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Theorem List for Metamath Proof Explorer - 31901-32000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdihjatcclem1 31901 Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  V  =  ( ( P  .\/  Q )  ./\  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( P 
 .\/  Q ) )  =  ( ( ( I `
  P )  .(+)  ( I `  Q ) )  .(+)  ( I `  V ) ) )
 
Theoremdihjatcclem2 31902 Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  V  =  ( ( P  .\/  Q )  ./\  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )   &    |-  ( ph  ->  ( I `  V )  C_  ( ( I `  P ) 
 .(+)  ( I `  Q ) ) )   =>    |-  ( ph  ->  ( I `  ( P 
 .\/  Q ) )  =  ( ( I `  P )  .(+)  ( I `
  Q ) ) )
 
Theoremdihjatcclem3 31903* Lemma for dihjatcc 31905. (Contributed by NM, 28-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  V  =  ( ( P  .\/  Q )  ./\  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )   &    |-  C  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  G  =  ( iota_ d  e.  T ( d `  C )  =  P )   &    |-  D  =  ( iota_ d  e.  T ( d `  C )  =  Q )   =>    |-  ( ph  ->  ( R `  ( G  o.  `' D ) )  =  V )
 
Theoremdihjatcclem4 31904* Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  V  =  ( ( P  .\/  Q )  ./\  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )   &    |-  C  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  G  =  ( iota_ d  e.  T ( d `  C )  =  P )   &    |-  D  =  ( iota_ d  e.  T ( d `  C )  =  Q )   &    |-  N  =  ( a  e.  E  |->  ( d  e.  T  |->  `' ( a `  d
 ) ) )   &    |-  .0.  =  ( d  e.  T  |->  (  _I  |`  B )
 )   &    |-  J  =  ( a  e.  E ,  b  e.  E  |->  ( d  e.  T  |->  ( ( a `
  d )  o.  ( b `  d
 ) ) ) )   =>    |-  ( ph  ->  ( I `  V )  C_  (
 ( I `  P )  .(+)  ( I `  Q ) ) )
 
Theoremdihjatcc 31905 Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( P 
 .\/  Q ) )  =  ( ( I `  P )  .(+)  ( I `
  Q ) ) )
 
Theoremdihjat 31906 Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  P  e.  A )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  =  ( ( I `  P )  .(+)  ( I `  Q ) ) )
 
Theoremdihprrnlem1N 31907 Lemma for dihprrn 31909, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |- 
 .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  ( ph  ->  Y  =/=  .0.  )   &    |-  ( ph  ->  ( `' I `  ( N `  { X } ) )  .<_  W )   &    |-  ( ph  ->  -.  ( `' I `  ( N `  { Y } ) )  .<_  W )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  e.  ran  I )
 
Theoremdihprrnlem2 31908 Lemma for dihprrn 31909. (Contributed by NM, 29-Sep-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  ( ph  ->  X  =/=  .0.  )   &    |-  ( ph  ->  Y  =/=  .0.  )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  e.  ran  I )
 
Theoremdihprrn 31909 The span of a vector pair belongs to the range of isomorphism H i.e. is a closed subspace. (Contributed by NM, 29-Sep-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  e.  ran  I )
 
Theoremdjhlsmat 31910 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 31909; should we directly use dihjat 31906? (Contributed by NM, 13-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .\/  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( N `  { X } )  .(+)  ( N `
  { Y }
 ) )  =  ( ( N `  { X } )  .\/  ( N `
  { Y }
 ) ) )
 
Theoremdihjat1lem 31911 Subspace sum of a closed subspace and an atom. (pmapjat1 30335 analog.) TODO: merge into dihjat1 31912? (Contributed by NM, 18-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .\/  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  .0.  =  ( 0g `  U )   &    |-  ( ph  ->  T  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( X  .\/  ( N ` 
 { T } )
 )  =  ( X 
 .(+)  ( N `  { T } ) ) )
 
Theoremdihjat1 31912 Subspace sum of a closed subspace and an atom. (pmapjat1 30335 analog.) (Contributed by NM, 1-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .\/  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  ( X  .\/  ( N `  { T } ) )  =  ( X  .(+)  ( N `  { T } ) ) )
 
Theoremdihsmsprn 31913 Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  ( X  .(+)  ( N `  { T } ) )  e.  ran  I )
 
Theoremdihjat2 31914 The subspace sum of a closed subspace and an atom is the same as their subspace join. (Contributed by NM, 1-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .\/  =  ( (joinH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( X  .\/  Q )  =  ( X  .(+)  Q ) )
 
