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Theorem List for Metamath Proof Explorer - 32001-32100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlclkrlem1 32001* 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 32002 Lemma for lclkr 32028. Use lshpat 29551 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 32003 Lemma for lclkr 32028. (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 32004 Lemma for lclkr 32028. (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 32005 Lemma for lclkr 32028. (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 32006 Lemma for lclkr 32028. 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 32007 Lemma for lclkr 32028. 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 32008 Lemma for lclkr 32028. 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 32009 Lemma for lclkr 32028. 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 32010 Lemma for lclkr 32028. 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 32011 Lemma for lclkr 32028. 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 32012 Lemma for lclkr 32028. 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 32013 Lemma for lclkr 32028. 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 32014 Lemma for lclkr 32028. 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 32015 Lemma for lclkr 32028. (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 32016 Lemma for lclkr 32028. 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 32017 Lemma for lclkr 32028. 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 32018 Lemma for lclkr 32028. 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 32019 Lemma for lclkr 32028. 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 32020 Lemma for lclkr 32028. 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 32021 Lemma for lclkr 32028. 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 32022 Lemma for lclkr 32028. lclkrlem2t 32021 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 32023 Lemma for lclkr 32028. When the hypotheses of lclkrlem2u 32022 and lclkrlem2u 32022 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 31963, which requires the orthomodular law dihoml4 31872 (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 32024 Lemma for lclkr 32028. This is the same as lclkrlem2u 32022 and lclkrlem2u 32022 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 32025 Lemma for lclkr 32028. Eliminate by cases the hypotheses of lclkrlem2u 32022, lclkrlem2u 32022 and lclkrlem2w 32024. (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 32026 Lemma for lclkr 32028. 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 32027* The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 32002 through lclkrlem2y 32026 are used for the proof. Here we express lclkrlem2y 32026 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 32028* 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 32029* 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 32030* 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 32031* 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 32032* 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 32033* 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 32034* 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 32028 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 32028 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 32035* 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 32123. (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 32036* 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 32037 Lemma for lcfr 32080. 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 32038 Lemma for lcfr 32080. (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 32039 Lemma for lcfr 32080. (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 32040* Lemma for lcfr 32080. (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 32041* Lemma for lcfr 32080. 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 32042* Lemma for lcfr 32080. 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 32043* Lemma for lcfr 32080. 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 32044* Lemma for lcf1o 32046 and lcfr 32080. (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 32045* Lemma for lcf1o 32046. (This part has undesirable $d's on  J and  ph that we remove in lcf1o 32046.) 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 32046* 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 32047* Lemma for lcfr 32080. (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 32048* Lemma for lcfr 32080. (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 32049* Lemma for lcfr 32080. (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 32050* Lemma for lcfr 32080. (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 32051* Lemma for lcfr 32080. (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 32052* Lemma for lcfr 32080. (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 32053* Lemma for lcfr 32080. (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 32054 Lemma for lcfr 32080. 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 32055 Lemma for lcfr 32080. (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 32056 Lemma for lcfr 32080. (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 32057 Lemma for lcfr 32080. (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 32058 Lemma for lcfr 32080. (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 32059 Lemma for lcfr 32080. (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 32060 Lemma for lcfr 32080. 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  (  ._|_  `  { ( X  .+  Y ) }
 ) )   &    |-  .(+)  =  ( LSSum `  U )   =>    |-  ( ph  ->  (
 (  ._|_  `  { X ,  Y } )  .(+)  B )  =  (  ._|_  `  { ( X  .+  Y ) }
 ) )
 
Theoremlcfrlem24 32061* Lemma for lcfr 32080. (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 ) }
 ) )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |-  R  =  ( Base `  S )   &    |-  J  =  ( x  e.  ( V 
 \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  I  e.  B )   &    |-  L  =  (LKer `  U )   =>    |-  ( ph  ->  (  ._|_  `  { X ,  Y } )  =  ( ( L `  ( J `  X ) )  i^i  ( L `  ( J `  Y ) ) ) )
 
