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Theorem List for Metamath Proof Explorer - 24501-24600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
16.11.16  Groups and related structures
 
Theoremridlideq 24501* If a magma has a left identity element and a right identity element, they are equal. (Contributed by FL, 25-Sep-2011.)
 |-  (
 ( U  e.  X  /\  V  e.  X ) 
 ->  ( A. x  e.  X  ( ( U G x )  =  x  /\  ( x G V )  =  x )  ->  U  =  V ) )
 
Theoremrzrlzreq 24502* If a magma has a left zero element and a right zero element, they are equal. (Contributed by FL, 25-Dec-2011.)
 |-  (
 ( U  e.  X  /\  V  e.  X ) 
 ->  ( A. x  e.  X  ( ( U G x )  =  U  /\  ( x G V )  =  V )  ->  U  =  V ) )
 
Theoremmgmlion 24503* If a magma has a left identity element, it is onto. (Contributed by FL, 25-Sep-2011.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  Magma  /\  U  e.  X  /\  A. x  e.  X  ( U G x )  =  x )  ->  G : ( X  X.  X ) -onto-> X )
 
Theoremrrisgrp 24504  RR is a group for addition. (Contributed by FL, 22-Dec-2008.)
 |-  (  +  |`  ( RR  X.  RR ) )  e.  GrpOp
 
Theoremdmrngrp 24505 A way to express the domain of a group. (Contributed by FL, 9-Jan-2011.)
 |-  ( G  e.  GrpOp  ->  dom  G  =  ( ran  G  X.  ran  G ) )
 
Theorembsmgrli 24506 The base set of an operation with a right and left identity element is not empty. (Contributed by FL, 18-Feb-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  X  =/=  (/) )
 
Theoremsmgrpass2 24507 A semi-group is associative. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  SemiGrp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Theoremablocomgrp 24508 An abelian group is a commutative group. (Contributed by FL, 14-Sep-2010.)
 |-  ( G  e.  AbelOp  ->  G  e.  ( GrpOp  i^i  Com1 )
 )
 
Theoremreacomsmgrp1 24509 Rearrangement of three terms in a commutative semi-group. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  ( SemiGrp  i^i  Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A G ( B G C ) )  =  ( B G ( A G C ) ) )
 
Theoremreacomsmgrp2 24510 Rearrangement of three terms in a commutative semi-group. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  ( SemiGrp  i^i  Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A G ( B G C ) )  =  ( C G ( B G A ) ) )
 
Theoremreacomsmgrp3 24511 Rearrangement of three terms in a commutative semi-group. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  ( SemiGrp  i^i  Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )
 
Theoremreacomsmgrp4 24512 Rearrangement of terms in a commutative semi-group. (Contributed by FL, 18-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  ( SemiGrp  i^i  Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A G ( B G C ) )  =  ( ( C G B ) G A ) )
 
Theoremclfsebs 24513* Closure of a finite composite of elements of the base set of an internal operation. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 26-May-2014.)
 |-  X  =  dom  dom  G   =>    |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  Magma  /\  A. k  e.  ( M ... N ) A  e.  X )  ->  prod_ k  e.  ( M ... N ) G A  e.  X )
 
Theoremclfsebsg 24514* Closure of a finite composite of elements of the base set of an internal operation. (Closed version.) (Contributed by FL, 14-Sep-2010.)
 |-  (
 ( N  e.  ( ZZ>=
 `  M )  /\  G  e.  Magma  /\  A. k  e.  ( M ... N ) A  e.  dom 
 dom  G )  ->  prod_ k  e.  ( M ... N ) G A  e.  dom  dom 
 G )
 
Theoremclfsebsr 24515* Closure of a finite composite of elements of the base set of an internal operation. (Case of a magma with a right and left identity element.) (Contributed by FL, 14-Sep-2010.)
 |-  X  =  ran  G   =>    |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  ( Magma  i^i  ExId  )  /\  A. k  e.  ( M ... N ) A  e.  X )  ->  prod_ k  e.  ( M ... N ) G A  e.  X )
 
