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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdochocsn 30701 The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  { X } ) )  =  ( N `  { X } ) )
 
Theoremdochsncom 30702 Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( X  e.  (  ._|_  ` 
 { Y } )  <->  Y  e.  (  ._|_  `  { X } ) ) )
 
Theoremdochsat 30703 The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  Q  e.  S )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  Q ) )  e.  A  <->  Q  e.  A ) )
 
Theoremdochshpncl 30704 If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  Y  =  (LSHyp `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  Y )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  X ) )  =/= 
 X 
 <->  (  ._|_  `  (  ._|_  `  X ) )  =  V ) )
 
Theoremdochlkr 30705 Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  Y  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  e.  Y  <->  ( (  ._|_  `  (  ._|_  `  ( L `
  G ) ) )  =  ( L `
  G )  /\  ( L `  G )  e.  Y ) ) )
 
Theoremdochkrshp 30706 The closure of a kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  Y  =  (LSHyp `  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.  Y ) )
 
Theoremdochkrshp2 30707 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  Y  =  (LSHyp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =/= 
 V 
 <->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G )  /\  ( L `
  G )  e.  Y ) ) )
 
Theoremdochkrshp3 30708 Properties of the closure of the kernel of a functional. (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 )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =/= 
 V 
 <->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G )  /\  ( L `
  G )  =/= 
 V ) ) )
 
Theoremdochkrshp4 30709 Properties of the closure of the kernel of a functional. (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 )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) 
 <->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =/=  V  \/  ( L `  G )  =  V ) ) )
 
Theoremdochdmj1 30710 DeMorgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V  /\  Y  C_  V )  ->  (  ._|_  `  ( X  u.  Y ) )  =  ( (  ._|_  `  X )  i^i  (  ._|_  `  Y ) ) )
 
Theoremdochnoncon 30711 Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .0.  =  ( 0g `  U )   &    |- 
 ._|_  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  S )  ->  ( X  i^i  (  ._|_  `  X ) )  =  {  .0.  } )
 
Theoremdochnel2 30712 A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .0.  =  ( 0g `  U )   &    |- 
 ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  X  e.  ( T  \  {  .0.  } ) )   =>    |-  ( ph  ->  -.  X  e.  (  ._|_  `  T )
 )
 
Theoremdochnel 30713 A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)
 |-  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  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  -.  X  e.  (  ._|_  `  { X } ) )
 
Syntaxcdjh 30714 Extend class notation with subspace join for  DVecH vector space.
 class joinH
 
Definitiondf-djh 30715* Define (closed) subspace join for  DVecH vector space. (Contributed by NM, 19-Jul-2014.)
 |- joinH  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  k
 ) `  w )
 ) ,  y  e. 
 ~P ( Base `  (
 ( DVecH `  k ) `  w ) )  |->  ( ( ( ocH `  k
 ) `  w ) `  ( ( ( ( ocH `  k ) `  w ) `  x )  i^i  ( ( ( ocH `  k ) `  w ) `  y
 ) ) ) ) ) )
 
Theoremdjhffval 30716* Subspace join for  DVecH vector space. (Contributed by NM, 19-Jul-2014.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  X  ->  (joinH `  K )  =  ( w  e.  H  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  K ) `  w ) ) ,  y  e.  ~P ( Base `  (
 ( DVecH `  K ) `  w ) )  |->  ( ( ( ocH `  K ) `  w ) `  ( ( ( ( ocH `  K ) `  w ) `  x )  i^i  ( ( ( ocH `  K ) `  w ) `  y
 ) ) ) ) ) )
 
Theoremdjhfval 30717* Subspace join for  DVecH vector space. (Contributed by NM, 19-Jul-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  .\/  =  ( (joinH `  K ) `  W )   =>    |-  ( ( K  e.  X  /\  W  e.  H )  ->  .\/  =  ( x  e.  ~P V ,  y  e.  ~P V  |->  (  ._|_  `  (
 (  ._|_  `  x )  i^i  (  ._|_  `  y ) ) ) ) )
 