Theoremlcfrlem25 32062* Lemma for lcfr 32080. Special case of lcfrlem35 32072 when  ( ( J `
 Y ) `  I ) is zero. (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 }
 ) )   &    |-  B  =  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) }
 ) )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |-  R  =  ( Base `  S )   &    |-  J  =  ( x  e.  ( V 
 \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  I  e.  B )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  ( ph  ->  ( ( J `  Y ) `  I )  =  Q )   &    |-  ( ph  ->  I  =/=  .0.  )   =>    |-  ( ph  ->  ( 
 ._|_  `  { ( X 
 .+  Y ) }
 )  =  ( L `
  ( J `  Y ) ) )
 
Theoremlcfrlem26 32063* Lemma for lcfr 32080. Special case of lcfrlem36 32073 when  ( ( J `
 Y ) `  I ) is zero. (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 }
 ) )   &    |-  B  =  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) }
 ) )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |-  R  =  ( Base `  S )   &    |-  J  =  ( x  e.  ( V 
 \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  I  e.  B )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  ( ph  ->  ( ( J `  Y ) `  I )  =  Q )   &    |-  ( ph  ->  I  =/=  .0.  )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  (  ._|_  `  ( L `  ( J `  Y ) ) ) )
 
Theoremlcfrlem27 32064* Lemma for lcfr 32080. Special case of lcfrlem37 32074 when  ( ( J `
 Y ) `  I ) is zero. (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 }
 ) )   &    |-  B  =  ( ( N `  { X ,  Y } )  i^i  (  ._|_  `  { ( X  .+  Y ) }
 ) )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |-  R  =  ( Base `  S )   &    |-  J  =  ( x  e.  ( V 
 \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  I  e.  B )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  ( ph  ->  ( ( J `  Y ) `  I )  =  Q )   &    |-  ( ph  ->  I  =/=  .0.  )   &    |-  ( ph  ->  G  e.  ( LSubSp `
  D ) )   &    |-  ( ph  ->  G  C_  { f  e.  (LFnl `  U )  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) } )   &    |-  E  =  U_ g  e.  G  (  ._|_  `  ( L `  g ) )   &    |-  ( ph  ->  X  e.  E )   &    |-  ( ph  ->  Y  e.  E )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  E )
 
Theoremlcfrlem28 32065* Lemma for lcfr 32080. TODO: This can be a hypothesis since the zero version of  ( J `  Y ) `  I needs it. (Contributed by NM, 9-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  (  ._|_  `  { ( X  .+  Y ) }
 ) )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |-  R  =  ( Base `  S )   &    |-  J  =  ( x  e.  ( V 
 \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  I  e.  B )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  ( ph  ->  ( ( J `  Y ) `  I )  =/= 
 Q )   =>    |-  ( ph  ->  I  =/=  .0.  )
 
Theoremlcfrlem29 32066* Lemma for lcfr 32080. (Contributed by NM, 9-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  (  ._|_  `  { ( X  .+  Y ) }
 ) )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |-  R  =  ( Base `  S )   &    |-  J  =  ( x  e.  ( V 
 \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  I  e.  B )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  ( ph  ->  ( ( J `  Y ) `  I )  =/= 
 Q )   &    |-  F  =  (
 invr `  S )   =>    |-  ( ph  ->  ( ( F `  (
 ( J `  Y ) `  I ) ) ( .r `  S ) ( ( J `
  X ) `  I ) )  e.  R )
 
Theoremlcfrlem30 32067* Lemma for lcfr 32080. (Contributed by NM, 6-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  (  ._|_  `  { ( X  .+  Y ) }
 ) )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |-  R  =  ( Base `  S )   &    |-  J  =  ( x  e.  ( V 
 \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  I  e.  B )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  ( ph  ->  ( ( J `  Y ) `  I )  =/= 
 Q )   &    |-  F  =  (
 invr `  S )   &    |-  .-  =  ( -g `  D )   &    |-  C  =  ( ( J `  X )  .-  ( ( ( F `
  ( ( J `
  Y ) `  I ) ) ( .r `  S ) ( ( J `  X ) `  I
 ) ) ( .s
 `  D ) ( J `  Y ) ) )   =>    |-  ( ph  ->  C  e.  (LFnl `  U )
 )
 
Theoremlcfrlem31 32068* Lemma for lcfr 32080. (Contributed by NM, 10-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =