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Theorem List for Metamath Proof Explorer - 15701-15800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlssvsubcl 15701 Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  .-  =  ( -g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( X  e.  U  /\  Y  e.  U )
 )  ->  ( X  .-  Y )  e.  U )
 
Theoremlssvancl1 15702 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. TODO: notice similarity to lspindp3 15889. Can it be used along with lspsnne1 15870, lspsnne2 15871 to shorten this proof? (Contributed by NM, 14-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  -.  Y  e.  U )   =>    |-  ( ph  ->  -.  ( X  .+  Y )  e.  U )
 
Theoremlssvancl2 15703 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. (Contributed by NM, 20-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  -.  Y  e.  U )   =>    |-  ( ph  ->  -.  ( Y  .+  X )  e.  U )
 
Theoremlss0cl 15704 The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  .0.  e.  U )
 
Theoremlsssn0 15705 The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( W  e.  LMod 
 ->  {  .0.  }  e.  S )
 
Theoremlss0ss 15706 The zero subspace is included in every subspace. (sh0le 22019 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  S )  ->  {  .0.  } 
 C_  X )
 
Theoremlssle0 15707 No subspace is smaller than the zero subspace. (shle0 22021 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  S )  ->  ( X 
 C_  {  .0.  }  <->  X  =  {  .0.  } ) )
 
Theoremlssne0 15708* A nonzero subspace has a nonzero vector. (shne0i 22027 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( X  e.  S  ->  ( X  =/=  {  .0.  }  <->  E. y  e.  X  y  =/=  .0.  ) )
 
Theoremlssneln0 15709 A vector which doesn't belong to a subspace is nonzero. (Contributed by NM, 14-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  U )   =>    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )
 
Theoremlssssr 15710* Conclude subspace ordering from nonzero vector membership. (ssrdv 3185 analog.) (Contributed by NM, 17-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T 
 C_  V )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ( ph  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( x  e.  T  ->  x  e.  U ) )   =>    |-  ( ph  ->  T  C_  U )
 
Theoremlssvacl 15711 Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .+  =  ( +g  `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( X  e.  U  /\  Y  e.  U )
 )  ->  ( X  .+  Y )  e.  U )
 
Theoremlssvscl 15712 Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  B  =  (
 Base `  F )   &    |-  S  =  ( LSubSp `  W )   =>    |-  (
 ( ( W  e.  LMod  /\  U  e.  S ) 
 /\  ( X  e.  B  /\  Y  e.  U ) )  ->  ( X 
 .x.  Y )  e.  U )
 
Theoremlssvnegcl 15713 Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  ( inv g `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  X )  e.  U )
 
Theoremlsssubg 15714 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S ) 
 ->  U  e.  (SubGrp `  W ) )
 
Theoremlsssssubg 15715 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
 
Theoremislss3 15716 A linear subspace of a module is a subset which is a module in its own right. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  X  =  ( Ws  U )   &    |-  V  =  (
 Base `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( W  e.  LMod  ->  ( U  e.  S  <->  ( U  C_  V  /\  X  e.  LMod ) ) )
 
Theoremlsslmod 15717 A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod
 )
 
Theoremlsslss 15718 The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   &    |-  T  =  ( LSubSp `  X )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  S ) 
 ->  ( V  e.  T  <->  ( V  e.  S  /\  V  C_  U ) ) )
 
Theoremislss4 15719* A linear subspace is a subgroup which respects scalar multiplication. (Contributed by Stefan O'Rear, 11-Dec-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( W  e.  LMod  ->  ( U  e.  S  <->  ( U  e.  (SubGrp `  W )  /\  A. a  e.  B  A. b  e.  U  ( a  .x.  b )  e.  U ) ) )
 
Theoremlss1d 15720* One-dimensional subspace (or zero-dimensional if  X is the zero vector). (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  S  =  ( LSubSp `  W )   =>    |-  (
 ( W  e.  LMod  /\  X  e.  V ) 
 ->  { v  |  E. k  e.  K  v  =  ( k  .x.  X ) }  e.  S )
 
Theoremlssintcl 15721 The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  e.  S )
 
Theoremlssincl 15722 The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
 
Theoremlssmre 15723 The subspaces of a module comprise a Moore system on the vectors of the module. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  B  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( W  e.  LMod 
 ->  S  e.  (Moore `  B ) )
 
