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Theorem List for Metamath Proof Explorer - 30501-30600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdicopelval2 30501* Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 20-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  G  =  (
 iota_ g  e.  T ( g `  P )  =  Q )   &    |-  F  e.  _V   &    |-  S  e.  _V   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  Q ) 
 <->  ( F  =  ( S `  G ) 
 /\  S  e.  E ) ) )
 
Theoremdicelval2N 30502* Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  G  =  (
 iota_ g  e.  T ( g `  P )  =  Q )   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( Y  e.  ( I `  Q )  <->  ( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y )  =  ( ( 2nd `  Y ) `  G )  /\  ( 2nd `  Y )  e.  E ) ) ) )
 
TheoremdicfnN 30503* Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { p  e.  A  |  -.  p  .<_  W }
 )
 
TheoremdicdmN 30504* Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  { p  e.  A  |  -.  p  .<_  W }
 )
 
TheoremdicvalrelN 30505 The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )
 
Theoremdicssdvh 30506 The partial isomorphism C maps to a set of vectors in full vector space H. (Contributed by NM, 19-Jan-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  C_  V )
 
Theoremdicelval1sta 30507* Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 16-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q ) )  ->  ( 1st `  Y )  =  ( ( 2nd `  Y ) `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) )
 
Theoremdicelval1stN 30508 Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 16-Feb-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  Y  e.  ( I `  Q ) ) 
 ->  ( 1st `  Y )  e.  T )
 
Theoremdicelval2nd 30509 Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 16-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  Y  e.  ( I `  Q ) ) 
 ->  ( 2nd `  Y )  e.  E )
 
Theoremdicvaddcl 30510 Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q ) ) )  ->  ( X  .+  Y )  e.  ( I `  Q ) )
 
Theoremdicvscacl 30511 Membership in value of the partial isomorphism C is closed under scalar product. (Contributed by NM, 16-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `
  Q ) ) )  ->  ( X  .x.  Y )  e.  ( I `  Q ) )
 
Theoremdicn0 30512 The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  =/=  (/) )
 
Theoremdiclss 30513 The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  e.  S )
 
Theoremdiclspsn 30514* The value of isomorphism C is spanned by vector  F. Part of proof of Lemma N of [Crawley] p. 121 line 29. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  =  ( N `  { <. F ,  (  _I  |`  T )
 >. } ) )
 
Theoremcdlemn2 30515* Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  F  =  ( iota_ h  e.  T ( h `  Q )  =  S )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  S  .<_  ( Q 
 .\/  X ) )  ->  ( R `  F ) 
 .<_  X )
 
Theoremcdlemn2a 30516* Part of proof of Lemma N of [Crawley] p. 121. (Contributed by NM, 24-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoB `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  Q )  =  S )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  S  .<_  ( Q 
 .\/  X ) )  ->  ( N `  { <. F ,  O >. } )  C_  ( I `  X ) )
 
Theoremcdlemn3 30517* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   &    |-  J  =  ( iota_ h  e.  T ( h `  Q )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( J  o.  F )  =  G )
 
Theoremcdlemn4 30518* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  F  =  (
 iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   &    |-  J  =  ( iota_ h  e.  T ( h `  Q )  =  R )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 ->  <. G ,  (  _I  |`  T ) >.  =  ( <. F ,  (  _I  |`  T ) >.  .+ 
 <. J ,  O >. ) )
 
Theoremcdlemn4a 30519* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 24-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  F  =  (
 iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   &    |-  J  =  ( iota_ h  e.  T ( h `  Q )  =  R )   &    |-  N  =  ( LSpan `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( N `  { <. G ,  (  _I  |`  T ) >. } )  C_  (
 ( N `  { <. F ,  (  _I  |`  T )
 >. } )  .(+)  ( N `
  { <. J ,  O >. } ) ) )
 
Theoremcdlemn5pre 30520* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  N  =  ( LSpan `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   &    |-  M  =  ( iota_ h  e.  T ( h `  Q )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  R  .<_  ( Q 
 .\/  X ) )  ->  ( J `  R ) 
 C_  ( ( J `
  Q )  .(+)  ( I `  X ) ) )
 
Theoremcdlemn5 30521 Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  R  .<_  ( Q 
 .\/  X ) )  ->  ( J `  R ) 
 C_  ( ( J `
  Q )  .(+)  ( I `  X ) ) )
 
Theoremcdlemn6 30522* Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T ) )  ->  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )  =  <. ( ( s `  F )  o.  g ) ,  s >. )
 
