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Theorem List for Metamath Proof Explorer - 29301-29400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlcvexchlem5 29301 Lemma for lcvexch 29302. (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 29302 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 22951 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 29303 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22957 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 29304 Covering property of a subspace plus an atom. (chcv1 22937 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 29305 Covering property of a subspace plus an atom. (chcv2 22938 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 29306 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 22963 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 29307 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 22958 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 29308 The meet of distinct atoms is the zero subspace. (atnemeq0 22959 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 29309 The atom exch1ange property. (hlatexch1 29657 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 29310 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 22961 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 29311 Two atoms covering the zero subspace are equal. (atcv1 22962 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 29312 Lemma for lsatcvat 29313. (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 29313 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 22968 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 29314 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 22969 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 29315 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22978 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 29316 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 29317 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 29737 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 29318 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 29738 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 29319 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 30305 analog.) TODO: This changes  U C V in l1cvpat 29317 and l1cvat 29318 to  U  e.  H, which in turn change  U  e.  H in islshpcv 29316 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 )
 
18.27.4  Functionals and kernels of a left vector space (or module)
 
Syntaxclfn 29320 Extend class notation with all linear functionals of a left module or left vector space.
 class LFnl
 
Definitiondf-lfl 29321* 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 29322* 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 29323* 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 29324 Property of a linear functional. (lnfnli 22622 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 29325* Properties that determine a linear functional. TODO: use this in place of islfl 29323 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 29326 A linear functional is a function from vectors to scalars. (lnfnfi 22623 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 29327 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 29328 A linear functional is zero at the zero vector. (lnfn0i 22624 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 29329 Property of a linear functional. (lnfnaddi 22625 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 29330 Property of a linear functional. (lnfnaddi 22625 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 29331 Property of a linear functional. (lnfnmuli 22626 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 29332 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 29333* 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 29334 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 29335 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 29336 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 29337 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 29338* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 29409, 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 29339* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 29409, 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 29340 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 29341 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 29342 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 29343 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 29344 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 29345 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 29346 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 29347 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 29348 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 29349* 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 29350* 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 29351 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 29352 Membership in the kernel of a functional. (elnlfn 22510 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 29353* 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 29354 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 29355 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 29356 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 29357 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 29358 The kernel of a linear functional is a subspace. (nlelshi 22642 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 29359 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 29360 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 29361 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 29362* 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 29363* 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 29364 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 29365 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 29352) 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 29366 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 29367 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 29368 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 29369 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 29370 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 29371 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 29372* Lemma for lshpkrex 29381. 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 2767 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 29373* Lemma for lshpkrex 29381. 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 29374* Lemma for lshpkrex 29381. 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 29375* Lemma for lshpkrex 29381. 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 29376* Lemma for lshpkrex 29381. 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 29377* Lemma for lshpkrex 29381. 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 29378* Lemma for lshpkrex 29381. 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 29379* 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 29380* 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 29381* 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 29382* 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 29383* 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 29381? 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 29384* 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 29385* Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 29384 may be more compatible with lspsn 15761. (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 } ) ) }
 )
 
18.27.5  Opposite rings and dual vector spaces
 
Syntaxcld 29386 Extend class notation with left dualvector space.
 class LDual
 
Definitiondf-ldual 29387* 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 6118. 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 29388* 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 29389 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 29390 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 29391 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 29392 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 ) )
 
Theoremldualvaddcl 29393 The value of vector addition in the dual of a vector space is a functional. (Contributed by NM, 21-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( G  .+  H )  e.  F )
 
Theoremldualvaddval 29394 The value of the value of vector addition in the dual of a vector space. (Contributed by NM, 7-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .+b  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  (
 ( G  .+b  H ) `
  X )  =  ( ( G `  X )  .+  ( H `
  X ) ) )
 
Theoremldualsca 29395 The ring of scalars of the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
 |-  F  =  (Scalar `  W )   &    |-  O  =  (oppr `  F )   &    |-  D  =  (LDual `  W )   &    |-  R  =  (Scalar `  D )   &    |-  ( ph  ->  W  e.  X )   =>    |-  ( ph  ->  R  =  O )
 
Theoremldualsbase 29396 Base set of scalar ring for the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
 |-  F  =  (Scalar `  W )   &    |-  L  =  ( Base `  F )   &    |-  D  =  (LDual `  W )   &    |-  R  =  (Scalar `  D )   &    |-  K  =  ( Base `  R )   &    |-  ( ph  ->  W  e.  V )   =>    |-  ( ph  ->  K  =  L )
 
TheoremldualsaddN 29397 Scalar addition for the dual of a vector space. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
 |-  F  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  F )   &    |-  D  =  (LDual `  W )   &    |-  R  =  (Scalar `  D )   &    |-  .+b  =  ( +g  `  R )   &    |-  ( ph  ->  W  e.  V )   =>    |-  ( ph  ->  .+b  =  .+  )
 
Theoremldualsmul 29398 Scalar multiplication for the dual of a vector space. (Contributed by NM, 19-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .r `  F )   &    |-  D  =  (LDual `  W )   &    |-  R  =  (Scalar `  D )   &    |-  .xb  =  ( .r `  R )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   =>    |-  ( ph  ->  ( X  .xb  Y )  =  ( Y  .x.  X ) )
 
Theoremldualfvs 29399* Scalar product operation for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  D  =  (LDual `  W )   &    |-  .xb  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  Y )   &    |-  .x.  =  ( k  e.  K ,  f  e.  F  |->  ( f  o F  .X.  ( V  X.  {
 k } ) ) )   =>    |-  ( ph  ->  .xb  =  .x.  )
 
Theoremldualvs 29400 Scalar product operation value (which is a functional) for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  D  =  (LDual `  W )   &    |-  .xb  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  Y )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( X  .xb  G )  =  ( G  o F  .X.  ( V  X.  { X } ) ) )
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