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Theorem List for Metamath Proof Explorer - 30601-30700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdih0bN 30601 A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  Z  =  ( 0g `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  =  .0.  <->  ( I `  X )  =  { Z } ) )
 
Theoremdih0vbN 30602 A vector is zero iff its span is the isomorphism of lattice zero. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   &    |-  Z  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( X  =  Z  <->  ( N `  { X } )  =  ( I `  .0.  ) ) )
 
Theoremdih0cnv 30603 The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  Z  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( `' I `  { Z } )  =  .0.  )
 
Theoremdih0rn 30604 The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  {  .0.  }  e.  ran 
 I )
 
Theoremdih0sb 30605 A subspace is zero iff the converse of its isomorphism is lattice zero. (Contributed by NM, 17-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   &    |-  Z  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ran  I )   =>    |-  ( ph  ->  ( X  =  { Z } 
 <->  ( `' I `  X )  =  .0.  ) )
 
Theoremdih1 30606 The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)
 |-  .1.  =  ( 1. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .1.  )  =  V )
 
Theoremdih1rn 30607 The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  V  e.  ran  I
 )
 
Theoremdih1cnv 30608 The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( `' I `  V )  =  .1.  )
 
TheoremdihwN 30609* Value of isomorphism H at the fiducial hyperplane  W. (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   =>    |-  ( ph  ->  ( I `  W )  =  ( T  X.  {  .0.  } ) )
 
Theoremdihmeetlem1N 30610* Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  q )   &    |-  .0.  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `
  X )  i^i  ( I `  Y ) ) )
 
Theoremdihglblem5apreN 30611* A conjunction property of isomorphism H. TODO: reduce antecedent size; general review for shorter proof. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  q )   &    |-  .0.  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 ->  ( I `  ( X  ./\  W ) )  =  ( ( I `
  X )  i^i  ( I `  W ) ) )
 
Theoremdihglblem5aN 30612 A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B ) 
 ->  ( I `  ( X  ./\  W ) )  =  ( ( I `
  X )  i^i  ( I `  W ) ) )
 
Theoremdihglblem2aN 30613* Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\ 
 W ) }   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) ) 
 ->  T  =/=  (/) )
 
Theoremdihglblem2N 30614* The GLB of a set of lattice elements  S is the same as that of the set  T with elements of  S cut down to be under  W. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\ 
 W ) }   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  C_  B  /\  ( G `  S ) 
 .<_  W )  ->  ( G `  S )  =  ( G `  T ) )
 
Theoremdihglblem3N 30615* Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\ 
 W ) }   &    |-  J  =  ( ( DIsoB `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
 .<_  W )  ->  ( I `  ( G `  T ) )  = 
 |^|_ x  e.  T  ( I `  x ) )
 
Theoremdihglblem3aN 30616* Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\ 
 W ) }   &    |-  J  =  ( ( DIsoB `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
 .<_  W )  ->  ( I `  ( G `  S ) )  = 
 |^|_ x  e.  T  ( I `  x ) )
 
Theoremdihglblem4 30617* Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\ 
 W ) }   &    |-  J  =  ( ( DIsoB `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) ) 
 ->  ( I `  ( G `  S ) ) 
 C_  |^|_ x  e.  S  ( I `  x ) )
 
Theoremdihglblem5 30618* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  S  =  (
 LSubSp `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( T  C_  B  /\  T  =/=  (/) ) ) 
 ->  |^|_ x  e.  T  ( I `  x )  e.  S )
 
Theoremdihmeetlem2N 30619 Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  q )   &    |-  .0.  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `
  X )  i^i  ( I `  Y ) ) )
 
TheoremdihglbcpreN 30620* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane  W. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  F  =  ( iota_ g  e.  T ( g `  P )  =  q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  -.  ( G `  S )  .<_  W )  ->  ( I `  ( G `
  S ) )  =  |^|_ x  e.  S  ( I `  x ) )
 
TheoremdihglbcN 30621* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane  W. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/= 
 (/) )  /\  -.  ( G `  S ) 
 .<_  W )  ->  ( I `  ( G `  S ) )  = 
 |^|_ x  e.  S  ( I `  x ) )
 
TheoremdihmeetcN 30622 Isomorphism H of a lattice meet when the meet is not under the fiducial hyperplane  W. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  -.  ( X  ./\  Y )  .<_  W )  ->  ( I `  ( X 
 ./\  Y ) )  =  ( ( I `  X )  i^i  ( I `
  Y ) ) )
 
TheoremdihmeetbN 30623 Isomorphism H of a lattice meet when one element is under the fiducial hyperplane  W. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  ( I `  ( X 
 ./\  Y ) )  =  ( ( I `  X )  i^i  ( I `
  Y ) ) )
 