Theoremcdlemn7 30523* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  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 ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. G ,  (  _I  |`  T ) >.  =  ( <. ( s `
  F ) ,  s >.  .+  <. g ,  O >. ) ) ) 
 ->  ( G  =  ( ( s `  F )  o.  g )  /\  (  _I  |`  T )  =  s ) )
 
Theoremcdlemn8 30524* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  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 ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. G ,  (  _I  |`  T ) >.  =  ( <. ( s `
  F ) ,  s >.  .+  <. g ,  O >. ) ) ) 
 ->  g  =  ( G  o.  `' F ) )
 
Theoremcdlemn9 30525* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  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 ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. G ,  (  _I  |`  T ) >.  =  ( <. ( s `
  F ) ,  s >.  .+  <. g ,  O >. ) ) ) 
 ->  ( g `  Q )  =  R )
 
Theoremcdlemn10 30526 Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( g  e.  T  /\  ( g `
  Q )  =  S  /\  ( R `
  g )  .<_  X ) )  ->  S  .<_  ( Q  .\/  X ) )
 
Theoremcdlemn11a 30527* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( J `  N )  C_  ( ( J `  Q ) 
 .(+)  ( I `  X ) ) )  ->  <. G ,  (  _I  |`  T ) >.  e.  ( J `  N ) )
 
Theoremcdlemn11b 30528* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( J `  N )  C_  ( ( J `  Q ) 
 .(+)  ( I `  X ) ) )  ->  <. G ,  (  _I  |`  T ) >.  e.  (
 ( J `  Q )  .(+)  ( I `  X ) ) )
 
Theoremcdlemn11c 30529* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( J `  N )  C_  ( ( J `  Q ) 
 .(+)  ( I `  X ) ) )  ->  E. y  e.  ( J `  Q ) E. z  e.  ( I `  X ) <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z ) )
 
Theoremcdlemn11pre 30530* Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 30527, cdlemn11b 30528, cdlemn11c 30529, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( J `  N )  C_  ( ( J `  Q ) 
 .(+)  ( I `  X ) ) )  ->  N  .<_  ( Q  .\/  X ) )
 
Theoremcdlemn11 30531 Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( J `  R )  C_  ( ( J `  Q ) 
 .(+)  ( I `  X ) ) )  ->  R  .<_  ( Q  .\/  X ) )
 
Theoremcdlemn 30532 Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) )  ->  ( R  .<_  ( Q  .\/  X ) 
 <->  ( J `  R )  C_  ( ( J `
  Q )  .(+)  ( I `  X ) ) ) )
 
Theoremdihordlem6 30533* Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T ) )  ->  ( <. ( s `  G ) ,  s >.  .+  <. g ,  O >. )  =  <. ( ( s `  G )  o.  g ) ,  s >. )
 
Theoremdihordlem7 30534* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G ) ,  s >.  .+ 
 <. g ,  O >. ) ) )  ->  (
 f  =  ( ( s `  G )  o.  g )  /\  O  =  s )
 )
 
Theoremdihordlem7b 30535* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G ) ,  s >.  .+ 
 <. g ,  O >. ) ) )  ->  (
 f  =  g  /\  O  =  s )
 )
 
Theoremdihjustlem 30536 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B )  /\  ( Q  .\/  ( X  ./\  W ) )  =  ( R  .\/  ( X  ./\  W ) ) )  ->  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  (
 ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) )
 
Theoremdihjust 30537 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B )  /\  ( Q  .\/  ( X  ./\  W ) )  =  ( R  .\/  ( X  ./\  W ) ) )  ->  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  =  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) )
 
Theoremdihord1 30538 Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change  ( Q  .\/  ( X  ./\  W ) )  =  X to  Q  .<_  X using lhpmcvr3 29344, here and all theorems below. (Contributed by NM, 2-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( Q 
 .\/  ( X  ./\  W ) )  =  X  /\  ( R  .\/  ( Y  ./\  W ) )  =  Y  /\  X  .<_  Y ) )  ->  ( ( J `  Q )  .(+)  ( I `
  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `
  ( Y  ./\  W ) ) ) )
 
Theoremdihord2a 30539 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( Q 
 .\/  ( X  ./\  W ) )  =  X  /\  ( R  .\/  ( Y  ./\  W ) )  =  Y  /\  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  (
 ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) ) ) 
 ->  Q  .<_  ( R  .\/  ( Y  ./\  W ) ) )
 
Theoremdihord2b 30540 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( J `
  Q )  .(+)  ( I `  ( X 
 ./\  W ) ) ) 
 C_  ( ( J `
  R )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) )  ->  ( I `  ( X  ./\  W ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) )
 
