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Theorem List for Metamath Proof Explorer - 36001-36100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempwfi2en 36001* Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.)
 |-  S  =  { y  e.  ( 2o  ^m  A )  |  y finSupp  (/) }   =>    |-  ( A  e.  V  ->  S  ~~  ( ~P A  i^i  Fin )
 )
 
Theoremfrlmpwfi 36002 Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Proof shortened by AV, 14-Jun-2020.)
 |-  R  =  (ℤ/n `  2 )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  Y )   =>    |-  ( I  e.  V  ->  B  ~~  ( ~P I  i^i  Fin )
 )
 
Theoremgicabl 36003 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  ( G  ~=g𝑔 
 H  ->  ( G  e.  Abel 
 <->  H  e.  Abel )
 )
 
Theoremimasgim 36004 A relabeling of the elements of a group induces an isomorphism to the relabeled group. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) (Revised by Mario Carneiro, 11-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R ) )   &    |-  ( ph  ->  V  =  (
 Base `  R ) )   &    |-  ( ph  ->  F : V
 -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  F  e.  ( R GrpIso  U ) )
 
Theorembasfn 36005 Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  Base  Fn 
 _V
 
Theoremisnumbasgrplem1 36006 A set which is equipollent to the base set of a definable Abelian group is the base set of some (relabeled) Abelian group. (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  B  =  ( Base `  R )   =>    |-  (
 ( R  e.  Abel  /\  C  ~~  B ) 
 ->  C  e.  ( Base "
 Abel ) )
 
Theoremharn0 36007 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( S  e.  V  ->  (har `  S )  =/=  (/) )
 
Theoremnuminfctb 36008 A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  (
 ( S  e.  dom  card  /\  -.  S  e.  Fin )  ->  om  ~<_  S )
 
Theoremisnumbasgrplem2 36009 If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  (
 ( S  u.  (har `  S ) )  e.  ( Base " Grp )  ->  S  e.  dom  card )
 
Theoremisnumbasgrplem3 36010 Every nonempty numerable set can be given the structure of an Abelian group, either a finite cyclic group or a vector space over Z/2Z. (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  (
 ( S  e.  dom  card  /\  S  =/=  (/) )  ->  S  e.  ( Base "
 Abel ) )
 
Theoremisnumbasabl 36011 A set is numerable iff it and its Hartogs number can be jointly given the structure of an Abelian group. (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( S  e.  dom  card  <->  ( S  u.  (har `  S ) )  e.  ( Base " Abel ) )
 
Theoremisnumbasgrp 36012 A set is numerable iff it and its Hartogs number can be jointly given the structure of a group. (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( S  e.  dom  card  <->  ( S  u.  (har `  S ) )  e.  ( Base " Grp ) )
 
Theoremdfacbasgrp 36013 A choice equivalent in abstract algebra: All nonempty sets admit a group structure. From http://mathoverflow.net/a/12988. (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  (CHOICE  <->  ( Base " Grp )  =  ( _V  \  { (/) } ) )
 
21.23.42  Noetherian rings and left modules II
 
Syntaxclnr 36014 Extend class notation with the class of left Noetherian rings.
 class LNoeR
 
Definitiondf-lnr 36015 A ring is left-Noetherian iff it is Noetherian as a left module over itself. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |- LNoeR  =  {
 a  e.  Ring  |  (ringLMod `  a )  e. LNoeM }
 
Theoremislnr 36016 Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  <->  ( A  e.  Ring  /\  (ringLMod `  A )  e. LNoeM ) )
 
Theoremlnrring 36017 Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  ->  A  e.  Ring
 )
 
Theoremlnrlnm 36018 Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  ->  (ringLMod `  A )  e. LNoeM )
 
Theoremislnr2 36019* Property of being a left-Noetherian ring in terms of finite generation of ideals (the usual "pure ring theory" definition). (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (LIdeal `  R )   &    |-  N  =  (RSpan `  R )   =>    |-  ( R  e. LNoeR  <->  ( R  e.  Ring  /\  A. i  e.  U  E. g  e.  ( ~P B  i^i  Fin )
 i  =  ( N `
  g ) ) )
 
