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Theorem List for Metamath Proof Explorer - 26601-26700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuvcfval 26601* Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  W )  ->  U  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) )
 
Theoremuvcval 26602* Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I ) 
 ->  ( U `  J )  =  ( k  e.  I  |->  if (
 k  =  J ,  .1.  ,  .0.  ) ) )
 
Theoremuvcvval 26603 Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  /\  K  e.  I )  ->  ( ( U `  J ) `
  K )  =  if ( K  =  J ,  .1.  ,  .0.  ) )
 
Theoremuvcvvcl 26604 A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  /\  K  e.  I )  ->  ( ( U `  J ) `
  K )  e. 
 {  .0.  ,  .1.  } )
 
Theoremuvcvvcl2 26605 A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  J  e.  I )   &    |-  ( ph  ->  K  e.  I
 )   =>    |-  ( ph  ->  (
 ( U `  J ) `  K )  e.  B )
 
Theoremuvcvv1 26606 The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  J  e.  I )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ph  ->  (
 ( U `  J ) `  J )  =  .1.  )
 
Theoremuvcvv0 26607 The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  J  e.  I )   &    |-  ( ph  ->  K  e.  I
 )   &    |-  ( ph  ->  J  =/=  K )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ph  ->  ( ( U `  J ) `  K )  =  .0.  )
 
Theoremuvcff 26608 Domain and range of the unit vector generator; ring condition required to be sure 1 and 0 are actually in the ring. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  W ) 
 ->  U : I --> B )
 
Theoremuvcf1 26609 In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   =>    |-  (
 ( R  e. NzRing  /\  I  e.  W )  ->  U : I -1-1-> B )
 
Theoremuvcresum 26610 Any element of a free module can be expressed as a finite linear combination of unit vectors. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Jul-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   &    |-  .x.  =  ( .s `  Y )   =>    |-  ( ( R  e.  Ring  /\  I  e.  W  /\  X  e.  B ) 
 ->  X  =  ( Y 
 gsumg  ( X  o F  .x.  U ) ) )
 
Theoremfrlmsplit2 26611* Restriction is homomoprhic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  Y  =  ( R freeLMod  U )   &    |-  Z  =  ( R freeLMod  V )   &    |-  B  =  ( Base `  Y )   &    |-  C  =  ( Base `  Z )   &    |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y LMHom  Z ) )
 
Theoremfrlmsslss 26612* A subset of a free module obtained by restricting the support set is a submodule.  J is the set of forbidden unit vectors. (Contributed by Stefan O'Rear, 4-Feb-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  U  =  ( LSubSp `  Y )   &    |-  B  =  ( Base `  Y )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( x  |`  J )  =  ( J  X.  {  .0.  } ) }   =>    |-  ( ( R  e.  Ring  /\  I  e.  V  /\  J  C_  I )  ->  C  e.  U )
 
Theoremfrlmsslss2 26613* A subset of a free module obtained by restricting the support set is a submodule.  J is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  U  =  ( LSubSp `  Y )   &    |-  B  =  ( Base `  Y )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( `' x " ( _V  \  {  .0.  } )
 )  C_  J }   =>    |-  (
 ( R  e.  Ring  /\  I  e.  V  /\  J  C_  I )  ->  C  e.  U )
 
Theoremfrlmssuvc1 26614* A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  B  =  ( Base `  F )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  F )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( `' x " ( _V  \  {  .0.  } )
 )  C_  J }   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  J 
 C_  I )   &    |-  ( ph  ->  L  e.  J )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  ( X  .x.  ( U `  L ) )  e.  C )
 
Theoremfrlmssuvc2 26615* A nonzero scalar multiple of a unit vector not included in a support-restriction subspace is not included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  B  =  ( Base `  F )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  F )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( `' x " ( _V  \  {  .0.  } )
 )  C_  J }   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  J 
 C_  I )   &    |-  ( ph  ->  L  e.  ( I  \  J ) )   &    |-  ( ph  ->  X  e.  ( K  \  {  .0.  } ) )   =>    |-  ( ph  ->  -.  ( X  .x.  ( U `  L ) )  e.  C )
 
Theoremfrlmsslsp 26616* A subset of a free module obtained by restricting the support set is spanned by the relevant unit vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  K  =  ( LSpan `  Y )   &    |-  B  =  ( Base `  Y )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( `' x " ( _V  \  {  .0.  } )
 )  C_  J }   =>    |-  (
 ( R  e.  Ring  /\  I  e.  V  /\  J  C_  I )  ->  ( K `  ( U
 " J ) )  =  C )
 