Theoremdihord2cN 30541* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f )  .<_  ( X  ./\  W )
 ) )  ->  <. f ,  O >.  e.  ( I `  ( X  ./\  W ) ) )
 
Theoremdihord11b 30542* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( ( oc `  K ) `
  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( J `
  Q )  .(+)  ( I `  ( X 
 ./\  W ) ) ) 
 C_  ( ( J `
  N )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) )  /\  ( f  e.  T  /\  ( R `  f )  .<_  ( X  ./\  W )
 ) )  ->  <. f ,  O >.  e.  (
 ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )
 
Theoremdihord10 30543* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( ( oc `  K ) `
  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( f  e.  T  /\  ( R `
  f )  .<_  ( X  ./\  W )
 )  /\  ( (
 s  e.  E  /\  g  e.  T )  /\  ( R `  g
 )  .<_  ( Y  ./\  W )  /\  <. f ,  O >.  =  ( <. ( s `  G ) ,  s >.  .+ 
 <. g ,  O >. ) ) )  ->  ( R `  f )  .<_  ( Y  ./\  W )
 )
 
Theoremdihord11c 30544* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( ( oc `  K ) `
  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( ( J `  Q ) 
 .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) 
 /\  f  e.  T  /\  ( R `  f
 )  .<_  ( X  ./\  W ) ) )  ->  E. y  e.  ( J `  N ) E. z  e.  ( I `  ( Y  ./\  W ) ) <. f ,  O >.  =  ( y  .+  z ) )
 
Theoremdihord2pre 30545* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( ( oc `  K ) `
  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( J `
  Q )  .(+)  ( I `  ( X 
 ./\  W ) ) ) 
 C_  ( ( J `
  N )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) )  ->  ( X  ./\ 
 W )  .<_  ( Y 
 ./\  W ) )
 
Theoremdihord2pre2 30546* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( ( oc `  K ) `
  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( Q 
 .\/  ( X  ./\  W ) )  =  X  /\  ( N  .\/  ( Y  ./\  W ) )  =  Y  /\  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  (
 ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) ) ) 
 ->  ( Q  .\/  ( X  ./\  W ) ) 
 .<_  ( N  .\/  ( Y  ./\  W ) ) )
 
Theoremdihord2 30547 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. Todo: do we need 
-.  X  .<_  W and  -.  Y  .<_  W? (Contributed by NM, 4-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( Q 
 .\/  ( X  ./\  W ) )  =  X  /\  ( N  .\/  ( Y  ./\  W ) )  =  Y  /\  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  (
 ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) ) ) 
 ->  X  .<_  Y )
 
Syntaxcdih 30548 Extend class notation with isomorphism H.
 class  DIsoH
 
Definitiondf-dih 30549* Define isomorphism H. (Contributed by NM, 28-Jan-2014.)
 |-  DIsoH  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ( Base `  k
 )  |->  if ( x ( le `  k ) w ,  ( ( ( DIsoB `  k ) `  w ) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  k ) `  w ) ) A. q  e.  ( Atoms `  k ) ( ( -.  q ( le `  k ) w  /\  ( q ( join `  k ) ( x ( meet `  k ) w ) )  =  x )  ->  u  =  ( ( ( (
 DIsoC `  k ) `  w ) `  q
 ) ( LSSum `  (
 ( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
 ) `  w ) `  ( x ( meet `  k ) w ) ) ) ) ) ) ) ) )
 
Theoremdihffval 30550* The isomorphism H for a lattice  K. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 DIsoH `  K )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( ( DIsoB `  K ) `  w ) `  x ) ,  ( iota_ u  e.  ( LSubSp `
  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  ( ( -.  q  .<_  w  /\  (
 q  .\/  ( x  ./\ 
 w ) )  =  x )  ->  u  =  ( ( ( (
 DIsoC `  K ) `  w ) `  q
 ) ( LSSum `  (
 ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K ) `  w ) `  ( x  ./\  w ) ) ) ) ) ) ) ) )
 
Theoremdihfval 30551* Isomorphism H for a lattice  K. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  B  |->  if ( x  .<_  W ,  ( D `  x ) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  (
 q  .\/  ( x  ./\ 
 W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
  ( x  ./\  W ) ) ) ) ) ) ) )
 
Theoremdihval 30552* Value of isomorphism H for a lattice  K. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  B ) 
 ->  ( I `  X )  =  if ( X  .<_  W ,  ( D `  X ) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
 .\/  ( X  ./\  W ) )  =  X )  ->  u  =  ( ( C `  q
 )  .(+)  ( D `  ( X  ./\  W ) ) ) ) ) ) )
 
