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Theorem List for Metamath Proof Explorer - 31601-31700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdihatexv 31601* 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 31602* 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 31603* 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 31604* 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 31605* 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 31606 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 31607 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 31608 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 31609 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 31610 Extend class notation with subspace orthocomplement for  DVecH vector space.
 class  ocH
 
Definitiondf-doch 31611* 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 31612* 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 31613* 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 31614* 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 31615* 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 31616 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 31617 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 31618 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 31619 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 31620 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 31621 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 31622 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 31623 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 31624 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 31625 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 31626 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 31627* 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 31628 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 31629 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 31630 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 31631 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 31632 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 31633 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 31634 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 31635 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 31636 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 31637 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 31638 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 31639 Version of dihoml4 31640 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 31640 Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 30215 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 31641 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 31642 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 31643 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 31644 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 31645 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 31646 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 31647 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 31648 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 31649 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 31650 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 31651 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 31652 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 31653 De Morgan-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 31654 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 31655 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 31656 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 31657 Extend class notation with subspace join for  DVecH vector space.
 class joinH
 
Definitiondf-djh 31658* 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 31659* 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 31660* 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 31661 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 31662 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 31663 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 31664 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 31665 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 31666  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 31667  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 31668 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 31669 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 31670 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 31671 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 31672 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 31673 De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  .\/  =  ( (joinH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  (  ._|_  `  ( X  i^i  Y ) )  =  ( (  ._|_  `  X ) 
 .\/  (  ._|_  `  Y ) ) )
 
Theoremdjhexmid 31674 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 31675 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 31676 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 31677 A closed subspace sum equals subspace join. (shjshseli 22074 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 31678* A covering property. (cvrat42 29706 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 31679 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 31680 Isomorphism H of lattice join of an element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( X  e.  B  /\  X  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( X 
 .\/  P ) )  =  ( ( I `  X )  .(+)  ( I `
  P ) ) )
 
Theoremdihjatcclem1 31681 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 31682 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 31683* Lemma for dihjatcc 31685. (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 31684* 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 31685 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 31686 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 31687 Lemma for dihprrn 31689, 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 31688 Lemma for dihprrn 31689. (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 31689 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 31690 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 31689; should we directly use dihjat 31686? (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 31691 Subspace sum of a closed subspace and an atom. (pmapjat1 30115 analog.) TODO: merge into dihjat1 31692? (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 31692 Subspace sum of a closed subspace and an atom. (pmapjat1 30115 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 31693 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 31694 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 31695 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 31696 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 31697 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 31698 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 31699 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 31694. (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 31700 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 ) ) ) )
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