Theoremdihjat3 31915 Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  P  e.  A )   =>    |-  ( ph  ->  ( I `  ( X  .\/  P ) )  =  ( ( I `  X )  .(+)  ( I `  P ) ) )
 
Theoremdihjat4 31916 Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
 |-  .\/  =  ( join `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( X  .(+)  Q )  =  ( I `  (
 ( `' I `  X )  .\/  ( `' I `  Q ) ) ) )
 
Theoremdihjat6 31917 Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
 |-  .\/  =  ( join `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( `' I `  ( X 
 .(+)  Q ) )  =  ( ( `' I `  X )  .\/  ( `' I `  Q ) ) )
 
Theoremdihsmsnrn 31918 The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( N `  { X } )  .(+)  ( N `
  { Y }
 ) )  e.  ran  I )
 
Theoremdihsmatrn 31919 The subspace sum of a closed subspace and an atom is closed. TODO: see if proof at http://math.stackexchange.com/a/1233211/50776 and Mon, 13 Apr 2015 20:44:07 -0400 email could be used instead of this and dihjat2 31914. (Contributed by NM, 15-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( X  .(+)  Q )  e. 
 ran  I )
 
Theoremdihjat5N 31920 Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  P  e.  A )   =>    |-  ( ph  ->  ( X  .\/  P )  =  ( `' I `  ( ( I `  X )  .(+)  ( I `
  P ) ) ) )
 
Theoremdvh4dimat 31921* There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  P  e.  A )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   =>    |-  ( ph  ->  E. s  e.  A  -.  s  C_  ( ( P  .(+)  Q )  .(+)  R )
 )
 
Theoremdvh3dimatN 31922* There is an atom that is outside the subspace sum of 2 others. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  P  e.  A )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  E. s  e.  A  -.  s  C_  ( P  .(+)  Q ) )
 
Theoremdvh2dimatN 31923* Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  P  e.  A )   =>    |-  ( ph  ->  E. s  e.  A  s  =/=  P )
 
Theoremdvh1dimat 31924* There exists an atom. (Contributed by NM, 25-Apr-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  E. s  s  e.  A )
 
Theoremdvh1dim 31925* There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  E. z  e.  V  z  =/=  .0.  )
 
Theoremdvh4dimlem 31926* Lemma for dvh4dimN 31930. (Contributed by NM, 22-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  .0.  =  ( 0g `  U )   &    |-  ( ph  ->  X  =/=  .0.  )   &    |-  ( ph  ->  Y  =/=  .0.  )   &    |-  ( ph  ->  Z  =/=  .0.  )   =>    |-  ( ph  ->  E. z  e.  V  -.  z  e.  ( N `  { X ,  Y ,  Z }
 ) )
 
Theoremdvhdimlem 31927* Lemma for dvh2dim 31928 and dvh3dim 31929. TODO: make this obsolete and use dvh4dimlem 31926 directly? (Contributed by NM, 24-Apr-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  ( ph  ->  X  =/=  .0.  )   &    |-  ( ph  ->  Y  =/=  .0.  )   =>    |-  ( ph  ->  E. z  e.  V  -.  z  e.  ( N `  { X ,  Y } ) )
 
Theoremdvh2dim 31928* There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  E. z  e.  V  -.  z  e.  ( N ` 
 { X } )
 )
 
Theoremdvh3dim 31929* There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  E. z  e.  V  -.  z  e.  ( N `  { X ,  Y } ) )
 
Theoremdvh4dimN 31930* There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   =>    |-  ( ph  ->  E. z  e.  V  -.  z  e.  ( N `  { X ,  Y ,  Z }
 ) )
 
Theoremdvh3dim2 31931* There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   =>    |-  ( ph  ->  E. z  e.  V  ( -.  z  e.  ( N `  { X ,  Y } )  /\  -.  z  e.  ( N `
  { X ,  Z } ) ) )
 
Theoremdvh3dim3N 31932* There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 31931 everywhere. If this one is needed, make dvh3dim2 31931 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  E. z  e.  V  ( -.  z  e.  ( N `  { X ,  Y } )  /\  -.  z  e.  ( N ` 
 { Z ,  T } ) ) )
 
Theoremdochsnnz 31933 The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  (  ._|_  `  { X }
 )  =/=  {  .0.  } )
 
Theoremdochsatshp 31934 The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 27-Jul-2014.) (Revised by Mario Carneiro, 1-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   &    |-  Y  =  (LSHyp `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( 
 ._|_  `  Q )  e.  Y )
 
