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Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhbtlem6 26701* There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  N  =  (RSpan `  P )   &    |-  ( ph  ->  R  e. LNoeR )   &    |-  ( ph  ->  I  e.  U )   &    |-  ( ph  ->  X  e.  NN0 )   =>    |-  ( ph  ->  E. k  e.  ( ~P I  i^i  Fin ) ( ( S `
  I ) `  X )  C_  ( ( S `  ( N `
  k ) ) `
  X ) )
 
Theoremhbt 26702 The Hilbert Basis Theorem - the ring of univariate polynomials over a Noetherian ring is a Noetherian ring. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e. LNoeR  ->  P  e. LNoeR )
 
18.16.50  Additional material on polynomials [DEPRECATED]
 
Syntaxcmnc 26703 Extend class notation with the class of monic polynomials.
 class  Monic
 
Syntaxcplylt 26704 Extend class notatin with the class of limited-degree polynomials.
 class Poly<
 
Definitiondf-mnc 26705* Define the class of monic polynomials. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  Monic  =  ( s  e.  ~P CC  |->  { p  e.  (Poly `  s )  |  (
 (coeff `  p ) `  (deg `  p )
 )  =  1 } )
 
Definitiondf-plylt 26706* Define the class of limited-degree polynomials. (Contributed by Stefan O'Rear, 8-Dec-2014.)
 |- Poly<  =  (
 s  e.  ~P CC ,  x  e.  NN0  |->  { p  e.  (Poly `  s )  |  ( p  =  0 p  \/  (deg `  p )  <  x ) } )
 
Theoremdgrsub2 26707 Subtracting two polynomials with the same degree and top coefficient gives a polynomial of strictly lower degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  N  =  (deg `  F )   =>    |-  (
 ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T ) )  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F ) `  N )  =  ( (coeff `  G ) `  N ) ) ) 
 ->  (deg `  ( F  o F  -  G ) )  <  N )
 
Theoremdgrnznn 26708 A nonzero polynomial with a root has positive degree. TODO: use in aaliou2 19683. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  (
 ( ( P  e.  (Poly `  S )  /\  P  =/=  0 p ) 
 /\  ( A  e.  CC  /\  ( P `  A )  =  0
 ) )  ->  (deg `  P )  e.  NN )
 
Theoremelmnc 26709 Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  <->  ( P  e.  (Poly `  S )  /\  ( (coeff `  P ) `  (deg `  P )
 )  =  1 ) )
 
Theoremmncply 26710 A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  P  e.  (Poly `  S )
 )
 
Theoremmnccoe 26711 A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  (
 (coeff `  P ) `  (deg `  P )
 )  =  1 )
 
Theoremmncn0 26712 A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  P  =/=  0 p )
 
18.16.51  Degree and minimal polynomial of algebraic numbers
 
Syntaxcdgraa 26713 Extend class notation to include the degree function for algebraic numbers.
 class degAA
 
Syntaxcmpaa 26714 Extend class notation to include the minimal polynomial for an algebraic number.
 class minPolyAA
 
Definitiondf-dgraa 26715* Define the degree of an algebraic number as the smallest degree of any nonzero polynomial which has said number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |- degAA  =  ( x  e.  AA  |->  sup ( { d  e.  NN  |  E. p  e.  (
 (Poly `  QQ )  \  { 0 p }
 ) ( (deg `  p )  =  d  /\  ( p `  x )  =  0 ) } ,  RR ,  `'  <  ) )
 
Definitiondf-mpaa 26716* Define the minimal polynomial of an algebraic number as the unique monic polynomial which achieves the minimum of degAA. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |- minPolyAA  =  ( x  e.  AA  |->  (
 iota_ p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  x )  /\  ( p `  x )  =  0  /\  ( (coeff `  p ) `  (degAA `  x ) )  =  1 ) ) )
 
Theoremdgraaval 26717* Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (degAA `  A )  =  sup ( { d  e.  NN  |  E. p  e.  (
 (Poly `  QQ )  \  { 0 p }
 ) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  ) )
 
Theoremdgraalem 26718* Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  ( (degAA `  A )  e. 
 NN  /\  E. p  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg `  p )  =  (degAA `  A )  /\  ( p `  A )  =  0 ) ) )
 
