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Theorem List for Metamath Proof Explorer - 32301-32400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremlpolpolsatN 32301 Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  A  =  (LSAtoms `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  e.  P )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  Q ) )  =  Q )
 
TheoremdochpolN 32302 The subspace orthocomplement for the  DVecH vector space is a polarity. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  P  =  (LPol `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ._|_  e.  P )
 
Theoremlcfl1lem 32303* Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)
 |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   =>    |-  ( G  e.  C 
 <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) ) )
 
Theoremlcfl1 32304* Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.)
 |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) ) )
 
Theoremlcfl2 32305* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  (  ._|_  `  ( L `
  G ) ) )  =/=  V  \/  ( L `  G )  =  V ) ) )
 
Theoremlcfl3 32306* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  ( L `  G ) )  e.  A  \/  ( L `  G )  =  V )
 ) )
 
Theoremlcfl4N 32307* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  Y  =  (LSHyp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  (  ._|_  `  ( L `
  G ) ) )  e.  Y  \/  ( L `  G )  =  V ) ) )
 
Theoremlcfl5 32308* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( L `  G )  e.  ran  I ) )
 
Theoremlcfl5a 32309 Property of a functional with a closed kernel. TODO: Make lcfl5 32308 etc. obsolete and rewrite w/out 
C hypothesis? (Contributed by NM, 29-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) 
 <->  ( L `  G )  e.  ran  I ) )
 
Theoremlcfl6lem 32310* Lemma for lcfl6 32312. A functional  G (whose kernel is closed by dochsnkr 32284) is comletely determined by a vector  X in the orthocomplement in its kernel at which the functional value is 1. Note that the  \  {  .0.  } in the  X hypothesis is redundant by the last hypothesis but allows easier use of other theorems. (Contributed by NM, 3-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .1.  =  ( 1r `  S )   &    |-  R  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  ( (  ._|_  `  ( L `  G ) )  \  {  .0.  } ) )   &    |-  ( ph  ->  ( G `  X )  =  .1.  )   =>    |-  ( ph  ->  G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
 k  .x.  X )
 ) ) ) )
 
Theoremlcfl7lem 32311* Lemma for lcfl7N 32313. If two functionals  G and  J are equal, they are determined by the same vector. (Contributed by NM, 4-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
 k  .x.  X )
 ) ) )   &    |-  J  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { Y }
 ) v  =  ( w  .+  ( k 
 .x.  Y ) ) ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  G  =  J )   =>    |-  ( ph  ->  X  =  Y )
 
Theoremlcfl6 32312* Property of a functional with a closed kernel. Note that  ( L `  G )  =  V means the functional is zero by lkr0f 29906. (Contributed by NM, 3-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  {
 f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( ( L `
  G )  =  V  \/  E. x  e.  ( V  \  {  .0.  } ) G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x }
 ) v  =  ( w  .+  ( k 
 .x.  x ) ) ) ) ) ) )
 
Theoremlcfl7N 32313* Property of a functional with a closed kernel. Every nonzero functional is determined by a unique nonzero vector. Note that  ( L `  G )  =  V means the functional is zero by lkr0f 29906. (Contributed by NM, 4-Jan-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  {
 f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( ( L `
  G )  =  V  \/  E! x  e.  ( V  \  {  .0.  } ) G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x }
 ) v  =  ( w  .+  ( k 
 .x.  x ) ) ) ) ) ) )
 
Theoremlcfl8 32314* Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  E. x  e.  V  ( L `  G )  =  (  ._|_  `  { x } ) ) )
 
Theoremlcfl8a 32315* Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) 
 <-> 
 E. x  e.  V  ( L `  G )  =  (  ._|_  `  { x } ) ) )
 
Theoremlcfl8b 32316* Property of a nonzero functional with a closed kernel. (Contributed by NM, 4-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Y  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  ( C  \  { Y } ) )   =>    |-  ( ph  ->  E. x  e.  ( V  \  {  .0.  } ) (  ._|_  `  ( L `  G ) )  =  ( N `  { x }
 ) )
 
