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Theorem List for Metamath Proof Explorer - 29901-30000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlcvat 29901* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 23874 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T C U )   =>    |-  ( ph  ->  E. q  e.  A  ( T  .(+)  q )  =  U )
 
Theoremlsatcv0 29902 An atom covers the zero subspace. (atcv0 23850 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  {  .0.  } C Q )
 
Theoremlsatcveq0 29903 A subspace covered by an atom must be the zero subspace. (atcveq0 23856 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  ( 
 <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( U C Q  <->  U  =  {  .0.  } ) )
 
Theoremlsat0cv 29904 A subspace is an atom iff it covers the zero subspace. This could serve as an alternate definition of an atom. TODO: this is a quick-and-dirty proof that could probably be more efficient. (Contributed by NM, 14-Mar-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  ( 
 <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( U  e.  A  <->  {  .0.  } C U ) )
 
Theoremlcvexchlem1 29905 Lemma for lcvexch 29910. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( T  C.  ( T 
 .(+)  U )  <->  ( T  i^i  U )  C.  U ) )
 
Theoremlcvexchlem2 29906 Lemma for lcvexch 29910. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  ( T  i^i  U )  C_  R )   &    |-  ( ph  ->  R 
 C_  U )   =>    |-  ( ph  ->  ( ( R  .(+)  T )  i^i  U )  =  R )
 
Theoremlcvexchlem3 29907 Lemma for lcvexch 29910. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  T  C_  R )   &    |-  ( ph  ->  R 
 C_  ( T  .(+)  U ) )   =>    |-  ( ph  ->  (
 ( R  i^i  U )  .(+)  T )  =  R )
 
Theoremlcvexchlem4 29908 Lemma for lcvexch 29910. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T C ( T  .(+)  U )
 )   =>    |-  ( ph  ->  ( T  i^i  U ) C U )
 
Theoremlcvexchlem5 29909 Lemma for lcvexch 29910. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  ( T  i^i  U ) C U )   =>    |-  ( ph  ->  T C ( T  .(+)  U ) )
 
Theoremlcvexch 29910 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 23877 analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( ( T  i^i  U ) C U  <->  T C ( T 
 .(+)  U ) ) )
 
Theoremlcvp 29911 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 23883 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  (
 ( U  i^i  Q )  =  {  .0.  }  <->  U C ( U  .(+)  Q ) ) )
 
Theoremlcv1 29912 Covering property of a subspace plus an atom. (chcv1 23863 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( -.  Q  C_  U  <->  U C ( U  .(+)  Q ) ) )
 
Theoremlcv2 29913 Covering property of a subspace plus an atom. (chcv2 23864 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( U  C.  ( U 
 .(+)  Q )  <->  U C ( U 
 .(+)  Q ) ) )
 
Theoremlsatexch 29914 The atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem was originally proved by Hermann Grassmann in 1862. (atexch 23889 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  Q 
 C_  ( U  .(+)  R ) )   &    |-  ( ph  ->  ( U  i^i  Q )  =  {  .0.  }
 )   =>    |-  ( ph  ->  R  C_  ( U  .(+)  Q ) )
 
Theoremlsatnle 29915 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 23884 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( -.  Q  C_  U  <->  ( U  i^i  Q )  =  {  .0.  } ) )
 
Theoremlsatnem0 29916 The meet of distinct atoms is the zero subspace. (atnemeq0 23885 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   =>    |-  ( ph  ->  ( Q  =/=  R  <->  ( Q  i^i  R )  =  {  .0.  } ) )
 
Theoremlsatexch1 29917 The atom exch1ange property. (hlatexch1 30265 analog.) (Contributed by NM, 14-Jan-2015.)
 |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  S  e.  A )   &    |-  ( ph  ->  Q  C_  ( S  .(+)  R ) )   &    |-  ( ph  ->  Q  =/=  S )   =>    |-  ( ph  ->  R 
 C_  ( S  .(+)  Q ) )
 
Theoremlsatcv0eq 29918 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 23887 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   =>    |-  ( ph  ->  ( {  .0.  } C ( Q  .(+)  R )  <->  Q  =  R ) )
 
Theoremlsatcv1 29919 Two atoms covering the zero subspace are equal. (atcv1 23888 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  ( 
 <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  U C ( Q  .(+)  R ) )   =>    |-  ( ph  ->  ( U  =  {  .0.  }  <->  Q  =  R )
 )
 