Theoremdgraacl 26719 Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
 
Theoremdgraaf 26720 Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |- degAA : AA --> NN
 
Theoremdgraaub 26721 Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  (
 ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p ) 
 /\  ( A  e.  CC  /\  ( P `  A )  =  0
 ) )  ->  (degAA `  A )  <_  (deg `  P ) )
 
Theoremdgraa0p 26722 A rational polynomial of degree less than an algebraic number cannot be zero at that number unless it is the zero polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  (
 ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  ->  ( ( P `  A )  =  0  <->  P  =  0 p ) )
 
Theoremmpaaeu 26723* An algebraic number has exactly one monic polynomial of least degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  E! p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  ( p `
  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 ) )
 
Theoremmpaaval 26724* Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (minPolyAA `  A )  =  (
 iota_ p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  ( p `  A )  =  0  /\  ( (coeff `  p ) `  (degAA `  A ) )  =  1 ) ) )
 
Theoremmpaalem 26725 Properties of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  ( (minPolyAA `  A )  e.  (Poly `  QQ )  /\  ( (deg `  (minPolyAA `  A ) )  =  (degAA `  A )  /\  ( (minPolyAA `  A ) `  A )  =  0  /\  ( (coeff `  (minPolyAA `  A ) ) `  (degAA `  A ) )  =  1 ) ) )
 
Theoremmpaacl 26726 Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (minPolyAA `  A )  e.  (Poly `  QQ ) )
 
Theoremmpaadgr 26727 Minimal polynomial has degree the degree of the number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (deg `  (minPolyAA `  A ) )  =  (degAA `  A ) )
 
Theoremmpaaroot 26728 The minimal polynomial of an algebraic number has the number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  ( (minPolyAA `  A ) `  A )  =  0
 )
 
Theoremmpaamn 26729 Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  ( (coeff `  (minPolyAA `  A ) ) `  (degAA `  A ) )  =  1 )
 
18.16.52  Algebraic integers I
 
Syntaxcitgo 26730 Extend class notation with the integral-over predicate.
 class IntgOver
 
Syntaxcza 26731 Extend class notation with the class of algebraic integers.
 class
 
Definitiondf-itgo 26732* A complex number is said to be integral over a subset if it is the root of a monic polynomial with coefficients from the subset. This definition is typically not used for fields but it works there, see aaitgo 26735. This definition could work for subsets of an arbitrary ring with a more general definition of polynomials. TODO: use  Monic (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |- IntgOver  =  ( s  e.  ~P CC  |->  { x  e.  CC  |  E. p  e.  (Poly `  s ) ( ( p `  x )  =  0  /\  (
 (coeff `  p ) `  (deg `  p )
 )  =  1 ) } )
 
Definitiondf-za 26733 Define an algebraic integer as a complex number which is the root of a monic integer polynomial. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  =  (IntgOver `  ZZ )
 
Theoremitgoval 26734* Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  ( S  C_  CC  ->  (IntgOver `  S )  =  { x  e.  CC  |  E. p  e.  (Poly `  S ) ( ( p `
  x )  =  0  /\  ( (coeff `  p ) `  (deg `  p ) )  =  1 ) } )
 
Theoremaaitgo 26735 The standard algebraic numbers  AA are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  AA  =  (IntgOver `  QQ )
 
Theoremitgoss 26736 An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  (
 ( S  C_  T  /\  T  C_  CC )  ->  (IntgOver `  S )  C_  (IntgOver `  T )
 )
 
Theoremitgocn 26737 All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  (IntgOver `  S )  C_  CC
 
Theoremcnsrexpcl 26738 Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  S  e.  (SubRing ` fld ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  NN0 )   =>    |-  ( ph  ->  ( X ^ Y )  e.  S )
 
Theoremfsumcnsrcl 26739* Finite sums are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  S  e.  (SubRing ` fld ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  S )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  S )
 
Theoremcnsrplycl 26740 Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  S  e.  (SubRing ` fld ) )   &    |-  ( ph  ->  P  e.  (Poly `  C ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  C  C_  S )   =>    |-  ( ph  ->  ( P `  X )  e.  S )
 