Theoremislnr3 36020 Relate left-Noetherian rings to Noetherian-type closure property of the left ideal system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e. LNoeR  <->  ( R  e.  Ring  /\  U  e.  (NoeACS `  B ) ) )
 
Theoremlnr2i 36021* Given an ideal in a left-Noetherian ring, there is a finite subset which generates it. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  N  =  (RSpan `  R )   =>    |-  (
 ( R  e. LNoeR  /\  I  e.  U )  ->  E. g  e.  ( ~P I  i^i  Fin ) I  =  ( N `  g ) )
 
Theoremlpirlnr 36022 Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( R  e. LPIR  ->  R  e. LNoeR )
 
Theoremlnrfrlm 36023 Finite-dimensional free modules over a Noetherian ring are Noetherian. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  Y  =  ( R freeLMod  I )   =>    |-  (
 ( R  e. LNoeR  /\  I  e.  Fin )  ->  Y  e. LNoeM )
 
Theoremlnrfg 36024 Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.)
 |-  S  =  (Scalar `  M )   =>    |-  (
 ( M  e. LFinGen  /\  S  e. LNoeR )  ->  M  e. LNoeM )
 
Theoremlnrfgtr 36025 A submodule of a finitely generated module over a Noetherian ring is finitely generated. Often taken as the definition of Noetherian ring. (Contributed by Stefan O'Rear, 7-Feb-2015.)
 |-  S  =  (Scalar `  M )   &    |-  U  =  ( LSubSp `  M )   &    |-  N  =  ( Ms  P )   =>    |-  ( ( M  e. LFinGen  /\  S  e. LNoeR  /\  P  e.  U )  ->  N  e. LFinGen )
 
21.23.43  Hilbert's Basis Theorem
 
Syntaxcldgis 36026 The leading ideal sequence used in the Hilbert Basis Theorem.
 class ldgIdlSeq
 
Definitiondf-ldgis 36027* Define a function which carries polynomial ideals to the sequence of coefficient ideals of leading coefficients of degree-  x elements in the polynomial ideal. The proof that this map is strictly monotone is the core of the Hilbert Basis Theorem hbt 36035. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |- ldgIdlSeq  =  ( r  e.  _V  |->  ( i  e.  (LIdeal `  (Poly1 `  r ) )  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( ( deg1  `  r
 ) `  k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) }
 ) ) )
 
Theoremhbtlem1 36028* Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  D  =  ( deg1  `  R )   =>    |-  (
 ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  ->  ( ( S `  I ) `  X )  =  { j  |  E. k  e.  I  ( ( D `  k )  <_  X  /\  j  =  ( (coe1 `  k ) `  X ) ) } )
 
Theoremhbtlem2 36029 Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  T  =  (LIdeal `  R )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  U  /\  X  e.  NN0 )  ->  ( ( S `  I ) `  X )  e.  T )
 
Theoremhbtlem7 36030 Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  T  =  (LIdeal `  R )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  U ) 
 ->  ( S `  I
 ) : NN0 --> T )
 
Theoremhbtlem4 36031 The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  U )   &    |-  ( ph  ->  X  e.  NN0 )   &    |-  ( ph  ->  Y  e.  NN0 )   &    |-  ( ph  ->  X 
 <_  Y )   =>    |-  ( ph  ->  (
 ( S `  I
 ) `  X )  C_  ( ( S `  I ) `  Y ) )
 
Theoremhbtlem3 36032 The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  U )   &    |-  ( ph  ->  J  e.  U )   &    |-  ( ph  ->  I  C_  J )   &    |-  ( ph  ->  X  e.  NN0 )   =>    |-  ( ph  ->  (
 ( S `  I
 ) `  X )  C_  ( ( S `  J ) `  X ) )
 
Theoremhbtlem5 36033* The leading ideal function is strictly monotone. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  U )   &    |-  ( ph  ->  J  e.  U )   &    |-  ( ph  ->  I  C_  J )   &    |-  ( ph  ->  A. x  e.  NN0  ( ( S `
  J ) `  x )  C_  ( ( S `  I ) `
  x ) )   =>    |-  ( ph  ->  I  =  J )
 
Theoremhbtlem6 36034* 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 36035 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 )
 