Theoremlssacs 15724 Submodules are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  B  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( W  e.  LMod 
 ->  S  e.  (ACS `  B ) )
 
Theoremprdsvscacl 15725* Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .s `  Y )   &    |-  K  =  ( Base `  S )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R : I --> LMod )   &    |-  ( ph  ->  F  e.  K )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ( ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x ) )  =  S )   =>    |-  ( ph  ->  ( F  .x.  G )  e.  B )
 
Theoremprdslmodd 15726* The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R : I --> LMod )   &    |-  ( ( ph  /\  y  e.  I ) 
 ->  (Scalar `  ( R `  y ) )  =  S )   =>    |-  ( ph  ->  Y  e.  LMod )
 
Theorempwslmod 15727 The product of a family of left modules is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  LMod  /\  I  e.  V )  ->  Y  e.  LMod )
 
Syntaxclspn 15728 Extend class notation with span of a set of vectors.
 class  LSpan
 
Definitiondf-lsp 15729* Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
 |- 
 LSpan  =  ( w  e.  _V  |->  ( s  e. 
 ~P ( Base `  w )  |->  |^| { t  e.  ( LSubSp `  w )  |  s  C_  t }
 ) )
 
Theoremlspfval 15730* The span function for a left vector space (or a left module). (df-span 21888 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( W  e.  X  ->  N  =  ( s  e. 
 ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) )
 
Theoremlspf 15731 The span operator on a left module maps subsets to subsets. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( W  e.  LMod  ->  N : ~P V --> S )
 
Theoremlspval 15732* The span of a set of vectors (in a left module). (spanval 21912 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  |^| { t  e.  S  |  U  C_  t } )
 
Theoremlspcl 15733 The span of a set of vectors is a subspace. (spancl 21915 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  e.  S )
 
Theoremlspsncl 15734 The span of a singleton is a subspace (frequently used special case of lspcl 15733). (Contributed by NM, 17-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( N `  { X } )  e.  S )
 
Theoremlspprcl 15735 The span of a pair is a subspace (frequently used special case of lspcl 15733). (Contributed by NM, 11-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  e.  S )
 
Theoremlsptpcl 15736 The span of an unordered triple is a subspace (frequently used special case of lspcl 15733). (Contributed by NM, 22-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   =>    |-  ( ph  ->  ( N `  { X ,  Y ,  Z }
 )  e.  S )
 
Theoremlspsnsubg 15737 The span of a singleton is an additive subgroup (frequently used special case of lspcl 15733). (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `
  { X }
 )  e.  (SubGrp `  W ) )
 
Theorem00lsp 15738 fvco4i 5597 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  (/)  =  ( LSpan `  (/) )
 
Theoremlspid 15739 The span of a subspace is itself. (spanid 21926 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `
  U )  =  U )
 
Theoremlspssv 15740 A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `
  U )  C_  V )
 
Theoremlspss 15741 Span preserves subset ordering. (spanss 21927 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  ( N `  T )  C_  ( N `
  U ) )
 
Theoremlspssid 15742 A set of vectors is a subset of its span. (spanss2 21924 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  C_  ( N `  U ) )
 
Theoremlspidm 15743 The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `
  ( N `  U ) )  =  ( N `  U ) )
 
Theoremlspun 15744 The span of union is the span of the union of spans. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  T  C_  V  /\  U  C_  V )  ->  ( N `  ( T  u.  U ) )  =  ( N `  ( ( N `
  T )  u.  ( N `  U ) ) ) )
 
Theoremlspssp 15745 If a set of vectors is a subset of a subspace, then the span of those vectors is also contained in the subspace. (Contributed by Mario Carneiro, 4-Sep-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S  /\  T  C_  U )  ->  ( N `  T )  C_  U )
 
Theoremmrclsp 15746 Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  U  =  ( LSubSp `  W )   &    |-  K  =  (
 LSpan `  W )   &    |-  F  =  (mrCls `  U )   =>    |-  ( W  e.  LMod  ->  K  =  F )
 
Theoremlspsnss 15747 The span of the singleton of a subspace member is included in the subspace. (spansnss 22150 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U )
 
Theoremlspsnel3 15748 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 22151 analog.) (Contributed by NM, 4-Jul-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  ( N `  { X }
 ) )   =>    |-  ( ph  ->  Y  e.  U )
 
Theoremlspprss 15749 The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  C_  U )
 
Theoremlspsnid 15750 A vector belongs to the span of its singleton. (spansnid 22142 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
 