Theoremfincmpzer 24516* Finite composite of identity elements. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2014.)
 |-  U  =  (GId `  G )   =>    |-  (
 ( N  e.  ( ZZ>=
 `  M )  /\  G  e.  ( Magma  i^i  ExId  ) )  ->  prod_ k  e.  ( M ... N ) G U  =  U )
 
Theoremresgcom 24517 Rearrangement of four terms in a commutative, associative magma. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  ( SemiGrp  i^i  Com1 )  /\  ( ( 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 ) ) )
 
Theoremfprodadd 24518* The composite of two finite composites. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 30-May-2014.)
 |-  X  =  dom  dom  G   =>    |-  ( ( N  e.  ( ZZ>= `  M )  /\  A. k  e.  ( M ... N ) ( A  e.  X  /\  B  e.  X )  /\  G  e.  ( SemiGrp  i^i  Com1 ) )  ->  prod_ k  e.  ( M ... N ) G ( A G B )  =  ( prod_ k  e.  ( M
 ... N ) G A G prod_ k  e.  ( M ... N ) G B ) )
 
Theoremabloinvop 24519 The inverse of the abelian group operation doesn't reverse the arguments. cf grpoinvop 20738. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  (
 ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `
  A ) G ( N `  B ) ) )
 
Theoremisppm 24520 The sequence of partial composites of elements of a magma is a sequence of elements of this magma. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 30-May-2014.)
 |-  X  =  dom  dom  G   &    |-  Z  =  (
 ZZ>= `  M )   =>    |-  ( ( G  e.  Magma  /\  M  e.  ZZ  /\  F : Z --> X )  ->  seq  M ( G ,  F ) : Z --> X )
 
Theoremseqzp2 24521 Value of the arbitrary-based recursive sequence builder at a successor value when the operation  G is associative. Compare with seqp1 10939. (Contributed by FL, 24-Jan-2010.)
 |-  X  =  dom  dom  G   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  G  e. 
 SemiGrp   &    |-  F : Z --> X   =>    |-  ( N  e.  Z  ->  (  seq  M ( G ,  F ) `
  ( N  +  1 ) )  =  ( ( F `  M ) G ( 
 seq  ( M  +  1 ) ( G ,  F ) `  ( N  +  1
 ) ) ) )
 
Theoremmndoisass 24522 A monoid is associative. (Contributed by FL, 2-Nov-2009.)
 |-  ( G  e. MndOp  ->  G  e.  Ass )
 
Theoremmndoid 24523* A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   =>    |-  ( G  e. MndOp  ->  E. x  e.  X  A. y  e.  X  (
 ( x G y )  =  y  /\  ( y G x )  =  y ) )
 
Theoremmndoio 24524 A monoid is an internal operation. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   =>    |-  ( G  e. MndOp  ->  G : ( X  X.  X ) --> X )
 
Theoremmndoass 24525* A monoid is associative. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   =>    |-  ( G  e. MndOp  ->  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )
 
Theoremmndoass2 24526 A monoid is associative. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  ran  G   =>    |-  ( ( G  e. MndOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Syntaxclsg 24527 Extend class notation with monoid exponentiation.
 class  ^ md
 
Definitiondf-expsg 24528* Define the exponentiation of an element of a monoid. Experimental. I define exponentiation on a monoid (and not on a semi-group or a magma ) because I need an identity element for the basis hypothesis and associativity for interesting properties such as the composite of two exponentiated elements.  ZZ is used in df-gx 20692 here I used  NN0 because the inverse is not defined in a monoid. (Contributed by FL, 2-Sep-2013.)
 |-  ^ md  =  ( g  e. MndOp  |->  ( x  e.  ran  g ,  y  e.  NN0  |->  if (
 y  =  0 ,  (GId `  g ) ,  (  seq  1 ( g ,  ( NN 
 x.  { x } )
 ) `  y )
 ) ) )
 
Theoremexpmiz 24529 Value of a member of a monoid (or any other structure where GId is defined ) raised to the 0th power. (Contributed by FL, 12-Dec-2009.)
 |-  F  =  ( rec ( ( a  e.  _V  |->  ( a G A ) ) ,  (GId `  G ) )  |`  om )   =>    |-  ( F `  (/) )  =  (GId `  G )
 