Theoremlsatcvatlem 29920 Lemma for lsatcvat 29921. (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  U  =/=  {  .0.  }
 )   &    |-  ( ph  ->  U  C.  ( Q  .(+)  R ) )   &    |-  ( ph  ->  -.  Q  C_  U )   =>    |-  ( ph  ->  U  e.  A )
 
Theoremlsatcvat 29921 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 23894 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  U  =/=  {  .0.  }
 )   &    |-  ( ph  ->  U  C.  ( Q  .(+)  R ) )   =>    |-  ( ph  ->  U  e.  A )
 
Theoremlsatcvat2 29922 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 23895 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  Q  =/=  R )   &    |-  ( ph  ->  U C ( Q  .(+)  R ) )   =>    |-  ( ph  ->  U  e.  A )
 
Theoremlsatcvat3 29923 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23904 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  Q  =/=  R )   &    |-  ( ph  ->  -.  R  C_  U )   &    |-  ( ph  ->  Q  C_  ( U  .(+)  R ) )   =>    |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  A )
 
Theoremislshpcv 29924 Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   =>    |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U C V ) ) )
 
Theoreml1cvpat 29925 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 30345 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  U C V )   &    |-  ( ph  ->  -.  Q  C_  U )   =>    |-  ( ph  ->  ( U  .(+)  Q )  =  V )
 
Theoreml1cvat 29926 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 30346 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  Q  =/=  R )   &    |-  ( ph  ->  U C V )   &    |-  ( ph  ->  -.  Q  C_  U )   =>    |-  ( ph  ->  (
 ( Q  .(+)  R )  i^i  U )  e.  A )
 
Theoremlshpat 29927 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 30913 analog.) TODO: This changes  U C V in l1cvpat 29925 and l1cvat 29926 to  U  e.  H, which in turn change  U  e.  H in islshpcv 29924 to  U C V, with a couple of conversions of span to atom. Seems convoluted. Would a direct proof be better? (Contributed by NM, 11-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  Q  =/=  R )   &    |-  ( ph  ->  -.  Q  C_  U )   =>    |-  ( ph  ->  ( ( Q 
 .(+)  R )  i^i  U )  e.  A )
 
19.26.4  Functionals and kernels of a left vector space (or module)
 
Syntaxclfn 29928 Extend class notation with all linear functionals of a left module or left vector space.
 class LFnl
 
Definitiondf-lfl 29929* Define the set of all linear functionals (maps from vectors to the ring) of a left module or left vector space. (Contributed by NM, 15-Apr-2014.)
 |- LFnl  =  ( w  e.  _V  |->  { f  e.  ( (
 Base `  (Scalar `  w ) )  ^m  ( Base `  w ) )  | 
 A. r  e.  ( Base `  (Scalar `  w ) ) A. x  e.  ( Base `  w ) A. y  e.  ( Base `  w ) ( f `  ( ( r ( .s `  w ) x ) ( +g  `  w ) y ) )  =  ( ( r ( .r `  (Scalar `  w ) ) ( f `  x ) ) ( +g  `  (Scalar `  w ) ) ( f `  y ) ) } )
 
Theoremlflset 29930* The set of linear functionals in a left module or left vector space. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  D )   &    |-  .+^  =  ( +g  `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   =>    |-  ( W  e.  X  ->  F  =  { f  e.  ( K  ^m  V )  |  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( f `  ( ( r  .x.  x )  .+  y ) )  =  ( ( r  .X.  ( f `  x ) )  .+^  ( f `  y
 ) ) } )
 
Theoremislfl 29931* The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  D )   &    |-  .+^  =  ( +g  `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   =>    |-  ( W  e.  X  ->  ( G  e.  F  <->  ( G : V
 --> K  /\  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( G `  ( ( r  .x.  x )  .+  y ) )  =  ( ( r  .X.  ( G `  x ) )  .+^  ( G `  y ) ) ) ) )
 
Theoremlfli 29932 Property of a linear functional. (lnfnli 23548 analog.) (Contributed by NM, 16-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  D )   &    |-  .+^  =  ( +g  `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  Z  /\  G  e.  F  /\  ( R  e.  K  /\  X  e.  V  /\  Y  e.  V )
 )  ->  ( G `  ( ( R  .x.  X )  .+  Y ) )  =  ( ( R  .X.  ( G `  X ) )  .+^  ( G `  Y ) ) )
 