Theoremrgspnval 26741* Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  N  =  (RingSpan `  R ) )   &    |-  ( ph  ->  U  =  ( N `  A ) )   =>    |-  ( ph  ->  U  =  |^| { t  e.  (SubRing `  R )  |  A  C_  t }
 )
 
Theoremrgspncl 26742 The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  N  =  (RingSpan `  R ) )   &    |-  ( ph  ->  U  =  ( N `  A ) )   =>    |-  ( ph  ->  U  e.  (SubRing `  R ) )
 
Theoremrgspnssid 26743 The ring-span of a set contains the set. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  N  =  (RingSpan `  R ) )   &    |-  ( ph  ->  U  =  ( N `  A ) )   =>    |-  ( ph  ->  A 
 C_  U )
 
Theoremrgspnmin 26744 The ring-span is contained in all subspaces which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  N  =  (RingSpan `  R ) )   &    |-  ( ph  ->  U  =  ( N `  A ) )   &    |-  ( ph  ->  S  e.  (SubRing `  R ) )   &    |-  ( ph  ->  A  C_  S )   =>    |-  ( ph  ->  U  C_  S )
 
Theoremrgspnid 26745 The span of a subring is itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  A  e.  (SubRing `  R ) )   &    |-  ( ph  ->  S  =  ( (RingSpan `  R ) `  A ) )   =>    |-  ( ph  ->  S  =  A )
 
Theoremrngunsnply 26746* Adjoining one element to a ring results in a set of polynomial evaluations. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  ( ph  ->  B  e.  (SubRing ` fld ) )   &    |-  ( ph  ->  X  e.  CC )   &    |-  ( ph  ->  S  =  ( (RingSpan ` fld ) `  ( B  u.  { X }
 ) ) )   =>    |-  ( ph  ->  ( V  e.  S  <->  E. p  e.  (Poly `  B ) V  =  ( p `  X ) ) )
 
Theoremflcidc 26747* Finite linear combinations with an indicator function. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( ph  ->  F  =  ( j  e.  S  |->  if ( j  =  K ,  1 ,  0 ) ) )   &    |-  ( ph  ->  S  e.  Fin )   &    |-  ( ph  ->  K  e.  S )   &    |-  ( ( ph  /\  i  e.  S ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ i  e.  S  ( ( F `  i )  x.  B )  =  [_ K  /  i ]_ B )
 
18.16.53  Finite cardinality [SO]
 
Theoremen1uniel 26748 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  ( S  ~~  1o  ->  U. S  e.  S )
 
Theoremen2eleq 26749 Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  (
 ( X  e.  P  /\  P  ~~  2o )  ->  P  =  { X ,  U. ( P  \  { X } ) }
 )
 
Theoremen2other2 26750 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  (
 ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { U. ( P  \  { X } ) }
 )  =  X )
 
18.16.54  Words in monoids and ordered group sum

One important use of words is as formal composites in cases where order is significant, using the general sum operator df-gsum 13368. If order is not significant, it is simpler to use families instead.

 
Theoremissubmd 26751* Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  .0.  =  ( 0g `  M )   &    |-  ( ph  ->  M  e.  Mnd )   &    |-  ( ph  ->  ch )   &    |-  ( ( ph  /\  ( ( x  e.  B  /\  y  e.  B )  /\  ( th  /\  ta ) ) )  ->  et )   &    |-  (
 z  =  .0.  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  x  ->  ( ps  <->  th ) )   &    |-  (
 z  =  y  ->  ( ps  <->  ta ) )   &    |-  (
 z  =  ( x 
 .+  y )  ->  ( ps  <->  et ) )   =>    |-  ( ph  ->  { z  e.  B  |  ps }  e.  (SubMnd `  M ) )
 
18.16.55  Transpositions in the symmetric group
 
Syntaxcpmtr 26752 Syntax for the transposition generator function.
 class pmTrsp
 
Definitiondf-pmtr 26753* Define a function that generates the transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |- pmTrsp  =  ( d  e.  _V  |->  ( p  e.  { y  e.  ~P d  |  y 
 ~~  2o }  |->  ( z  e.  d  |->  if (
 z  e.  p ,  U. ( p  \  {
 z } ) ,  z ) ) ) )
 