Theoremdihvalc 30553* Value of isomorphism H for a lattice  K when  -.  X  .<_  W. (Contributed by NM, 4-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  ( I `  X )  =  (
 iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
 .\/  ( X  ./\  W ) )  =  X )  ->  u  =  ( ( C `  q
 )  .(+)  ( D `  ( X  ./\  W ) ) ) ) ) )
 
Theoremdihlsscpre 30554 Closure of isomorphism H for a lattice  K when  -.  X  .<_  W. (Contributed by NM, 6-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  ( I `  X )  e.  S )
 
Theoremdihvalcqpre 30555 Value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. (Contributed by NM, 6-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .\/  ( X 
 ./\  W ) )  =  X ) )  ->  ( I `  X )  =  ( ( C `
  Q )  .(+)  ( D `  ( X 
 ./\  W ) ) ) )
 
Theoremdihvalcq 30556 Value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. TODO: Use dihvalcq2 30567 (with lhpmcvr3 29344 for  ( Q  .\/  ( X  ./\  W ) )  =  X simplification) that changes  C and  D to  I and make this obsolete. Do to other theorems as well. (Contributed by NM, 6-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .\/  ( X 
 ./\  W ) )  =  X ) )  ->  ( I `  X )  =  ( ( C `
  Q )  .(+)  ( D `  ( X 
 ./\  W ) ) ) )
 
Theoremdihvalb 30557 Value of isomorphism H for a lattice  K when  X  .<_  W. (Contributed by NM, 4-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  =  ( D `  X ) )
 
TheoremdihopelvalbN 30558* Ordered pair member of the partial isomorphism H for argument under  W. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( g  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( <. F ,  S >.  e.  ( I `  X )  <->  ( ( F  e.  T  /\  ( R `  F )  .<_  X )  /\  S  =  O ) ) )
 
Theoremdihvalcqat 30559 Value of isomorphism H for a lattice  K at an atom not under  W. (Contributed by NM, 27-Mar-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  =  ( J `  Q ) )
 
Theoremdih1dimb 30560* Two expressions for a 1-dimensional subspace of vector space H (when  F is a nonzero vector i.e. non-identity translation). (Contributed by NM, 27-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( I `  ( R `  F ) )  =  ( N `  { <. F ,  O >. } ) )
 
Theoremdih1dimb2 30561* Isomorphism H at an atom under  W. (Contributed by NM, 27-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) ) 
 ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. f ,  O >. } ) ) )
 
Theoremdih1dimc 30562* Isomorphism H at an atom not under 
W. (Contributed by NM, 27-Apr-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  =  ( N `  { <. F ,  (  _I  |`  T )
 >. } ) )
 
Theoremdib2dim 30563 Extend dia2dim 30397 to partial isomorphism B. (Contributed by NM, 22-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  P  .<_  W ) )   &    |-  ( ph  ->  ( Q  e.  A  /\  Q  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  C_  (
 ( I `  P )  .(+)  ( I `  Q ) ) )
 
Theoremdih2dimb 30564 Extend dib2dim 30563 to isomorphism H. (Contributed by NM, 22-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  P  .<_  W ) )   &    |-  ( ph  ->  ( Q  e.  A  /\  Q  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  C_  (
 ( I `  P )  .(+)  ( I `  Q ) ) )
 
Theoremdih2dimbALTN 30565 Extend dia2dim 30397 to isomorphism H. (This version combines dib2dim 30563 and dih2dimb 30564 for shorter overall proof, but may be less easy to understand. TODO: decide which to use.) (Contributed by NM, 22-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  P  .<_  W ) )   &    |-  ( ph  ->  ( Q  e.  A  /\  Q  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  C_  (
 ( I `  P )  .(+)  ( I `  Q ) ) )
 
Theoremdihopelvalcqat 30566* Ordered pair member of the partial isomorphism H for atom argument not under  W. TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  G  =  (
 iota_ g  e.  T ( g `  P )  =  Q )   &    |-  F  e.  _V   &    |-  S  e.  _V   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  Q ) 
 <->  ( F  =  ( S `  G ) 
 /\  S  e.  E ) ) )
 
Theoremdihvalcq2 30567 Value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. (Contributed by NM, 26-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) ) 
 ->  ( I `  X )  =  ( ( I `  Q )  .(+)  ( I `  ( X 
 ./\  W ) ) ) )
 