Theoremislfld 29933* Properties that determine a linear functional. TODO: use this in place of islfl 29931 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
 |-  ( ph  ->  V  =  (
 Base `  W ) )   &    |-  ( ph  ->  .+  =  (
 +g  `  W )
 )   &    |-  ( ph  ->  D  =  (Scalar `  W )
 )   &    |-  ( ph  ->  .x.  =  ( .s `  W ) )   &    |-  ( ph  ->  K  =  ( Base `  D ) )   &    |-  ( ph  ->  .+^  =  ( +g  `  D ) )   &    |-  ( ph  ->  .X. 
 =  ( .r `  D ) )   &    |-  ( ph  ->  F  =  (LFnl `  W ) )   &    |-  ( ph  ->  G : V --> K )   &    |-  ( ( ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  ( G `
  ( ( r 
 .x.  x )  .+  y ) )  =  ( ( r  .X.  ( G `  x ) )  .+^  ( G `  y ) ) )   &    |-  ( ph  ->  W  e.  X )   =>    |-  ( ph  ->  G  e.  F )
 
Theoremlflf 29934 A linear functional is a function from vectors to scalars. (lnfnfi 23549 analog.) (Contributed by NM, 15-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  X  /\  G  e.  F ) 
 ->  G : V --> K )
 
Theoremlflcl 29935 A linear functional value is a scalar. (Contributed by NM, 15-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  Y  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  K )
 
Theoremlfl0 29936 A linear functional is zero at the zero vector. (lnfn0i 23550 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  Z  =  ( 0g
 `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  LMod  /\  G  e.  F ) 
 ->  ( G `  Z )  =  .0.  )
 
Theoremlfladd 29937 Property of a linear functional. (lnfnaddi 23551 analog.) (Contributed by NM, 18-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .+^  =  (
 +g  `  D )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  LMod  /\  G  e.  F  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  ( G `  ( X  .+  Y ) )  =  (
 ( G `  X )  .+^  ( G `  Y ) ) )
 
Theoremlflsub 29938 Property of a linear functional. (lnfnaddi 23551 analog.) (Contributed by NM, 18-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  M  =  ( -g `  D )   &    |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  LMod  /\  G  e.  F  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  ( G `  ( X  .-  Y ) )  =  (
 ( G `  X ) M ( G `  Y ) ) )
 
Theoremlflmul 29939 Property of a linear functional. (lnfnmuli 23552 analog.) (Contributed by NM, 16-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  LMod  /\  G  e.  F  /\  ( R  e.  K  /\  X  e.  V ) )  ->  ( G `  ( R  .x.  X ) )  =  ( R  .X.  ( G `  X ) ) )
 
Theoremlfl0f 29940 The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  } )  e.  F )
 
Theoremlfl1 29941* A non-zero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |- 
 .1.  =  ( 1r `  D )   &    |-  V  =  (
 Base `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) ) 
 ->  E. x  e.  V  ( G `  x )  =  .1.  )
 
Theoremlfladdcl 29942 Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
 |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( G  o F  .+  H )  e.  F )
 
Theoremlfladdcom 29943 Commutativity of functional addition. (Contributed by NM, 19-Oct-2014.)
 |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( G  o F  .+  H )  =  ( H  o F  .+  G ) )
 
Theoremlfladdass 29944 Associativity of functional addition. (Contributed by NM, 19-Oct-2014.)
 |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  I  e.  F )   =>    |-  ( ph  ->  (
 ( G  o F  .+  H )  o F  .+  I )  =  ( G  o F  .+  ( H  o F  .+  I ) ) )
 
Theoremlfladd0l 29945 Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( ( V  X.  {  .0.  } )  o F  .+  G )  =  G )
 
Theoremlflnegcl 29946* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 30017, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  I  =  ( inv g `  R )   &    |-  N  =  ( x  e.  V  |->  ( I `  ( G `
  x ) ) )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  N  e.  F )
 
Theoremlflnegl 29947* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 30017, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  I  =  ( inv g `  R )   &    |-  N  =  ( x  e.  V  |->  ( I `  ( G `
  x ) ) )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  .+  =  ( +g  `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ph  ->  ( N  o F  .+  G )  =  ( V  X.  {  .0.  } )
 )
 
