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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.)
 |-  W  =  ran  H   =>    |-  ( H  e.  ( SubGrpOp `  G )  ->  H  =  ( G  |`  ( W  X.  W ) ) )
 
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.)
 |-  W  =  ran  H   =>    |-  (
 ( H  e.  ( SubGrpOp `  G )  /\  ( A  e.  W  /\  B  e.  W )
 )  ->  ( A H B )  =  ( A G B ) )
 
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.)
 |-  X  =  ran  G   &    |-  W  =  ran  H   =>    |-  ( H  e.  ( SubGrpOp `  G )  ->  W  C_  X )
 
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.)
 |-  U  =  (GId `  G )   &    |-  T  =  (GId `  H )   =>    |-  ( H  e.  ( SubGrpOp `  G )  ->  T  =  U )
 
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.)
 |-  W  =  ran  H   &    |-  M  =  ( inv `  G )   &    |-  N  =  ( inv `  H )   =>    |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W ) 
 ->  ( N `  A )  =  ( M `  A ) )
 
Theoremissubgoilem 20806* Lemma for issubgoi 20807. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |-  ( ( x  e.  Y  /\  y  e.  Y )  ->  ( x H y )  =  ( x G y ) )   =>    |-  ( ( A  e.  Y  /\  B  e.  Y )  ->  ( A H B )  =  ( A G B ) )
 
Theoremissubgoi 20807* Properties that determine a subgroup. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |-  G  e.  GrpOp   &    |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  Y  C_  X   &    |-  H  =  ( G  |`  ( Y  X.  Y ) )   &    |-  (
 ( x  e.  Y  /\  y  e.  Y )  ->  ( x G y )  e.  Y )   &    |-  U  e.  Y   &    |-  ( x  e.  Y  ->  ( N `  x )  e.  Y )   =>    |-  H  e.  ( SubGrpOp `  G )
 
Theoremsubgoablo 20808 A subgroup of an Abelian group is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.) (New usage is discouraged.)
 |-  ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G ) )  ->  H  e.  AbelOp )
 
15.1.4  Operation properties
 
Syntaxcass 20809 Extend class notation with a device to add associativity to internal operations.
 class  Ass
 
Definitiondf-ass 20810* A device to add associativity to various sorts of internal operations. The definition is meaningful when  g is a magma at least. (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)
 |- 
 Ass  =  { g  |  A. x  e.  dom  dom  g A. y  e. 
 dom  dom  g A. z  e.  dom  dom  g (
 ( x g y ) g z )  =  ( x g ( y g z ) ) }
 
Syntaxcexid 20811 Extend class notation with the class of all the internal operations with an identity element.
 class  ExId
 
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.)
 |- 
 ExId  =  { g  |  E. x  e.  dom  dom  g A. y  e. 
 dom  dom  g ( ( x g y )  =  y  /\  (
 y g x )  =  y ) }
 
Theoremisass 20813* The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  A  ->  ( G  e.  Ass  <->  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) ) )
 
Theoremisexid 20814* The predicate  G has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  A  ->  ( G  e.  ExId  <->  E. x  e.  X  A. y  e.  X  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) )
 
15.1.5  Group-like structures
 
Syntaxcmagm 20815 Extend class notation with the class of all magmas.
 class  Magma
 
Definitiondf-mgm 20816* A magma is a binary internal operation. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |- 
 Magma  =  { g  |  E. t  g : ( t  X.  t
 ) --> t }
 
Theoremismgm 20817 The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  A  ->  ( G  e.  Magma  <->  G : ( X  X.  X ) --> X ) )
 
Theoremclmgm 20818 Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  Magma  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
 
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.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X ) -onto-> X )
 
Theoremrngopid 20820 Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran 
 G  =  dom  dom  G )
 
Theoremopidon2 20821 An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G :
 ( X  X.  X ) -onto-> X )
 
Theoremisexid2 20822* If  G  e.  ( Magma  i^i  ExId  ) 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.)
 |-  X  =  ran  G   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
 
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.)
 |-  X  =  ran  G   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E! u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
 
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.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  ( G  e.  A  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
 
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.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  X )
 
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.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  ( Magma  i^i  ExId  )  /\  A  e.  X )  ->  ( ( U G A )  =  A  /\  ( A G U )  =  A )
 )
 
Syntaxcsem 20827 Extend class notation with the class of all semi-groups.
 class  SemiGrp
 
Definitiondf-sgr 20828 A semi-group is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  SemiGrp  =  ( Magma  i^i  Ass )
 
Theoremsmgrpismgm 20829 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e.  SemiGrp  ->  G  e.  Magma )
 
Theoremsmgrpisass 20830 A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e.  SemiGrp  ->  G  e.  Ass )
 