Theoremfrlmlbs 26617 The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  J  =  (LBasis `  F )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  V ) 
 ->  ran  U  e.  J )
 
Theoremfrlmup1 26618* Any assignment of unit vectors to target vectors can be extended (uniquely) to a homomorphism from a free module to an arbitrary other module on the same base ring. (Contributed by Stefan O'Rear, 7-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  ->  E  e.  ( F LMHom  T ) )
 
Theoremfrlmup2 26619* The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-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  ->  Y  e.  I )   &    |-  U  =  ( R unitVec  I )   =>    |-  ( ph  ->  ( E `  ( U `  Y ) )  =  ( A `  Y ) )
 
Theoremfrlmup3 26620* The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-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 )   &    |-  K  =  ( LSpan `  T )   =>    |-  ( ph  ->  ran  E  =  ( K `  ran  A ) )
 
Theoremfrlmup4 26621* Universal propery of the free module by existential uniquenes. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  R  =  (Scalar `  T )   &    |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  C  =  ( Base `  T )   =>    |-  (
 ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C ) 
 ->  E! m  e.  ( F LMHom  T ) ( m  o.  U )  =  A )
 
Theoremellspd 26622* The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.)
 |-  N  =  ( LSpan `  M )   &    |-  B  =  ( Base `  M )   &    |-  K  =  ( Base `  S )   &    |-  S  =  (Scalar `  M )   &    |-  .0.  =  ( 0g `  S )   &    |- 
 .x.  =  ( .s `  M )   &    |-  ( ph  ->  F : I --> B )   &    |-  ( ph  ->  M  e.  LMod
 )   &    |-  ( ph  ->  I  e.  _V )   =>    |-  ( ph  ->  ( X  e.  ( N `  ( F " I
 ) )  <->  E. f  e.  ( K  ^m  I ) ( ( `' f "
 ( _V  \  {  .0.  } ) )  e. 
 Fin  /\  X  =  ( M  gsumg  ( f  o F  .x.  F ) ) ) ) )
 
Theoremelfilspd 26623* Simplified version of ellspd 26622 when the spanning set is finite: all linear combinations are then acceptable. (Contributed by Stefan O'Rear, 7-Feb-2015.)
 |-  N  =  ( LSpan `  M )   &    |-  B  =  ( Base `  M )   &    |-  K  =  ( Base `  S )   &    |-  S  =  (Scalar `  M )   &    |-  .0.  =  ( 0g `  S )   &    |- 
 .x.  =  ( .s `  M )   &    |-  ( ph  ->  F : I --> B )   &    |-  ( ph  ->  M  e.  LMod
 )   &    |-  ( ph  ->  I  e.  Fin )   =>    |-  ( ph  ->  ( X  e.  ( N `  ( F " I
 ) )  <->  E. f  e.  ( K  ^m  I ) X  =  ( M  gsumg  ( f  o F  .x.  F ) ) ) )
 
18.16.45  Every set admits a group structure iff choice
 
Theoremunxpwdom3 26624* Weaker version of unxpwdom 7271 where a function is required only to be cancellative, not an injection.  D and  B are to be thought of as "large" "horizonal" sets, the others as "small". Because the operator is row-wise injective, but the whole row cannot inject into  A, each row must hit an element of 
B; by column injectivity, each row can be identified in at least one way by the  B element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  D  e.  X )   &    |-  (
 ( ph  /\  a  e.  C  /\  b  e.  D )  ->  (
 a  .+  b )  e.  ( A  u.  B ) )   &    |-  ( ( (
 ph  /\  a  e.  C )  /\  ( b  e.  D  /\  c  e.  D ) )  ->  ( ( a  .+  b )  =  (
 a  .+  c )  <->  b  =  c ) )   &    |-  ( ( ( ph  /\  d  e.  D ) 
 /\  ( a  e.  C  /\  c  e.  C ) )  ->  ( ( c  .+  d )  =  (
 a  .+  d )  <->  c  =  a ) )   &    |-  ( ph  ->  -.  D  ~<_  A )   =>    |-  ( ph  ->  C  ~<_*  ( D  X.  B ) )
 
Theoremenfixsn 26625* Given two equipollent sets, a bijection can always be chosen which fixes a single point. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  (
 ( A  e.  X  /\  B  e.  Y  /\  X  ~~  Y )  ->  E. f ( f : X -1-1-onto-> Y  /\  ( f `
  A )  =  B ) )
 