21.23.44  Additional material on polynomials [DEPRECATED]
 
Syntaxcmnc 36036 Extend class notation with the class of monic polynomials.
 class  Monic
 
Syntaxcplylt 36037 Extend class notatin with the class of limited-degree polynomials.
 class Poly<
 
Definitiondf-mnc 36038* 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 36039* 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  =  0p  \/  (deg `  p )  <  x ) } )
 
Theoremdgrsub2 36040 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  oF  -  G ) )  <  N )
 
Theoremelmnc 36041 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 36042 A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  P  e.  (Poly `  S )
 )
 
Theoremmnccoe 36043 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 36044 A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  P  =/=  0p )
 
21.23.45  Degree and minimal polynomial of algebraic numbers
 
Syntaxcdgraa 36045 Extend class notation to include the degree function for algebraic numbers.
 class degAA
 
Syntaxcdgraaold 36046 Extend class notation to include the degree function for algebraic numbers (old version).
 class degAA
 
Syntaxcmpaa 36047 Extend class notation to include the minimal polynomial for an algebraic number.
 class minPolyAA
 
Definitiondf-dgraa 36048* 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.) (Revised by AV, 29-Sep-2020.)
 |- degAA  =  ( x  e.  AA  |-> inf ( {
 d  e.  NN  |  E. p  e.  (
 (Poly `  QQ )  \  { 0p }
 ) ( (deg `  p )  =  d  /\  ( p `  x )  =  0 ) } ,  RR ,  <  ) )
 
Definitiondf-dgraaOLD 36049* 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.) Obsolete version of df-dgraa 36048 as of 29-Sep-2020. (New usage is discouraged.)
 |- degAA  =  ( x  e.  AA  |->  sup ( { d  e.  NN  |  E. p  e.  (
 (Poly `  QQ )  \  { 0p }
 ) ( (deg `  p )  =  d  /\  ( p `  x )  =  0 ) } ,  RR ,  `'  <  ) )
 
Definitiondf-mpaa 36050* 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 36051* Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.)
 |-  ( A  e.  AA  ->  (degAA `  A )  = inf ( { d  e.  NN  |  E. p  e.  (
 (Poly `  QQ )  \  { 0p }
 ) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  <  ) )
 
TheoremdgraavalOLD 36052* Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of dgraaval 36051 as of 29-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A  e.  AA  ->  (degAA `  A )  =  sup ( { d  e.  NN  |  E. p  e.  (
 (Poly `  QQ )  \  { 0p }
 ) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  ) )
 
Theoremdgraalem 36053* Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.)
 |-  ( A  e.  AA  ->  ( (degAA `  A )  e. 
 NN  /\  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg `  p )  =  (degAA `  A )  /\  ( p `  A )  =  0 ) ) )
 
TheoremdgraalemOLD 36054* Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of dgraalem 36053 as of 29-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A  e.  AA  ->  ( (degAA `  A )  e. 
 NN  /\  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg `  p )  =  (degAA `  A )  /\  ( p `  A )  =  0 ) ) )
 
Theoremdgraacl 36055 Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
 
TheoremdgraaclOLD 36056 Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of dgraacl 36055 as of 29-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
 
Theoremdgraaf 36057 Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.)
 |- degAA : AA --> NN
 
TheoremdgraafOLD 36058 Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of dgraaf 36057 as of 29-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |- degAA : AA --> NN
 
Theoremdgraaub 36059 Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.)
 |-  (
 ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0p ) 
 /\  ( A  e.  CC  /\  ( P `  A )  =  0
 ) )  ->  (degAA `  A )  <_  (deg `  P ) )
 
TheoremdgraaubOLD 36060 Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of dgraaub 36059 as of 29-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0p ) 
 /\  ( A  e.  CC  /\  ( P `  A )  =  0
 ) )  ->  (degAA `  A )  <_  (deg `  P ) )
 
Theoremdgraa0p 36061 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  =  0p ) )
 
Theoremmpaaeu 36062* An algebraic number has exactly one monic polynomial of the 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 36063* 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 36064 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 36065 Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (minPolyAA `  A )  e.  (Poly `  QQ ) )
 