Theoremissmgrp 20831* The predicate "is a semi-group". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  A  ->  ( G  e.  SemiGrp  <->  ( G :
 ( X  X.  X )
 --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) ) ) )
 
Theoremsmgrpmgm 20832 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  SemiGrp  ->  G : ( X  X.  X ) --> X )
 
Theoremsmgrpass 20833* A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  SemiGrp  ->  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )
 
Syntaxcmndo 20834 Extend class notation with the class of all monoids.
 class 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  =  ( SemiGrp  i^i  ExId  )
 
Theoremmndoissmgrp 20836 A monoid is a semi-group. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e. MndOp  ->  G  e.  SemiGrp )
 
Theoremmndoisexid 20837 A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e. MndOp  ->  G  e.  ExId  )
 
Theoremmndoismgm 20838 A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e. MndOp  ->  G  e.  Magma )
 
Theoremmndomgmid 20839 A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
 |-  ( G  e. MndOp  ->  G  e.  ( Magma  i^i  ExId  ) )
 
Theoremismndo 20840* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  A  ->  ( G  e. MndOp  <->  ( G  e.  SemiGrp  /\ 
 E. x  e.  X  A. y  e.  X  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) ) )
 
Theoremismndo1 20841* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  A  ->  ( G  e. MndOp  <->  ( G :
 ( X  X.  X )
 --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) 
 /\  E. x  e.  X  A. y  e.  X  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) ) )
 
Theoremismndo2 20842* The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  A  ->  ( G  e. MndOp  <->  ( G :
 ( X  X.  X )
 --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) 
 /\  E. x  e.  X  A. y  e.  X  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) ) )
 
Theoremgrpomndo 20843 A group is a monoid. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  ( G  e.  GrpOp  ->  G  e. 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.)
 |-  A  e.  _V   =>    |-  { <. <. A ,  A >. ,  A >. }  e.  AbelOp
 
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.)
 |-  A  e.  _V   =>    |-  (GId `  { <. <. A ,  A >. ,  A >. } )  =  A
 
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.)
 |-  A  e.  _V   =>    |-  ( inv `  { <. <. A ,  A >. ,  A >. } )  =  (  _I  |`  { A } )
 
Theoremcnaddablo 20847 Complex number addition is an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
 |- 
 +  e.  AbelOp
 
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.)
 |-  0  =  (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.)
 |-  ( A  e.  CC  ->  ( ( inv `  +  ) `  A )  =  -u A )
 
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.)
 |-  (  +  |`  ( RR 
 X.  RR ) )  e.  ( SubGrpOp `  +  )
 
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.)
 |-  (  +  |`  ( ZZ 
 X.  ZZ ) )  e.  ( SubGrpOp `  +  )
 
Theoremablomul 20852 Nonzero complex number multiplication is an Abelian group operation. (Contributed by Steve Rodriguez, 12-Feb-2007.) (New usage is discouraged.)
 |-  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  { 0 } ) ) )  e.  AbelOp
 
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 `  (  x.  |`  ( ( CC  \  { 0 } )  X.  ( CC  \  {
 0 } ) ) ) )  =  1
 
15.1.7  Group homomorphism and isomorphism
 
Syntaxcghom 20854 Extend class notation to include the class of group homomorphisms.
 class GrpOpHom
 
Definitiondf-ghom 20855* Define the set of group homomorphisms from  g to  h. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |- GrpOpHom  =  ( g  e.  GrpOp ,  h  e.  GrpOp  |->  { f  |  ( f : ran  g
 --> ran  h  /\  A. x  e.  ran  g A. y  e.  ran  g ( ( f `  x ) h ( f `  y ) )  =  ( f `  ( x g y ) ) ) } )
 
Syntaxcgiso 20856 Extend class notation to include the class of group isomorphisms.
 class  GrpOpIso
 
Definitiondf-giso 20857* Define the set of group isomorphisms from  g to  h. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |-  GrpOpIso 
 =  ( g  e. 
 GrpOp ,  h  e.  GrpOp  |->  { f  e.  ( g GrpOpHom  h )  |  f : ran  g -1-1-onto-> ran  h } )
 
Theoremelghomlem1 20858* Lemma for elghom 20860. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |-  S  =  { f  |  ( f : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x ) H ( f `  y ) )  =  ( f `  ( x G y ) ) ) }   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G GrpOpHom  H )  =  S )
 
Theoremelghomlem2 20859* Lemma for elghom 20860. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |-  S  =  { f  |  ( f : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x ) H ( f `  y ) )  =  ( f `  ( x G y ) ) ) }   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
  ( x G y ) ) ) ) )
 
Theoremelghom 20860* Membership in the set of group homomorphisms from  G to  H. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  W  =  ran  H   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : X
 --> W  /\  A. x  e.  X  A. y  e.  X  ( ( F `
  x ) H ( F `  y
 ) )  =  ( F `  ( x G y ) ) ) ) )
 