Theoremmapfien2 26626* Equinumerousity relation for sets of finitely supported functions. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  S  =  { x  e.  ( B  ^m  A )  |  ( `' x "
 ( _V  \  {  .0.  } ) )  e. 
 Fin }   &    |-  T  =  { x  e.  ( D  ^m  C )  |  ( `' x " ( _V  \  { W } )
 )  e.  Fin }   &    |-  ( ph  ->  A  ~~  C )   &    |-  ( ph  ->  B  ~~  D )   &    |-  ( ph  ->  .0. 
 e.  B )   &    |-  ( ph  ->  W  e.  D )   =>    |-  ( ph  ->  S  ~~  T )
 
Theoremfsuppeq 26627 Two ways of writing the support of a function with known codomain. MOVABLE SHORTEN nn0supp (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( F : I --> S  ->  ( `' F " ( _V  \  { X } )
 )  =  ( `' F " ( S 
 \  { X }
 ) ) )
 
Theorempwfi2f1o 26628* The pw2f1o 6935 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  S  =  { y  e.  ( 2o  ^m  A )  |  ( `' y "
 ( _V  \  { (/)
 } ) )  e. 
 Fin }   &    |-  F  =  ( x  e.  S  |->  ( `' x " { 1o } ) )   =>    |-  ( A  e.  V  ->  F : S -1-1-onto-> ( ~P A  i^i  Fin ) )
 
Theorempwfi2en 26629* Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  S  =  { y  e.  ( 2o  ^m  A )  |  ( `' y "
 ( _V  \  { (/)
 } ) )  e. 
 Fin }   =>    |-  ( A  e.  V  ->  S  ~~  ( ~P A  i^i  Fin )
 )
 
Theoremfrlmpwfi 26630 Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  R  =  (ℤ/n `  2 )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  Y )   =>    |-  ( I  e.  V  ->  B  ~~  ( ~P I  i^i  Fin )
 )
 
Theoremgicabl 26631 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  ( G  ~=ph𝑔 
 H  ->  ( G  e.  Abel 
 <->  H  e.  Abel )
 )
 
Theoremimasgim 26632 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 26633 Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  Base  Fn 
 _V
 
Theoremisnumbasgrplem1 26634 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 26635 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( S  e.  V  ->  (har `  S )  =/=  (/) )
 
Theoremnuminfctb 26636 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 26637 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 26638 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 26639 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 26640 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 26641 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  \  { (/) } ) )
 
18.16.46  Independent sets and families
 
Syntaxclindf 26642 The class relationship of independent families in a module.
 class LIndF
 
Syntaxclinds 26643 The class generator of independent sets in a module.
 class LIndS
 
Definitiondf-lindf 26644* 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 26664, 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 26676) and only one representation for each element of the range (islindf5 26677). (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 26645* An independent set is a set which is independent as a family. See also islinds3 26672 and islinds4 26673. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- LIndS  =  ( w  e.  _V  |->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s ) LIndF 
 w } )
 
Theoremrellindf 26646 The independent-family predicate is a proper relation and can be used with brrelexi 4717. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  Rel LIndF
 
Theoremislinds 26647 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 26648 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 26649 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 26650* 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 26651* 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 26652* 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 26653 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 26654 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 26655 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 26656 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 26657 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 26658 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 26659 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 26660 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 26661 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 26662 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 26663 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 26664 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 26665 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 26666 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 26667 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 26668 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 26669 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 26670 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 26671 A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  J  =  (LBasis `  W )   =>    |-  J  C_  (LIndS `  W )
 
Theoremislinds3 26672 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 26673* 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 ) )
 
18.16.47  Characterization of free modules
 
Theoremlmimlbs 26674 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 26675 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 26676* 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 26677* 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 26678* 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 26679 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 26680* 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 15881 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 ) ) )
 
18.16.48  Noetherian rings and left modules II
 
Syntaxclnr 26681 Extend class notation with the class of left Noetherian rings.
 class LNoeR
 
Definitiondf-lnr 26682 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 26683 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 26684 Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  ->  A  e.  Ring
 )
 
Theoremlnrlnm 26685 Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  ->  (ringLMod `  A )  e. LNoeM )
 
Theoremislnr2 26686* 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 26687 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 26688* 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 26689 Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( R  e. LPIR  ->  R  e. LNoeR )
 
Theoremlnrfrlm 26690 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 26691 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 26692 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 )
 
18.16.49  Hilbert's Basis Theorem
 
Syntaxcldgis 26693 The leading ideal sequence used in the Hilbert Basis Theorem.
 class ldgIdlSeq
 
Definitiondf-ldgis 26694* 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 26702. (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 26695* 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 26696 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 26697 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 26698 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 26699 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 26700* 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 )
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