Theoremexpm 24530* Exponentiation of a monoid. The value at a successor. What I am calculating is  ( A ( ^ md `  G ) N ). (Contributed by FL, 12-Dec-2009.)
 |-  F  =  ( rec ( ( a  e.  _V  |->  ( a G A ) ) ,  (GId `  G ) )  |`  om )   =>    |-  ( N  e.  om  ->  ( F `  suc  N )  =  ( ( F `  N ) G A ) )
 
Theoremexpus 24531* The exponentiation of a member of a monoid belongs to the underlying set. (Contributed by FL, 12-Dec-2009.)
 |-  F  =  ( rec ( ( a  e.  _V  |->  ( a G A ) ) ,  (GId `  G ) )  |`  om )   &    |-  G  e. MndOp   &    |-  A  e.  X   &    |-  X  =  ran  G   =>    |-  ( x  e.  om  ->  ( F `  x )  e.  X )
 
Theoremmgmrddd 24532 The range of the domain of a magma equals the domain of the domain. (Contributed by FL, 17-May-2010.)
 |-  ( G  e.  Magma  ->  ran  dom  G  =  dom  dom  G )
 
Theoremunsgrp 24533 The underlying set of a group is a set. (Contributed by FL, 17-May-2010.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  X  e.  _V )
 
Theoremsymgfo 24534 The operation of a symetry group is onto. (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  X  =  ( Base `  G )   &    |-  P  =  ( +g  `  G )   =>    |-  ( A  e.  V  ->  P : ( X  X.  X ) -onto-> X )
 
Theoremgapm2 24535 The action of a particular group element is a permutation of the base set. gapm 14595 expressed with the currying operator. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  X  =  ( Base `  G )   =>    |-  (
 (  .(+)  e.  ( G 
 GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  ( ( cur1 `  .(+)  ) `  A ) : Y -1-1-onto-> Y )
 
Theoremrngapm 24536 The range of the action of a particular group element equals the range of the action. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  X  =  ( Base `  G )   =>    |-  (
 (  .(+)  e.  ( G 
 GrpAct  Y )  /\  ( A  e.  X  /\  Y  =/=  (/) ) )  ->  ran  ( ( cur1 `  .(+)  ) `  A )  =  ran  .(+) 
 )
 
Theoremfnopabco2b 24537* Composition of a function with a function abstraction. Adapted from fnopabco 25554. (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  G  =  ( x  e.  A  |->  ( H `  B ) )   =>    |-  ( ( A. x  e.  A  B  e.  C  /\  H  Fn  C ) 
 ->  G  =  ( H  o.  F ) )
 
Theoremcurgrpact 24538 The currying of a group action is a group homomorphism between the group  G and the symmetry group  ( SymGrp `  Y
). (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
 |-  (
 (  .(+)  e.  ( G 
 GrpAct  Y )  /\  Y  =/= 
 (/) )  ->  ( cur1 `  .(+)  )  e.  ( G  GrpHom  ( SymGrp `  Y ) ) )
 
Theoremgrpodivone 24539 "Division" by the neutral element of a group. (Contributed by FL, 21-Jun-2010.)
 |-  X  =  ran  G   &    |-  D  =  ( 
 /g  `  G )   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A D U )  =  A )
 
Theoremgrpodivfo 24540 A "division" maps onto the group's underlying set. (Contributed by FL, 21-Jun-2010.)
 |-  X  =  ran  G   &    |-  D  =  ( 
 /g  `  G )   =>    |-  ( G  e.  GrpOp  ->  D : ( X  X.  X ) -onto-> X )
 
Theoremgrpodrcan 24541 Right cancellation law for group "subtraction" (or "division"). (Contributed by FL, 14-Sep-2010.)
 |-  X  =  ran  G   &    |-  D  =  ( 
 /g  `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D C )  =  ( B D C ) 
 <->  A  =  B ) )
 
Theoremgrpodlcan 24542 Left cancellation law for group "subtraction" (or "division"). (Contributed by FL, 14-Sep-2010.)
 |-  X  =  ran  G   &    |-  D  =  ( 
 /g  `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( C D A )  =  ( C D B ) 
 <->  A  =  B ) )
 