Theoremghomlin 20861 Linearity of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( ( F `  A ) H ( F `  B ) )  =  ( F `
  ( A G B ) ) )
 
Theoremghomid 20862 A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
 |-  U  =  (GId `  G )   &    |-  T  =  (GId `  H )   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  ( F `  U )  =  T )
 
Theoremghgrplem1 20863* Lemma for ghgrp 20865. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ( ph  /\  w  e.  X ) 
 ->  ps )   &    |-  ( C  =  ( F `  w ) 
 ->  ( ch  <->  ps ) )   =>    |-  ( ( ph  /\  C  e.  Y ) 
 ->  ch )
 
Theoremghgrplem2 20864* Lemma for ghgrp 20865. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( F `
  ( x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )   &    |-  H  =  ( O  |`  ( Y  X.  Y ) )   =>    |-  ( ( ph  /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( F `  ( C G D ) )  =  (
 ( F `  C ) H ( F `  D ) ) )
 
Theoremghgrp 20865* The image of a group  G under a group homomorphism  F is a group. This is a stronger result than that usually found in the literature, since the target of the homomorphism (operator  O 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.)
 |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( F `
  ( x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )   &    |-  H  =  ( O  |`  ( Y  X.  Y ) )   &    |-  X  =  ran  G   &    |-  ( ph  ->  Y 
 C_  A )   &    |-  ( ph  ->  O  Fn  ( A  X.  A ) )   &    |-  ( ph  ->  G  e.  GrpOp
 )   =>    |-  ( ph  ->  H  e.  GrpOp )
 
Theoremghablo 20866* The image of a Abelian group  G under a group homomorphism  F is an Abelian group (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( F `
  ( x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )   &    |-  H  =  ( O  |`  ( Y  X.  Y ) )   &    |-  X  =  ran  G   &    |-  ( ph  ->  Y 
 C_  A )   &    |-  ( ph  ->  O  Fn  ( A  X.  A ) )   &    |-  ( ph  ->  G  e.  AbelOp )   =>    |-  ( ph  ->  H  e.  AbelOp )
 
Theoremghsubgolem 20867* The image of a subgroup  S of group  G under a group homomorphism  F on  G is a group, and furthermore is Abelian if  S is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  S  e.  ( SubGrpOp `  G )
 )   &    |-  X  =  ran  G   &    |-  ( ph  ->  F : X --> Y )   &    |-  ( ph  ->  Y 
 C_  A )   &    |-  ( ph  ->  O  Fn  ( A  X.  A ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( F `  ( x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )   &    |-  Z  =  ran  S   &    |-  W  =  ( F " Z )   &    |-  H  =  ( O  |`  ( W  X.  W ) )   =>    |-  ( ph  ->  ( H  e.  GrpOp  /\  ( S  e.  AbelOp  ->  H  e.  AbelOp ) ) )
 
Theoremghsubgo 20868* The image of a subgroup  S of group  G under a group homomorphism  F on  G is a group. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  S  e.  ( SubGrpOp `  G )
 )   &    |-  X  =  ran  G   &    |-  ( ph  ->  F : X --> Y )   &    |-  ( ph  ->  Y 
 C_  A )   &    |-  ( ph  ->  O  Fn  ( A  X.  A ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( F `  ( x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )   &    |-  Z  =  ran  S   &    |-  W  =  ( F " Z )   &    |-  H  =  ( O  |`  ( W  X.  W ) )   =>    |-  ( ph  ->  H  e.  GrpOp )
 
Theoremghsubablo 20869* The image of an Abelian subgroup  S of group  G under a group homomorphism  F on  G is an Abelian group. (Contributed by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  S  e.  ( SubGrpOp `  G )
 )   &    |-  X  =  ran  G   &    |-  ( ph  ->  F : X --> Y )   &    |-  ( ph  ->  Y 
 C_  A )   &    |-  ( ph  ->  O  Fn  ( A  X.  A ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( F `  ( x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )   &    |-  Z  =  ran  S   &    |-  W  =  ( F " Z )   &    |-  H  =  ( O  |`  ( W  X.  W ) )   &    |-  ( ph  ->  S  e.  AbelOp )   =>    |-  ( ph  ->  H  e.  AbelOp )
 
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.)
 |-  S  =  { y  |  E. x  e.  X  y  =  ( exp `  ( A  x.  x ) ) }   &    |-  G  =  (  x.  |`  ( S  X.  S ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  X 
 C_  CC )   &    |-  (  +  |`  ( X  X.  X ) )  e.  ( SubGrpOp `  +  )   =>    |-  ( ph  ->  G  e.  AbelOp )
 
