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Theorem List for Metamath Proof Explorer - 28401-28500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlflvsass 28401 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 28402 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 28403 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 28404 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 28405 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 28406* 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 28407* 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 28408 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 28409 Membership in the kernel of a functional. (elnlfn 22433 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 28410* 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 28411 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 28412 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 28413 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 28414 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 28415 The kernel of a linear functional is a subspace. (nlelshi 22565 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 28416 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 28417 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 28418 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 28419* 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 28420* 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 28421 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 28422 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 28409) 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 28423 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 28424 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 28425 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 28426 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 28427 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 28428 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 28429* Lemma for lshpkrex 28438. 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 2718 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 28430* Lemma for lshpkrex 28438. 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 28431* Lemma for lshpkrex 28438. 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 28432* Lemma for lshpkrex 28438. 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 28433* Lemma for lshpkrex 28438. 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 28434* Lemma for lshpkrex 28438. 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 28435* Lemma for lshpkrex 28438. 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 28436* 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 28437* 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 28438* 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 28439* 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 28440* 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 28438? 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 28441* 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 28442* Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 28441 may be more compatible with lspsn 15686. (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 } ) ) }
 )
 
16.24.9  Opposite rings and dual vector spaces
 
Syntaxcld 28443 Extend class notation with left dualvector space.
 class LDual
 
Definitiondf-ldual 28444* 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 6015. 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 28445* 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 28446 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 28447 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 28448 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 28449 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 28450 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 28451 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 28452 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 28453 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 28454 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 28455 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 28456* 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 28457 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 28458 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 28459 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 28460 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 28461 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 28462 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 28463 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 ) ) )
 
Theoremldualvsdi2 28464 Reverse distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  K  =  ( Base `  R )   &    |-  D  =  (LDual `  W )   &    |-  .+b  =  ( +g  `  D )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LMod
 )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (
 ( X  .+  Y )  .x.  G )  =  ( ( X  .x.  G )  .+b  ( Y  .x.  G ) ) )
 
Theoremldualgrplem 28465 Lemma for ldualgrp 28466. (Contributed by NM, 22-Oct-2014.)
 |-  D  =  (LDual `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  o F ( +g  `  W )   &    |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  (
 Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .x.  =  ( .s `  D )   =>    |-  ( ph  ->  D  e.  Grp )
 
Theoremldualgrp 28466 The dual of a vector space is a group. (Contributed by NM, 21-Oct-2014.)
 |-  D  =  (LDual `  W )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  D  e.  Grp )
 
Theoremldual0 28467 The zero scalar of the dual of a vector space. (Contributed by NM, 28-Dec-2014.)
 |-  R  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  (LDual `  W )   &    |-  S  =  (Scalar `  D )   &    |-  O  =  ( 0g `  S )   &    |-  ( ph  ->  W  e.  LMod
 )   =>    |-  ( ph  ->  O  =  .0.  )
 
Theoremldual1 28468 The unit scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
 |-  R  =  (Scalar `  W )   &    |-  .1.  =  ( 1r `  R )   &    |-  D  =  (LDual `  W )   &    |-  S  =  (Scalar `  D )   &    |-  I  =  ( 1r `  S )   &    |-  ( ph  ->  W  e.  LMod
 )   =>    |-  ( ph  ->  I  =  .1.  )
 
Theoremldualneg 28469 The negative of a scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
 |-  R  =  (Scalar `  W )   &    |-  M  =  ( inv g `  R )   &    |-  D  =  (LDual `  W )   &    |-  S  =  (Scalar `  D )   &    |-  N  =  ( inv g `  S )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  N  =  M )
 
Theoremldual0v 28470 The zero vector of the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  (LDual `  W )   &    |-  O  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod
 )   =>    |-  ( ph  ->  O  =  ( V  X.  {  .0.  } ) )
 
Theoremldual0vcl 28471 The dual zero vector is a functional. (Contributed by NM, 5-Mar-2015.)
 |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  .0.  e.  F )
 
Theoremlduallmodlem 28472 Lemma for lduallmod 28473. (Contributed by NM, 22-Oct-2014.)
 |-  D  =  (LDual `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  o F ( +g  `  W )   &    |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  (
 Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .x.  =  ( .s `  D )   =>    |-  ( ph  ->  D  e.  LMod )
 
Theoremlduallmod 28473 The dual of a left module is also a left module. (Contributed by NM, 22-Oct-2014.)
 |-  D  =  (LDual `  W )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  D  e.  LMod
 )
 
Theoremlduallvec 28474 The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 15332; otherwise, the dual would be a right vector space as is sometimes the case in the literature. (Contributed by NM, 22-Oct-2014.)
 |-  D  =  (LDual `  W )   &    |-  ( ph  ->  W  e.  LVec )   =>    |-  ( ph  ->  D  e.  LVec
 )
 
Theoremldualvsub 28475 The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
 |-  R  =  (Scalar `  W )   &    |-  N  =  ( inv g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .+  =  ( +g  `  D )   &    |-  .x.  =  ( .s `  D )   &    |-  .-  =  ( -g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( G  .-  H )  =  ( G  .+  (
 ( N `  .1.  )  .x.  H ) ) )
 
Theoremldualvsubcl 28476 Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
 |-  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 )
 
Theoremldualvsubval 28477 The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 28475? (Requires  D to oppr conversion.) (Contributed by NM, 26-Feb-2015.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  S  =  ( -g `  R )   &    |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .-  =  ( -g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  (
 ( G  .-  H ) `  X )  =  ( ( G `  X ) S ( H `  X ) ) )
 
Theoremldualssvscl 28478 Closure of scalar product in a dual subspace.) (Contributed by NM, 5-Feb-2015.)
 |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  S  =  ( LSubSp `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  U )
 