Theoremlflvscl 29948 Closure of a scalar product with a functional. Note that this is the scalar product for a right vector space with the scalar after the vector; reversing these fails closure. (Contributed by NM, 9-Oct-2014.) (Revised by Mario Carneiro, 22-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  R  e.  K )   =>    |-  ( ph  ->  ( G  o F  .x.  ( V  X.  { R }
 ) )  e.  F )
 
Theoremlflvsdi1 29949 Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  (
 ( G  o F  .+  H )  o F  .x.  ( V  X.  { X } ) )  =  ( ( G  o F  .x.  ( V  X.  { X } ) )  o F  .+  ( H  o F  .x.  ( V  X.  { X }
 ) ) ) )
 
Theoremlflvsdi2 29950 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  o F  .x.  (
 ( V  X.  { X } )  o F  .+  ( V  X.  { Y } ) ) )  =  ( ( G  o F  .x.  ( V  X.  { X }
 ) )  o F  .+  ( G  o F  .x.  ( V  X.  { Y } ) ) ) )
 
Theoremlflvsdi2a 29951 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  o F  .x.  ( V  X.  { ( X 
 .+  Y ) }
 ) )  =  ( ( G  o F  .x.  ( V  X.  { X } ) )  o F  .+  ( G  o F  .x.  ( V  X.  { Y }
 ) ) ) )
 
Theoremlflvsass 29952 Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  o F  .x.  ( V  X.  { ( X 
 .x.  Y ) } )
 )  =  ( ( G  o F  .x.  ( V  X.  { X } ) )  o F  .x.  ( V  X.  { Y } )
 ) )
 
Theoremlfl0sc 29953 The (right vector space) scalar product of a functional with zero is the zero functional. Note that the first occurrence of  ( V  X.  {  .0.  }
) represents the zero scalar, and the second is the zero functional. (Contributed by NM, 7-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  o F  .x.  ( V  X.  {  .0.  } ) )  =  ( V  X.  {  .0.  } ) )
 
Theoremlflsc0N 29954 The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  ( ( V  X.  {  .0.  } )  o F  .x.  ( V  X.  { X } ) )  =  ( V  X.  {  .0.  } ) )
 
Theoremlfl1sc 29955 The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  .1.  =  ( 1r `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  o F  .x.  ( V  X.  {  .1.  } ) )  =  G )
 
Syntaxclk 29956 Extend class notation with the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space.
 class LKer
 
Definitiondf-lkr 29957* Define the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space. (Contributed by NM, 15-Apr-2014.)
 |- LKer  =  ( w  e.  _V  |->  ( f  e.  (LFnl `  w )  |->  ( `' f " { ( 0g `  (Scalar `  w ) ) } )
 ) )
 
Theoremlkrfval 29958* The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( W  e.  X  ->  K  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) ) )
 
Theoremlkrval 29959 Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
 
Theoremellkr 29960 Membership in the kernel of a functional. (elnlfn 23436 analog.) (Contributed by NM, 16-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
 ( X  e.  V  /\  ( G `  X )  =  .0.  )
 ) )
 
Theoremlkrval2 29961* Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G )  =  { x  e.  V  |  ( G `  x )  =  .0.  } )
 
Theoremellkr2 29962 Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  Y )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( X  e.  ( K `  G )  <->  ( G `  X )  =  .0.  ) )
 
Theoremlkrcl 29963 A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  (
 ( W  e.  Y  /\  G  e.  F  /\  X  e.  ( K `  G ) )  ->  X  e.  V )
 
Theoremlkrf0 29964 The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  Y  /\  G  e.  F  /\  X  e.  ( K `
  G ) ) 
 ->  ( G `  X )  =  .0.  )
 
Theoremlkr0f 29965 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  LMod  /\  G  e.  F ) 
 ->  ( ( K `  G )  =  V  <->  G  =  ( V  X.  {  .0.  } ) ) )
 
Theoremlkrlss 29966 The kernel of a linear functional is a subspace. (nlelshi 23568 analog.) (Contributed by NM, 16-Apr-2014.)
 |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  (
 ( W  e.  LMod  /\  G  e.  F ) 
 ->  ( K `  G )  e.  S )
 
