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Theorem List for Metamath Proof Explorer - 26601-26700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisnumbasgrplem2 26601 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 26602 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 26603 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 26604 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 26605 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  \  { (/) } ) )
 
16.16.46  Independent sets and families
 
Syntaxclindf 26606 The class relationship of independent families in a module.
 class LIndF
 
Syntaxclinds 26607 The class generator of independent sets in a module.
 class LIndS
 
Definitiondf-lindf 26608* An independent family is a family of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements of the family. This is almost, but not quite, the same as a function into an independent set.

This is a defined concept because it matters in many cases whether independence is taken at a set or family level. For instance, a number is transcedental iff its nonzero powers are linearly independent. Is 1 transcedental? It has only one nonzero power.

We can almost define family independence as a family of unequal elements with independent range, as islindf3 26628, but taking that as primitive would lead to unpleasant corner case behavior with the zero ring.

This is equivalent to the common definition of having no nontrivial representations of zero (islindf4 26640) and only one representation for each element of the range (islindf5 26641). (Contributed by Stefan O'Rear, 24-Feb-2015.)

 |- LIndF  =  { <. f ,  w >.  |  ( f : dom  f
 --> ( Base `  w )  /\  [. (Scalar `  w )  /  s ]. A. x  e.  dom  f A. k  e.  ( ( Base `  s
 )  \  { ( 0g `  s ) }
 )  -.  ( k
 ( .s `  w ) ( f `  x ) )  e.  ( ( LSpan `  w ) `  ( f "
 ( dom  f  \  { x } ) ) ) ) }
 
Definitiondf-linds 26609* An independent set is a set which is independent as a family. See also islinds3 26636 and islinds4 26637. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- LIndS  =  ( w  e.  _V  |->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s ) LIndF 
 w } )
 
Theoremrellindf 26610 The independent-family predicate is a proper relation and can be used with brrelexi 4682. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  Rel LIndF
 
Theoremislinds 26611 Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   =>    |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W )  <->  ( X  C_  B  /\  (  _I  |`  X ) LIndF  W ) ) )
 
Theoremlinds1 26612 An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   =>    |-  ( X  e.  (LIndS `  W )  ->  X  C_  B )
 
Theoremlinds2 26613 An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( X  e.  (LIndS `  W )  ->  (  _I  |`  X ) LIndF  W )
 
Theoremislindf 26614* Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( LSpan `  W )   &    |-  S  =  (Scalar `  W )   &    |-  N  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ( W  e.  Y  /\  F  e.  X )  ->  ( F LIndF  W  <->  ( F : dom  F --> B  /\  A. x  e. 
 dom  F A. k  e.  ( N  \  {  .0.  } )  -.  (
 k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } )
 ) ) ) ) )
 
Theoremislinds2 26615* Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( LSpan `  W )   &    |-  S  =  (Scalar `  W )   &    |-  N  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( W  e.  Y  ->  ( F  e.  (LIndS `  W )  <->  ( F  C_  B  /\  A. x  e.  F  A. k  e.  ( N  \  {  .0.  } )  -.  (
 k  .x.  x )  e.  ( K `  ( F  \  { x }
 ) ) ) ) )
 
Theoremislindf2 26616* Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( LSpan `  W )   &    |-  S  =  (Scalar `  W )   &    |-  N  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B ) 
 ->  ( F LIndF  W  <->  A. x  e.  I  A. k  e.  ( N  \  {  .0.  }
 )  -.  ( k  .x.  ( F `  x ) )  e.  ( K `  ( F "
 ( I  \  { x } ) ) ) ) )
 
Theoremlindff 26617 Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   =>    |-  (
 ( F LIndF  W  /\  W  e.  Y )  ->  F : dom  F --> B )
 
Theoremlindfind 26618 A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  L  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  L )   &    |-  K  =  ( Base `  L )   =>    |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  -.  ( A  .x.  ( F `  E ) )  e.  ( N `  ( F " ( dom 
 F  \  { E } ) ) ) )
 
Theoremlindsind 26619 A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  L  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  L )   &    |-  K  =  ( Base `  L )   =>    |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  -.  ( A  .x.  E )  e.  ( N `  ( F  \  { E }
 ) ) )
 
Theoremlindfind2 26620 In a linearly independent family in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  K  =  ( LSpan `  W )   &    |-  L  =  (Scalar `  W )   =>    |-  (
 ( ( W  e.  LMod  /\  L  e. NzRing )  /\  F LIndF  W 
 /\  E  e.  dom  F )  ->  -.  ( F `  E )  e.  ( K `  ( F " ( dom  F  \  { E } )
 ) ) )
 
Theoremlindsind2 26621 In a linearly independent set in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  K  =  ( LSpan `  W )   &    |-  L  =  (Scalar `  W )   =>    |-  (
 ( ( W  e.  LMod  /\  L  e. NzRing )  /\  F  e.  (LIndS `  W )  /\  E  e.  F ) 
 ->  -.  E  e.  ( K `  ( F  \  { E } ) ) )
 
