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Theorem List for Metamath Proof Explorer - 24701-24800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgapm2 24701 The action of a particular group element is a permutation of the base set. gapm 14687 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 24702 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 24703* Composition of a function with a function abstraction. Adapted from fnopabco 25720. (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 24704 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 24705 "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 24706 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 24707 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 24708 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 24709 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 24710* 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 24711* 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 24712* 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 24713* 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 24714* 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 )
 
16.12.17  Free structures
 
Syntaxcsubsmg 24715 Extend class notation to include the class of subsemigroups.
 class  SubSemiGrp
 
Definitiondf-subsmg 24716* 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 24717 Extend class notation with a function that returns the subsemigroup of a group generated by a set.
 class  subSemiGrpGen
 
Definitiondf-sggen 24718* 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 24719 Extend class notation to include the class of semigroup homomorphisms.
 class  SemiGrpHom
 
Definitiondf-gsmhom 24720* 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 24721 Extend class notation to include the class of free semigroup.
 class  FreeSemiGrp
 
Definitiondf-frsmgrp 24722* 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 ) }
 
16.12.18  Translations
 
Theoremtrdom2 24723* 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 24724* 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 24725* 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 24726* 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 24727* 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 24728* Composite of two right translations. The terms  A and 
B are constant. Don't use. See cmprtr2 24729. (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 24729* Composite of two right translations. (cmprtr 24728 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 24730* 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 24731* 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 24732* "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 24733* 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 24734* 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 24735* 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 24736* 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 24737* 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 24738* 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 24739* 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 24740* Composite of two left translations. The terms  A and 
B are constant. Don't use. See cmpltr2 24739. (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 24741* 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 24742* Composite of two right and left translations. Note that  x and  y can't be the same. See cmprltr2 24743 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 24743* 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 24744* 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 24745* 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 24746* 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 24747* 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 )
 
16.12.19  Fields and Rings
 
Theoremcom2i 24748* 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 24749* 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 24750 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 24751 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 24752* 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 24753 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 24754 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 24755 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 24756 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 24757* 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 24758* 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 24759* 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 24760 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 24761 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 24762* 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 24763* 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 24764* 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 24765 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 24766* 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 ) )
 
Theoremzrfld 24767 The zero ring is not a field. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  A  e.  _V   =>    |- 
 -.  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
 >.  e.  Fld
 
Theoremzerdivemp1 24768* In a unitary ring a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  ( X 
 \  Z )  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) )
 
Theoremrngoridfz 24769* In a unitary ring a left invertible element is different from zero iff  1  =/=  0. (Contributed by FL, 18-Apr-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( A  =/=  Z  <->  U  =/=  Z ) )
 
Theoremzintdom 24770  ZZ is a commutative ring. (Contributed by FL, 18-Apr-2010.)
 |-  <.  +  ,  x.  >.  e.  ( RingOps  i^i  Com2 )
 
Syntaxctofld 24771 Extend class notation with the class of all totally ordered fields.
 class  Tofld
 
Definitiondf-tofld 24772* Definition of a totally ordered field. Experimental. (Contributed by FL, 27-Jun-2011.)
 |-  Tofld  =  { <.
 <. g ,  h >. ,  r >.  |  ( <. g ,  h >.  e. 
 Fld  /\  ( r  e.  TosetRel  /\  U. U. r  = 
 ran  g )  /\  A. x  e.  U. U. r A. y  e.  U. U. r A. z  e. 
 U. U. r ( ( x r y  ->  ( x g z ) r ( y g z ) )  /\  ( (GId `  g )
 r z  ->  ( x h z ) r ( y h z ) ) ) ) }
 
Syntaxczerodiv 24773 Extend class notation with the class of all the zero divisors.
 class  zeroDiv
 
Definitiondf-zd 24774* Definition of the zero divisors of a ring. Experimental. (Contributed by FL, 27-Jun-2011.)
 |-  zeroDiv  =  { <. r ,  y >.  |  y  =  { a  e.  ran  ( 1st `  r
 )  |  ( a  =/=  (GId `  ( 1st `  r ) ) 
 /\  E. b  e.  ran  ( 1st `  r )
 ( b  =/=  (GId `  ( 1st `  r
 ) )  /\  (
 a ( 2nd `  r
 ) b )  =  (GId `  ( 1st `  r ) ) ) ) } }
 
16.12.20  Ideals
 
Syntaxcidln 24775 Extend class notation with the class of ideals.
 class IdlNEW
 
Definitiondf-idlNEW 24776* Define the class of (two-sided) ideals of a ring  R. A subset of  R is an ideal if it contains  0, is closed under addition, and is closed under multiplication on either side by any element of  R. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
 |- IdlNEW  =  ( r  e.  Ring  |->  { i  e.  ~P ( Base `  r
 )  |  ( ( 0g `  r )  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x ( +g  `  r
 ) y )  e.  i  /\  A. z  e.  ( Base `  r )
 ( ( z ( .r `  r ) x )  e.  i  /\  ( x ( .r
 `  r ) z )  e.  i ) ) ) } )
 