Theoremdihopelvalcpre 30568* Member of value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. TODO: refactor to be shorter and more understandable; add lemmas? (Contributed by NM, 13-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  G  =  ( iota_ g  e.  T ( g `  P )  =  Q )   &    |-  F  e.  _V   &    |-  S  e.  _V   &    |-  Z  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  N  =  ( (
 DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  V  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  O  =  ( a  e.  E ,  b  e.  E  |->  ( h  e.  T  |->  ( ( a `  h )  o.  (
 b `  h )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .\/  ( X 
 ./\  W ) )  =  X ) )  ->  ( <. F ,  S >.  e.  ( I `  X )  <->  ( ( F  e.  T  /\  S  e.  E )  /\  ( R `  ( F  o.  `' ( S `  G ) ) )  .<_  X ) ) )
 
Theoremdihopelvalc 30569* Member of value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. (Contributed by NM, 13-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  G  =  ( iota_ g  e.  T ( g `  P )  =  Q )   &    |-  F  e.  _V   &    |-  S  e.  _V   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .\/  ( X 
 ./\  W ) )  =  X ) )  ->  ( <. F ,  S >.  e.  ( I `  X )  <->  ( ( F  e.  T  /\  S  e.  E )  /\  ( R `  ( F  o.  `' ( S `  G ) ) )  .<_  X ) ) )
 
Theoremdihlss 30570 The value of isomorphism H is a subspace. (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 )  /\  X  e.  B )  ->  ( I `  X )  e.  S )
 
Theoremdihss 30571 The value of isomorphism H is a set of vectors. (Contributed by NM, 14-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  ->  ( I `  X )  C_  V )
 
Theoremdihssxp 30572 An isomorphism H value is included in the vector space (expressed as  T  X.  E). (Contributed by NM, 26-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( I `  X ) 
 C_  ( T  X.  E ) )
 
Theoremdihopcl 30573 Closure of an ordered pair (vector) member of a value of isomorphism H. (Contributed by NM, 26-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  <. F ,  S >.  e.  ( I `  X ) )   =>    |-  ( ph  ->  ( F  e.  T  /\  S  e.  E )
 )
 
TheoremxihopellsmN 30574* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  A  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  L  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( <. F ,  S >.  e.  ( ( I `  X )  .(+)  ( I `
  Y ) )  <->  E. g E. t E. h E. u ( (
 <. g ,  t >.  e.  ( I `  X )  /\  <. h ,  u >.  e.  ( I `  Y ) )  /\  ( F  =  (
 g  o.  h ) 
 /\  S  =  ( t A u ) ) ) ) )
 
Theoremdihopellsm 30575* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  A  =  ( v  e.  E ,  w  e.  E  |->  ( i  e.  T  |->  ( ( v `  i )  o.  ( w `  i ) ) ) )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  L  =  (
 LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( <. F ,  S >.  e.  ( ( I `  X )  .(+)  ( I `
  Y ) )  <->  E. g E. t E. h E. u ( (
 <. g ,  t >.  e.  ( I `  X )  /\  <. h ,  u >.  e.  ( I `  Y ) )  /\  ( F  =  (
 g  o.  h ) 
 /\  S  =  ( t A u ) ) ) ) )
 
Theoremdihord6apre 30576* Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  q )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  /\  ( I `  X ) 
 C_  ( I `  Y ) )  ->  X  .<_  Y )
 
Theoremdihord3 30577 The isomorphism H for a lattice  K is order-preserving in the region under co-atom  W. (Contributed by NM, 6-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 ) )
 
Theoremdihord4 30578 The isomorphism H for a lattice  K is order-preserving in the region not under co-atom  W. TODO: reformat q e. A /\ -. q .<_ W to eliminate adant*. (Contributed by NM, 6-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 ) )
 
Theoremdihord5b 30579 Part of proof that isomorphism H is order-preserving. TODO: eliminate 3ad2ant1; combine w/ other way to have one lhpmcvr2 (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 ) )  /\  X  .<_  Y )  ->  ( I `  X )  C_  ( I `  Y ) )
 
Theoremdihord6b 30580 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 ) )  /\  X  .<_  Y )  ->  ( I `  X )  C_  ( I `  Y ) )
 
Theoremdihord6a 30581 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 30582 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 30583 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 30584 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 30585 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 30586 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 30587 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 30588 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 30589 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 30590 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 30591 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 30592 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 30593 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 30594 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 30595 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 30596 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 30597 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 30598 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 30599 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 30600 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 } )
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