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Theorem List for Metamath Proof Explorer - 31201-31300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvamulr 31201 Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .x.  =  ( .r `  F )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  S  e.  E )
 )  ->  ( R  .x.  S )  =  ( R  o.  S ) )
 
Theoremdvavbase 31202 The vectors (vector base set) of the constructed partial vector space A are all translations (for a fiducial co-atom  W). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  V  =  T )
 
Theoremdvafvadd 31203* The vector sum operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  .+  =  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) )
 
Theoremdvavadd 31204 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) )  ->  ( F 
 .+  G )  =  ( F  o.  G ) )
 
Theoremdvafvsca 31205* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  f  e.  T  |->  ( s `
  f ) ) )
 
Theoremdvavsca 31206 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  F  e.  T )
 )  ->  ( R  .x.  F )  =  ( R `  F ) )
 
Theoremtendospid 31207 Identity property of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  ( F  e.  T  ->  ( (  _I  |`  T ) `
  F )  =  F )
 
Theoremtendospcl 31208 Closure of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T )  ->  ( U `  F )  e.  T )
 
Theoremtendospass 31209 Associative law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  X  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  F  e.  T )
 )  ->  ( ( U  o.  V ) `  F )  =  ( U `  ( V `  F ) ) )
 
Theoremtendospdi1 31210 Forward distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T )
 )  ->  ( U `  ( F  o.  G ) )  =  (
 ( U `  F )  o.  ( U `  G ) ) )
 
Theoremtendocnv 31211 Converse of a trace-preserving endomorphism value. (Contributed by NM, 7-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  F  e.  T )  ->  `' ( S `  F )  =  ( S `  `' F ) )
 
Theoremtendospdi2 31212* Reverse distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U P V ) `
  F )  =  ( ( U `  F )  o.  ( V `  F ) ) )
 
TheoremtendospcanN 31213* Cancellation law for trace-perserving endomorphism values (used as scalar product). (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  E  /\  S  =/=  O )  /\  ( F  e.  T  /\  G  e.  T ) )  ->  ( ( S `  F )  =  ( S `  G ) 
 <->  F  =  G ) )
 
Theoremdvaabl 31214 The constructed partial vector space A for a lattice  K is an abelian group. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Abel )
 
Theoremdvalveclem 31215 Lemma for dvalvec 31216. (Contributed by NM, 11-Oct-2013.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  B  =  ( Base `  K )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  .x. 
 =  ( .s `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
 
Theoremdvalvec 31216 The constructed partial vector space A for a lattice  K is a left vector space. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
 
Theoremdva0g 31217 The zero vector of partial vector space A. (Contributed by NM, 9-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  (  _I  |`  B ) )
 
Syntaxcdia 31218 Extend class notation with partial isomorphism A.
 class  DIsoA
 
Definitiondf-disoa 31219* Define partial isomorphism A. (Contributed by NM, 15-Oct-2013.)
 |-  DIsoA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  { y  e.  ( Base `  k )  |  y ( le `  k
 ) w }  |->  { f  e.  ( (
 LTrn `  k ) `  w )  |  (
 ( ( trL `  k
 ) `  w ) `  f ) ( le `  k ) x }
 ) ) )
 
Theoremdiaffval 31220* The partial isomorphism A for a lattice  K. (Contributed by NM, 15-Oct-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  ( DIsoA `  K )  =  ( w  e.  H  |->  ( x  e.  { y  e.  B  |  y  .<_  w }  |->  { f  e.  (
 ( LTrn `  K ) `  w )  |  ( ( ( trL `  K ) `  w ) `  f )  .<_  x }
 ) ) )
 
Theoremdiafval 31221* The partial isomorphism A for a lattice  K. (Contributed by NM, 15-Oct-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  {
 y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
 .<_  x } ) )
 
Theoremdiaval 31222* The partial isomorphism A for a lattice  K. Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  =  { f  e.  T  |  ( R `
  f )  .<_  X } )
 
Theoremdiaelval 31223 Member of the partial isomorphism A for a lattice  K. (Contributed by NM, 3-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( F  e.  ( I `  X )  <->  ( F  e.  T  /\  ( R `  F )  .<_  X ) ) )
 
Theoremdiafn 31224* Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
 
Theoremdiadm 31225* Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  { x  e.  B  |  x  .<_  W }
 )
 
Theoremdiaeldm 31226 Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom 
 I 
 <->  ( X  e.  B  /\  X  .<_  W ) ) )
 
TheoremdiadmclN 31227 A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  dom  I
 )  ->  X  e.  B )
 
TheoremdiadmleN 31228 A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X  .<_  W )
 
Theoremdian0 31229 The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  =/=  (/) )
 
Theoremdia0eldmN 31230 The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  dom  I )
 
Theoremdia1eldmN 31231 The fiducial hyperplane (largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  dom  I )
 
Theoremdiass 31232 The value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  C_  T )
 
