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Theorem List for Metamath Proof Explorer - 27401-27500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlindsmm 27401 Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  ( Base `  T )   =>    |-  (
 ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  ( F  e.  (LIndS `  S ) 
 <->  ( G " F )  e.  (LIndS `  T ) ) )
 
Theoremlindsmm2 27402 The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  ( Base `  T )   =>    |-  (
 ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  e.  (LIndS `  S ) ) 
 ->  ( G " F )  e.  (LIndS `  T ) )
 
Theoremlsslindf 27403 Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  U  =  ( LSubSp `  W )   &    |-  X  =  ( Ws  S )   =>    |-  ( ( W  e.  LMod  /\  S  e.  U  /\  ran 
 F  C_  S )  ->  ( F LIndF  X  <->  F LIndF  W ) )
 
Theoremlsslinds 27404 Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  U  =  ( LSubSp `  W )   &    |-  X  =  ( Ws  S )   =>    |-  ( ( W  e.  LMod  /\  S  e.  U  /\  F  C_  S )  ->  ( F  e.  (LIndS `  X )  <->  F  e.  (LIndS `  W ) ) )
 
Theoremislbs4 27405 A basis is an independent spanning set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  K  =  ( LSpan `  W )   =>    |-  ( X  e.  J  <->  ( X  e.  (LIndS `  W )  /\  ( K `  X )  =  B ) )
 
Theoremlbslinds 27406 A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  J  =  (LBasis `  W )   =>    |-  J  C_  (LIndS `  W )
 
Theoremislinds3 27407 A subset is linearly independent iff it is a basis of its span. (Contributed by Stefan O'Rear, 25-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  K  =  ( LSpan `  W )   &    |-  X  =  ( Ws  ( K `  Y ) )   &    |-  J  =  (LBasis `  X )   =>    |-  ( W  e.  LMod  ->  ( Y  e.  (LIndS `  W )  <->  Y  e.  J ) )
 
Theoremislinds4 27408* A set is independent in a vector space iff it is a subset of some basis. (AC equivalent) (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  J  =  (LBasis `  W )   =>    |-  ( W  e.  LVec  ->  ( Y  e.  (LIndS `  W ) 
 <-> 
 E. b  e.  J  Y  C_  b ) )
 
18.17.47  Characterization of free modules
 
Theoremlmimlbs 27409 The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  J  =  (LBasis `  S )   &    |-  K  =  (LBasis `  T )   =>    |-  (
 ( F  e.  ( S LMIso  T )  /\  B  e.  J )  ->  ( F " B )  e.  K )
 
Theoremlmiclbs 27410 Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  J  =  (LBasis `  S )   &    |-  K  =  (LBasis `  T )   =>    |-  ( S  ~=ph𝑚 
 T  ->  ( J  =/= 
 (/)  ->  K  =/=  (/) ) )
 
Theoremislindf4 27411* A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  Y  =  ( 0g `  R )   &    |-  L  =  ( Base `  ( R freeLMod  I )
 )   =>    |-  ( ( W  e.  LMod  /\  I  e.  X  /\  F : I --> B ) 
 ->  ( F LIndF  W  <->  A. x  e.  L  ( ( W  gsumg  ( x  o F  .x.  F ) )  =  .0.  ->  x  =  ( I  X.  { Y }
 ) ) ) )
 
Theoremislindf5 27412* A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  F )   &    |-  C  =  ( Base `  T )   &    |-  .x.  =  ( .s `  T )   &    |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  o F  .x.  A ) ) )   &    |-  ( ph  ->  T  e.  LMod )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  R  =  (Scalar `  T ) )   &    |-  ( ph  ->  A : I --> C )   =>    |-  ( ph  ->  ( A LIndF  T  <->  E : B -1-1-> C ) )
 
Theoremindlcim 27413* An independent, spanning family extends to an isomorphism from a free module. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  F )   &    |-  C  =  ( Base `  T )   &    |-  .x.  =  ( .s `  T )   &    |-  N  =  ( LSpan `  T )   &    |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  o F  .x.  A ) ) )   &    |-  ( ph  ->  T  e.  LMod
 )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  R  =  (Scalar `  T ) )   &    |-  ( ph  ->  A : I -onto-> J )   &    |-  ( ph  ->  A LIndF  T )   &    |-  ( ph  ->  ( N `  J )  =  C )   =>    |-  ( ph  ->  E  e.  ( F LMIso  T ) )
 