Theoremlindff1 26622 A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  L  =  (Scalar `  W )   =>    |-  (
 ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  F : dom  F
 -1-1-> B )
 
Theoremlindfrn 26623 The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  (
 ( W  e.  LMod  /\  F LIndF  W )  ->  ran  F  e.  (LIndS `  W ) )
 
Theoremf1lindf 26624 Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  (
 ( W  e.  LMod  /\  F LIndF  W  /\  G : K -1-1-> dom  F )  ->  ( F  o.  G ) LIndF  W )
 
Theoremlindfres 26625 Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  (
 ( W  e.  LMod  /\  F LIndF  W )  ->  ( F  |`  X ) LIndF  W )
 
Theoremlindsss 26626 Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  (
 ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  G  e.  (LIndS `  W ) )
 
Theoremf1linds 26627 A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  (
 ( W  e.  LMod  /\  S  e.  (LIndS `  W )  /\  F : D -1-1-> S )  ->  F LIndF  W )
 
Theoremislindf3 26628 In a nonzero ring, independent families can be equivalently characterized as renamings of independent sets. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  L  =  (Scalar `  W )   =>    |-  (
 ( W  e.  LMod  /\  L  e. NzRing )  ->  ( F LIndF  W  <->  ( F : dom  F -1-1-> _V  /\  ran  F  e.  (LIndS `  W )
 ) ) )
 
Theoremlindfmm 26629 Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  ( Base `  T )   =>    |-  (
 ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F : I
 --> B )  ->  ( F LIndF  S  <->  ( G  o.  F ) LIndF  T ) )
 
Theoremlindsmm 26630 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 26631 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 26632 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 26633 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 26634 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 26635 A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  J  =  (LBasis `  W )   =>    |-  J  C_  (LIndS `  W )
 
Theoremislinds3 26636 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 26637* 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 ) )
 
16.16.47  Characterization of free modules
 
Theoremlmimlbs 26638 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 26639 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 26640* 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 26641* 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 26642* 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 26643 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 26644* 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 15845 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 ) ) )
 
16.16.48  Noetherian rings and left modules II
 
Syntaxclnr 26645 Extend class notation with the class of left Noetherian rings.
 class LNoeR
 
Definitiondf-lnr 26646 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 26647 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 26648 Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  ->  A  e.  Ring
 )
 
Theoremlnrlnm 26649 Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  ->  (ringLMod `  A )  e. LNoeM )
 
Theoremislnr2 26650* 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 26651 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 26652* 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 26653 Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( R  e. LPIR  ->  R  e. LNoeR )
 
Theoremlnrfrlm 26654 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 26655 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 26656 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 )
 
16.16.49  Hilbert's Basis Theorem
 
Syntaxcldgis 26657 The leading ideal sequence used in the Hilbert Basis Theorem.
 class ldgIdlSeq
 
Definitiondf-ldgis 26658* 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 26666. (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 26659* 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 26660 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 26661 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 26662 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 26663 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 26664* 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 26665* 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 26666 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 )
 
16.16.50  Additional material on polynomials [DEPRECATED]
 
Syntaxcmnc 26667 Extend class notation with the class of monic polynomials.
 class  Monic
 
Syntaxcplylt 26668 Extend class notatin with the class of limited-degree polynomials.
 class Poly<
 
Definitiondf-mnc 26669* 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 26670* 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 26671 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 26672 A nonzero polynomial with a root has positive degree. TODO: use in aaliou2 19647. (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 26673 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 26674 A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  P  e.  (Poly `  S )
 )
 
Theoremmnccoe 26675 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 26676 A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)
 |-  ( P  e.  (  Monic  `  S )  ->  P  =/=  0 p )
 
16.16.51  Degree and minimal polynomial of algebraic numbers
 
Syntaxcdgraa 26677 Extend class notation to include the degree function for algebraic numbers.
 class degAA
 
Syntaxcmpaa 26678 Extend class notation to include the minimal polynomial for an algebraic number.
 class minPolyAA
 
Definitiondf-dgraa 26679* 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 26680* 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 26681* 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 26682* 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 26683 Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
 
Theoremdgraaf 26684 Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |- degAA : AA --> NN
 
Theoremdgraaub 26685 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 26686 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 26687* 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 26688* 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 26689 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 26690 Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  (minPolyAA `  A )  e.  (Poly `  QQ ) )
 
Theoremmpaadgr 26691 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 26692 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 26693 Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)
 |-  ( A  e.  AA  ->  ( (coeff `  (minPolyAA `  A ) ) `  (degAA `  A ) )  =  1 )
 
16.16.52  Algebraic integers I
 
Syntaxcitgo 26694 Extend class notation with the integral-over predicate.
 class IntgOver
 
Syntaxcza 26695 Extend class notation with the class of algebraic integers.
 class
 
Definitiondf-itgo 26696* 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 26699. 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 26697 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 26698* 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 26699 The standard algebraic numbers  AA are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  AA  =  (IntgOver `  QQ )
 
Theoremitgoss 26700 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 )
 )
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