Theoremdochsatshpb 31935 The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  Y  =  (LSHyp `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  Q  e.  S )   =>    |-  ( ph  ->  ( Q  e.  A  <->  (  ._|_  `  Q )  e.  Y )
 )
 
Theoremdochsnshp 31936 The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  Y  =  (LSHyp `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  { X }
 )  e.  Y )
 
Theoremdochshpsat 31937 A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   &    |-  Y  =  (LSHyp `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  Y )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  X ) )  =  X  <->  (  ._|_  `  X )  e.  A )
 )
 
Theoremdochkrsat 31938 The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (
 (  ._|_  `  ( L `  G ) )  =/=  {  .0.  }  <->  (  ._|_  `  ( L `  G ) )  e.  A ) )
 
Theoremdochkrsat2 31939 The orthocomplement of a kernel is an atom iff the double orthocomplement is not the vector space. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =/= 
 V 
 <->  (  ._|_  `  ( L `
  G ) )  e.  A ) )
 
Theoremdochsat0 31940 The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (
 (  ._|_  `  ( L `  G ) )  e.  A  \/  (  ._|_  `  ( L `  G ) )  =  {  .0.  } ) )
 
Theoremdochkrsm 31941 The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 31897 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( X  .(+)  (  ._|_  `  ( L `  G ) ) )  e.  ran  I
 )
 
Theoremdochexmidat 31942 Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( (  ._|_  `  { X } )  .(+)  ( N `
  { X }
 ) )  =  V )
 
Theoremdochexmidlem1 31943 Lemma for dochexmid 31951. Holland's proof implicitly requires  q  =/=  r, which we prove here. (Contributed by NM, 14-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  p  e.  A )   &    |-  ( ph  ->  q  e.  A )   &    |-  ( ph  ->  r  e.  A )   &    |-  ( ph  ->  q  C_  (  ._|_  `  X ) )   &    |-  ( ph  ->  r  C_  X )   =>    |-  ( ph  ->  q  =/=  r )
 
Theoremdochexmidlem2 31944 Lemma for dochexmid 31951. (Contributed by NM, 14-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  p  e.  A )   &    |-  ( ph  ->  q  e.  A )   &    |-  ( ph  ->  r  e.  A )   &    |-  ( ph  ->  q  C_  (  ._|_  `  X ) )   &    |-  ( ph  ->  r  C_  X )   &    |-  ( ph  ->  p  C_  ( r  .(+)  q ) )   =>    |-  ( ph  ->  p  C_  ( X  .(+)  (  ._|_  `  X ) ) )
 
Theoremdochexmidlem3 31945 Lemma for dochexmid 31951. Use atom exchange lsatexch1 29529 to swap  p and  q. (Contributed by NM, 14-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  p  e.  A )   &    |-  ( ph  ->  q  e.  A )   &    |-  ( ph  ->  r  e.  A )   &    |-  ( ph  ->  q  C_  (  ._|_  `  X ) )   &    |-  ( ph  ->  r  C_  X )   &    |-  ( ph  ->  q  C_  ( r  .(+)  p ) )   =>    |-  ( ph  ->  p  C_  ( X  .(+)  (  ._|_  `  X ) ) )
 
Theoremdochexmidlem4 31946 Lemma for dochexmid 31951. (Contributed by NM, 15-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  p  e.  A )   &    |-  ( ph  ->  q  e.  A )   &    |-  .0.  =  ( 0g `  U )   &    |-  M  =  ( X  .(+) 
 p )   &    |-  ( ph  ->  X  =/=  {  .0.  }
 )   &    |-  ( ph  ->  q  C_  ( (  ._|_  `  X )  i^i  M ) )   =>    |-  ( ph  ->  p  C_  ( X  .(+)  (  ._|_  `  X ) ) )
 
Theoremdochexmidlem5 31947 Lemma for dochexmid 31951. (Contributed by NM, 15-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  p  e.  A )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  M  =  ( X  .(+)  p )   &    |-  ( ph  ->  X  =/=  {  .0.  } )   &    |-  ( ph  ->  -.  p  C_  ( X  .(+) 
 (  ._|_  `  X )
 ) )   =>    |-  ( ph  ->  (
 (  ._|_  `  X )  i^i  M )  =  {  .0.  } )
 