Theoremlkrssv 29967 The kernel of a linear functional is a set of vectors. (Contributed by NM, 1-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( K `  G )  C_  V )
 
Theoremlkrsc 29968 The kernel of a non-zero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 9-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  R  e.  K )   &    |- 
 .0.  =  ( 0g `  D )   &    |-  ( ph  ->  R  =/=  .0.  )   =>    |-  ( ph  ->  ( L `  ( G  o F  .x.  ( V  X.  { R }
 ) ) )  =  ( L `  G ) )
 
Theoremlkrscss 29969 The kernel of a scalar product of a functional includes the kernel of the functional. (The inclusion is proper for the zero product and equality otherwise.) (Contributed by NM, 9-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  R  e.  K )   =>    |-  ( ph  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { R }
 ) ) ) )
 
Theoremeqlkr 29970* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 18-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   =>    |-  ( ( W  e.  LVec  /\  ( G  e.  F  /\  H  e.  F ) 
 /\  ( L `  G )  =  ( L `  H ) ) 
 ->  E. r  e.  K  A. x  e.  V  ( H `  x )  =  ( ( G `
  x )  .x.  r ) )
 
Theoremeqlkr2 29971* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 10-Oct-2014.)
 |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   =>    |-  ( ( W  e.  LVec  /\  ( G  e.  F  /\  H  e.  F ) 
 /\  ( L `  G )  =  ( L `  H ) ) 
 ->  E. r  e.  K  H  =  ( G  o F  .x.  ( V  X.  { r }
 ) ) )
 
Theoremeqlkr3 29972 Two functionals with the same kernel are equal if they are equal at any nonzero value. (Contributed by NM, 2-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (Scalar `  W )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  ( K `  G )  =  ( K `  H ) )   &    |-  ( ph  ->  ( G `  X )  =  ( H `  X ) )   &    |-  ( ph  ->  ( G `  X )  =/=  .0.  )   =>    |-  ( ph  ->  G  =  H )
 
Theoremlkrlsp 29973 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 29960) is the whole vector space. (Contributed by NM, 19-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  (
 ( W  e.  LVec  /\  ( X  e.  V  /\  G  e.  F ) 
 /\  ( G `  X )  =/=  .0.  )  ->  ( ( K `  G )  .(+)  ( N `
  { X }
 ) )  =  V )
 
Theoremlkrlsp2 29974 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 12-May-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  (
 ( W  e.  LVec  /\  ( X  e.  V  /\  G  e.  F ) 
 /\  -.  X  e.  ( K `  G ) )  ->  ( ( K `  G )  .(+)  ( N `  { X } ) )  =  V )
 
Theoremlkrlsp3 29975 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 29-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  (
 ( W  e.  LVec  /\  ( X  e.  V  /\  G  e.  F ) 
 /\  -.  X  e.  ( K `  G ) )  ->  ( N `  ( ( K `  G )  u.  { X } ) )  =  V )
 
Theoremlkrshp 29976 The kernel of a nonzero functional is a hyperplane. (Contributed by NM, 29-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  H  =  (LSHyp `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) ) 
 ->  ( K `  G )  e.  H )
 
Theoremlkrshp3 29977 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 17-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  H  =  (LSHyp `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( ( K `  G )  e.  H  <->  G  =/=  ( V  X.  {  .0.  }
 ) ) )
 
Theoremlkrshpor 29978 The kernel of a functional is either a hyperplane or the full vector space. (Contributed by NM, 7-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (
 ( K `  G )  e.  H  \/  ( K `  G )  =  V ) )
 
Theoremlkrshp4 29979 A kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)
 |-  V  =  ( Base `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (
 ( K `  G )  =/=  V  <->  ( K `  G )  e.  H ) )
 
Theoremlshpsmreu 29980* Lemma for lshpkrex 29989. Show uniqueness of ring multiplier  k when a vector  X is broken down into components, one in a hyperplane and the other outside of it . TODO: do we need the cbvrexv 2935 for 
a to  c? (Contributed by NM, 4-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { Z } ) )  =  V )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .s `  W )   =>    |-  ( ph  ->  E! k  e.  K  E. y  e.  U  X  =  ( y  .+  ( k 
 .x.  Z ) ) )
 
