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Theorem List for Metamath Proof Explorer - 34801-34900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdihord 34801 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 34802 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 34803 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 34804 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 34805 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 34806 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 34807 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 34808 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 34809 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 34810 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 34811 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 34812 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 34813 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 34814 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 34815 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 34816 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 34817 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 34818 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 34819 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 34820 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 34821 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 34822 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 34823 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 34824 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 34825 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 34826* 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 34827* 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 34828* 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 34829 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 34830* 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 34831* 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 34832* 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 34833* 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 34834* 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 34835* 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 34836 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 34837* 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 34838* 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 34839 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 34840 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 34841 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 34842 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 34843* 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 34844 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 34845 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 34846 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 34847 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 34848 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 34849 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 34850 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 34851 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 34852 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 34853 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 34854 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 34855 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 34856* 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 34857 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 34858 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 34859 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 34860 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 34861 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 34862 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 34863 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 34864 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 34865* Lemma for dih1dimat 34867. (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 34866* Lemma for dih1dimat 34867. (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 34867 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 34868 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 34869 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 34870 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 34871 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 34872 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 34873* The value of isomorphism H at the fiducial atom  P is determined by the vector  <. 0 ,  S >. (the zero translation ltrnid 33669 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 34874 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 34875* 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 34876* 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 34877* 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 34878* 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 34879* 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 34880 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 34881 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 34882 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 34883 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 34884 Extend class notation with subspace orthocomplement for  DVecH vector space.
 class  ocH
 
Definitiondf-doch 34885* 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 34886* 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 34887* 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 34888* 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 34889* 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 34890 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 34891 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 34892 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 34893 Domain and codomain of the subspace orthocomplement for the  DVecH vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ._|_  : ~P V
 --> ran  I )
 
Theoremdochvalr 34894 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 34895 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 34896 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 34897 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 34898 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 34899 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 34900 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 ) ) )
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