Theoremlspsnel6 15751 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( X  e.  U  <->  ( X  e.  V  /\  ( N `  { X } )  C_  U ) ) )
 
Theoremlspsnel5 15752 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( X  e.  U  <->  ( N `  { X } )  C_  U ) )
 
Theoremlspsnel5a 15753 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   =>    |-  ( ph  ->  ( N `  { X } )  C_  U )
 
Theoremlspprid1 15754 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  X  e.  ( N `  { X ,  Y }
 ) )
 
Theoremlspprid2 15755 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  Y  e.  ( N `  { X ,  Y }
 ) )
 
Theoremlspprvacl 15756 The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  ( N `  { X ,  Y }
 ) )
 
Theoremlssats2 15757* A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  U  =  U_ x  e.  U  ( N `  { x } ) )
 
Theoremlspsneli 15758 A scalar product with a vector belongs to the span of its singleton. (spansnmul 22143 analog.) (Contributed by NM, 2-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( A  .x.  X )  e.  ( N `  { X } ) )
 
Theoremlspsn 15759* Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( N `  { X } )  =  {
 v  |  E. k  e.  K  v  =  ( k  .x.  X ) } )
 
Theoremlspsnel 15760* Member of span of the singleton of a vector. (elspansn 22145 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( U  e.  ( N `  { X }
 ) 
 <-> 
 E. k  e.  K  U  =  ( k  .x.  X ) ) )
 
Theoremlspsnvsi 15761 Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  V )  ->  ( N `  { ( R  .x.  X ) }
 )  C_  ( N ` 
 { X } )
 )
 
Theoremlspsnss2 15762* Comparable spans of singletons must have proportional vectors. See lspsneq 15875 for equal span version. (Contributed by NM, 7-Jun-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (Scalar `  W )   &    |-  K  =  (
 Base `  S )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( N `  { X } )  C_  ( N `
  { Y }
 ) 
 <-> 
 E. k  e.  K  X  =  ( k  .x.  Y ) ) )
 
Theoremlspsnneg 15763 Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  M  =  ( inv g `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( N `  { ( M `  X ) }
 )  =  ( N `
  { X }
 ) )
 
Theoremlspsnsub 15764 Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( N `  { ( X  .-  Y ) }
 )  =  ( N `
  { ( Y 
 .-  X ) }
 ) )
 
Theoremlspsn0 15765 Span of the singleton of the zero vector. (spansn0 22120 analog.) (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( W  e.  LMod 
 ->  ( N `  {  .0.  } )  =  {  .0.  } )
 
Theoremlsp0 15766 Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( W  e.  LMod 
 ->  ( N `  (/) )  =  {  .0.  } )
 
Theoremlspuni0 15767 Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( W  e.  LMod 
 ->  U. ( N `  (/) )  =  .0.  )
 
Theoremlspun0 15768 The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X 
 C_  V )   =>    |-  ( ph  ->  ( N `  ( X  u.  {  .0.  }
 ) )  =  ( N `  X ) )
 
Theoremlspsneq0 15769 Span of the singleton is the zero subspace iff the vector is zero. (Contributed by NM, 27-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( ( N `  { X } )  =  {  .0.  }  <->  X  =  .0.  ) )
 
Theoremlspsneq0b 15770 Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( N `  { X }
 )  =  ( N `
  { Y }
 ) )   =>    |-  ( ph  ->  ( X  =  .0.  <->  Y  =  .0.  ) )
 
Theoremlmodindp1 15771 Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   =>    |-  ( ph  ->  ( X  .+  Y )  =/= 
 .0.  )
 
Theoremlsslsp 15772 Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap  M `  G and  N `
 G since we are computing a property of  N `  G? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015.
 |-  X  =  ( Ws  U )   &    |-  M  =  (
 LSpan `  W )   &    |-  N  =  ( LSpan `  X )   &    |-  L  =  ( LSubSp `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  =  ( N `  G ) )
 
Theoremlss0v 15773 The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.)
 |-  X  =  ( Ws  U )   &    |-  .0.  =  ( 0g `  W )   &    |-  Z  =  ( 0g `  X )   &    |-  L  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  Z  =  .0.  )
 