TheoremidlvalNEW 24777* The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
 |-  P  =  ( +g  `  R )   &    |-  T  =  ( .r
 `  R )   &    |-  B  =  ( Base `  R )   &    |-  Z  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  (IdlNEW `  R )  =  { i  e.  ~P B  |  ( Z  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x P y )  e.  i  /\  A. z  e.  B  ( ( z T x )  e.  i  /\  ( x T z )  e.  i ) ) ) } )
 
TheoremisidlNEW 24778* The predicate "is an ideal of the ring  R." (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
 |-  P  =  ( +g  `  R )   &    |-  T  =  ( .r
 `  R )   &    |-  B  =  ( Base `  R )   &    |-  Z  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( I  e.  (IdlNEW `  R )  <->  ( I  C_  B  /\  Z  e.  I  /\  A. x  e.  I  ( A. y  e.  I  ( x P y )  e.  I  /\  A. z  e.  B  (
 ( z T x )  e.  I  /\  ( x T z )  e.  I ) ) ) ) )
 
16.12.21  Generic modules and vector spaces (New Structure builder)
 
Syntaxcact 24779 Extend class notation to include actions.
 class  Action
 
Definitiondf-act 24780* Definition of an action law. The action is the function ( k ^m ( v ^m v ). Definitions equivalent through currying. (Contributed by FL, 24-Dec-2013.)
 |-  Action  =  {
 f  |  E. k E. v E. s ( ( k  =  (
 Base `  (Scalar `  f
 ) )  /\  v  =  ( Base `  f )  /\  s  =  ( .s `  f ) ) 
 /\  A. r  e.  k  A. w  e.  v  ( r s w )  e.  v ) }
 
16.12.22  Generic modules and vector spaces
 
Syntaxcvec 24781 Extend class notation with the class of all generic vector spaces and modules.
 class  Vec
 
Definitiondf-vec 24782* Definition of a vector space (
<. g ,  h >. is a field ), or of a module ( <. g ,  h >. is a ring ). (Contributed by FL, 12-Jul-2010.)
 |-  Vec  =  { z  |  E. g E. h E. a E. b ( z  = 
 <. <. g ,  h >. ,  <. a ,  b >.
 >.  /\  ( a  e.  AbelOp 
 /\  b : ( ran  g  X.  ran  a ) --> ran  a  /\  A. u  e.  ran  a ( ( (GId `  h ) b u )  =  u  /\  A. x  e.  ran  g
 ( A. v  e.  ran  a ( x b ( u a v ) )  =  ( ( x b u ) a ( x b v ) ) 
 /\  A. y  e.  ran  g ( ( ( x g y ) b u )  =  ( ( x b u ) a ( y b u ) )  /\  ( ( x h y ) b u )  =  ( x b ( y b u ) ) ) ) ) ) ) }
 
Theoremvecval1b 24783* The predicate "is a vector space" or "is a module". (Contributed by FL, 12-Jul-2010.)
 |-  X  =  ran  G   &    |-  W  =  ran  A   =>    |-  ( ( ( G  e.  M  /\  H  e.  N  /\  A  e.  O )  /\  B  e.  P )  ->  ( <. G ,  H >.  Vec  <. A ,  B >. 
 <->  ( A  e.  AbelOp  /\  B : ( X  X.  W ) --> W  /\  A. u  e.  W  ( ( (GId `  H ) B u )  =  u  /\  A. x  e.  X  ( A. v  e.  W  ( x B ( u A v ) )  =  ( ( x B u ) A ( x B v ) ) 
 /\  A. y  e.  X  ( ( ( x G y ) B u )  =  ( ( x B u ) A ( y B u ) ) 
 /\  ( ( x H y ) B u )  =  ( x B ( y B u ) ) ) ) ) ) ) )
 
Theoremvecval3b 24784* The "axioms" of a vector space or module. (Contributed by FL, 12-Jul-2010.)
 |-  X  =  ran  G   &    |-  G  =  ( 1st `  ( 1st `  R ) )   &    |-  H  =  ( 2nd `  ( 1st `  R ) )   &    |-  A  =  ( 1st `  ( 2nd `  R ) )   &    |-  B  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  A   =>    |-  ( R  e.  Vec  ->  ( A  e.  AbelOp  /\  B : ( X  X.  W ) --> W  /\  A. u  e.  W  ( ( (GId `  H ) B u )  =  u  /\  A. x  e.  X  ( A. v  e.  W  ( x B ( u A v ) )  =  ( ( x B u ) A ( x B v ) ) 
 /\  A. y  e.  X  ( ( ( x G y ) B u )  =  ( ( x B u ) A ( y B u ) ) 
 /\  ( ( x H y ) B u )  =  ( x B ( y B u ) ) ) ) ) ) )
 
