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Theorem List for Metamath Proof Explorer - 30601-30700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlclkrlem2a 30601 Lemma for lclkr 30627. Use lshpat 28150 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 30602 Lemma for lclkr 30627. (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 30603 Lemma for lclkr 30627. (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 30604 Lemma for lclkr 30627. (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 30605 Lemma for lclkr 30627. 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 30606 Lemma for lclkr 30627. 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 30607 Lemma for lclkr 30627. 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 30608 Lemma for lclkr 30627. 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 30609 Lemma for lclkr 30627. 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 30610 Lemma for lclkr 30627. 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 30611 Lemma for lclkr 30627. 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 30612 Lemma for lclkr 30627. 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 30613 Lemma for lclkr 30627. 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 30614 Lemma for lclkr 30627. (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 30615 Lemma for lclkr 30627. 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 30616 Lemma for lclkr 30627. 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 30617 Lemma for lclkr 30627. 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 30618 Lemma for lclkr 30627. 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 30619 Lemma for lclkr 30627. 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 30620 Lemma for lclkr 30627. 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 30621 Lemma for lclkr 30627. lclkrlem2t 30620 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 30622 Lemma for lclkr 30627. When the hypotheses of lclkrlem2u 30621 and lclkrlem2u 30621 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 30562, which requires the orthomodular law dihoml4 30471 (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 30623 Lemma for lclkr 30627. This is the same as lclkrlem2u 30621 and lclkrlem2u 30621 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 30624 Lemma for lclkr 30627. Eliminate by cases the hypotheses of lclkrlem2u 30621, lclkrlem2u 30621 and lclkrlem2w 30623. (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 30625 Lemma for lclkr 30627. 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 30626* The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 30601 through lclkrlem2y 30625 are used for the proof. Here we express lclkrlem2y 30625 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 30627* 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 30628* 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 30629* 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 30630* 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 30631* 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 30632* 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 30633* 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 30627 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 30627 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 30634* 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 30722. (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 30635* 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 30636 Lemma for lcfr 30679. 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 30637 Lemma for lcfr 30679. (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 30638 Lemma for lcfr 30679. (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 30639* Lemma for lcfr 30679. (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 30640* Lemma for lcfr 30679. 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 30641* Lemma for lcfr 30679. 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 30642* Lemma for lcfr 30679. 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 30643* Lemma for lcf1o 30645 and lcfr 30679. (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 30644* Lemma for lcf1o 30645. (This part has undesirable $d's on  J and  ph that we remove in lcf1o 30645.) 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 30645* 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 30646* Lemma for lcfr 30679. (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 30647* Lemma for lcfr 30679. (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 30648* Lemma for lcfr 30679. (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 30649* Lemma for lcfr 30679. (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 30650* Lemma for lcfr 30679. (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 30651* Lemma for lcfr 30679. (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 30652* Lemma for lcfr 30679. (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 30653 Lemma for lcfr 30679. 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 30654 Lemma for lcfr 30679. (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 30655 Lemma for lcfr 30679. (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 30656 Lemma for lcfr 30679. (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 30657 Lemma for lcfr 30679. (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 30658 Lemma for lcfr 30679. (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 30659 Lemma for lcfr 30679. 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 30660* Lemma for lcfr 30679. (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 30661* Lemma for lcfr 30679. Special case of lcfrlem35 30671 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 30662* Lemma for lcfr 30679. Special case of lcfrlem36 30672 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 30663* Lemma for lcfr 30679. Special case of lcfrlem37 30673 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 30664* Lemma for lcfr 30679. 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 30665* Lemma for lcfr 30679. (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 30666* Lemma for lcfr 30679. (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 30667* Lemma for lcfr 30679. (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  =  (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 )