Theoremf1omvdmvd 26754 A permutation of any class moves a point which is moved to a different point which is moved. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  (
 ( F : A -1-1-onto-> A  /\  X  e.  dom  (  F  \  _I  ) ) 
 ->  ( F `  X )  e.  ( dom  (  F  \  _I  )  \  { X } )
 )
 
Theoremf1omvdcnv 26755 A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  ( F : A -1-1-onto-> A  ->  dom  ( `' F  \  _I  )  =  dom  (  F  \  _I  ) )
 
Theoremmvdco 26756 Composing two permutations moves at most the union of the points. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  dom  ( ( F  o.  G )  \  _I  )  C_  ( dom  (  F 
 \  _I  )  u. 
 dom  (  G  \  _I  ) )
 
Theoremf1omvdconj 26757 Conjugation of a permutation takes the image of the moved subclass. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  (
 ( F : A --> A  /\  G : A -1-1-onto-> A )  ->  dom  ( (
 ( G  o.  F )  o.  `' G ) 
 \  _I  )  =  ( G " dom  (  F  \  _I  )
 ) )
 
Theoremf1otrspeq 26758 A transposition is characterized by the points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  (
 ( ( F : A
 -1-1-onto-> A  /\  G : A -1-1-onto-> A )  /\  ( dom  (  F  \  _I  )  ~~  2o  /\  dom  (  G  \  _I  )  =  dom  (  F  \  _I  )
 ) )  ->  F  =  G )
 
Theoremf1omvdco2 26759 If exactly one of two permutations is limited to a set of points, then the composition will not be. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  (
 ( F : A -1-1-onto-> A  /\  G : A -1-1-onto-> A  /\  ( dom  (  F  \  _I  )  C_  X \/_ dom  (  G  \  _I  )  C_  X ) )  ->  -.  dom  ( ( F  o.  G )  \  _I  )  C_  X )
 
Theoremf1omvdco3 26760 If a point is moved by exactly one of two permutations, then it will be moved by their composite. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  (
 ( F : A -1-1-onto-> A  /\  G : A -1-1-onto-> A  /\  ( X  e.  dom  (  F  \  _I  ) \/_ X  e.  dom  (  G  \  _I  ) ) )  ->  X  e.  dom  ( ( F  o.  G ) 
 \  _I  ) )
 
Theorempmtrfval 26761* The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  ( D  e.  V  ->  T  =  ( p  e. 
 { y  e.  ~P D  |  y  ~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  p ,  U. ( p  \  { z }
 ) ,  z ) ) ) )
 
Theorempmtrval 26762* A generated transposition, expressed in a symmetric form. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  (
 ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z }
 ) ,  z ) ) )
 
Theorempmtrfv 26763 General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  (
 ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D ) 
 ->  ( ( T `  P ) `  Z )  =  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z ) )
 
Theorempmtrprfv 26764 In a transposition of two given points, each maps to the other. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  (
 ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y ) ) 
 ->  ( ( T `  { X ,  Y }
 ) `  X )  =  Y )
 
Theorempmtrf 26765 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  (
 ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P ) : D --> D )
 
Theorempmtrmvd 26766 A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   =>    |-  (
 ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  dom  ( ( T `  P )  \  _I  )  =  P )
 
Theorempmtrrn 26767 Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  R )
 
Theorempmtrfrn 26768 A transposition (as a kind of function) is the function transposing the two points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   &    |-  P  =  dom  (  F  \  _I  )   =>    |-  ( F  e.  R  ->  ( ( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) )
 
Theorempmtrffv 26769 Mapping of a point under a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   &    |-  P  =  dom  (  F  \  _I  )   =>    |-  (
 ( F  e.  R  /\  Z  e.  D ) 
 ->  ( F `  Z )  =  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z ) )
 
Theorempmtrfinv 26770 A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  ( F  o.  F )  =  (  _I  |`  D ) )
 
Theorempmtrfmvdn0 26771 A transpositon moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  dom  (  F  \  _I  )  =/=  (/) )
 
Theorempmtrff1o 26772 A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  F : D -1-1-onto-> D )
 
Theorempmtrfcnv 26773 A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  ->  `' F  =  F )
 
Theorempmtrfb 26774 An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( F  e.  R  <->  ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  (  F  \  _I  )  ~~  2o ) )
 