Theoremlcfl9a 32317 Property implying that a functional has a closed kernel. (Contributed by NM, 16-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  (  ._|_  `  { X }
 )  C_  ( L `  G ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `
  G ) )
 
Theoremlclkrlem1 32318* The set of functionals having closed kernels is closed under scalar product. (Contributed by NM, 28-Dec-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .s `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  G  e.  C )   =>    |-  ( ph  ->  ( X  .x.  G )  e.  C )
 
Theoremlclkrlem2a 32319 Lemma for lclkr 32345. Use lshpat 29868 to show that the intersection of a hyperplane with a noncomparable sum of atoms is an atom. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  (  ._|_  ` 
 { X } )  =/=  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  -.  X  e.  (  ._|_  `  { B }
 ) )   =>    |-  ( ph  ->  (
 ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  { B } ) )  e.  A )
 
Theoremlclkrlem2b 32320 Lemma for lclkr 32345. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  (  ._|_  ` 
 { X } )  =/=  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  `  { B } )  \/  -.  Y  e.  (  ._|_  `  { B } ) ) )   =>    |-  ( ph  ->  ( (
 ( N `  { X } )  .(+)  ( N `
  { Y }
 ) )  i^i  (  ._|_  `  { B }
 ) )  e.  A )
 
Theoremlclkrlem2c 32321 Lemma for lclkr 32345. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  (  ._|_  ` 
 { X } )  =/=  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  `  { B } )  \/  -.  Y  e.  (  ._|_  `  { B } ) ) )   &    |-  J  =  (LSHyp `  U )   =>    |-  ( ph  ->  (
 ( (  ._|_  `  { X } )  i^i  (  ._|_  ` 
 { Y } )
 )  .(+)  ( N `  { B } ) )  e.  J )
 
Theoremlclkrlem2d 32322 Lemma for lclkr 32345. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  (  ._|_  ` 
 { X } )  =/=  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  `  { B } )  \/  -.  Y  e.  (  ._|_  `  { B } ) ) )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ph  ->  (
 ( (  ._|_  `  { X } )  i^i  (  ._|_  ` 
 { Y } )
 )  .(+)  ( N `  { B } ) )  e.  ran  I )
 
Theoremlclkrlem2e 32323 Lemma for lclkr 32345. The kernel of the sum is closed when the kernels of the summands are equal and closed. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  E )  =  ( L `  G ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2f 32324 Lemma for lclkr 32345. Construct a closed hyperplane under the kernel of the sum. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( L `  E )  =/=  ( L `  G ) )   &    |-  ( ph  ->  ( L `  ( E  .+  G ) )  e.  J )   =>    |-  ( ph  ->  (
 ( ( L `  E )  i^i  ( L `
  G ) ) 
 .(+)  ( N `  { B } ) )  C_  ( L `  ( E 
 .+  G ) ) )
 
Theoremlclkrlem2g 32325 Lemma for lclkr 32345. Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( L `  E )  =/=  ( L `  G ) )   &    |-  ( ph  ->  ( L `  ( E  .+  G ) )  e.  J )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2h 32326 Lemma for lclkr 32345. Eliminate the  ( L `  ( E 
.+  G ) )  e.  J hypothesis. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( L `  E )  =/=  ( L `  G ) )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `
  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2i 32327 Lemma for lclkr 32345. Eliminate the  ( L `  E )  =/=  ( L `  G ) hypothesis. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `
  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2j 32328 Lemma for lclkr 32345. Kernel closure when  Y is zero. (Contributed by NM, 18-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  =  .0.  )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2k 32329 Lemma for lclkr 32345. Kernel closure when  X is zero. (Contributed by NM, 18-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  =  .0.  )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2l 32330 Lemma for lclkr 32345. Eliminate the  X  =/=  .0.,  Y  =/=  .0. hypotheses. (Contributed by NM, 18-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2m 32331 Lemma for lclkr 32345. Construct a vector  B that makes the sum of functionals zero. Combine with  B  e.  V to shorten overall proof. (Contributed by NM, 17-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  U  e.  LVec )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   =>    |-  ( ph  ->  ( B  e.  V  /\  ( ( E  .+  G ) `  B )  =  .0.  )
 )
 