Theoremdjhval 30718 Subspace join for  DVecH vector space. (Contributed by NM, 19-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  /\  Y  C_  V )
 )  ->  ( X  .\/  Y )  =  ( 
 ._|_  `  ( (  ._|_  `  X )  i^i  (  ._|_  `  Y ) ) ) )
 
Theoremdjhval2 30719 Value of subspace join for 
DVecH vector space. (Contributed by NM, 6-Aug-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  /\  Y  C_  V )  ->  ( X  .\/  Y )  =  (  ._|_  `  (  ._|_  `  ( X  u.  Y ) ) ) )
 
Theoremdjhcl 30720 Closure of subspace join for 
DVecH vector space. (Contributed by NM, 19-Jul-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .\/  =  ( (joinH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  V  /\  Y  C_  V )
 )  ->  ( X  .\/  Y )  e.  ran  I )
 
Theoremdjhlj 30721 Transfer lattice join to  DVecH vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  J  =  ( (joinH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( I `  ( X  .\/  Y ) )  =  (
 ( I `  X ) J ( I `  Y ) ) )
 
TheoremdjhljjN 30722 Lattice join in terms of  DVecH vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  J  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .\/  Y )  =  ( `' I `  ( ( I `  X ) J ( I `  Y ) ) ) )
 
Theoremdjhjlj 30723  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 30724  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 30725 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 30726 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 30727 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 30728 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 30729 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 30730 DeMorgan-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 30731 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 30732 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 30733 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 30734 A closed subspace sum equals subspace join. (shjshseli 21997 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 30735* A covering property. (cvrat42 28763 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 30736 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 30737 Isomorphism H of lattice join of a 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 30738 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 30739 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 30740* Lemma for dihjatcc 30742. (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 30741* 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 30742 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 30743 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 30744 Lemma for dihprrn 30746, 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 30745 Lemma for dihprrn 30746. (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 30746 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 30747 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 30746; should we directly use dihjat 30743? (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 30748 Subspace sum of a closed subspace and an atom. (pmapjat1 29172 analog.) TODO: merge into dihjat1 30749? (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 30749 Subspace sum of a closed subspace and an atom. (pmapjat1 29172 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 30750 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 30751 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 30752 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 30753 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 30754 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 30755 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 30756 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 30751. (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 30757 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 30758* 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 30759* 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 30760* 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 30761* 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 30762* 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 30763* Lemma for dvh4dimN 30767. (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 30764* Lemma for dvh2dim 30765 and dvh3dim 30766. TODO: make this obsolete and use dvh4dimlem 30763 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 30765* 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 30766* 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 30767* 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 30768* 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 30769* There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 30768 everywhere. If this one is needed, make dvh3dim2 30768 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 30770 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 30771 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 30772 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 30773 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 30774 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 30775 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 30776 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 30777 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 30778 The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 30734 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 30779 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 30780 Lemma for dochexmid 30788. 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 30781 Lemma for dochexmid 30788. (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 30782 Lemma for dochexmid 30788. Use atom exchange lsatexch1 28366 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 30783 Lemma for dochexmid 30788. (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 30784 Lemma for dochexmid 30788. (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 30785 Lemma for dochexmid 30788. (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 30786 Lemma for dochexmid 30788. Contradict dochexmidlem6 30785. (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 30787 Lemma for dochexmid 30788. The contradiction of dochexmidlem6 30785 and dochexmidlem7 30786 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 30788 Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 30697. 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 29297 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 30789 Lemma for dochsnkr 30792. (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 30790 Lemma for dochsnkr 30792. (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 30791 Lemma for dochsnkr 30792. (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 30792 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 30793* Kernel of the explicit functional 
G determined by a nonzero vector  X. Compare the more general lshpkr 28437. (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 30794* 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 30795* Closure of the explicit functional 
G determined by a nonzero vector  X. Compare the more general lshpkrcl 28436. (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 30796* 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 30797 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 30798* 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 28390. (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 30799* 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 28390. (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.  )
 
16.24.15  Construction of involution and inner product from a Hilbert lattice
 
SyntaxclpoN 30800 Extend class notation with all polarities of a left module or left vector space.
 class LPol
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