Theoremlshpkrlem1 29981* Lemma for lshpkrex 29989. The value of tentative functional  G is zero iff its argument belongs to hyperplane  U. (Contributed by NM, 14-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { Z } ) )  =  V )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  G  =  ( x  e.  V  |->  (
 iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z ) ) ) )   =>    |-  ( ph  ->  ( X  e.  U  <->  ( G `  X )  =  .0.  ) )
 
Theoremlshpkrlem2 29982* Lemma for lshpkrex 29989. The value of tentative functional  G is a scalar. (Contributed by NM, 16-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { Z } ) )  =  V )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  G  =  ( x  e.  V  |->  (
 iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z ) ) ) )   =>    |-  ( ph  ->  ( G `  X )  e.  K )
 
Theoremlshpkrlem3 29983* Lemma for lshpkrex 29989. Defining property of  G `  X. (Contributed by NM, 15-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { Z } ) )  =  V )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  G  =  ( x  e.  V  |->  (
 iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z ) ) ) )   =>    |-  ( ph  ->  E. z  e.  U  X  =  ( z  .+  ( ( G `  X ) 
 .x.  Z ) ) )
 
Theoremlshpkrlem4 29984* Lemma for lshpkrex 29989. Part of showing linearity of  G. (Contributed by NM, 16-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { Z } ) )  =  V )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  G  =  ( x  e.  V  |->  (
 iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z ) ) ) )   =>    |-  ( ( ( ph  /\  l  e.  K  /\  u  e.  V )  /\  ( v  e.  V  /\  r  e.  V  /\  s  e.  V )  /\  ( u  =  ( r  .+  (
 ( G `  u )  .x.  Z ) ) 
 /\  v  =  ( s  .+  ( ( G `  v ) 
 .x.  Z ) ) ) )  ->  ( (
 l  .x.  u )  .+  v )  =  ( ( ( l  .x.  r )  .+  s ) 
 .+  ( ( ( l ( .r `  D ) ( G `
  u ) ) ( +g  `  D ) ( G `  v ) )  .x.  Z ) ) )
 
Theoremlshpkrlem5 29985* Lemma for lshpkrex 29989. Part of showing linearity of  G. (Contributed by NM, 16-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { Z } ) )  =  V )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  G  =  ( x  e.  V  |->  (
 iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z ) ) ) )   =>    |-  ( ( ( ph  /\  l  e.  K  /\  u  e.  V )  /\  ( v  e.  V  /\  r  e.  U  /\  ( s  e.  U  /\  z  e.  U ) )  /\  ( u  =  ( r  .+  ( ( G `  u )  .x.  Z ) )  /\  v  =  ( s  .+  (
 ( G `  v
 )  .x.  Z )
 )  /\  ( (
 l  .x.  u )  .+  v )  =  ( z  .+  ( ( G `  ( ( l  .x.  u )  .+  v ) )  .x.  Z ) ) ) ) 
 ->  ( G `  (
 ( l  .x.  u )  .+  v ) )  =  ( ( l ( .r `  D ) ( G `  u ) ) (
 +g  `  D )
 ( G `  v
 ) ) )
 
Theoremlshpkrlem6 29986* Lemma for lshpkrex 29989. Show linearlity of  G. (Contributed by NM, 17-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { Z } ) )  =  V )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  G  =  ( x  e.  V  |->  (
 iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  ( k  .x.  Z ) ) ) )   =>    |-  ( ( ph  /\  (
 l  e.  K  /\  u  e.  V  /\  v  e.  V )
 )  ->  ( G `  ( ( l  .x.  u )  .+  v ) )  =  ( ( l ( .r `  D ) ( G `
  u ) ) ( +g  `  D ) ( G `  v ) ) )
 
Theoremlshpkrcl 29987* The set  G defined by hyperplane  U is a linear functional. (Contributed by NM, 17-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { Z } ) )  =  V )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  (
 Base `  D )   &    |-  .x.  =  ( .s `  W )   &    |-  G  =  ( x  e.  V  |->  ( iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  ( k 
 .x.  Z ) ) ) )   &    |-  F  =  (LFnl `  W )   =>    |-  ( ph  ->  G  e.  F )
 
Theoremlshpkr 29988* The kernel of functional  G is the hyperplane defining it. (Contributed by NM, 17-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { Z } ) )  =  V )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  (
 Base `  D )   &    |-  .x.  =  ( .s `  W )   &    |-  G  =  ( x  e.  V  |->  ( iota_ k  e.  K E. y  e.  U  x  =  ( y  .+  ( k 
 .x.  Z ) ) ) )   &    |-  L  =  (LKer `  W )   =>    |-  ( ph  ->  ( L `  G )  =  U )
 