Theoremlclkrlem2n 32332 Lemma for lclkr 32345. (Contributed by NM, 12-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  U  e.  LVec )   &    |-  ( ph  ->  ( ( E  .+  G ) `  X )  =  .0.  )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =  .0.  )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  C_  ( L `  ( E  .+  G ) ) )
 
Theoremlclkrlem2o 32333 Lemma for lclkr 32345. When  B is nonzero, the vectors  X and  Y can't both belong to the hyperplane generated by  B. (Contributed by NM, 17-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  B  =/=  ( 0g `  U ) )   =>    |-  ( ph  ->  ( -.  X  e.  (  ._|_  `  { B }
 )  \/  -.  Y  e.  (  ._|_  `  { B } ) ) )
 
Theoremlclkrlem2p 32334 Lemma for lclkr 32345. When  B is zero,  X and  Y must colinear, so their orthocomplements must be comparable. (Contributed by NM, 17-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  B  =  ( 0g `  U ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  { Y }
 )  C_  (  ._|_  ` 
 { X } )
 )
 
Theoremlclkrlem2q 32335 Lemma for lclkr 32345. The sum has a closed kernel when  B is nonzero. (Contributed by NM, 18-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  B  =/=  ( 0g `  U ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2r 32336 Lemma for lclkr 32345. When  B is zero, i.e. when  X and  Y are colinear, the intersection of the kernels of  E and  G equal the kernel of  G, so the kernels of  G and the sum are comparable. (Contributed by NM, 18-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  B  =  ( 0g `  U ) )   =>    |-  ( ph  ->  ( L `  G ) 
 C_  ( L `  ( E  .+  G ) ) )
 
Theoremlclkrlem2s 32337 Lemma for lclkr 32345. Thus, the sum has a closed kernel when  B is zero. (Contributed by NM, 18-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  B  =  ( 0g `  U ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2t 32338 Lemma for lclkr 32345. We eliminate all hypotheses with  B here. (Contributed by NM, 18-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2u 32339 Lemma for lclkr 32345. lclkrlem2t 32338 with  X and  Y swapped. (Contributed by NM, 18-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  X )  =/= 
 .0.  )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2v 32340 Lemma for lclkr 32345. When the hypotheses of lclkrlem2u 32339 and lclkrlem2u 32339 are negated, the functional sum must be zero, so the kernel is the vector space. We make use of the law of excluded middle, dochexmid 32280, which requires the orthomodular law dihoml4 32189 (Lemma 3.3 of [Holland95] p. 214). (Contributed by NM, 16-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  X )  =  .0.  )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =  .0.  )   =>    |-  ( ph  ->  ( L `  ( E  .+  G ) )  =  V )
 
Theoremlclkrlem2w 32341 Lemma for lclkr 32345. This is the same as lclkrlem2u 32339 and lclkrlem2u 32339 with the inequality hypotheses negated. When the sum of two functionals is zero at each generating vector, the kernel is the vector space and therefore closed. (Contributed by NM, 16-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  X )  =  .0.  )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =  .0.  )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2x 32342 Lemma for lclkr 32345. Eliminate by cases the hypotheses of lclkrlem2u 32339, lclkrlem2u 32339 and lclkrlem2w 32341. (Contributed by NM, 18-Jan-2015.)
 |-  L  =  (LKer `  U )   &    |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2y 32343 Lemma for lclkr 32345. Restate the hypotheses for  E and  G to say their kernels are closed, in order to eliminate the generating vectors  X and  Y. (Contributed by NM, 18-Jan-2015.)
 |-  L  =  (LKer `  U )   &    |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  E ) ) )  =  ( L `
  E ) )   &    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `
  G ) ) )  =  ( L `
  G ) )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `
  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2 32344* The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 32319 through lclkrlem2y 32343 are used for the proof. Here we express lclkrlem2y 32343 in terms of membership in the set  C of functionals with closed kernels. (Contributed by NM, 18-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  E  e.  C )   &    |-  ( ph  ->  G  e.  C )   =>    |-  ( ph  ->  ( E  .+  G )  e.  C )
 
Theoremlclkr 32345* The set of functionals with closed kernels is a subspace. Part of proof of Theorem 3.6 of [Holland95] p. 218, line 20, stating "The fM that arise this way generate a subspace F of E'". Our proof was suggested by Mario Carneiro, 5-Jan-2015. (Contributed by NM, 18-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  S  =  ( LSubSp `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  C  e.  S )
 
Theoremlcfls1lem 32346* Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)
 |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f )  /\  (  ._|_  `  ( L `  f
 ) )  C_  Q ) }   =>    |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `
  G )  /\  (  ._|_  `  ( L `  G ) )  C_  Q ) )
 
Theoremlcfls1N 32347* Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
 |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f )  /\  (  ._|_  `  ( L `  f
 ) )  C_  Q ) }   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  (  ._|_  `  ( L `
  G ) ) )  =  ( L `
  G )  /\  (  ._|_  `  ( L `  G ) )  C_  Q ) ) )
 
Theoremlcfls1c 32348* Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015.)
 |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f )  /\  (  ._|_  `  ( L `  f
 ) )  C_  Q ) }   &    |-  D  =  {
 f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f ) }   =>    |-  ( G  e.  C  <->  ( G  e.  D  /\  (  ._|_  `  ( L `  G ) ) 
 C_  Q ) )
 
Theoremlclkrslem1 32349* The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace  Q is closed under scalar product. (Contributed by NM, 27-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .s `  D )   &    |-  C  =  { f  e.  F  |  ( ( 
 ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  Q  e.  S )   &    |-  ( ph  ->  G  e.  C )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  .x.  G )  e.  C )
 
Theoremlclkrslem2 32350* The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace  Q is closed under scalar product. (Contributed by NM, 28-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .s `  D )   &    |-  C  =  { f  e.  F  |  ( ( 
 ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  Q  e.  S )   &    |-  ( ph  ->  G  e.  C )   &    |- 
 .+  =  ( +g  `  D )   &    |-  ( ph  ->  E  e.  C )   =>    |-  ( ph  ->  ( E  .+  G )  e.  C )
 
Theoremlclkrs 32351* The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace  R is a subspace of the dual space. TODO: This proof repeats large parts of the lclkr 32345 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 32345 a special case of this? (Contributed by NM, 29-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  T  =  ( LSubSp `  D )   &    |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f )  /\  (  ._|_  `  ( L `  f
 ) )  C_  R ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  R  e.  S )   =>    |-  ( ph  ->  C  e.  T )
 
Theoremlclkrs2 32352* The set of functionals with closed kernels and majorizing the orthocomplement of a given subspace  Q is a subspace of the dual space containing functionals with closed kernels. Note that  R is the value given by mapdval 32440. (Contributed by NM, 12-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  T  =  ( LSubSp `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  R  =  { g  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  g ) ) )  =  ( L `  g )  /\  (  ._|_  `  ( L `  g
 ) )  C_  Q ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  Q  e.  S )   =>    |-  ( ph  ->  ( R  e.  T  /\  R  C_  C ) )
 
TheoremlcfrvalsnN 32353* Reconstruction from the dual space span of a singleton. (Contributed by NM, 19-Feb-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  N  =  ( LSpan `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  Q  =  U_ f  e.  R  (  ._|_  `  ( L `  f ) )   &    |-  R  =  ( N `  { G } )   =>    |-  ( ph  ->  Q  =  (  ._|_  `  ( L `  G ) ) )
 
Theoremlcfrlem1 32354 Lemma for lcfr 32397. Note that  X is z in Mario's notes. (Contributed by NM, 27-Feb-2015.)
 |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  (
 invr `  S )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .x.  =  ( .s `  D )   &    |-  .-  =  ( -g `  D )   &    |-  ( ph  ->  U  e.  LVec )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( G `  X )  =/=  .0.  )   &    |-  H  =  ( E  .-  (
 ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )   =>    |-  ( ph  ->  ( H `  X )  =  .0.  )
 
Theoremlcfrlem2 32355 Lemma for lcfr 32397. (Contributed by NM, 27-Feb-2015.)
 |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  (
 invr `  S )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .x.  =  ( .s `  D )   &    |-  .-  =  ( -g `  D )   &    |-  ( ph  ->  U  e.  LVec )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( G `  X )  =/=  .0.  )   &    |-  H  =  ( E  .-  (
 ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )   &    |-  L  =  (LKer `  U )   =>    |-  ( ph  ->  (
 ( L `  E )  i^i  ( L `  G ) )  C_  ( L `  H ) )
 
Theoremlcfrlem3 32356 Lemma for lcfr 32397. (Contributed by NM, 27-Feb-2015.)
 |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  (
 invr `  S )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .x.  =  ( .s `  D )   &    |-  .-  =  ( -g `  D )   &    |-  ( ph  ->  U  e.  LVec )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( G `  X )  =/=  .0.  )   &    |-  H  =  ( E  .-  (
 ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )   &    |-  L  =  (LKer `  U )   =>    |-  ( ph  ->  X  e.  ( L `  H ) )
 
Theoremlcfrlem4 32357* Lemma for lcfr 32397. (Contributed by NM, 10-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( LSubSp `  D )   &    |-  E  =  U_ g  e.  G  (  ._|_  `  ( L `  g ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  Q )   &    |-  ( ph  ->  X  e.  E )   =>    |-  ( ph  ->  X  e.  V )
 
Theoremlcfrlem5 32358* Lemma for lcfr 32397. The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace  Q is closed under scalar product. TODO: share hypotheses with others. Use more consistent variable names here or elsewhere when possible. (Contributed by NM, 5-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  S  =  ( LSubSp `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  R  e.  S )   &    |-  Q  =  U_ f  e.  R  (  ._|_  `  ( L `  f ) )   &    |-  ( ph  ->  X  e.  Q )   &    |-  C  =  (Scalar `  U )   &    |-  B  =  ( Base `  C )   &    |-  .x.  =  ( .s `  U )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  ( A  .x.  X )  e.  Q )
 
Theoremlcfrlem6 32359* Lemma for lcfr 32397. Closure of vector sum with colinear vectors. TODO: Move down  N definition so top hypotheses can be shared. (Contributed by NM, 10-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( LSubSp `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  Q )   &    |-  E  =  U_ g  e.  G  (  ._|_  `  ( L `  g ) )   &    |-  ( ph  ->  X  e.  E )   &    |-  ( ph  ->  Y  e.  E )   &    |-  ( ph  ->  ( N `  { X } )  =  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  E )
 
Theoremlcfrlem7 32360* Lemma for lcfr 32397. Closure of vector sum when one vector is zero. TODO: share hypotheses with others. (Contributed by NM, 11-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( LSubSp `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  Q )   &    |-  E  =  U_ g  e.  G  (  ._|_  `  ( L `  g ) )   &    |-  ( ph  ->  X  e.  E )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  ( ph  ->  Y  =  .0.  )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  E )
 
Theoremlcfrlem8 32361* Lemma for lcf1o 32363 and lcfr 32397. (Contributed by NM, 21-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( J `  X )  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
 ) v  =  ( w  .+  ( k 
 .x.  X ) ) ) ) )
 