Theoremdochexmidlem6 31948 Lemma for dochexmid 31951. (Contributed by NM, 15-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  p  e.  A )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  M  =  ( X  .(+)  p )   &    |-  ( ph  ->  X  =/=  {  .0.  } )   &    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  X ) )  =  X )   &    |-  ( ph  ->  -.  p  C_  ( X  .(+)  (  ._|_  `  X ) ) )   =>    |-  ( ph  ->  M  =  X )
 
Theoremdochexmidlem7 31949 Lemma for dochexmid 31951. Contradict dochexmidlem6 31948. (Contributed by NM, 15-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  p  e.  A )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  M  =  ( X  .(+)  p )   &    |-  ( ph  ->  X  =/=  {  .0.  } )   &    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  X ) )  =  X )   &    |-  ( ph  ->  -.  p  C_  ( X  .(+)  (  ._|_  `  X ) ) )   =>    |-  ( ph  ->  M  =/=  X )
 
Theoremdochexmidlem8 31950 Lemma for dochexmid 31951. The contradiction of dochexmidlem6 31948 and dochexmidlem7 31949 shows that there can be no atom  p that is not in  X  +  ( 
._|_  `  X ), which is therefore the whole atom space. (Contributed by NM, 15-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  ( ph  ->  X  =/=  {  .0.  } )   &    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )   =>    |-  ( ph  ->  ( X  .(+)  (  ._|_  `  X ) )  =  V )
 
Theoremdochexmid 31951 Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 31860. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables  X,  M,  p,  q,  r in place of Hollands' l, m, P, Q, L respectively. (pexmidALTN 30460 analog.) (Contributed by NM, 15-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )   =>    |-  ( ph  ->  ( X  .(+)  (  ._|_  `  X ) )  =  V )
 
Theoremdochsnkrlem1 31952 Lemma for dochsnkr 31955. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  ( (  ._|_  `  ( L `  G ) )  \  {  .0.  } ) )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =/=  V )
 
Theoremdochsnkrlem2 31953 Lemma for dochsnkr 31955. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  ( (  ._|_  `  ( L `  G ) )  \  {  .0.  } ) )   &    |-  A  =  (LSAtoms `  U )   =>    |-  ( ph  ->  (  ._|_  `  ( L `  G ) )  e.  A )
 
Theoremdochsnkrlem3 31954 Lemma for dochsnkr 31955. (Contributed by NM, 2-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  ( (  ._|_  `  ( L `  G ) )  \  {  .0.  } ) )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `
  G ) )
 
Theoremdochsnkr 31955 A (closed) kernel expressed in terms of a nonzero vector in its orthocomplement. TODO: consolidate lemmas unless they're needed for something else (in which case break out as theorems) (Contributed by NM, 2-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  ( (  ._|_  `  ( L `  G ) )  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { X } ) )
 
Theoremdochsnkr2 31956* Kernel of the explicit functional 
G determined by a nonzero vector  X. Compare the more general lshpkr 29600. (Contributed by NM, 27-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |- 
 .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (Scalar `  U )   &    |-  R  =  ( Base `  D )   &    |-  G  =  ( 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 `  G )  =  (  ._|_  `  { X } ) )
 
Theoremdochsnkr2cl 31957* The  X determining functional  G belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |- 
 .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (Scalar `  U )   &    |-  R  =  ( Base `  D )   &    |-  G  =  ( 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 `  G ) )  \  {  .0.  } ) )
 
Theoremdochflcl 31958* Closure of the explicit functional 
G determined by a nonzero vector  X. Compare the more general lshpkrcl 29599. (Contributed by NM, 27-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |- 
 .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (Scalar `  U )   &    |-  R  =  ( Base `  D )   &    |-  G  =  ( 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  ->  G  e.  F )
 
Theoremdochfl1 31959* The value of the explicit functional  G is 1 at the  X that determines it. (Contributed by NM, 27-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  D  =  (Scalar `  U )   &    |-  R  =  (
 Base `  D )   &    |-  .1.  =  ( 1r `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  G  =  ( v  e.  V  |->  (
 iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
 ) v  =  ( w  .+  ( k 
 .x.  X ) ) ) )   =>    |-  ( ph  ->  ( G `  X )  =  .1.  )
 
Theoremdochfln0 31960 The value of a functional is nonzero at a nonzero vector in the orthocomplement of its kernel. (Contributed by NM, 2-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  N  =  ( 0g `  R )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  ( (  ._|_  `  ( L `  G ) )  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( G `  X )  =/= 
 N )
 