Theorempmtrfconj 26775 Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  (pmTrsp `  D )   &    |-  R  =  ran  T   =>    |-  ( ( F  e.  R  /\  G : D -1-1-onto-> D )  ->  ( ( G  o.  F )  o.  `' G )  e.  R )
 
Theoremsymgsssg 26776* The symmetric group has subgroups restricting the set of non-fixed points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  ( Base `  G )   =>    |-  ( D  e.  V  ->  { x  e.  B  |  dom  (  x  \  _I  )  C_  X }  e.  (SubGrp `  G ) )
 
Theoremsymgfisg 26777* The symmetric group has a subgroup of permutations that move finitely many points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  ( Base `  G )   =>    |-  ( D  e.  V  ->  { x  e.  B  |  dom  (  x  \  _I  )  e.  Fin }  e.  (SubGrp `  G ) )
 
Theoremsymgtrf 26778 Transpositions are elements of the symmetric group. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   =>    |-  T  C_  B
 
Theoremsymggen 26779* The span of the transpositions is the subgroup that moves finitely many points. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   &    |-  K  =  (mrCls `  (SubMnd `  G ) )   =>    |-  ( D  e.  V  ->  ( K `  T )  =  { x  e.  B  |  dom  (  x  \  _I  )  e. 
 Fin } )
 
Theoremsymggen2 26780 A finite permutation group is generated by the transpositions. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  ( SymGrp `  D )   &    |-  B  =  (
 Base `  G )   &    |-  K  =  (mrCls `  (SubMnd `  G ) )   =>    |-  ( D  e.  Fin  ->  ( K `  T )  =  B )
 
Theoremsymgtrinv 26781 To invert a permutation represented as a sequence of transpositions, reverse the sequence. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  G  =  ( SymGrp `  D )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I `  ( G  gsumg 
 W ) )  =  ( G  gsumg  (reverse `  W )
 ) )
 
18.16.56  The sign of a permutation
 
Syntaxcpsgn 26782 Syntax for the sign of a permutation.
 class pmSgn
 
Definitiondf-psgn 26783* Define a function which takes the value  1 for even permutations and  -u 1 for odd. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |- pmSgn  =  ( d  e.  _V  |->  ( x  e.  { p  e.  ( Base `  ( SymGrp `  d
 ) )  |  dom  (  p  \  _I  )  e.  Fin }  |->  ( iota
 s E. w  e. Word  ran  (pmTrsp `  d )
 ( x  =  ( ( SymGrp `  d )  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) ) )
 
Theorempsgnunilem1 26784* Lemma for psgnuni 26790. Given two consequtive transpositions in a representation of a permutation, either they are equal and therefore equivalent to the identity, or they are not and it is possible to commute them such that a chosen point in the left transposition is preserved in the right. By repeating this process, a point can be removed from a representation of the identity. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  P  e.  T )   &    |-  ( ph  ->  Q  e.  T )   &    |-  ( ph  ->  A  e.  dom  (  P  \  _I  ) )   =>    |-  ( ph  ->  (
 ( P  o.  Q )  =  (  _I  |`  D )  \/  E. r  e.  T  E. s  e.  T  ( ( P  o.  Q )  =  ( r  o.  s
 )  /\  A  e.  dom  (  s  \  _I  )  /\  -.  A  e.  dom  (  r  \  _I  ) ) ) )
 
Theorempsgnunilem5 26785* Lemma for psgnuni 26790. It is impossible to shift a transposition off the end because if the active transposition is at the right end, it is the only transposition moving  A in contradiction to this being a representation of the identity. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D ) )   &    |-  ( ph  ->  ( # `  W )  =  L )   &    |-  ( ph  ->  I  e.  ( 0..^ L ) )   &    |-  ( ph  ->  A  e.  dom  ( ( W `  I )  \  _I  ) )   &    |-  ( ph  ->  A. k  e.  ( 0..^ I )  -.  A  e.  dom  ( ( W `
  k )  \  _I  ) )   =>    |-  ( ph  ->  ( I  +  1 )  e.  ( 0..^ L ) )
 
