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Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremdihlspsnssN 30801 A subspace included in a 1-dim subspace belongs to the range of isomorphism H. (Contributed by NM, 26-Apr-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  T  C_  ( N `  { X } ) ) 
 ->  ( T  e.  S  <->  T  e.  ran  I )
 )
 
Theoremdihlspsnat 30802 The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)
 |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( `' I `  ( N `  { X } ) )  e.  A )
 
Theoremdihatlat 30803 The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
 |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  L  =  (LSAtoms `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  A )  ->  ( I `  Q )  e.  L )
 
Theoremdihat 30804 There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   =>    |-  ( ph  ->  ( I `  P )  e.  A )
 
TheoremdihpN 30805* The value of isomorphism H at the fiducial atom  P is determined by the vector  <. 0 ,  S >. (the zero translation ltrnid 29603 and a nonzero member of the endomorphism ring). In particular,  S can be replaced with the ring unit  (  _I  |`  T ). (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( S  e.  E  /\  S  =/=  O ) )   =>    |-  ( ph  ->  ( I `  P )  =  ( N `  { <. (  _I  |`  B ) ,  S >. } ) )
 
Theoremdihlatat 30806 The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.)
 |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  L  =  (LSAtoms `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  Q  e.  L )  ->  ( `' I `  Q )  e.  A )
 
Theoremdihatexv 30807* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  Q  e.  B )   =>    |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( V  \  {  .0.  }
 ) ( I `  Q )  =  ( N `  { x }
 ) ) )
 
Theoremdihatexv2 30808* There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)
 |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   =>    |-  ( ph  ->  ( Q  e.  A  <->  E. x  e.  ( V  \  {  .0.  }
 ) Q  =  ( `' I `  ( N `
  { x }
 ) ) ) )
 
Theoremdihglblem6 30809* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  P  =  ( LSubSp `  U )   &    |-  D  =  (LSAtoms `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) ) 
 ->  ( I `  ( G `  S ) )  =  |^|_ x  e.  S  ( I `  x ) )
 
Theoremdihglb 30810* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/= 
 (/) ) )  ->  ( I `  ( G `
  S ) )  =  |^|_ x  e.  S  ( I `  x ) )
 
Theoremdihglb2 30811* Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  C_  V )  ->  ( I `  ( G `  { x  e.  B  |  S  C_  ( I `  x ) } ) )  = 
 |^| { y  e.  ran  I  |  S  C_  y } )
 
Theoremdihmeet 30812 Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `
  X )  i^i  ( I `  Y ) ) )
 
Theoremdihintcl 30813 The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) ) 
 ->  |^| S  e.  ran  I )
 
Theoremdihmeetcl 30814 Closure of closed subspace meet for  DVecH vector space. (Contributed by NM, 5-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  ran  I 
 /\  Y  e.  ran  I ) )  ->  ( X  i^i  Y )  e. 
 ran  I )
 
Theoremdihmeet2 30815 Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)
 |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  ( `' I `  ( X  i^i  Y ) )  =  ( ( `' I `  X ) 
 ./\  ( `' I `  Y ) ) )
 
Syntaxcoch 30816 Extend class notation with subspace orthocomplement for  DVecH vector space.
 class  ocH
 
Definitiondf-doch 30817* Define subspace orthocomplement for  DVecH vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 14-Mar-2014.)
 |-  ocH  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ~P ( Base `  ( ( DVecH `  k ) `  w ) )  |->  ( ( ( DIsoH `  k ) `  w ) `  (
 ( oc `  k
 ) `  ( ( glb `  k ) `  { y  e.  ( Base `  k )  |  x  C_  ( (
 ( DIsoH `  k ) `  w ) `  y
 ) } ) ) ) ) ) )
 