Theoremgrpodivzer 24543 Condition for a "subtraction" (or "division") value to be equal to the identity element. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  D  =  ( 
 /g  `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A D B )  =  U  <->  A  =  B ) )
 
Theoremfprodneg 24544* The inverse of a finite composite in the case of an abelian group. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 30-May-2014.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  (
 ( M 2  e.  ( ZZ>= `  M1 )  /\  A. k  e.  ( M1 ...
 M 2 ) A  e.  X  /\  G  e.  AbelOp )  ->  ( N `  prod_ k  e.  ( M1
 ... M 2 ) G A )  =  prod_ k  e.  ( M1 ... M 2 ) G ( N `  A ) )
 
Theoremfprodsub 24545* The "difference" (or "quotient") of two finite composites. (Contributed by FL, 14-Sep-2010.) (Revised by Mario Carneiro, 30-May-2014.)
 |-  X  =  ran  G   &    |-  D  =  ( 
 /g  `  G )   =>    |-  (
 ( N  e.  ( ZZ>=
 `  M )  /\  A. k  e.  ( M
 ... N ) ( A  e.  X  /\  B  e.  X )  /\  G  e.  AbelOp )  ->  prod_ k  e.  ( M
 ... N ) G ( A D B )  =  ( prod_ k  e.  ( M ... N ) G A D prod_ k  e.  ( M
 ... N ) G B ) )
 
Theoremclfsebs3 24546* Closure of a finite composite of elements of the base set of an internal operation. (When the operation is a monoid.) (Contributed by FL, 22-Nov-2010.)
 |-  X  =  ran  G   =>    |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e. MndOp  /\  A. k  e.  ( M ... N ) A  e.  X )  ->  prod_ k  e.  ( M ... N ) G A  e.  X )
 
Theoremclfsebs4 24547* Closure of a finite composite of elements of the base set of an internal operation. (When the operation is a group.) (Contributed by FL, 22-Nov-2010.)
 |-  X  =  ran  G   =>    |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  GrpOp  /\  A. k  e.  ( M ... N ) A  e.  X )  ->  prod_ k  e.  ( M ... N ) G A  e.  X )
 
Theoremclfsebs5 24548* Closure of a finite composite of elements of the base set of an internal operation. (When the operation is an abelian group.) (Contributed by FL, 22-Nov-2010.)
 |-  X  =  ran  G   =>    |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  AbelOp  /\  A. k  e.  ( M ... N ) A  e.  X )  ->  prod_ k  e.  ( M ... N ) G A  e.  X )
 
16.11.17  Free structures
 
Syntaxcsubsmg 24549 Extend class notation to include the class of subsemigroups.
 class  SubSemiGrp
 
Definitiondf-subsmg 24550* Define the set of subsemigroups of  g. Experimental. (Contributed by FL, 2-Sep-2013.)
 |-  SubSemiGrp  =  (
 g  e.  SemiGrp  |->  { h  e. 
 SemiGrp  |  h  C_  g } )
 
Syntaxcsbsgrg 24551 Extend class notation with a function that returns the subsemigroup of a group generated by a set.
 class  subSemiGrpGen
 
Definitiondf-sggen 24552* the subsemigroup of  g generated by  x. Experimental. (Contributed by FL, 15-Jul-2012.)
 |-  subSemiGrpGen  =  ( g  e.  SemiGrp ,  x  e.  ~P dom  dom  g  |->  |^| { h  e.  ( SubSemiGrp `  g )  |  x  C_  dom  dom  h } )
 
Syntaxcsmhom 24553 Extend class notation to include the class of semigroup homomorphisms.
 class  SemiGrpHom
 
Definitiondf-gsmhom 24554* Define the set of semigroup homomorphisms from  g to  h. Experimental. (Contributed by FL, 15-Jul-2012.)
 |-  SemiGrpHom  =  (
 g  e.  SemiGrp ,  h  e. 
 SemiGrp 
 |->  { f  |  ( f : dom  dom  g
 --> dom  dom  h  /\  A. x  e.  dom  dom  g A. y  e.  dom  dom  g ( ( f `
  x ) h ( f `  y
 ) )  =  ( f `  ( x g y ) ) ) } )
 
Syntaxcfsm 24555 Extend class notation to include the class of free semigroup.
 class  FreeSemiGrp
 