Theoremlshpkrex 29989* There exists a functional whose kernel equals a given hyperplane. Part of Th. 1.27 of Barbu and Precupanu, Convexity and Optimization in Banach Spaces. (Contributed by NM, 17-Jul-2014.)
 |-  H  =  (LSHyp `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  (
 ( W  e.  LVec  /\  U  e.  H ) 
 ->  E. g  e.  F  ( K `  g )  =  U )
 
Theoremlshpset2N 29990* The set of all hyperplanes of a left module or left vector space equals the set of all kernels of nonzero functionals. (Contributed by NM, 17-Jul-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  H  =  (LSHyp `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( W  e.  LVec  ->  H  =  { s  |  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
 )  /\  s  =  ( K `  g ) ) } )
 
TheoremislshpkrN 29991* The predicate "is a hyperplane" (of a left module or left vector space). TODO: should it be 
U  =  ( K `
 g ) or  ( K `  g )  =  U as in lshpkrex 29989? Both standards seem to be used randomly throughout set.mm; we should decide on a preferred one. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  H  =  (LSHyp `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( W  e.  LVec  ->  ( U  e.  H  <->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
 )  /\  U  =  ( K `  g ) ) ) )
 
Theoremlfl1dim 29992* Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  ( ph  ->  W  e.  LVec
 )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  { g  e.  F  |  ( L `
  G )  C_  ( L `  g ) }  =  { g  |  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k }
 ) ) } )
 
Theoremlfl1dim2N 29993* Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 29992 may be more compatible with lspsn 16083. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  ( ph  ->  W  e.  LVec
 )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  { g  e.  F  |  ( L `
  G )  C_  ( L `  g ) }  =  { g  e.  F  |  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) }
 )
 
19.26.5  Opposite rings and dual vector spaces
 
Syntaxcld 29994 Extend class notation with left dualvector space.
 class LDual
 
Definitiondf-ldual 29995* Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. The restriction on  o F ( +g  `  v
) allows it to be a set; see ofmres 6346. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
 |- LDual  =  ( v  e.  _V  |->  ( { <. ( Base `  ndx ) ,  (LFnl `  v
 ) >. ,  <. ( +g  ` 
 ndx ) ,  (  o F ( +g  `  (Scalar `  v ) )  |`  ( (LFnl `  v )  X.  (LFnl `  v )
 ) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  (Scalar `  v ) ) >. }  u.  { <. ( .s
 `  ndx ) ,  (
 k  e.  ( Base `  (Scalar `  v )
 ) ,  f  e.  (LFnl `  v )  |->  ( f  o F
 ( .r `  (Scalar `  v ) ) ( ( Base `  v )  X.  { k } )
 ) ) >. } )
 )
 
Theoremldualset 29996* Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  (  o F  .+  |`  ( F  X.  F ) )   &    |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .xb  =  (
 k  e.  K ,  f  e.  F  |->  ( f  o F  .x.  ( V  X.  { k }
 ) ) )   &    |-  ( ph  ->  W  e.  X )   =>    |-  ( ph  ->  D  =  ( { <. ( Base ` 
 ndx ) ,  F >. ,  <. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
 
Theoremldualvbase 29997 The vectors of a dual space are functionals of the original space. (Contributed by NM, 18-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  V  =  ( Base `  D )   &    |-  ( ph  ->  W  e.  X )   =>    |-  ( ph  ->  V  =  F )
 
Theoremldualelvbase 29998 Utility theorem for converting a functional to a vector of the dual space in order to use standard vector theorems. (Contributed by NM, 6-Jan-2015.)
 |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  V  =  ( Base `  D )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  G  e.  V )
 
Theoremldualfvadd 29999 Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  D  =  (LDual `  W )   &    |-  .+b  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  X )   &    |-  .+^  =  (  o F  .+  |`  ( F  X.  F ) )   =>    |-  ( ph  ->  .+b 
 =  .+^  )
 
Theoremldualvadd 30000 Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  D  =  (LDual `  W )   &    |-  .+b  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( G  .+b  H )  =  ( G  o F  .+  H ) )
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