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Theorem List for Metamath Proof Explorer - 21801-21900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremablonnncan 21801 Cancellation law for group division. (nnncan 9282 analog.) (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremablonncan 21802 Cancellation law for group division. (nncan 9276 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)

Theoremablonnncan1 21803 Cancellation law for group division. (nnncan1 9283 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)

Theoremgxdi 21804 Distribution of group power over group operation for abelian groups. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)

Theoremisgrpda 21805* Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)

Theoremisgrpod 21806* Properties that determine a group operation. (Renamed from isgrpd 14771 to isgrpod 21806 to prevent naming conflict. -NM 5-Jun-2013) (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)

Theoremisabloda 21807* Properties that determine an Abelian group operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (New usage is discouraged.)

Theoremisablod 21808* Properties that determine an Abelian group operation. (Changed label from isabld 15366 to isablod 21808-NM 6-Aug-2013.) (Contributed by Jeff Madsen, 5-Dec-2009.) (New usage is discouraged.)

16.1.3  Subgroups

Syntaxcsubgo 21809 Extend class notation to include the class of subgroups.

Definitiondf-subgo 21810 Define the set of subgroups of . (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)

Theoremissubgo 21811 The predicate "is a subgroup of ." (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 12-Jul-2014.) (New usage is discouraged.)

Theoremsubgores 21812 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 21813 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 21814 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 21815 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 21816 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 21817* Lemma for issubgoi 21818. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)

Theoremissubgoi 21818* Properties that determine a subgroup. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GId

Theoremsubgoablo 21819 A subgroup of an Abelian group is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.) (New usage is discouraged.)

16.1.4  Operation properties

Syntaxcass 21820 Extend class notation with a device to add associativity to internal operations.

Definitiondf-ass 21821* 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 21822 Extend class notation with the class of all the internal operations with an identity element.

Definitiondf-exid 21823* A device to add an identity element to various sorts of internal operations. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremisass 21824* The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)

Theoremisexid 21825* 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 21826 Extend class notation with the class of all magmas.

Definitiondf-mgm 21827* A magma is a binary internal operation. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremismgm 21828 The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremclmgm 21829 Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.)

Theoremopidon 21830 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 21831 Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremopidon2 21832 An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremisexid2 21833* 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 21834* 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 21835* 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 21836 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 21837 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 21838 Extend class notation with the class of all semi-groups.

Definitiondf-sgr 21839 A semi-group is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpismgm 21840 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpisass 21841 A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremissmgrp 21842* The predicate "is a semi-group". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpmgm 21843 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Theoremsmgrpass 21844* A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Syntaxcmndo 21845 Extend class notation with the class of all monoids.
MndOp

Definitiondf-mndo 21846 A monoid is a semi-group with an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoissmgrp 21847 A monoid is a semi-group. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoisexid 21848 A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndoismgm 21849 A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp

Theoremmndomgmid 21850 A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
MndOp

Theoremismndo 21851* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremismndo1 21852* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremismndo2 21853* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremgrpomndo 21854 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 21855 The Abelian group operation for the singleton group. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremgidsn 21856 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 21857 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 21858 Complex number addition is an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

Theoremcnid 21859 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 21860 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 21861 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 21862 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 21863 Nonzero complex number multiplication is an Abelian group operation. (Contributed by Steve Rodriguez, 12-Feb-2007.) (New usage is discouraged.)

Theoremmulid 21864 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 21865 Extend class notation to include the class of group homomorphisms.
GrpOpHom

Definitiondf-ghom 21866* Define the set of group homomorphisms from to . (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Syntaxcgiso 21867 Extend class notation to include the class of group isomorphisms.

Definitiondf-giso 21868* Define the set of group isomorphisms from to . (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Theoremelghomlem1 21869* Lemma for elghom 21871. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Theoremelghomlem2 21870* Lemma for elghom 21871. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
GrpOpHom

Theoremelghom 21871* Membership in the set of group homomorphisms from to . (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
GrpOpHom

Theoremghomlin 21872 Linearity of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
GrpOpHom

Theoremghomid 21873 A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
GId       GId       GrpOpHom

Theoremghgrplem1 21874* Lemma for ghgrp 21876. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremghgrplem2 21875* Lemma for ghgrp 21876. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)

Theoremghgrp 21876* 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 21877* 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 21878* 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 21879* 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 21880* 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 21881* 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 21882 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 21883 Extend class notation with the class of all unital rings.

Definitiondf-rngo 21884* Define the class of all unital rings. (Contributed by Jeffrey Hankins, 21-Nov-2006.) (New usage is discouraged.)

Theoremrelrngo 21885 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 21886* 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 21887* Conditions that determine a ring. (Changed label from isrngd 15639 to isrngod 21887-NM 2-Aug-2013.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremrngoi 21888* 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 21889 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 21890 Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngoid 21891* 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 21892* 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 21893 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 21894 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 21895 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 21896* 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 21897 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 21898 A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngogcl 21899 Closure law for the addition (group) operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Theoremrngocom 21900 The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

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