Theoremdiael 31233 A member of the value of the partial isomorphism A is a translation i.e. a vector. (Contributed by NM, 17-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  F  e.  ( I `  X ) )  ->  F  e.  T )
 
Theoremdiatrl 31234 Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  F  e.  ( I `  X ) )  ->  ( R `  F ) 
 .<_  X )
 
TheoremdiaelrnN 31235 Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  S  e.  ran  I
 )  ->  S  C_  T )
 
Theoremdialss 31236 The value of partial isomorphism A is a subspace of partial vector space A. Part of Lemma M of [Crawley] p. 120 line 26. (Contributed by NM, 17-Jan-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  S  =  (
 LSubSp `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  e.  S )
 
Theoremdiaord 31237 The partial isomorphism A for a lattice  K is order-preserving in the region under co-atom  W. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  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 ) )
 
Theoremdia11N 31238 The partial isomorphism A for a lattice  K is one-to-one in the region under co-atom  W. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( ( I `  X )  =  ( I `  Y )  <->  X  =  Y ) )
 
Theoremdiaf11N 31239 The partial isomorphism A for a lattice  K is a one-to-one function. . Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
 
TheoremdiaclN 31240 Closure of partial isomorphism A for a lattice  K. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I
 )  ->  ( I `  X )  e.  ran  I )
 
TheoremdiacnvclN 31241 Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I
 )  ->  ( `' I `  X )  e. 
 dom  I )
 
Theoremdia0 31242 The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .0.  )  =  { (  _I  |`  B ) }
 )
 
Theoremdia1N 31243 The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  W )  =  T )
 
Theoremdia1elN 31244 The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  T  e.  ran  I )
 
TheoremdiaglbN 31245* Partial isomorphism A of a lattice glb. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
 |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  dom  I  /\  S  =/=  (/) ) )  ->  ( I `  ( G `
  S ) )  =  |^|_ x  e.  S  ( I `  x ) )
 
TheoremdiameetN 31246 Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  dom  I 
 /\  Y  e.  dom  I ) )  ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `  X )  i^i  ( I `  Y ) ) )
 
TheoremdiainN 31247 Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  ran  I 
 /\  Y  e.  ran  I ) )  ->  ( X  i^i  Y )  =  ( I `  (
 ( `' I `  X )  ./\  ( `' I `  Y ) ) ) )
 
TheoremdiaintclN 31248 The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) ) 
 ->  |^| S  e.  ran  I )
 
TheoremdiasslssN 31249 The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  ran  I  C_  S )
 
TheoremdiassdvaN 31250 The partial isomorphism A maps to a set of vectors in partial vector space A. (Contributed by NM, 1-Jan-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  V  =  (
 Base `  U )   =>    |-  ( ( ( K  e.  Y  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  C_  V )
 
Theoremdia1dim 31251* Two expressions for the 1-dimensional subspaces of partial vector space A (when  F is a nonzero vector i.e. non-identity translation). Remark after Lemma L in [Crawley] p. 120 line 21. (Contributed by NM, 15-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( I `  ( R `  F ) )  =  { g  | 
 E. s  e.  E  g  =  ( s `  F ) } )
 
Theoremdia1dim2 31252 Two expressions for a 1-dimensional subspace of partial vector space A (when  F is a nonzero vector i.e. non-identity translation). (Contributed by NM, 15-Jan-2014.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  N  =  ( LSpan `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  ->  ( I `  ( R `  F ) )  =  ( N `
  { F }
 ) )
 
Theoremdia1dimid 31253 A vector (translation) belongs to the 1-dim subspace it generates. (Contributed by NM, 8-Sep-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  F  e.  ( I `
  ( R `  F ) ) )
 
Theoremdia2dimlem1 31254 Lemma for dia2dim 31267. Show properties of the auxiliary atom  Q. Part of proof of Lemma M in [Crawley] p. 121 line 3. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( F  e.  T  /\  ( F `
  P )  =/= 
 P ) )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/= 
 U )   =>    |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
 
Theoremdia2dimlem2 31255 Lemma for dia2dim 31267. Define a translation  G whose trace is atom  U. Part of proof of Lemma M in [Crawley] p. 121 line 4. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( F  e.  T  /\  ( F `
  P )  =/= 
 P ) )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   &    |-  ( ph  ->  G  e.  T )   &    |-  ( ph  ->  ( G `  P )  =  Q )   =>    |-  ( ph  ->  ( R `  G )  =  U )
 
Theoremdia2dimlem3 31256 Lemma for dia2dim 31267. Define a translation  D whose trace is atom  V. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( F  e.  T  /\  ( F `
  P )  =/= 
 P ) )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/= 
 U )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   &    |-  ( ph  ->  D  e.  T )   &    |-  ( ph  ->  ( D `  Q )  =  ( F `  P ) )   =>    |-  ( ph  ->  ( R `  D )  =  V )
 