Theoremldualssvsubcl 28479 Closure of vector subtraction in a dual subspace.) (Contributed by NM, 9-Mar-2015.)
 |-  D  =  (LDual `  W )   &    |-  .-  =  ( -g `  D )   &    |-  S  =  ( LSubSp `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( X  .-  Y )  e.  U )
 
Theoremldual0vs 28480 Scalar zero times a functional is the zero functional. (Contributed by NM, 17-Feb-2015.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  O  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (  .0.  .x.  G )  =  O )
 
Theoremlkr0f2 28481 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 4-Feb-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( ( K `  G )  =  V  <->  G  =  .0.  ) )
 
Theoremlduallkr3 28482 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 22-Feb-2015.)
 |-  H  =  (LSHyp `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( ( K `  G )  e.  H  <->  G  =/=  .0.  )
 )
 
TheoremlkrpssN 28483 Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)
 |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  (
 ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  .0.  /\  H  =  .0.  )
 ) )
 
Theoremlkrin 28484 Intersection of the kernels of 2 functionals is included in the kernel of their sum. (Contributed by NM, 7-Jan-2015.)
 |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( ( K `  G )  i^i  ( K `  H ) )  C_  ( K `  ( G 
 .+  H ) ) )
 
Theoremeqlkr4 28485* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 4-Feb-2015.)
 |-  S  =  (Scalar `  W )   &    |-  R  =  ( Base `  S )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  ( K `  G )  =  ( K `  H ) )   =>    |-  ( ph  ->  E. r  e.  R  H  =  ( r  .x.  G )
 )
 
Theoremldual1dim 28486* Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  N  =  ( LSpan `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( N `  { G }
 )  =  { g  e.  F  |  ( L `
  G )  C_  ( L `  g ) } )
 
Theoremldualkrsc 28487 The kernel of a non-zero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014.)
 |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  X  =/=  .0.  )   =>    |-  ( ph  ->  ( L `  ( X  .x.  G ) )  =  ( L `  G ) )
 
Theoremlkrss 28488 The kernel of a scalar product of a functional includes the kernel of the functional. (Contributed by NM, 27-Jan-2015.)
 |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  ( L `  G ) 
 C_  ( L `  ( X  .x.  G ) ) )
 
Theoremlkrss2N 28489* Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.)
 |-  S  =  (Scalar `  W )   &    |-  R  =  ( Base `  S )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( ( K `  G )  C_  ( K `  H )  <->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
 
TheoremlkreqN 28490 Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
 |-  S  =  (Scalar `  W )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  A  e.  ( R  \  {  .0.  } ) )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  G  =  ( A  .x.  H )
 )   =>    |-  ( ph  ->  ( K `  G )  =  ( K `  H ) )
 
TheoremlkrlspeqN 28491 Condition for colinear functionals to have equal kernels. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
 |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  N  =  ( LSpan `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  G  e.  (
 ( N `  { H } )  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( L `  G )  =  ( L `  H ) )
 
16.24.10  Ortholattices and orthomodular lattices
 
Syntaxcops 28492 Extend class notation with orthoposets.
 class  OP
 
SyntaxccmtN 28493 Extend class notation with the commutes relation.
 class  cm
 
Syntaxcol 28494 Extend class notation with orthlattices.
 class  OL
 
Syntaxcoml 28495 Extend class notation with orthomodular lattices.
 class  OML
 
Definitiondf-oposet 28496* Define the class of orthoposets. (Contributed by NM, 20-Oct-2011.)
 |-  OP  =  { p  e.  Poset  |  ( ( ( 0. `  p )  e.  ( Base `  p )  /\  ( 1. `  p )  e.  ( Base `  p ) )  /\  E. o
 ( o  =  ( oc `  p ) 
 /\  A. a  e.  ( Base `  p ) A. b  e.  ( Base `  p ) ( ( ( o `  a
 )  e.  ( Base `  p )  /\  (
 o `  ( o `  a ) )  =  a  /\  ( a ( le `  p ) b  ->  ( o `
  b ) ( le `  p ) ( o `  a
 ) ) )  /\  ( a ( join `  p ) ( o `
  a ) )  =  ( 1. `  p )  /\  ( a (
 meet `  p ) ( o `  a ) )  =  ( 0. `  p ) ) ) ) }
 
Definitiondf-cmtN 28497* Define the commutes relation for orthoposets. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Nov-2011.)
 |-  cm  =  ( p  e.  _V  |->  {
 <. x ,  y >.  |  ( x  e.  ( Base `  p )  /\  y  e.  ( Base `  p )  /\  x  =  ( ( x (
 meet `  p ) y ) ( join `  p ) ( x (
 meet `  p ) ( ( oc `  p ) `  y ) ) ) ) } )
 
Definitiondf-ol 28498 Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
 |-  OL  =  ( Lat  i^i  OP )
 
Definitiondf-oml 28499* Define the class of orthomodular lattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
 |-  OML  =  { l  e.  OL  |  A. a  e.  ( Base `  l ) A. b  e.  ( Base `  l ) ( a ( le `  l
 ) b  ->  b  =  ( a ( join `  l ) ( b ( meet `  l )
 ( ( oc `  l ) `  a
 ) ) ) ) }
 
Theoremisopos 28500* The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( K  e.  OP  <->  (
 ( K  e.  Poset  /\ 
 .0.  e.  B  /\  .1.  e.  B )  /\  A. x  e.  B  A. y  e.  B  (
 ( (  ._|_  `  x )  e.  B  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x  /\  ( x  .<_  y  ->  (  ._|_  `  y )  .<_  (  ._|_  `  x ) ) )  /\  ( x  .\/  (  ._|_  `  x ) )  =  .1.  /\  ( x  ./\  (  ._|_  `  x ) )  =  .0.  ) ) )
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