Definitiondf-frsmgrp 24556* Definition of a free semigroup. The definition is somewhat cryptic. Let's say it guarantees the elements of the semigroup can be decomposed into elementary components and that the decomposition is unique. As a consequence you define the elements of the semigroup with nice recursive function  h by giving the value  h ( x ) for every elementary component  x and the recursive equation  h ( x  +  y )  =  h ( x )  +  h ( y ). This is not true in every semigroup. For intance if you take the semigroup of strings generated by the elementary components "ab", "c", "a", "bc", the string "abc" is equal to "ab"  + "c" or to "a"  + "bc" and those beautiful recursive function can't exist. (See a nice explanation in Gallier p. 20.) Experimental. (Contributed by FL, 15-Jul-2012.)
 |-  FreeSemiGrp  =  { <. g ,  x >.  |  (
 g  =  (  subSemiGrpGen  `  <. g ,  x >. )  /\  A. h  e. 
 SemiGrp  A. a  e.  ( dom  dom  h  ^m  x ) E. u  e.  ( SemiGrpHom ` 
 <. g ,  h >. ) ( u  |`  x )  =  a ) }
 
16.11.18  Translations
 
Theoremtrdom2 24557* The domain of a right translation. The term  A is a constant:  x is not present. (Contributed by FL, 21-Jun-2010.)
 |-  F  =  ( x  e.  X  |->  ( x G A ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  dom  F  =  X )
 
Theoremtrset 24558* A right translation is a set. (Contributed by FL, 19-Sep-2010.)
 |-  F  =  ( x  e.  X  |->  ( x G A ) )   &    |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  F  e.  _V )
 
Theoremtrran2 24559* The range of a right translation. The term  A is a constant:  x is not present. (Contributed by FL, 21-Jun-2010.)
 |-  F  =  ( x  e.  X  |->  ( x G A ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ran  F  =  X )
 
Theoremtrooo 24560* A right translation is a bijection. The term  A is a constant. (Contributed by FL, 21-Jun-2010.)
 |-  F  =  ( x  e.  X  |->  ( x G A ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  F : X -1-1-onto-> X )
 
Theoremtrinv 24561* The converse of a right translation. The term  A is a constant. (Contributed by FL, 21-Jun-2010.)
 |-  F  =  ( x  e.  X  |->  ( x G A ) )   &    |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  `' F  =  ( x  e.  X  |->  ( x D A ) ) )
 
Theoremcmprtr 24562* Composite of two right translations. The terms  A and 
B are constant. Don't use. See cmprtr2 24563. (Contributed by FL, 17-Oct-2010.)
 |-  F  =  ( x  e.  X  |->  ( x G A ) )   &    |-  X  =  ran  G   &    |-  H  =  ( x  e.  X  |->  ( x G B ) )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( x  e.  X  |->  ( x G ( B G A ) ) ) )
 
Theoremcmprtr2 24563* Composite of two right translations. (cmprtr 24562 with a distinct variable condition relaxed.) (Contributed by FL, 1-Jan-2011.)
 |-  F  =  ( x  e.  X  |->  ( x G A ) )   &    |-  X  =  ran  G   &    |-  H  =  ( x  e.  X  |->  ( x G B ) )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( x  e.  X  |->  ( x G ( B G A ) ) ) )
 
Theoremimtr 24564* The image of a set through a translation. (Contributed by FL, 30-Dec-2010.)
 |-  F  =  ( x  e.  X  |->  ( x G A ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  ~P X )  ->  ( F " B )  =  {
 a  |  E. u  e.  B  a  =  ( F `  u ) } )
 
Theoremprsubrtr 24565* The product of a subset  B of  X by an element of  X is the image of  B by a right translation. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  ( x  e.  X  |->  ( x G A ) )   &    |-  X  =  ran  G   &    |-  H  =  ( cset `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  ~P X )  ->  ( B H { A } )  =  ( F " B ) )
 
