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Theorem List for Metamath Proof Explorer - 36601-36700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdjhjlj 36601  DVecH vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
 |-  .\/  =  ( join `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  J  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  ( X J Y )  =  ( I `  (
 ( `' I `  X )  .\/  ( `' I `  Y ) ) ) )
 
Theoremdjhj 36602  DVecH vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
 |-  .\/  =  ( join `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  J  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  ( `' I `  ( X J Y ) )  =  ( ( `' I `  X ) 
 .\/  ( `' I `  Y ) ) )
 
Theoremdjhcom 36603 Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .\/  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X 
 C_  V )   &    |-  ( ph  ->  Y  C_  V )   =>    |-  ( ph  ->  ( X  .\/  Y )  =  ( Y  .\/  X ) )
 
Theoremdjhspss 36604 Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .\/  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X 
 C_  V )   &    |-  ( ph  ->  Y  C_  V )   =>    |-  ( ph  ->  ( N `  ( X  u.  Y ) )  C_  ( X  .\/  Y ) )
 
Theoremdjhsumss 36605 Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  .\/  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X 
 C_  V )   &    |-  ( ph  ->  Y  C_  V )   =>    |-  ( ph  ->  ( X  .(+)  Y )  C_  ( X  .\/  Y ) )
 
Theoremdihsumssj 36606 The subspace sum of two isomorphisms of lattice elements is less than the isomorphism of their lattice join. (Contributed by NM, 23-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( I `  X )  .(+)  ( I `  Y ) )  C_  ( I `  ( X 
 .\/  Y ) ) )
 
TheoremdjhunssN 36607 Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .\/  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X 
 C_  V )   &    |-  ( ph  ->  Y  C_  V )   =>    |-  ( ph  ->  ( X  u.  Y )  C_  ( X  .\/  Y ) )
 
Theoremdochdmm1 36608 De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  .\/  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  (  ._|_  `  ( X  i^i  Y ) )  =  ( (  ._|_  `  X ) 
 .\/  (  ._|_  `  Y ) ) )
 
Theoremdjhexmid 36609 Excluded middle property of 
DVecH vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  .\/  =  ( (joinH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V )  ->  ( X  .\/  (  ._|_  `  X ) )  =  V )
 
Theoremdjh01 36610 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  .\/  =  (
 (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ran  I )   =>    |-  ( ph  ->  ( X  .\/  {  .0.  } )  =  X )
 
Theoremdjh02 36611 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  .\/  =  (
 (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ran  I )   =>    |-  ( ph  ->  ( {  .0.  }  .\/  X )  =  X )
 
Theoremdjhlsmcl 36612 A closed subspace sum equals subspace join. (shjshseli 26234 analog.) (Contributed by NM, 13-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .\/  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  (
 ( X  .(+)  Y )  e.  ran  I  <->  ( X  .(+)  Y )  =  ( X 
 .\/  Y ) ) )
 
Theoremdjhcvat42 36613* A covering property. (cvrat42 34641 analog.) (Contributed by NM, 17-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .\/  =  (
 (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  S  e.  ran  I )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( ( S  =/=  {  .0.  }  /\  ( N `  { X } )  C_  ( S 
 .\/  ( N `  { Y } ) ) )  ->  E. z  e.  ( V  \  {  .0.  } ) ( ( N `  { z } )  C_  S  /\  ( N `  { X } )  C_  ( ( N `  { z } )  .\/  ( N `
  { Y }
 ) ) ) ) )
 
Theoremdihjatb 36614 Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-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 ) ) )
 
Theoremdihjatc 36615 Isomorphism H of lattice join of an element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( 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  ->  ( X  e.  B  /\  X  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( X 
 .\/  P ) )  =  ( ( I `  X )  .(+)  ( I `
  P ) ) )
 
Theoremdihjatcclem1 36616 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 36617 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 36618* Lemma for dihjatcc 36620. (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 36619* 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 36620 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 36621 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 36622 Lemma for dihprrn 36624, 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 36623 Lemma for dihprrn 36624. (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 36624 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 36625 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 36624; should we directly use dihjat 36621? (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 36626 Subspace sum of a closed subspace and an atom. (pmapjat1 35050 analog.) TODO: merge into dihjat1 36627? (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 36627 Subspace sum of a closed subspace and an atom. (pmapjat1 35050 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 36628 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 36629 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 36630 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 36631 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 36632 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 36633 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 36634 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 36629. (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 36635 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 36636* 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 36637* 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 36638* 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 36639* 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 36640* 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 36641* Lemma for dvh4dimN 36645. (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 36642* Lemma for dvh2dim 36643 and dvh3dim 36644. TODO: make this obsolete and use dvh4dimlem 36641 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 36643* 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 36644* 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 36645* 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 36646* 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 36647* There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 36646 everywhere. If this one is needed, make dvh3dim2 36646 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 36648 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 36649 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 36650 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 36651 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 36652 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 36653 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 36654 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 36655 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 36656 The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 36612 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 36657 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 36658 Lemma for dochexmid 36666. 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 36659 Lemma for dochexmid 36666. (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 36660 Lemma for dochexmid 36666. Use atom exchange lsatexch1 34244 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 36661 Lemma for dochexmid 36666. (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 36662 Lemma for dochexmid 36666. (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 36663 Lemma for dochexmid 36666. (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 36664 Lemma for dochexmid 36666. Contradict dochexmidlem6 36663. (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 36665 Lemma for dochexmid 36666. The contradiction of dochexmidlem6 36663 and dochexmidlem7 36664 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 36666 Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 36575. 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 35175 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 36667 Lemma for dochsnkr 36670. (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 36668 Lemma for dochsnkr 36670. (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 36669 Lemma for dochsnkr 36670. (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 36670 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 36671* Kernel of the explicit functional 
G determined by a nonzero vector  X. Compare the more general lshpkr 34315. (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 36672* 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 36673* Closure of the explicit functional 
G determined by a nonzero vector  X. Compare the more general lshpkrcl 34314. (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 36674* 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 36675 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 36676* 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 34268. (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 36677* 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 34268. (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.  )
 
21.30.11  Construction of involution and inner product from a Hilbert lattice
 
SyntaxclpoN 36678 Extend class notation with all polarities of a left module or left vector space.
 class LPol
 
Definitiondf-lpolN 36679* 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 36680* 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 36681* 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 36682* 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 36683 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 36684 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 36685 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 36686 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 36687 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 36688 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 36689* 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 36690* 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 36691* 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 36692* 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 36693* 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 36694* 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 36695 Property of a functional with a closed kernel. TODO: Make lcfl5 36694 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 36696* Lemma for lcfl6 36698. A functional  G (whose kernel is closed by dochsnkr 36670) 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 36697* Lemma for lcfl7N 36699. 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 36698* Property of a functional with a closed kernel. Note that  ( L `  G )  =  V means the functional is zero by lkr0f 34292. (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 36699* 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 34292. (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 36700* 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 } ) ) )
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