Theoremdochkr1 31961* A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 29553. (Contributed by NM, 2-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  .0.  =  ( 0g `  U )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  G ) ) )  =/=  V )   =>    |-  ( ph  ->  E. x  e.  ( (  ._|_  `  ( L `  G ) ) 
 \  {  .0.  }
 ) ( G `  x )  =  .1.  )
 
Theoremdochkr1OLDN 31962* A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 29553. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  G ) ) )  =/=  V )   =>    |-  ( ph  ->  E. x  e.  (  ._|_  `  ( L `  G ) ) ( G `  x )  =  .1.  )
 
19.26.11  Construction of involution and inner product from a Hilbert lattice
 
SyntaxclpoN 31963 Extend class notation with all polarities of a left module or left vector space.
 class LPol
 
Definitiondf-lpolN 31964* Define the set of all polarities of a left module or left vector space. A polarity is a kind of complementation operation on a subspace. The double polarity of a subspace is a closure operation. Based on Definition 3.2 of [Holland95] p. 214 for projective geometry polarities. For convenience, we open up the domain to include all vector subsets and not just subspaces, but any more restricted polarity can be converted to this one by taking the span of its argument. (Contributed by NM, 24-Nov-2014.)
 |- LPol  =  ( w  e.  _V  |->  { o  e.  ( (
 LSubSp `  w )  ^m  ~P ( Base `  w )
 )  |  ( ( o `  ( Base `  w ) )  =  { ( 0g `  w ) }  /\  A. x A. y ( ( x  C_  ( Base `  w )  /\  y  C_  ( Base `  w )  /\  x  C_  y
 )  ->  ( o `  y )  C_  (
 o `  x )
 )  /\  A. x  e.  (LSAtoms `  w )
 ( ( o `  x )  e.  (LSHyp `  w )  /\  (
 o `  ( o `  x ) )  =  x ) ) }
 )
 
TheoremlpolsetN 31965* The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  P  =  (LPol `  W )   =>    |-  ( W  e.  X  ->  P  =  { o  e.  ( S  ^m  ~P V )  |  (
 ( o `  V )  =  {  .0.  } 
 /\  A. x A. y
 ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y ) 
 ->  ( o `  y
 )  C_  ( o `  x ) )  /\  A. x  e.  A  ( ( o `  x )  e.  H  /\  ( o `  (
 o `  x )
 )  =  x ) ) } )
 
TheoremislpolN 31966* The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  P  =  (LPol `  W )   =>    |-  ( W  e.  X  ->  (  ._|_  e.  P  <->  ( 
 ._|_  : ~P V --> S  /\  ( (  ._|_  `  V )  =  {  .0.  } 
 /\  A. x A. y
 ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y ) 
 ->  (  ._|_  `  y
 )  C_  (  ._|_  `  x ) )  /\  A. x  e.  A  ( (  ._|_  `  x )  e.  H  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) ) ) ) )
 
TheoremislpoldN 31967* Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  : ~P V --> S )   &    |-  ( ph  ->  ( 
 ._|_  `  V )  =  {  .0.  } )   &    |-  (
 ( ph  /\  ( x 
 C_  V  /\  y  C_  V  /\  x  C_  y ) )  ->  (  ._|_  `  y )  C_  (  ._|_  `  x ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  (  ._|_  `  x )  e.  H )   &    |-  (
 ( ph  /\  x  e.  A )  ->  (  ._|_  `  (  ._|_  `  x ) )  =  x )   =>    |-  ( ph  ->  ._|_  e.  P )
 
TheoremlpolfN 31968 Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  e.  P )   =>    |-  ( ph  ->  ._|_  : ~P V
 --> S )
 
TheoremlpolvN 31969 The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  e.  P )   =>    |-  ( ph  ->  (  ._|_  `  V )  =  {  .0.  } )
 
TheoremlpolconN 31970 Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  e.  P )   &    |-  ( ph  ->  X  C_  V )   &    |-  ( ph  ->  Y 
 C_  V )   &    |-  ( ph  ->  X  C_  Y )   =>    |-  ( ph  ->  (  ._|_  `  Y )  C_  (  ._|_  `  X )
 )
 
TheoremlpolsatN 31971 The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  A  =  (LSAtoms `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  e.  P )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  (  ._|_  `  Q )  e.  H )
 
TheoremlpolpolsatN 31972 Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  A  =  (LSAtoms `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  e.  P )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  Q ) )  =  Q )
 
TheoremdochpolN 31973 The subspace orthocomplement for the  DVecH vector space is a polarity. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  P  =  (LPol `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ._|_  e.  P )
 
