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Theorem List for Metamath Proof Explorer - 28501-28600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlsatnle 28501 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 22948 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 28502 The meet of distinct atoms is the zero subspace. (atnemeq0 22949 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 28503 The atom exch1ange property. (hlatexch1 28851 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 28504 If the sum of two atoms cover the zero subspace, they are equal.. (atcv0eq 22951 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 28505 Two atoms covering the zero subspace are equal. (atcv1 22952 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 28506 Lemma for lsatcvat 28507. (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 28507 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 22958 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 28508 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 22959 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 28509 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22968 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 28510 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 28511 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 28931 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 28512 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 28932 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 28513 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 29499 analog.) TODO: This changes  U C V in l1cvpat 28511 and l1cvat 28512 to  U  e.  H, which in turn change  U  e.  H in islshpcv 28510 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 )
 
18.25.4  Functionals and kernels of a left vector space (or module)
 
Syntaxclfn 28514 Extend class notation with all linear functionals of a left module or left vector space.
 class LFnl
 
Definitiondf-lfl 28515* Define the set of all linear functionals (maps from vectors to 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 28516* 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 28517* 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 28518 Property of a linear functional. (lnfnli 22612 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 28519* Properties that determine a linear functional. TODO: use this in place of islfl 28517 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 28520 A linear functional is a function from vectors to scalars. (lnfnfi 22613 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 28521 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 28522 A linear functional is zero at the zero vector. (lnfn0i 22614 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 28523 Property of a linear functional. (lnfnaddi 22615 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 28524 Property of a linear functional. (lnfnaddi 22615 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 28525 Property of a linear functional. (lnfnmuli 22616 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 28526 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 28527* 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 28528 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 28529 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 28530 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 28531 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 28532* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 28603, 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 28533* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 28603, 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 28534 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 28535 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 28536 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 28537 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 28538 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 28539 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 28540 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 28541 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 28542 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 28543* 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 28544* 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 28545 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 28546 Membership in the kernel of a functional. (elnlfn 22500 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 28547* 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 28548 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 28549 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 28550 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 28551 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 28552 The kernel of a linear functional is a subspace. (nlelshi 22632 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 28553 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 28554 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 28555 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 28556* 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 28557* 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 28558 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 28559 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 28546) 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 28560 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 28561 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 28562 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 28563 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 28564 The kernels of a functionals 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 28565 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 28566* Lemma for lshpkrex 28575. 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 2766 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 28567* Lemma for lshpkrex 28575. 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 28568* Lemma for lshpkrex 28575. 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 28569* Lemma for lshpkrex 28575. 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 28570* Lemma for lshpkrex 28575. 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 28571* Lemma for lshpkrex 28575. 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 28572* Lemma for lshpkrex 28575. 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 28573* 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 28574* 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 28575* 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 28576* 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 28577* 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 28575? 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 28578* 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 28579* Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 28578 may be more compatible with lspsn 15753. (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 } ) ) }
 )
 
18.25.5  Opposite rings and dual vector spaces
 
Syntaxcld 28580 Extend class notation with left dualvector space.
 class LDual
 
Definitiondf-ldual 28581* 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 6077. 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 28582* 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 28583 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 28584 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 28585 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 28586 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 28587 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 28588 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 28589 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 28590 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 28591 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 28592 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 28593* 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 28594 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 } ) ) )
 
Theoremldualvsval 28595 Value of scalar product operation value 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  ->  A  e.  V )   =>    |-  ( ph  ->  ( ( X  .xb  G ) `
  A )  =  ( ( G `  A )  .X.  X ) )
 
Theoremldualvscl 28596 The scalar product operation value is a functional. (Contributed by NM, 18-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LMod
 )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( X  .x.  G )  e.  F )
 
Theoremldualvaddcom 28597 Commutative law for vector (functional) addition. (Contributed by NM, 17-Jan-2015.)
 |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  F )   &    |-  ( ph  ->  Y  e.  F )   =>    |-  ( ph  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremldualvsass 28598 Associative law for scalar product operation. (Contributed by NM, 20-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  D  =  (LDual `  W )   &    |- 
 .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (
 ( Y  .X.  X )  .x.  G )  =  ( X  .x.  ( Y  .x.  G ) ) )
 
Theoremldualvsass2 28599 Associative law for scalar product operation, using operations from the dual space. (Contributed by NM, 20-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  D  =  (LDual `  W )   &    |-  Q  =  (Scalar `  D )   &    |-  .X.  =  ( .r `  Q )   &    |-  .x. 
 =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (
 ( X  .X.  Y )  .x.  G )  =  ( X  .x.  ( Y  .x.  G ) ) )
 
Theoremldualvsdi1 28600 Distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  D  =  (LDual `  W )   &    |-  .+  =  ( +g  `  D )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LMod
 )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( X  .x.  G )  .+  ( X 
 .x.  H ) ) )
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