TheoremdihmeetbclemN 30624 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y ) 
 .<_  W )  ->  ( I `  ( X  ./\  Y ) )  =  ( ( ( I `  X )  i^i  ( I `
  Y ) )  i^i  ( I `  W ) ) )
 
Theoremdihmeetlem3N 30625 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y ) 
 .<_  W )  /\  (
 ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( Q  .\/  ( X  ./\  W ) )  =  X ) 
 /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( Y 
 ./\  W ) )  =  Y ) )  ->  Q  =/=  R )
 
Theoremdihmeetlem4preN 30626* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   &    |-  G  =  ( iota_ g  e.  T ( g `
  P )  =  Q )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( I `  Q )  i^i  ( I `  ( X  ./\  W ) ) )  =  {  .0.  } )
 
Theoremdihmeetlem4N 30627 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( I `  Q )  i^i  ( I `  ( X  ./\  W ) ) )  =  {  .0.  } )
 
Theoremdihmeetlem5 30628 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  ( X  ./\  ( Y  .\/  Q ) )  =  (
 ( X  ./\  Y ) 
 .\/  Q ) )
 
Theoremdihmeetlem6 30629 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) ) 
 ->  -.  ( X  ./\  ( Y  .\/  Q ) )  .<_  W )
 
Theoremdihmeetlem7N 30630 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( ( ( X 
 ./\  Y )  .\/  p )  ./\  Y )  =  ( X  ./\  Y ) )
 
Theoremdihjatc1 30631 Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change  .\/ order of  ( X  ./\  Y )  .\/  Q here and down? (Contributed by NM, 6-Apr-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 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q 
 .<_  X  /\  ( X 
 ./\  Y )  .<_  W ) )  ->  ( I `  ( ( X  ./\  Y )  .\/  Q )
 )  =  ( ( I `  Q ) 
 .(+)  ( I `  ( X  ./\  Y ) ) ) )
 
Theoremdihjatc2N 30632 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
 |-  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 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q 
 .<_  X  /\  ( X 
 ./\  Y )  .<_  W ) )  ->  ( I `  ( Q  .\/  ( X  ./\  Y ) ) )  =  ( ( I `  Q ) 
 .(+)  ( I `  ( X  ./\  Y ) ) ) )
 
Theoremdihjatc3 30633 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-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 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q 
 .<_  X  /\  ( X 
 ./\  Y )  .<_  W ) )  ->  ( I `  ( ( X  ./\  Y )  .\/  Q )
 )  =  ( ( I `  ( X 
 ./\  Y ) )  .(+)  ( I `  Q ) ) )
 
Theoremdihmeetlem8N 30634 Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change  .\/ order of  ( X  ./\  Y )  .\/  p here and down? (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  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 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( p  e.  A  /\  -.  p  .<_  W )  /\  ( p  .<_  X  /\  ( X  ./\  Y )  .<_  W ) )  ->  ( I `  ( ( X 
 ./\  Y )  .\/  p ) )  =  (
 ( I `  p )  .(+)  ( I `  ( X  ./\  Y ) ) ) )
 
Theoremdihmeetlem9N 30635 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  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 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  p  e.  A )  ->  ( ( ( I `  p ) 
 .(+)  ( I `  ( X  ./\  Y ) ) )  i^i  ( I `
  Y ) )  =  ( ( I `
  ( X  ./\  Y ) )  .(+)  ( ( I `  p )  i^i  ( I `  Y ) ) ) )
 
Theoremdihmeetlem10N 30636 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  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 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) ) 
 ->  ( I `  (
 ( X  ./\  Y ) 
 .\/  p ) )  =  ( ( I `
  X )  i^i  ( I `  ( Y  .\/  p ) ) ) )
 
Theoremdihmeetlem11N 30637 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  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 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) ) 
 ->  ( ( I `  ( ( X  ./\  Y )  .\/  p )
 )  i^i  ( I `  Y ) )  =  ( ( I `  X )  i^i  ( I `
  Y ) ) )
 
Theoremdihmeetlem12N 30638 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  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 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X  /\  ( X  ./\  Y )  .<_  W ) )  ->  (
 ( I `  ( X  ./\  Y ) ) 
 .(+)  ( ( I `  p )  i^i  ( I `
  Y ) ) )  =  ( ( I `  X )  i^i  ( I `  Y ) ) )
 
Theoremdihmeetlem13N 30639* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `
  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  ->  (
 ( I `  Q )  i^i  ( I `  R ) )  =  {  .0.  } )
 