Theoremlcfrlem9 32362* Lemma for lcf1o 32363. (This part has undesirable $d's on  J and  ph that we remove in lcf1o 32363.) TODO: ugly proof; maybe have better subtheorems or abbreviate some  iota_
k expansions with  J `  z? TODO: Some redundant $d's? (Contributed by NM, 22-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  J : ( V  \  {  .0.  } ) -1-1-onto-> ( C 
 \  { Q }
 ) )
 
Theoremlcf1o 32363* Define a function  J that provides a bijection from nonzero vectors  V to nonzero functionals with closed kernels  C. (Contributed by NM, 22-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  J : ( V  \  {  .0.  } ) -1-1-onto-> ( C 
 \  { Q }
 ) )
 
Theoremlcfrlem10 32364* Lemma for lcfr 32397. (Contributed by NM, 23-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( J `  X )  e.  F )
 
Theoremlcfrlem11 32365* Lemma for lcfr 32397. (Contributed by NM, 23-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( L `  ( J `  X ) )  =  (  ._|_  `  { X } ) )
 
Theoremlcfrlem12N 32366* Lemma for lcfr 32397. (Contributed by NM, 23-Feb-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  B  =  ( 0g `  S )   &    |-  ( ph  ->  Y  e.  (  ._|_  `  { X }
 ) )   =>    |-  ( ph  ->  (
 ( J `  X ) `  Y )  =  B )
 
Theoremlcfrlem13 32367* Lemma for lcfr 32397. (Contributed by NM, 8-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( J `  X )  e.  ( C  \  { Q } ) )
 
Theoremlcfrlem14 32368* Lemma for lcfr 32397. (Contributed by NM, 10-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  N  =  (
 LSpan `  U )   =>    |-  ( ph  ->  ( 
 ._|_  `  ( L `  ( J `  X ) ) )  =  ( N `  { X } ) )
 
Theoremlcfrlem15 32369* Lemma for lcfr 32397. (Contributed by NM, 9-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  ( J `  X ) ) ) )
 
Theoremlcfrlem16 32370* Lemma for lcfr 32397. (Contributed by NM, 8-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  P  =  ( LSubSp `  D )   &    |-  ( ph  ->  G  e.  P )   &    |-  ( ph  ->  G  C_  C )   &    |-  E  =  U_ g  e.  G  (  ._|_  `  ( L `  g ) )   &    |-  ( ph  ->  X  e.  ( E  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( J `  X )  e.  G )
 
Theoremlcfrlem17 32371 Lemma for lcfr 32397. Condition needed more than once. (Contributed by NM, 11-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  ( V  \  {  .0.  } ) )
 
Theoremlcfrlem18 32372 Lemma for lcfr 32397. (Contributed by NM, 24-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  =  ( (  ._|_  `  { X } )  i^i  (  ._|_  ` 
 { Y } )
 ) )
 
Theoremlcfrlem19 32373 Lemma for lcfr 32397. (Contributed by NM, 11-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { ( X  .+  Y ) } )  \/  -.  Y  e.  (  ._|_  `  { ( X 
 .+  Y ) }
 ) ) )
 
Theoremlcfrlem20 32374 Lemma for lcfr 32397. (Contributed by NM, 11-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  -.  X  e.  (  ._|_  ` 
 { ( X  .+  Y ) } )
 )   =>    |-  ( ph  ->  (
 ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) }
 ) )  e.  A )
 
Theoremlcfrlem21 32375 Lemma for lcfr 32397. (Contributed by NM, 11-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  (
 ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) }
 ) )  e.  A )
 
Theoremlcfrlem22 32376 Lemma for lcfr 32397. (Contributed by NM, 24-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  B  =  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) }
 ) )   =>    |-  ( ph  ->  B  e.  A )
 
Theoremlcfrlem23 32377 Lemma for lcfr 32397. TODO: this proof was built from other proof pieces that may change  N `  { X ,  Y } into subspace sum and back unnecessarily, or similar things. (Contributed by NM, 1-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  B  =  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  {