Theoremlcfl1lem 31974* Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)
 |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   =>    |-  ( G  e.  C 
 <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) ) )
 
Theoremlcfl1 31975* Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.)
 |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) ) )
 
Theoremlcfl2 31976* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  (  ._|_  `  ( L `
  G ) ) )  =/=  V  \/  ( L `  G )  =  V ) ) )
 
Theoremlcfl3 31977* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  ( L `  G ) )  e.  A  \/  ( L `  G )  =  V )
 ) )
 
Theoremlcfl4N 31978* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  Y  =  (LSHyp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  (  ._|_  `  ( L `
  G ) ) )  e.  Y  \/  ( L `  G )  =  V ) ) )
 
Theoremlcfl5 31979* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( L `  G )  e.  ran  I ) )
 
Theoremlcfl5a 31980 Property of a functional with a closed kernel. TODO: Make lcfl5 31979 etc. obsolete and rewrite w/out 
C hypothesis? (Contributed by NM, 29-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) 
 <->  ( L `  G )  e.  ran  I ) )
 
Theoremlcfl6lem 31981* Lemma for lcfl6 31983. A functional  G (whose kernel is closed by dochsnkr 31955) is comletely determined by a vector  X in the orthocomplement in its kernel at which the functional value is 1. Note that the  \  {  .0.  } in the  X hypothesis is redundant by the last hypothesis but allows easier use of other theorems. (Contributed by NM, 3-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .1.  =  ( 1r `  S )   &    |-  R  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  ( (  ._|_  `  ( L `  G ) )  \  {  .0.  } ) )   &    |-  ( ph  ->  ( G `  X )  =  .1.  )   =>    |-  ( ph  ->  G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
 k  .x.  X )
 ) ) ) )
 
Theoremlcfl7lem 31982* Lemma for lcfl7N 31984. If two functionals  G and  J are equal, they are determined by the same vector. (Contributed by NM, 4-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
 k  .x.  X )
 ) ) )   &    |-  J  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { Y }
 ) v  =  ( w  .+  ( k 
 .x.  Y ) ) ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  G  =  J )   =>    |-  ( ph  ->  X  =  Y )
 
Theoremlcfl6 31983* Property of a functional with a closed kernel. Note that  ( L `  G )  =  V means the functional is zero by lkr0f 29577. (Contributed by NM, 3-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  {
 f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( ( L `
  G )  =  V  \/  E. x  e.  ( V  \  {  .0.  } ) G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x }
 ) v  =  ( w  .+  ( k 
 .x.  x ) ) ) ) ) ) )
 
Theoremlcfl7N 31984* Property of a functional with a closed kernel. Every nonzero functional is determined by a unique nonzero vector. Note that  ( L `  G )  =  V means the functional is zero by lkr0f 29577. (Contributed by NM, 4-Jan-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  {
 f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( ( L `
  G )  =  V  \/  E! x  e.  ( V  \  {  .0.  } ) G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x }
 ) v  =  ( w  .+  ( k 
 .x.  x ) ) ) ) ) ) )
 
Theoremlcfl8 31985* Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  E. x  e.  V  ( L `  G )  =  (  ._|_  `  { x } ) ) )
 
Theoremlcfl8a 31986* Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) 
 <-> 
 E. x  e.  V  ( L `  G )  =  (  ._|_  `  { x } ) ) )
 
Theoremlcfl8b 31987* Property of a nonzero functional with a closed kernel. (Contributed by NM, 4-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Y  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  ( C  \  { Y } ) )   =>    |-  ( ph  ->  E. x  e.  ( V  \  {  .0.  } ) (  ._|_  `  ( L `  G ) )  =  ( N `  { x }
 ) )
 
Theoremlcfl9a 31988 Property implying that a functional has a closed kernel. (Contributed by NM, 16-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  (  ._|_  `  { X }
 )  C_  ( L `  G ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `
  G ) )
 
Theoremlclkrlem1 31989* 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 31990 Lemma for lclkr 32016. Use lshpat 29539 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 31991 Lemma for lclkr 32016. (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 31992 Lemma for lclkr 32016. (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 31993 Lemma for lclkr 32016. (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 31994 Lemma for lclkr 32016. 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 31995 Lemma for lclkr 32016. 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 31996 Lemma for lclkr 32016. 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 31997 Lemma for lclkr 32016. 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 31998 Lemma for lclkr 32016. 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 31999 Lemma for lclkr 32016. 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 32000 Lemma for lclkr 32016. 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 ) ) )
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