Theoremdochffval 30818* Subspace orthocomplement for 
DVecH vector space. (Contributed by NM, 14-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( K  e.  V  ->  ( ocH `  K )  =  ( w  e.  H  |->  ( x  e. 
 ~P ( Base `  (
 ( DVecH `  K ) `  w ) )  |->  ( ( ( DIsoH `  K ) `  w ) `  (  ._|_  `  ( G ` 
 { y  e.  B  |  x  C_  ( ( ( DIsoH `  K ) `  w ) `  y
 ) } ) ) ) ) ) )
 
Theoremdochfval 30819* Subspace orthocomplement for 
DVecH vector space. (Contributed by NM, 14-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( K  e.  X  /\  W  e.  H )  ->  N  =  ( x  e.  ~P V  |->  ( I `  (  ._|_  `  ( G ` 
 { y  e.  B  |  x  C_  ( I `
  y ) }
 ) ) ) ) )
 
Theoremdochval 30820* Subspace orthocomplement for 
DVecH vector space. (Contributed by NM, 14-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  Y  /\  W  e.  H )  /\  X  C_  V )  ->  ( N `  X )  =  ( I `  (  ._|_  `  ( G `  { y  e.  B  |  X  C_  ( I `  y ) } ) ) ) )
 
Theoremdochval2 30821* Subspace orthocomplement for 
DVecH vector space. (Contributed by NM, 14-Apr-2014.)
 |-  ._|_  =  ( oc `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   &    |-  N  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V )  ->  ( N `  X )  =  ( I `  (  ._|_  `  ( `' I `  |^| { z  e.  ran  I  |  X  C_  z } ) ) ) )
 
Theoremdochcl 30822 Closure of subspace orthocomplement for  DVecH vector space. (Contributed by NM, 9-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V )  ->  (  ._|_  `  X )  e.  ran  I )
 
Theoremdochlss 30823 A subspace orthocomplement is a subspace of the  DVecH vector space. (Contributed by NM, 22-Jul-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V )  ->  (  ._|_  `  X )  e.  S )
 
Theoremdochssv 30824 A subspace orthocomplement belongs to the  DVecH vector space. (Contributed by NM, 22-Jul-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V )  ->  (  ._|_  `  X ) 
 C_  V )
 
TheoremdochfN 30825 Domain and codomain of the subspace orthocomplement for the  DVecH vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ._|_  : ~P V
 --> ran  I )
 
Theoremdochvalr 30826 Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
 |-  ._|_  =  ( oc `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  N  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I
 )  ->  ( N `  X )  =  ( I `  (  ._|_  `  ( `' I `  X ) ) ) )
 
Theoremdoch0 30827 Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  ._|_  `  {  .0.  } )  =  V )
 
Theoremdoch1 30828 Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  ._|_  `  V )  =  {  .0.  } )
 
Theoremdochoc0 30829 The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  {  .0.  } ) )  =  {  .0.  } )
 
Theoremdochoc1 30830 The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  V ) )  =  V )
 
Theoremdochvalr2 30831 Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  N  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B ) 
 ->  ( N `  ( I `  X ) )  =  ( I `  (  ._|_  `  X )
 ) )
 
Theoremdochvalr3 30832 Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.)
 |-  ._|_  =  ( oc `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  N  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   =>    |-  ( ph  ->  (  ._|_  `  ( `' I `  X ) )  =  ( `' I `  ( N `
  X ) ) )
 
Theoremdoch2val2 30833* Double orthocomplement for 
DVecH vector space. (Contributed by NM, 26-Jul-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X 
 C_  V )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  X ) )  =  |^| { z  e.  ran  I  |  X  C_  z }
 )
 
Theoremdochss 30834 Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  (  ._|_  `  Y )  C_  (  ._|_  `  X ) )
 
Theoremdochocss 30835 Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
 
Theoremdochoc 30836 Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I
 )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
 
Theoremdochsscl 30837 If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X 
 C_  V )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  ( X  C_  Y  <->  (  ._|_  `  (  ._|_  `  X ) ) 
 C_  Y ) )
 
