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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdihord6a 30701 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( Y  e.  B  /\  Y  .<_  W ) )  /\  ( I `
  X )  C_  ( I `  Y ) )  ->  X  .<_  Y )
 
Theoremdihord5apre 30702 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-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  /\  X  .<_  W ) 
 /\  ( Y  e.  B  /\  -.  Y  .<_  W ) )  /\  ( I `  X )  C_  ( I `  Y ) )  ->  X  .<_  Y )
 
Theoremdihord5a 30703 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  -.  Y  .<_  W ) )  /\  ( I `
  X )  C_  ( I `  Y ) )  ->  X  .<_  Y )
 
Theoremdihord 30704 The isomorphism H is order-preserving. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( I `  X ) 
 C_  ( I `  Y )  <->  X  .<_  Y ) )
 
Theoremdih11 30705 The isomorphism H is one-to-one. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( I `  X )  =  ( I `  Y )  <->  X  =  Y ) )
 
Theoremdihf11lem 30706 Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  I : B --> S )
 
Theoremdihf11 30707 The isomorphism H for a lattice  K is a one-to-one function. . Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  I : B -1-1-> S )
 
Theoremdihfn 30708 Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  B )
 
Theoremdihdm 30709 Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  I  =  B )
 
Theoremdihcl 30710 Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B ) 
 ->  ( I `  X )  e.  ran  I )
 
Theoremdihcnvcl 30711 Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I
 )  ->  ( `' I `  X )  e.  B )
 
Theoremdihcnvid1 30712 The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B ) 
 ->  ( `' I `  ( I `  X ) )  =  X )
 
Theoremdihcnvid2 30713 The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I
 )  ->  ( I `  ( `' I `  X ) )  =  X )
 
Theoremdihcnvord 30714 Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.)
 |-  .<_  =  ( le `  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 `  Y )  <->  X  C_  Y ) )
 
Theoremdihcnv11 30715 The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014.)
 |-  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 `  Y )  <->  X  =  Y )
 )
 
Theoremdihsslss 30716 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  ran  I  C_  S )
 
Theoremdihrnlss 30717 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  X  e.  S )
 
Theoremdihrnss 30718 The isomorphism H maps to a set of vectors. (Contributed by NM, 14-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  X  C_  V )
 
Theoremdihvalrel 30719 The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  X ) )
 
Theoremdih0 30720 The value of isomorphism H at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 9-Mar-2014.)
 |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  O  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .0.  )  =  { O } )
 
Theoremdih0bN 30721 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 30722 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 30723 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 30724 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 30725 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 30726 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 30727 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 30728 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 30729* 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 30730* 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 30731* 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 30732 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 30733* 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 30734* 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 30735* 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 30736* 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 30737* 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 30738* 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 30739 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 30740* 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 30741* 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 30742 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 30743 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 30744 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 30745 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 30746* 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 30747 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 30748 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 30749 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 30750 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 30751 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 30752 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 30753 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 30754 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 30755 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 30756 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 30757 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 30758 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 30759* 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 30760 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 30761 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 30762 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 30763 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 30764 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 30765 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 30766 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 30767 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 30768* Lemma for dih1dimat 30770. (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 30769* Lemma for dih1dimat 30770. (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 30770 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 30771 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 30772 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 30773 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 30774 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 30775 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 30776* The value of isomorphism H at the fiducial atom  P is determined by the vector  <. 0 ,  S >. (the zero translation ltrnid 29574 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 30777 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 30778* 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 30779* 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 30780* 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 30781* 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 30782* 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 30783 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 30784 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 30785 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 30786 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 30787 Extend class notation with subspace orthocomplement for  DVecH vector space.
 class  ocH
 
Definitiondf-doch 30788* 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 30789* 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 30790* 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 30791* 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 30792* 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 30793 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 30794 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 30795 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 30796 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 30797 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 30798 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 30799 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 30800 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.  } )
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