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Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubgores 20801 A subgroup operation is the restriction of its parent group operation to its underlying set. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)

Theoremsubgoov 20802 The result of a subgroup operation is the same as the result of its parent operation. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 8-Jul-2014.) (New usage is discouraged.)

Theoremsubgornss 20803 The underlying set of a subgroup is a subset of its parent group's underlying set. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)

Theoremsubgoid 20804 The identity element of a subgroup is the same as its parent's. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
GId       GId

Theoremsubgoinv 20805 The inverse of a subgroup element is the same as its inverse in the parent group. (Contributed by Mario Carneiro, 8-Jul-2014.) (New usage is discouraged.)

Theoremissubgoilem 20806* Lemma for issubgoi 20807. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)

Theoremissubgoi 20807* Properties that determine a subgroup. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GId

Theoremsubgoablo 20808 A subgroup of an Abelian group is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.) (New usage is discouraged.)

15.1.4  Operation properties

Syntaxcass 20809 Extend class notation with a device to add associativity to internal operations.

Definitiondf-ass 20810* 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 20811 Extend class notation with the class of all the internal operations with an identity element.

Definitiondf-exid 20812* A device to add an identity element to various sorts of internal operations. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremisass 20813* The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)

Theoremisexid 20814* 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.)

15.1.5  Group-like structures

Syntaxcmagm 20815 Extend class notation with the class of all magmas.

Definitiondf-mgm 20816* A magma is a binary internal operation. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremismgm 20817 The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremclmgm 20818 Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.)

Theoremopidon 20819 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 20820 Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremopidon2 20821 An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremisexid2 20822* 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 20823* 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 20824* 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 20825 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 20826 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 20827 Extend class notation with the class of all semi-groups.

Definitiondf-sgr 20828 A semi-group is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpismgm 20829 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpisass 20830 A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremissmgrp 20831* The predicate "is a semi-group". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpmgm 20832 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpass 20833* A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Syntaxcmndo 20834 Extend class notation with the class of all monoids.
MndOp

Definitiondf-mndo 20835 A monoid is a semi-group with an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoissmgrp 20836 A monoid is a semi-group. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoisexid 20837 A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoismgm 20838 A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndomgmid 20839 A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
MndOp

Theoremismndo 20840* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremismndo1 20841* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremismndo2 20842* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremgrpomndo 20843 A group is a monoid. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

15.1.6  Examples of Abelian groups

Theoremablosn 20844 The Abelian group operation for the singleton group. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremgidsn 20845 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 20846 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 20847 Complex number addition is an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremcnid 20848 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 20849 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 20850 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 20851 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 20852 Nonzero complex number multiplication is an Abelian group operation. (Contributed by Steve Rodriguez, 12-Feb-2007.) (New usage is discouraged.)

Theoremmulid 20853 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

15.1.7  Group homomorphism and isomorphism

Syntaxcghom 20854 Extend class notation to include the class of group homomorphisms.
GrpOpHom

Definitiondf-ghom 20855* Define the set of group homomorphisms from to . (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Syntaxcgiso 20856 Extend class notation to include the class of group isomorphisms.

Definitiondf-giso 20857* Define the set of group isomorphisms from to . (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Theoremelghomlem1 20858* Lemma for elghom 20860. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Theoremelghomlem2 20859* Lemma for elghom 20860. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Theoremelghom 20860* Membership in the set of group homomorphisms from to . (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
GrpOpHom

Theoremghomlin 20861 Linearity of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
GrpOpHom

Theoremghomid 20862 A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
GId       GId       GrpOpHom

Theoremghgrplem1 20863* Lemma for ghgrp 20865. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremghgrplem2 20864* Lemma for ghgrp 20865. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremghgrp 20865* 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 20866* The image of a 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 20867* 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 20868* 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 20869* 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 20870* 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 20871 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.)

15.2  Additional material on Rings and Fields

15.2.1  Definition and basic properties

Syntaxcrngo 20872 Extend class notation with the class of all unital rings.

Definitiondf-rngo 20873* Define the class of all unital rings. (Contributed by Jeffrey Hankins, 21-Nov-2006.) (New usage is discouraged.)

Theoremrelrngo 20874 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 20875* 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 20876* Conditions that determine a ring. (Changed label from isrngd 15210 to isrngod 20876-NM 2-Aug-2013.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremrngoi 20877* 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 20878 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 20879 Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngoid 20880* 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 20881* 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 20882 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 20883 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 20884 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 20885* 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 20886 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 20887 A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngogcl 20888 Closure law for the addition (group) operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngocom 20889 The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngoaass 20890 The addition operation of a ring is associative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngoa32 20891 The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngoa4 20892 Rearrangement of 4 terms in a sum of ring elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngorcan 20893 Right cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngolcan 20894 Left cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngo0cl 20895 A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
GId

Theoremrngo0rid 20896 The additive identity of a ring is a right identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
GId

Theoremrngo0lid 20897 The additive identity of a ring is a left identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
GId

Theoremrngolz 20898 The zero of a unital ring is a left absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
GId

Theoremrngorz 20899 The zero of a unital ring is a right absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
GId

15.2.2  Examples of rings

Theoremcnrngo 20900 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.)

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