Theoremdihmeetlem14N 30640 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  Y  e.  B  /\  p  e.  B )  /\  ( ( r  e.  A  /\  -.  r  .<_  W )  /\  r  .<_  Y  /\  ( Y  ./\  p )  .<_  W ) )  ->  (
 ( I `  ( Y  ./\  p ) ) 
 .(+)  ( ( I `  r )  i^i  ( I `
  p ) ) )  =  ( ( I `  Y )  i^i  ( I `  p ) ) )
 
Theoremdihmeetlem15N 30641 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  B  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( ( r  e.  A  /\  -.  r  .<_  W )  /\  r  .<_  Y  /\  ( Y  ./\  p )  .<_  W ) )  ->  (
 ( I `  r
 )  i^i  ( I `  p ) )  =  {  .0.  } )
 
Theoremdihmeetlem16N 30642 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  Y  e.  B  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( ( r  e.  A  /\  -.  r  .<_  W )  /\  r  .<_  Y  /\  ( Y  ./\  p )  .<_  W ) )  ->  ( I `  ( Y  ./\  p ) )  =  ( ( I `  Y )  i^i  ( I `  p ) ) )
 
Theoremdihmeetlem17N 30643 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y ) 
 .<_  W  /\  p  .<_  X ) )  ->  ( Y  ./\  p )  =  .0.  )
 
Theoremdihmeetlem18N 30644 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  Y  e.  B )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  ( r  e.  A  /\  -.  r  .<_  W ) 
 /\  ( p  .<_  X 
 /\  r  .<_  Y  /\  ( X  ./\  Y ) 
 .<_  W ) ) ) 
 ->  ( ( I `  Y )  i^i  ( I `
  p ) )  =  {  .0.  }
 )
 
Theoremdihmeetlem19N 30645 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  Y  e.  B )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  ( r  e.  A  /\  -.  r  .<_  W ) 
 /\  ( p  .<_  X 
 /\  r  .<_  Y  /\  ( X  ./\  Y ) 
 .<_  W ) ) ) 
 ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `
  X )  i^i  ( I `  Y ) ) )
 
Theoremdihmeetlem20N 30646 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( ( Y  e.  B  /\  -.  Y  .<_  W )  /\  ( X  ./\  Y ) 
 .<_  W ) )  ->  ( I `  ( X 
 ./\  Y ) )  =  ( ( I `  X )  i^i  ( I `
  Y ) ) )
 
TheoremdihmeetALTN 30647 Isomorphism H of a lattice meet. This version does not depend on the atomisticity of the constructed vector space. TODO: Delete? (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 ) ) )
 
Theoremdih1dimatlem0 30648* Lemma for dih1dimat 30650. (Contributed by NM, 11-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   &    |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  F  =  (Scalar `  U )   &    |-  J  =  ( invr `  F )   &    |-  V  =  (
 Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `
  P )  =  ( ( ( J `
  s ) `  f ) `  P ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 f  e.  T  /\  s  e.  E )  /\  s  =/=  O ) 
 ->  ( ( i  =  ( p `  G )  /\  p  e.  E ) 
 <->  ( ( i  e.  T  /\  p  e.  E )  /\  E. t  e.  E  (
 i  =  ( t `
  f )  /\  p  =  ( t  o.  s ) ) ) ) )
 
Theoremdih1dimatlem 30649* Lemma for dih1dimat 30650. (Contributed by NM, 10-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   &    |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  F  =  (Scalar `  U )   &    |-  J  =  ( invr `  F )   &    |-  V  =  (
 Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `
  P )  =  ( ( ( J `
  s ) `  f ) `  P ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  A )  ->  D  e.  ran  I )
 
Theoremdih1dimat 30650 Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  ->  P  e.  ran  I )
 
Theoremdihlsprn 30651 The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V ) 
 ->  ( N `  { X } )  e.  ran  I )
 
TheoremdihlspsnssN 30652 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 30653 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 30654 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 30655 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 30656* The value of isomorphism H at the fiducial atom  P is determined by the vector  <. 0 ,  S >. (the zero translation ltrnid 29454 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 30657 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 30658* 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 30659* 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 30660* 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 30661* 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 30662* 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 30663 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 30664 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 30665 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 30666 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 30667 Extend class notation with subspace orthocomplement for  DVecH vector space.
 class  ocH
 
Definitiondf-doch 30668* 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 30669* 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 30670* 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 30671* 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 30672* 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 30673 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 30674 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 30675 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 30676 Domain and co-domain 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 30677 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 30678 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 30679 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 30680 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 30681 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 30682 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 30683 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 30684* 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 30685 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 30686 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 30687 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 30688 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 30689 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 30690 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 30691 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 30692 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 30693 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 30694 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 30695 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 30696 Version of dihoml4 30697 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 30697 Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 29272 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 30698 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 30699 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 30700 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 ) ) )
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