolem 21958* The image of a subgroup of group under a group homomorphism on is a group, and furthermore is Abelian if is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremghsubgo 21959* The image of a subgroup of group under a group homomorphism on is a group. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremghsubablo 21960* The image of an Abelian subgroup of group under a group homomorphism on is an Abelian group. (Contributed by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremefghgrp 21961* The image of a subgroup of the group , under the exponential function of a scaled complex number, is an Abelian group. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremcircgrp 21962 The circle group is an Abelian group. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)

16.2  Additional material on rings and fields

16.2.1  Definition and basic properties

Syntaxcrngo 21963 Extend class notation with the class of all unital rings.

Definitiondf-rngo 21964* Define the class of all unital rings. (Contributed by Jeffrey Hankins, 21-Nov-2006.) (New usage is discouraged.)

Theoremrelrngo 21965 The class of all unital rings is a relation. (Contributed by FL, 31-Aug-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremisrngo 21966* The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by Jeffrey Hankins, 21-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremisrngod 21967* Conditions that determine a ring. (Changed label from isrngd 15698 to isrngod 21967-NM 2-Aug-2013.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremrngoi 21968* The properties of a unital ring. (Contributed by Steve Rodriguez, 8-Sep-2007.) (Proof shortened by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremrngosm 21969 Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremrngocl 21970 Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngoid 21971* The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Theoremrngoideu 21972* The unit element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremrngodi 21973 Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremrngodir 21974 Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremrngoass 21975 Associative law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremrngo2 21976* A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Theoremrngoablo 21977 A ring's addition operation is an Abelian group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremrngogrpo 21978 A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngogcl 21979 Closure law for the addition (group) operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngocom 21980 The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngoaass 21981 The addition operation of a ring is associative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngoa32 21982 The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngoa4 21983 Rearrangement of 4 terms in a sum of ring elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngorcan 21984 Right cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngolcan 21985 Left cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngo0cl 21986 A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
GId

Theoremrngo0rid 21987 The additive identity of a ring is a right identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
GId

Theoremrngo0lid 21988 The additive identity of a ring is a left identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
GId

Theoremrngolz 21989 The zero of a unital ring is a left absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
GId

Theoremrngorz 21990 The zero of a unital ring is a right absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
GId

16.2.2  Examples of rings

Theoremcnrngo 21991 The set of complex numbers is a (unital) ring. (Contributed by Steve Rodriguez, 2-Feb-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Theoremrngosn 21992 The trivial or zero ring defined on a singleton set (see http://en.wikipedia.org/wiki/Trivial_ring). (Contributed by Steve Rodriguez, 10-Feb-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

16.2.3  Division Rings

Syntaxcdrng 21993 Extend class notation with the class of all division rings.

Definitiondf-drngo 21994* Define the class of all division rings (sometimes called skew fields). A division ring is a unital ring where every element except the additive identity has a multiplicative inverse. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
GId GId

Theoremdrngoi 21995 The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
GId

16.2.4  Star Fields

Syntaxcsfld 21996 Extend class notation with the class of all star fields.

Definitiondf-sfld 21997* Define the class of all star fields, which are all division rings with involutions. (Contributed by NM, 29-Aug-2010.) (New usage is discouraged.)

16.2.5  Fields and Rings

Syntaxccm2 21998 Extend class notation with a class that adds commutativity to various flavors of rings.

Definitiondf-com2 21999* A device to add commutativity to various sorts of rings. I use because I suppose has a neutral element and therefore is onto. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)

Theoremiscom2 22000* A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)

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