Theoremcaytr 24566* "It follows that if the entire group is multiplied by any one of the symbols, either as further or nearer factor, the effect is simply to reproduce the group... ." Cayley, On the theory of groups, as depending on the symbolic equation th^n = 1, 1854. (it is the original paper where the axiomatic definition of a group was given for the first time.) (Contributed by FL, 15-Oct-2012.)
 |-  F  =  ( x  e.  X  |->  ( x G A ) )   &    |-  X  =  ran  G   &    |-  H  =  ( cset `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( X H { A } )  =  X )
 
Theoremltrdom 24567* The domain of a left translation. The term  A is a constant. (Contributed by FL, 26-Apr-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  dom  F  =  X )
 
Theoremltrset 24568* A left translation is a set. (Contributed by FL, 28-Apr-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  F  e.  _V )
 
Theoremltrran2 24569* The range of a left translation. The term  A is a constant. (Contributed by FL, 28-Apr-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ran  F  =  X )
 
Theoremltrooo 24570* A left translation is a bijection. The term  A is a constant. (Contributed by FL, 29-Apr-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  F : X -1-1-onto-> X )
 
Theoremltrcmp 24571* Left translation expressed as a composite. (Contributed by FL, 3-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  F  =  ( G  o.  ( x  e.  X  |->  <. A ,  x >. ) ) )
 
Theoremltrinvlem 24572* The converse of a left translation. The term  A is a constant. (Contributed by FL, 30-Apr-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   &    |-  N  =  ( inv `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  `' F  =  ( x  e.  X  |->  ( ( N `  A ) G x ) ) )
 
Theoremcmpltr2 24573* Composite of two left translations. The terms  A and 
B are constant. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   &    |-  H  =  ( x  e.  X  |->  ( B G x ) )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( x  e.  X  |->  ( ( A G B ) G x ) ) )
 
Theoremcmpltr 24574* Composite of two left translations. The terms  A and 
B are constant. Don't use. See cmpltr2 24573. (Contributed by FL, 2-Jul-2012.) (Revised by Mario Carneiro, 2-Jun-2014.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   &    |-  H  =  ( x  e.  X  |->  ( B G x ) )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( x  e.  X  |->  ( ( A G B ) G x ) ) )
 
Theoremcmperltr 24575* A right and left translation expressed as a composite. Note that  x and  y can't be the same. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )   &    |-  E  =  ( y  e.  X  |->  ( A G y ) )   &    |-  H  =  ( x  e.  X  |->  ( x G B ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  F  =  ( E  o.  H ) )
 
Theoremcmprltr 24576* Composite of two right and left translations. Note that  x and  y can't be the same. See cmprltr2 24577 for a more general version. (Contributed by FL, 2-Jul-2012.) (Proof shortened by Mario Carneiro, 26-Jul-2014.)
 |-  F  =  ( y  e.  X  |->  ( ( A G y ) G C ) )   &    |-  E  =  ( x  e.  X  |->  ( ( B G x ) G D ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( F  o.  E )  =  ( x  e.  X  |->  ( ( ( A G B ) G x ) G ( D G C ) ) ) )
 
Theoremcmprltr2 24577* Composite of two right and left translations. No restriction:  x and  z can be equal. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( z  e.  X  |->  ( ( A G z ) G C ) )   &    |-  E  =  ( x  e.  X  |->  ( ( B G x ) G D ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( F  o.  E )  =  ( x  e.  X  |->  ( ( ( A G B ) G x ) G ( D G C ) ) ) )
 
Theoremrltrdom 24578* The domain of a right and left translation. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )   &    |-  X  =  ran  G   =>    |-  dom 
 F  =  X
 
Theoremrltrset 24579* A right and left translation is a set. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )   &    |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  F  e.  _V )
 
Theoremrltrran 24580* The range of a right and left translation. Note that  A and  B are constant. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ran  F  =  X )
 
Theoremrltrooo 24581* A right and left translation is a bijection. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  F : X -1-1-onto-> X )
 
16.11.19  Fields and Rings
 
Theoremcom2i 24582* Property of a commutative structure with two operation. (Contributed by FL, 14-Feb-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Com2  ->  A. a  e.  X  A. b  e.  X  (
 a H b )  =  ( b H a ) )
 
Theoremrngmgmbs3 24583* The domain of the first variable of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.)
 |-  (
 ( G : ( X  X.  X ) --> X  /\  E. u  e.  X  A. x  e.  X  ( ( x G u )  =  x  /\  ( u G x )  =  x ) )  ->  dom  dom  G  =  X )
 