Theorempsgnunilem2 26786* Lemma for psgnuni 26790. Induction step for moving a transposition as far to the right as possible. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D ) )   &    |-  ( ph  ->  ( # `  W )  =  L )   &    |-  ( ph  ->  I  e.  ( 0..^ L ) )   &    |-  ( ph  ->  A  e.  dom  ( ( W `  I )  \  _I  ) )   &    |-  ( ph  ->  A. k  e.  ( 0..^ I )  -.  A  e.  dom  ( ( W `
  k )  \  _I  ) )   &    |-  ( ph  ->  -. 
 E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
 ) )   =>    |-  ( ph  ->  E. w  e. Word  T ( ( ( G  gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w )  =  L )  /\  ( ( I  +  1 )  e.  (
 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1 ) ) 
 \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `
  j )  \  _I  ) ) ) )
 
Theorempsgnunilem3 26787* Lemma for psgnuni 26790. Any nonempty representation of the identity can be incrementally transformed into a representation two shorter. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( # `  W )  =  L )   &    |-  ( ph  ->  ( # `  W )  e.  NN )   &    |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D ) )   &    |-  ( ph  ->  -. 
 E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
 ) )   =>    |- 
 -.  ph
 
Theorempsgnunilem4 26788 Lemma for psgnuni 26790. An odd-length representation of the identity is impossible, as it could be repeatedly shortened to a length of 1, but a length 1 permutation must be a transposition. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D ) )   =>    |-  ( ph  ->  ( -u 1 ^ ( # `  W ) )  =  1 )
 
Theoremm1expaddsub 26789 Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  (
 ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ ( X  -  Y ) )  =  ( -u 1 ^ ( X  +  Y ) ) )
 
Theorempsgnuni 26790 If the same permutation can be written in more than one way as a product of transpositions, the parity of those products must agree; otherwise the product of one with the inverse of the other would be an odd representation of the identity. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  W  e. Word  T )   &    |-  ( ph  ->  X  e. Word  T )   &    |-  ( ph  ->  ( G  gsumg 
 W )  =  ( G  gsumg 
 X ) )   =>    |-  ( ph  ->  (
 -u 1 ^ ( # `
  W ) )  =  ( -u 1 ^ ( # `  X ) ) )
 
Theorempsgnfval 26791* Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  ( Base `  G )   &    |-  F  =  { p  e.  B  |  dom  (  p  \  _I  )  e.  Fin }   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  N  =  ( x  e.  F  |->  ( iota
 s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) )
 
Theorempsgnfn 26792* Functionality and domain of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  B  =  ( Base `  G )   &    |-  F  =  { p  e.  B  |  dom  (  p  \  _I  )  e.  Fin }   &    |-  N  =  (pmSgn `  D )   =>    |-  N  Fn  F
 
Theorempsgndmsubg 26793 The finitary permutations are a subgroup. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( D  e.  V  ->  dom 
 N  e.  (SubGrp `  G ) )
 
Theorempsgneldm 26794 Property of being a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   &    |-  B  =  ( Base `  G )   =>    |-  ( P  e.  dom  N  <->  ( P  e.  B  /\  dom  (  P  \  _I  )  e.  Fin ) )
 
Theorempsgneldm2 26795* The finitary permutations are the span of the transpositons. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( D  e.  V  ->  ( P  e.  dom  N  <->  E. w  e. Word  T P  =  ( G  gsumg  w ) ) )
 
Theorempsgneldm2i 26796 A sequence of transpositions describes a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( G  gsumg  W )  e.  dom  N )
 
Theorempsgneu 26797* A finitary permutation has exactly one parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( P  e.  dom  N 
 ->  E! s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) )
 
Theorempsgnval 26798* Value of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( P  e.  dom  N 
 ->  ( N `  P )  =  ( iota s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `
  w ) ) ) ) )
 
Theorempsgnvali 26799* A finitary permutation has at least one representation for its parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( P  e.  dom  N 
 ->  E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  ( N `  P )  =  ( -u 1 ^ ( # `
  w ) ) ) )
 
Theorempsgnvalii 26800 Any representation of a permutation is length matching the permutation sign. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  G  =  ( SymGrp `  D )   &    |-  T  =  ran  (pmTrsp `  D )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( N `  ( G  gsumg 
 W ) )  =  ( -u 1 ^ ( # `
  W ) ) )
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