Theoremlsspropd 15774* If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  B  C_  W )   &    |-  ( ( ph  /\  ( x  e.  W  /\  y  e.  W ) )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  e.  W )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  =  ( x ( .s
 `  L ) y ) )   &    |-  ( ph  ->  P  =  ( Base `  (Scalar `  K ) ) )   &    |-  ( ph  ->  P  =  ( Base `  (Scalar `  L ) ) )   =>    |-  ( ph  ->  (
 LSubSp `  K )  =  ( LSubSp `  L )
 )
 
Theoremlsppropd 15775* If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  B  C_  W )   &    |-  ( ( ph  /\  ( x  e.  W  /\  y  e.  W ) )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  e.  W )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  =  ( x ( .s
 `  L ) y ) )   &    |-  ( ph  ->  P  =  ( Base `  (Scalar `  K ) ) )   &    |-  ( ph  ->  P  =  ( Base `  (Scalar `  L ) ) )   &    |-  ( ph  ->  K  e.  _V )   &    |-  ( ph  ->  L  e.  _V )   =>    |-  ( ph  ->  ( LSpan `  K )  =  ( LSpan `  L )
 )
 
10.6.3  Homomorphisms and isomorphisms of left modules
 
Syntaxclmhm 15776 Extend class notation with the generator of left module hom-sets.
 class LMHom
 
Syntaxclmim 15777 The class of left module isomorphism sets.
 class LMIso
 
Syntaxclmic 15778 The class of the left module isomorphism relation.
 class  ~=ph𝑚
 
Definitiondf-lmhm 15779* A homomorphism of left modules is a group homomorphism which additionally preserves the scalar product. This requires both structures to be left modules over the same ring. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |- LMHom  =  ( s  e.  LMod ,  t  e.  LMod  |->  { f  e.  ( s  GrpHom  t )  |  [. (Scalar `  s
 )  /  w ]. (
 (Scalar `  t )  =  w  /\  A. x  e.  ( Base `  w ) A. y  e.  ( Base `  s ) ( f `  ( x ( .s `  s
 ) y ) )  =  ( x ( .s `  t ) ( f `  y
 ) ) ) }
 )
 
Definitiondf-lmim 15780* An isomorphism of modules is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group and scalar operations. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |- LMIso  =  ( s  e.  LMod ,  t  e.  LMod  |->  { g  e.  ( s LMHom  t )  |  g : (
 Base `  s ) -1-1-onto-> ( Base `  t ) } )
 
Definitiondf-lmic 15781 Two modules are said to be isomorphic iff they are connected by at least one isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |- 
 ~=ph𝑚  =  ( `' LMIso  " ( _V  \  1o ) )
 
Theoremreldmlmhm 15782 Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |- 
 Rel  dom LMHom
 
Theoremlmimfn 15783 Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |- LMIso  Fn  ( LMod  X.  LMod )
 
Theoremislmhm 15784* Property of being a homomorphism of left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
 |-  K  =  (Scalar `  S )   &    |-  L  =  (Scalar `  T )   &    |-  B  =  ( Base `  K )   &    |-  E  =  (
 Base `  S )   &    |-  .x.  =  ( .s `  S )   &    |-  .X. 
 =  ( .s `  T )   =>    |-  ( F  e.  ( S LMHom  T )  <->  ( ( S  e.  LMod  /\  T  e.  LMod
 )  /\  ( F  e.  ( S  GrpHom  T ) 
 /\  L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x 
 .x.  y ) )  =  ( x  .X.  ( F `  y ) ) ) ) )
 
Theoremislmhm3 15785* Property of a module homomorphism, similar to ismhm 14417. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  K  =  (Scalar `  S )   &    |-  L  =  (Scalar `  T )   &    |-  B  =  ( Base `  K )   &    |-  E  =  (
 Base `  S )   &    |-  .x.  =  ( .s `  S )   &    |-  .X. 
 =  ( .s `  T )   =>    |-  ( ( S  e.  LMod  /\  T  e.  LMod )  ->  ( F  e.  ( S LMHom  T )  <->  ( F  e.  ( S  GrpHom  T ) 
 /\  L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x 
 .x.  y ) )  =  ( x  .X.  ( F `  y ) ) ) ) )
 
Theoremlmhmlem 15786 Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  K  =  (Scalar `  S )   &    |-  L  =  (Scalar `  T )   =>    |-  ( F  e.  ( S LMHom  T )  ->  (
 ( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K )
 ) )
 