Theoremlbslcic 27414 A module with a basis is isomorphic to a free module with the same cardinality. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  F  =  (Scalar `  W )   &    |-  J  =  (LBasis `  W )   =>    |-  (
 ( W  e.  LMod  /\  B  e.  J  /\  I  ~~  B )  ->  W  ~=ph𝑚  ( F freeLMod  I )
 )
 
Theoremlmisfree 27415* A module has a basis iff it is isomorphic to a free module. In settings where isomorphic objects are not distinguished, it is common to define "free module" as any module with a basis; thus for instance lbsex 15934 might be described as "every vector space is free." (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  J  =  (LBasis `  W )   &    |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  LMod  ->  ( J  =/=  (/)  <->  E. k  W  ~=ph𝑚  ( F freeLMod  k ) ) )
 
18.17.48  Noetherian rings and left modules II
 
Syntaxclnr 27416 Extend class notation with the class of left Noetherian rings.
 class LNoeR
 
Definitiondf-lnr 27417 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 27418 Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  <->  ( A  e.  Ring  /\  (ringLMod `  A )  e. LNoeM ) )
 
Theoremlnrrng 27419 Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  ->  A  e.  Ring
 )
 
Theoremlnrlnm 27420 Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  ->  (ringLMod `  A )  e. LNoeM )
 
Theoremislnr2 27421* 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 27422 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 27423* 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 27424 Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( R  e. LPIR  ->  R  e. LNoeR )
 
Theoremlnrfrlm 27425 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 27426 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 27427 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 )
 
18.17.49  Hilbert's Basis Theorem
 
Syntaxcldgis 27428 The leading ideal sequence used in the Hilbert Basis Theorem.
 class ldgIdlSeq
 
Definitiondf-ldgis 27429* 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 27437. (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 27430* 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 27431 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 27432 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 27433 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 27434 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 27435* 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 27436* 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 27437 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 )
 
18.17.50  Additional material on polynomials [DEPRECATED]
 
Syntaxcmnc 27438 Extend class notation with the class of monic polynomials.
 class  Monic
 
Syntaxcplylt 27439 Extend class notatin with the class of limited-degree polynomials.
 class Poly<
 
Definitiondf-mnc 27440* 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 27441* 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 27442 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 27443 A nonzero polynomial with a root has positive degree. TODO: use in aaliou2 19736. (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 27444 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 27445 A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  P  e.  (Poly `  S )
 )
 
Theoremmnccoe 27446 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 27447 A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  P  =/=  0 p )
 
18.17.51  Degree and minimal polynomial of algebraic numbers
 
Syntaxcdgraa 27448 Extend class notation to include the degree function for algebraic numbers.
 class degAA
 
Syntaxcmpaa 27449 Extend class notation to include the minimal polynomial for an algebraic number.
 class minPolyAA
 
Definitiondf-dgraa 27450* 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 27451* 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 27452* 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 27453* 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 27454 Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
 
Theoremdgraaf 27455 Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |- degAA : AA --> NN
 
Theoremdgraaub 27456 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 27457 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 27458* 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 27459* 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 27460 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 27461 Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (minPolyAA `  A )  e.  (Poly `  QQ ) )
 
Theoremmpaadgr 27462 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 27463 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 27464 Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  ( (coeff `  (minPolyAA `  A ) ) `  (degAA `  A ) )  =  1 )
 
18.17.52  Algebraic integers I
 
Syntaxcitgo 27465 Extend class notation with the integral-over predicate.
 class IntgOver
 
Syntaxcza 27466 Extend class notation with the class of algebraic integers.
 class
 
Definitiondf-itgo 27467* 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 27470. 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 27468 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 27469* 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 27470 The standard algebraic numbers  AA are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  AA  =  (IntgOver `  QQ )
 
Theoremitgoss 27471 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 27472 All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  (IntgOver `  S )  C_  CC
 
Theoremcnsrexpcl 27473 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 27474* 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 27475 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 27476* 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 27477 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 27478 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 27479 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 27480 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 27481* 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 27482* 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 )
 
18.17.53  Finite cardinality [SO]
 
Theoremen1uniel 27483 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  ( S  ~~  1o  ->  U. S  e.  S )
 
Theoremen2eleq 27484 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 27485 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 )
 
18.17.54  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 13421. If order is not significant, it is simpler to use families instead.

 
Theoremissubmd 27486* 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 ) )
 
18.17.55  Transpositions in the symmetric group
 
Syntaxcpmtr 27487 Syntax for the transposition generator function.
 class pmTrsp
 
Definitiondf-pmtr 27488* 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 27489 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 27490 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 27491 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 27492 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 27493 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 27494 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 27495 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 27496* 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 27497* 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 27498 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 27499 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 27500 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 )
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