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Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
15.1.5  Group-like structures
 
Syntaxcmagm 21001 Extend class notation with the class of all magmas.
 class  Magma
 
Definitiondf-mgm 21002* 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 21003 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 21004 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 21005 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 21006 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 21007 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 21008* 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 21009* 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 21010* 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 21011 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 21012 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 21013 Extend class notation with the class of all semi-groups.
 class  SemiGrp
 
Definitiondf-sgr 21014 A semi-group is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  SemiGrp  =  ( Magma  i^i  Ass )
 
Theoremsmgrpismgm 21015 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e.  SemiGrp  ->  G  e.  Magma )
 
Theoremsmgrpisass 21016 A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e.  SemiGrp  ->  G  e.  Ass )
 
Theoremissmgrp 21017* 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 21018 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 21019* 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 21020 Extend class notation with the class of all monoids.
 class MndOp
 
Definitiondf-mndo 21021 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 21022 A monoid is a semi-group. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e. MndOp  ->  G  e.  SemiGrp )
 
Theoremmndoisexid 21023 A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e. MndOp  ->  G  e.  ExId  )
 
Theoremmndoismgm 21024 A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e. MndOp  ->  G  e.  Magma )
 
Theoremmndomgmid 21025 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 21026* 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 21027* 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 21028* 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 21029 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 21030 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 21031 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 21032 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 21033 Complex number addition is an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
 |- 
 +  e.  AbelOp
 
Theoremcnid 21034 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 21035 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 21036 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 21037 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 21038 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 21039 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 21040 Extend class notation to include the class of group homomorphisms.
 class GrpOpHom
 
Definitiondf-ghom 21041* 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 21042 Extend class notation to include the class of group isomorphisms.
 class  GrpOpIso
 
Definitiondf-giso 21043* 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 21044* Lemma for elghom 21046. (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 21045* Lemma for elghom 21046. (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 21046* 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 21047 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 21048 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 21049* Lemma for ghgrp 21051. (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 21050* Lemma for ghgrp 21051. (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 21051* 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 21052* The image of an 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 21053* 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 21054* 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 21055* 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 21056* 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 21057 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 21058 Extend class notation with the class of all unital rings.
 class  RingOps
 
Definitiondf-rngo 21059* 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 21060 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 21061* 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 21062* Conditions that determine a ring. (Changed label from isrngd 15391 to isrngod 21062-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 21063* 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 21064 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 21065 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 21066* 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 21067* 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 21068 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 21069 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 21070 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 21071* 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 21072 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 21073 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 21074 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 21075 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 21076 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 21077 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 21078 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 21079 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 21080 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 21081 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 21082 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 21083 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 21084 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 21085 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 21086 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
 
Theoremrngosn 21087 The trivial or zero ring defined on a singleton set  { A } (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.)
 |-  A  e.  _V   =>    |-  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
 >.  e.  RingOps
 
15.2.3  Division Rings
 
Syntaxcdrng 21088 Extend class notation with the class of all division rings.
 class  DivRingOps
 
Definitiondf-drngo 21089* 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.)
 |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g ) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
 
Theoremdrngoi 21090 The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  DivRingOps  ->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp ) )
 
15.2.4  Star Fields
 
Syntaxcsfld 21091 Extend class notation with the class of all star fields.
 class  *-Fld
 
Definitiondf-sfld 21092* Define the class of all star fields, which are all division rings with involutions. (Contributed by NM, 29-Aug-2010.) (New usage is discouraged.)
 |-  *-Fld  =  { <. r ,  n >.  |  (
 r  e.  DivRingOps  /\  n : ran  ( 1st `  r
 ) --> ran  ( 1st `  r )  /\  A. x  e.  dom  n A. y  e.  dom  n ( ( n `  ( x ( 1st `  r
 ) y ) )  =  ( ( n `
  x ) ( 1st `  r )
 ( n `  y
 ) )  /\  ( n `  ( x ( 2nd `  r )
 y ) )  =  ( ( n `  y ) ( 2nd `  r ) ( n `
  x ) ) 
 /\  ( n `  ( n `  x ) )  =  x ) ) }
 
15.2.5  Fields and Rings
 
Syntaxccm2 21093 Extend class notation with a class that adds commutativity to various flavors of rings.
 class  Com2
 
Definitiondf-com2 21094* A device to add commutativity to various sorts of rings. I use  ran  g because I suppose  g has a neutral element and therefore is onto. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
 |- 
 Com2  =  { <. g ,  h >.  |  A. a  e.  ran  g A. b  e.  ran  g ( a h b )  =  ( b h a ) }
 
Theoremiscom2 21095* A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
 |-  ( ( G  e.  A  /\  H  e.  B )  ->  ( <. G ,  H >.  e.  Com2  <->  A. a  e.  ran  G
 A. b  e.  ran  G ( a H b )  =  ( b H a ) ) )
 
Syntaxcfld 21096 Extend class notation with the class of all fields.
 class  Fld
 
Definitiondf-fld 21097 Definition of a field. A field is a commutative division ring. (Contributed by FL, 6-Sep-2009.) (Revised by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.)
 |- 
 Fld  =  ( DivRingOps  i^i  Com2 )
 
Theoremflddivrng 21098 A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  ( K  e.  Fld  ->  K  e.  DivRingOps )
 
Theoremrngon0 21099 The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  RingOps  ->  X  =/= 
 (/) )
 
Theoremrngmgmbs4 21100* The range of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  ( ( G :
 ( X  X.  X )
 --> X  /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )  ->  ran  G  =  X )
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