Theoremlmhmsca 15787 A homomorphism of left modules constrains both modules to the same ring of scalars. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  K  =  (Scalar `  S )   &    |-  L  =  (Scalar `  T )   =>    |-  ( F  e.  ( S LMHom  T )  ->  L  =  K )
 
Theoremlmghm 15788 A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( F  e.  ( S LMHom  T )  ->  F  e.  ( S  GrpHom  T ) )
 
Theoremlmhmlmod2 15789 A homomorphism of left modules has a left module as codomain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( F  e.  ( S LMHom  T )  ->  T  e.  LMod )
 
Theoremlmhmlmod1 15790 A homomorphism of left modules has a left module as domain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( F  e.  ( S LMHom  T )  ->  S  e.  LMod )
 
Theoremlmhmf 15791 A homomorphism of left modules is a function. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  (
 Base `  T )   =>    |-  ( F  e.  ( S LMHom  T )  ->  F : B --> C )
 
Theoremlmhmlin 15792 A homomorphism of left modules is 
K-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  K  =  (Scalar `  S )   &    |-  B  =  ( Base `  K )   &    |-  E  =  (
 Base `  S )   &    |-  .x.  =  ( .s `  S )   &    |-  .X. 
 =  ( .s `  T )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  X  e.  B  /\  Y  e.  E )  ->  ( F `  ( X  .x.  Y ) )  =  ( X  .X.  ( F `  Y ) ) )
 
Theoremlmodvsinv 15793 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `  W )   &    |-  M  =  ( inv g `  F )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( ( M `  R )  .x.  X )  =  ( N `  ( R  .x.  X ) ) )
 
Theoremlmodvsinv2 15794 Multiplying a negated vector by a scalar. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `  W )   &    |-  K  =  (
 Base `  F )   =>    |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( R  .x.  ( N `  X ) )  =  ( N `
  ( R  .x.  X ) ) )
 
Theoremislmhm2 15795* A one-equation proof of linearity of a left module homomorphism, similar to df-lss 15690. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  (
 Base `  T )   &    |-  K  =  (Scalar `  S )   &    |-  L  =  (Scalar `  T )   &    |-  E  =  ( Base `  K )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   &    |-  .x.  =  ( .s `  S )   &    |-  .X. 
 =  ( .s `  T )   =>    |-  ( ( S  e.  LMod  /\  T  e.  LMod )  ->  ( F  e.  ( S LMHom  T )  <->  ( F : B
 --> C  /\  L  =  K  /\  A. x  e.  E  A. y  e.  B  A. z  e.  B  ( F `  ( ( x  .x.  y )  .+  z ) )  =  ( ( x  .X.  ( F `  y ) )  .+^  ( F `  z ) ) ) ) )
 
Theoremislmhmd 15796* Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
 |-  X  =  ( Base `  S )   &    |-  .x.  =  ( .s `  S )   &    |-  .X.  =  ( .s `  T )   &    |-  K  =  (Scalar `  S )   &    |-  J  =  (Scalar `  T )   &    |-  N  =  ( Base `  K )   &    |-  ( ph  ->  S  e.  LMod )   &    |-  ( ph  ->  T  e.  LMod )   &    |-  ( ph  ->  J  =  K )   &    |-  ( ph  ->  F  e.  ( S  GrpHom  T ) )   &    |-  ( ( ph  /\  ( x  e.  N  /\  y  e.  X )
 )  ->  ( F `  ( x  .x.  y
 ) )  =  ( x  .X.  ( F `  y ) ) )   =>    |-  ( ph  ->  F  e.  ( S LMHom  T ) )
 
Theorem0lmhm 15797 The constant zero linear function between two modules. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- 
 .0.  =  ( 0g `  N )   &    |-  B  =  (
 Base `  M )   &    |-  S  =  (Scalar `  M )   &    |-  T  =  (Scalar `  N )   =>    |-  (
 ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  ( B  X.  {  .0.  }
 )  e.  ( M LMHom  N ) )
 
Theoremidlmhm 15798 The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  B  =  ( Base `  M )   =>    |-  ( M  e.  LMod  ->  (  _I  |`  B )  e.  ( M LMHom  M ) )
 
Theoreminvlmhm 15799 The negative function on a module is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  I  =  ( inv
 g `  M )   =>    |-  ( M  e.  LMod  ->  I  e.  ( M LMHom  M ) )
 
Theoremlmhmco 15800 The composition of two module-linear functions is module-linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o.  G )  e.  ( M LMHom  O ) )
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