Theoremmpaadgr 36066 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 36067 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 36068 Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  ( (coeff `  (minPolyAA `  A ) ) `  (degAA `  A ) )  =  1 )
 
21.23.46  Algebraic integers I
 
Syntaxcitgo 36069 Extend class notation with the integral-over predicate.
 class IntgOver
 
Syntaxcza 36070 Extend class notation with the class of algebraic integers.
 class
 
Definitiondf-itgo 36071* 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 36074. 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 36072 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 36073* 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 36074 The standard algebraic numbers  AA are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  AA  =  (IntgOver `  QQ )
 
Theoremitgoss 36075 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 36076 All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  (IntgOver `  S )  C_  CC
 
Theoremcnsrexpcl 36077 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 36078* 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 36079 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 36080* 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 36081 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 36082 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 36083 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 36084 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 36085* 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 36086* 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 )
 
21.23.47  Endomorphism algebra
 
Syntaxcmend 36087 Syntax for module endomorphism algebra.
 class MEndo
 
Definitiondf-mend 36088* Define the endomorphism algebra of a module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |- MEndo  =  ( m  e.  _V  |->  [_ ( m LMHom  m )  /  b ]_ ( { <. (
 Base `  ndx ) ,  b >. ,  <. ( +g  ` 
 ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  oF (
 +g  `  m )
 y ) ) >. , 
 <. ( .r `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  o.  y ) ) >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  m ) >. ,  <. ( .s
 `  ndx ) ,  ( x  e.  ( Base `  (Scalar `  m )
 ) ,  y  e.  b  |->  ( ( (
 Base `  m )  X.  { x } )  oF ( .s `  m ) y ) ) >. } ) )
 
Theoremalgstr 36089 Lemma to shorten proofs of algbase 36090 through algvsca 36094. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  A Struct  <. 1 ,  6
 >.
 
Theoremalgbase 36090 The base set of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  ( B  e.  V  ->  B  =  ( Base `  A ) )
 
Theoremalgaddg 36091 The additive operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .+  e.  V  ->  .+  =  ( +g  `  A ) )
 
Theoremalgmulr 36092 The multiplicative operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .X.  e.  V  -> 
 .X.  =  ( .r `  A ) )
 
Theoremalgsca 36093 The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  ( S  e.  V  ->  S  =  (Scalar `  A ) )
 
Theoremalgvsca 36094 The scalar product operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
 |-  A  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )   =>    |-  (  .x.  e.  V  ->  .x.  =  ( .s
 `  A ) )
 
Theoremmendval 36095* Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  B  =  ( M LMHom  M )   &    |-  .+  =  ( x  e.  B ,  y  e.  B  |->  ( x  oF ( +g  `  M ) y ) )   &    |-  .X. 
 =  ( x  e.  B ,  y  e.  B  |->  ( x  o.  y ) )   &    |-  S  =  (Scalar `  M )   &    |-  .x.  =  ( x  e.  ( Base `  S ) ,  y  e.  B  |->  ( ( ( Base `  M )  X.  { x }
 )  oF ( .s `  M ) y ) )   =>    |-  ( M  e.  X  ->  (MEndo `  M )  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } ) )
 
Theoremmendbas 36096 Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   =>    |-  ( M LMHom  M )  =  (
 Base `  A )
 
Theoremmendplusgfval 36097* Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   &    |-  .+  =  ( +g  `  M )   =>    |-  ( +g  `  A )  =  ( x  e.  B ,  y  e.  B  |->  ( x  oF  .+  y ) )
 
Theoremmendplusg 36098 A specific addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   &    |-  .+  =  ( +g  `  M )   &    |-  .+b  =  ( +g  `  A )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+b  Y )  =  ( X  oF  .+  Y ) )
 
Theoremmendmulrfval 36099* Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   =>    |-  ( .r `  A )  =  ( x  e.  B ,  y  e.  B  |->  ( x  o.  y
 ) )
 
Theoremmendmulr 36100 A specific multiplication in the module endormoprhism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
 |-  A  =  (MEndo `  M )   &    |-  B  =  ( Base `  A )   &    |-  .x.  =  ( .r `  A )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  =  ( X  o.  Y ) )
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