Theoremvecax1 24785 1st "axiom" of a vector space or module. The vector addition is an abelian group. (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by FL, 14-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   =>    |-  ( R  e.  Vec  ->  + w  e.  AbelOp )
 
Theoremvecax2 24786 2nd "axiom" of a vector space or module. Domain, codomain and functionality of the multiplication of a vector by a scalar. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   =>    |-  ( R  e.  Vec 
 ->  . w : ( X  X.  W ) --> W )
 
Theoremvecax3 24787* 3rd "axiom" of a vector space or module. Multiplication by 1. (Contributed by FL, 13-Sep-2010.)
 |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   &    |-  1 t  =  (GId `  ( 2nd `  ( 1st `  R ) ) )   =>    |-  ( R  e.  Vec 
 ->  A. u  e.  W  ( 1 t . w u )  =  u )
 
Theoremvecax4 24788* 4th "axiom" of a vector space or module. Multiplication by a scalar distributes over vector addition. (Contributed by FL, 13-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   =>    |-  ( R  e.  Vec  ->  A. u  e.  W  A. x  e.  X  A. v  e.  W  ( x . w ( u + w v ) )  =  ( ( x . w u ) + w ( x . w v ) ) )
 
Theoremvecax5 24789* 5th "axiom" of a vector space or module. Multiplication by a sum of scalars. (Contributed by FL, 13-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  + t  =  ( 1st `  ( 1st `  R ) )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   =>    |-  ( R  e.  Vec 
 ->  A. u  e.  W  A. x  e.  X  A. y  e.  X  (
 ( x + t
 y ) . w u )  =  (
 ( x . w u ) + w
 ( y . w u ) ) )
 
Theoremvecax6 24790* 6th "axiom" of a vector space or module. Relation between scalar multiplication and vector multiplication. (Contributed by FL, 13-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  . t  =  ( 2nd `  ( 1st `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   =>    |-  ( R  e.  Vec  ->  A. u  e.  W  A. x  e.  X  A. y  e.  X  (
 ( x . t
 y ) . w u )  =  ( x . w ( y . w u ) ) )
 
Theoremvecax5b 24791 Multiplication by a sum of scalars. (Contributed by FL, 13-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  + t  =  ( 1st `  ( 1st `  R ) )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   =>    |-  ( ( R  e.  Vec  /\  ( U  e.  W  /\  A  e.  X  /\  B  e.  X ) )  ->  ( ( A + t B ) . w U )  =  (
 ( A . w U ) + w
 ( B . w U ) ) )
 
Theoremcladdinvvec 24792 Closure of the additive inverse of a vector. (Contributed by FL, 13-Sep-2010.)
 |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   &    |-  ~ w  =  ( inv `  ( 1st `  ( 2nd `  R ) ) )   =>    |-  ( ( R  e.  Vec  /\  U  e.  W ) 
 ->  ( ~ w `  U )  e.  W )
 
Theoremvec2inv 24793 Double inverse law for vector additive inverse. (Contributed by FL, 13-Sep-2010.)
 |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   &    |-  ~ w  =  ( inv `  ( 1st `  ( 2nd `  R ) ) )   =>    |-  ( ( R  e.  Vec  /\  U  e.  W ) 
 ->  ( ~ w `  ( ~ w `  U ) )  =  U )
 
Theoremsum2vv 24794 The sum of two vectors is a vector. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  V1  e.  W  /\  V 2  e.  W ) 
 ->  ( V1 + w V 2 )  e.  W )
 
Theoremaddnull1 24795 Addition of the null vector. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   &    |-  0 w  =  (GId `  + w )   =>    |-  (
 ( R  e.  Vec  /\  U  e.  W ) 
 ->  ( U + w
 0 w )  =  U )
 
Theoremaddnull2 24796 Addition of the null vector. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   &    |-  0 w  =  (GId `  + w )   =>    |-  (
 ( R  e.  Vec  /\  U  e.  W ) 
 ->  ( 0 w + w U )  =  U )
 
Theoremaddvecass 24797 The addition of vectors is associative. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W ) )  ->  ( V1 + w ( V 2 + w V 3 )
 )  =  ( (
 V1 + w V 2 ) + w V 3 ) )
 
Theoremaddvecom 24798 The addition of vectors is associative. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W ) )  ->  ( V1 + w V 2 )  =  ( V 2 + w V1 ) )
 
Theoreminvaddvec 24799 Additive inverse of a sum of vectors. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   &    |-  ~ w  =  ( inv `  + w )   =>    |-  (
 ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W ) )  ->  ( ~ w `  ( V1 + w V 2 )
 )  =  ( ( ~ w `  V1 ) + w ( ~ w `  V 2 ) ) )
 
Theoremprodvs 24800 The product of a vector by a scalar is a vector. (Contributed by FL, 12-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   &    |-  X  =  ran  + t   &    |-  + t  =  ( 1st `  ( 1st `  R ) )   =>    |-  ( ( R  e.  Vec  /\  A  e.  X  /\  U  e.  W )  ->  ( A . w U )  e.  W )
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