Theoremrngodmdmrn 24584 In a unital ring the range of the multiplication equals the domain of the first variable. (Contributed by FL, 24-Jan-2010.)
 |-  H  =  ( 2nd `  R )   =>    |-  ( R  e.  RingOps  ->  dom  dom  H  =  ran  H )
 
Theoremrngodmeqrn 24585 In a unital ring the domain of the first operand of the addition equals the domain of the second operand of the addition. (Contributed by FL, 11-Feb-2010.)
 |-  G  =  ( 1st `  R )   =>    |-  ( R  e.  RingOps  ->  dom  dom  G  =  ran  dom 
 G )
 
Theoremununr 24586* The unit of a unital ring is unique. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 23-Dec-2013.)
 |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  ( 1st `  R )   =>    |-  ( R  e.  RingOps  ->  E! x  e.  X  A. y  e.  X  ( ( x H y )  =  y  /\  ( y H x )  =  y ) )
 
Theoremrngoinvcl 24587 The additive inverse of a unital ring element pertains to the unital ring. (Contributed by FL, 18-Apr-2010.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  (
 ( <. G ,  H >.  e.  RingOps  /\  A  e.  X )  ->  ( N `
  A )  e.  X )
 
Theoremmultinv 24588 Multiplication by an additive inverse. (Contributed by FL, 2-Sep-2009.)
 |-  X  =  ran  G   &    |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( inv `  G ) `  A ) H B )  =  ( ( inv `  G ) `  ( A H B ) ) )
 
Theoremmultinvb 24589 Multiplication by an additive inverse. (Contributed by FL, 6-Sep-2009.)
 |-  X  =  ran  G   &    |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H ( ( inv `  G ) `  B ) )  =  ( ( inv `  G ) `  ( A H B ) ) )
 
Theoremmult2inv 24590 Multiplication of two additive inverses. (Contributed by FL, 6-Sep-2009.)
 |-  X  =  ran  G   &    |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( inv `  G ) `  A ) H ( ( inv `  G ) `  B ) )  =  ( A H B ) )
 
Theoremrngounval2 24591* The value of the unit of a ring. (Contributed by FL, 12-Feb-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  RingOps  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
 
TheoremisfldOLD 24592* The predicate "is a field". (Contributed by FL, 6-Sep-2009.)
 |-  (
 ( G  e.  A  /\  H  e.  B ) 
 ->  ( <. G ,  H >.  e.  Fld  <->  ( <. G ,  H >.  e.  DivRingOps  /\  A. x  e.  ran  G A. y  e.  ran  G ( x H y )  =  ( y H x ) ) ) )
 
Theoremfldi 24593* The "axioms" of a field. (Contributed by FL, 15-Sep-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  Fld  ->  ( ( 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 ) )  /\  ( ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
 ) ) )  e. 
 GrpOp  /\  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) ) )
 
Theoremfldax1 24594 1st "axiom" of a field. The addition is an abelian group. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   =>    |-  ( R  e.  Fld  ->  G  e.  AbelOp )
 
Theoremfldax2 24595 2nd "axiom" of a field. The multiplication is an internal operation. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Fld  ->  H : ( X  X.  X ) --> X )
 
Theoremfldax3 24596* 3rd "axiom" of a field. The multiplication is associative. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Fld  ->  A. x  e.  X  A. y  e.  X  A. z  e.  X  (
 ( x H y ) H z )  =  ( x H ( y H z ) ) )
 
Theoremfldax4 24597* 4th "axiom" of a field. The multiplication is distributive. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Fld  ->  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) ) )
 
Theoremfldax5 24598* 5th "axiom" of a field. Existence of a neutral element. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Fld  ->  E. x  e.  X  A. y  e.  X  ( y H x )  =  y )
 
Theoremfldax6 24599 6th "axiom" of a field. The multiplication is a group on the underlying set deprived from zero. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  Fld  ->  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
 ) ) )  e. 
 GrpOp )
 
Theoremfldax7 24600* 7th "axiom" of a field. The multiplication is commutative. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Fld  ->  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) )
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