Theoremcircgrp 20871 The circle group  T is an Abelian group. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
 |-  C  =  ( `'
 abs " { 1 } )   &    |-  T  =  (  x.  |`  ( C  X.  C ) )   =>    |-  T  e.  AbelOp
 
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.
 class  RingOps
 
Definitiondf-rngo 20873* Define the class of all unital rings. (Contributed by Jeffrey Hankins, 21-Nov-2006.) (New usage is discouraged.)
 |-  RingOps  =  { <. g ,  h >.  |  (
 ( g  e.  AbelOp  /\  h : ( ran  g  X.  ran  g
 ) --> ran  g )  /\  ( A. x  e. 
 ran  g A. y  e.  ran  g A. z  e.  ran  g ( ( ( x h y ) h z )  =  ( x h ( y h z ) )  /\  ( x h ( y g z ) )  =  ( ( x h y ) g ( x h z ) )  /\  ( ( x g y ) h z )  =  ( ( x h z ) g ( y h z ) ) )  /\  E. x  e.  ran  g A. y  e.  ran  g ( ( x h y )  =  y  /\  ( y h x )  =  y ) ) ) }
 
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.)
 |- 
 Rel  RingOps
 
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.)
 |-  X  =  ran  G   =>    |-  ( H  e.  A  ->  (
 <. G ,  H >.  e.  RingOps  <->  ( ( G  e.  AbelOp  /\  H : ( X  X.  X ) --> X ) 
 /\  ( A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  /\  ( ( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )  /\  E. x  e.  X  A. y  e.  X  ( ( x H y )  =  y  /\  ( y H x )  =  y ) ) ) ) )
 
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.)
 |-  ( ph  ->  G  e.  AbelOp )   &    |-  ( ph  ->  X  =  ran  G )   &    |-  ( ph  ->  H :
 ( X  X.  X )
 --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( ( x H y ) H z )  =  ( x H ( y H z ) ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( ( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )   &    |-  ( ph  ->  U  e.  X )   &    |-  (
 ( ph  /\  y  e.  X )  ->  ( U H y )  =  y )   &    |-  ( ( ph  /\  y  e.  X ) 
 ->  ( y H U )  =  y )   =>    |-  ( ph  ->  <. G ,  H >.  e.  RingOps )
 
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.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  ( ( G  e.  AbelOp  /\  H : ( X  X.  X ) --> X ) 
 /\  ( A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  /\  ( ( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )  /\  E. x  e.  X  A. y  e.  X  ( ( x H y )  =  y  /\  ( y H x )  =  y ) ) ) )
 
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.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  H : ( X  X.  X ) --> X )
 
Theoremrngocl 20879 Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
 
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.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  E. u  e.  X  ( ( u H A )  =  A  /\  ( A H u )  =  A ) )
 
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.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  E! u  e.  X  A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
 
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.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A H ( B G C ) )  =  ( ( A H B ) G ( A H C ) ) )
 
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.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) H C )  =  ( ( A H C ) G ( B H C ) ) )
 
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.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A H B ) H C )  =  ( A H ( B H C ) ) )
 
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.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  E. x  e.  X  ( A G A )  =  ( ( x G x ) H A ) )
 
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.)
 |-  G  =  ( 1st `  R )   =>    |-  ( R  e.  RingOps  ->  G  e.  AbelOp )
 
Theoremrngogrpo 20887 A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   =>    |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
 
Theoremrngogcl 20888 Closure law for the addition (group) operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
 
Theoremrngocom 20889 The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
 
Theoremrngoaass 20890 The addition operation of a ring is associative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Theoremrngoa32 20891 The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )
 
Theoremrngoa4 20892 Rearrangement of 4 terms in a sum of ring elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( ( A G B ) G ( C G D ) )  =  ( ( A G C ) G ( B G D ) ) )
 
Theoremrngorcan 20893 Right cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G C )  =  ( B G C ) 
 <->  A  =  B ) )
 
Theoremrngolcan 20894 Left cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( C G A )  =  ( C G B ) 
 <->  A  =  B ) )
 
Theoremrngo0cl 20895 A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  RingOps  ->  Z  e.  X )
 
Theoremrngo0rid 20896 The additive identity of a ring is a right identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( A G Z )  =  A )
 
Theoremrngo0lid 20897 The additive identity of a ring is a left identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( Z G A )  =  A )
 
Theoremrngolz 20898 The zero of a unital ring is a left absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
 |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   =>    |-  (
 ( R  e.  RingOps  /\  A  e.  X )  ->  ( Z H A )  =  Z )
 
Theoremrngorz 20899 The zero of a unital ring is a right absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
 |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   =>    |-  (
 ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  =  Z )
 
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.)
 |- 
 <.  +  ,  x.  >.  e.  RingOps
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