Theoremdochoccl 30838 A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X 
 C_  V )   =>    |-  ( ph  ->  ( X  e.  ran  I  <->  ( 
 ._|_  `  (  ._|_  `  X ) )  =  X ) )
 
Theoremdochord 30839 Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  ( X  C_  Y  <->  (  ._|_  `  Y )  C_  (  ._|_  `  X ) ) )
 
Theoremdochord2N 30840 Ordering law for orthocomplement. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  (
 (  ._|_  `  X )  C_  Y  <->  (  ._|_  `  Y )  C_  X ) )
 
Theoremdochord3 30841 Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  ( X  C_  (  ._|_  `  Y ) 
 <->  Y  C_  (  ._|_  `  X ) ) )
 
Theoremdoch11 30842 Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  (
 (  ._|_  `  X )  =  (  ._|_  `  Y ) 
 <->  X  =  Y ) )
 
TheoremdochsordN 30843 Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  ( X  C.  Y  <->  (  ._|_  `  Y )  C.  (  ._|_  `  X ) ) )
 
Theoremdochn0nv 30844 An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-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  C_  V )   =>    |-  ( ph  ->  (
 (  ._|_  `  X )  =/=  {  .0.  }  <->  (  ._|_  `  (  ._|_  `  X ) )  =/=  V ) )
 
Theoremdihoml4c 30845 Version of dihoml4 30846 with closed subspaces. (Contributed by NM, 15-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   &    |-  ( ph  ->  X 
 C_  Y )   =>    |-  ( ph  ->  ( (  ._|_  `  ( ( 
 ._|_  `  X )  i^i 
 Y ) )  i^i 
 Y )  =  X )
 
Theoremdihoml4 30846 Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 29421 analog.) (Contributed by NM, 15-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   &    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  Y ) )  =  Y )   &    |-  ( ph  ->  X  C_  Y )   =>    |-  ( ph  ->  (
 (  ._|_  `  ( (  ._|_  `  X )  i^i 
 Y ) )  i^i 
 Y )  =  ( 
 ._|_  `  (  ._|_  `  X ) ) )
 
Theoremdochspss 30847 The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.)
 |-  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 
 C_  V )   =>    |-  ( ph  ->  ( N `  X ) 
 C_  (  ._|_  `  (  ._|_  `  X ) ) )
 
Theoremdochocsp 30848 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.)
 |-  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 
 C_  V )   =>    |-  ( ph  ->  ( 
 ._|_  `  ( N `  X ) )  =  (  ._|_  `  X ) )
 
TheoremdochspocN 30849 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.)
 |-  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 
 C_  V )   =>    |-  ( ph  ->  ( N `  (  ._|_  `  X ) )  =  (  ._|_  `  ( N `
  X ) ) )
 
Theoremdochocsn 30850 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 30851 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 30852 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 30853 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 30854 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 30855 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 30856 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 30857 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 30858 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 30859 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 30860 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 30861 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 30862 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 30863 Extend class notation with subspace join for  DVecH vector space.
 class joinH
 
Definitiondf-djh 30864* 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 30865* 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 30866* 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 30867 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 30868 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 30869 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 30870 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 30871 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 30872  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 30873  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 30874 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 30875 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 30876 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 30877 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 30878 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 30879 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 30880 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 30881 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 30882 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 30883 A closed subspace sum equals subspace join. (shjshseli 22068 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 30884* A covering property. (cvrat42 28912 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 30885 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 30886 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 30887 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 30888 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 30889* Lemma for dihjatcc 30891. (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 30890* 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 30891 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 30892 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 30893 Lemma for dihprrn 30895, 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 30894 Lemma for dihprrn 30895. (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 30895 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 30896 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 30895; should we directly use dihjat 30892? (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 30897 Subspace sum of a closed subspace and an atom. (pmapjat1 29321 analog.) TODO: merge into dihjat1 30898? (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 30898 Subspace sum of a closed subspace and an atom. (pmapjat1 29321 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 30899 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 30900 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 ) )
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