Theoremdia2dimlem4 31257 Lemma for dia2dim 31267. Show that the composition (sum) of translations (vectors)  G and  D equals  F. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  G  e.  T )   &    |-  ( ph  ->  ( G `  P )  =  Q )   &    |-  ( ph  ->  D  e.  T )   &    |-  ( ph  ->  ( D `  Q )  =  ( F `  P ) )   =>    |-  ( ph  ->  ( D  o.  G )  =  F )
 
Theoremdia2dimlem5 31258 Lemma for dia2dim 31267. The sum of vectors  G and  D belongs to the sum of the subspaces generated by them. Thus  F  =  ( G  o.  D ) belongs to the subspace sum. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( F  e.  T  /\  ( F `
  P )  =/= 
 P ) )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/= 
 U )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   &    |-  ( ph  ->  G  e.  T )   &    |-  ( ph  ->  ( G `  P )  =  Q )   &    |-  ( ph  ->  D  e.  T )   &    |-  ( ph  ->  ( D `  Q )  =  ( F `  P ) )   =>    |-  ( ph  ->  F  e.  ( ( I `  U )  .(+)  ( I `
  V ) ) )
 
Theoremdia2dimlem6 31259 Lemma for dia2dim 31267. Eliminate auxiliary translations  G and  D. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( F  e.  T  /\  ( F `
  P )  =/= 
 P ) )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/= 
 U )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   =>    |-  ( ph  ->  F  e.  ( ( I `
  U )  .(+)  ( I `  V ) ) )
 
Theoremdia2dimlem7 31260 Lemma for dia2dim 31267. Eliminate  ( F `  P )  =/=  P condition. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  ( R `  F ) 
 .<_  ( U  .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/=  U )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   =>    |-  ( ph  ->  F  e.  ( ( I `  U )  .(+)  ( I `
  V ) ) )
 
Theoremdia2dimlem8 31261 Lemma for dia2dim 31267. Eliminate no-longer used auxiliary atoms  P and  Q. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/= 
 U )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   =>    |-  ( ph  ->  F  e.  ( ( I `
  U )  .(+)  ( I `  V ) ) )
 
Theoremdia2dimlem9 31262 Lemma for dia2dim 31267. Eliminate  ( R `  F )  =/=  U,  V conditions. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   =>    |-  ( ph  ->  F  e.  ( ( I `  U )  .(+)  ( I `
  V ) ) )
 
Theoremdia2dimlem10 31263 Lemma for dia2dim 31267. Convert membership in closed subspace  ( I `  ( U  .\/  V ) ) to a lattice ordering. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  F  e.  ( I `  ( U  .\/  V ) ) )   =>    |-  ( ph  ->  ( R `  F ) 
 .<_  ( U  .\/  V ) )
 
Theoremdia2dimlem11 31264 Lemma for dia2dim 31267. Convert ordering hypothesis on  R `  F to subspace membership  F  e.  ( I `
 ( U  .\/  V ) ). (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  F  e.  ( I `  ( U  .\/  V ) ) )   =>    |-  ( ph  ->  F  e.  ( ( I `  U )  .(+)  ( I `
  V ) ) )
 
Theoremdia2dimlem12 31265 Lemma for dia2dim 31267. Obtain subset relation. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  U  =/=  V )   =>    |-  ( ph  ->  ( I `  ( U 
 .\/  V ) )  C_  ( ( I `  U )  .(+)  ( I `
  V ) ) )
 
Theoremdia2dimlem13 31266 Lemma for dia2dim 31267. Eliminate  U  =/=  V condition. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( U  .\/  V ) )  C_  (
 ( I `  U )  .(+)  ( I `  V ) ) )
 
Theoremdia2dim 31267 A two-dimensional subspace of partial vector space A is closed, or equivalently, the isomorphism of a join of two atoms is a subset of the subspace sum of the isomorphisms of each atom (and thus they are equal, as shown later for the full vector space H). (Contributed by NM, 9-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( U  .\/  V ) )  C_  (
 ( I `  U )  .(+)  ( I `  V ) ) )
 
Syntaxcdvh 31268 Extend class notation with constructed full vector space H.
 class  DVecH
 
Definitiondf-dvech 31269* Define constructed full vector space H. (Contributed by NM, 17-Oct-2013.)
 |-  DVecH  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( { <. ( Base `  ndx ) ,  ( ( ( LTrn `  k ) `  w )  X.  ( ( TEndo `  k ) `  w ) ) >. ,  <. (
 +g  `  ndx ) ,  ( f  e.  (
 ( ( LTrn `  k
 ) `  w )  X.  ( ( TEndo `  k
 ) `  w )
 ) ,  g  e.  ( ( ( LTrn `  k ) `  w )  X.  ( ( TEndo `  k ) `  w ) )  |->  <. ( ( 1st `  f )  o.  ( 1st `  g
 ) ) ,  ( h  e.  ( ( LTrn `  k ) `  w )  |->  ( ( ( 2nd `  f
 ) `  h )  o.  ( ( 2nd `  g
 ) `  h )
 ) ) >. ) >. , 
 <. (Scalar `  ndx ) ,  ( ( EDRing `  k
 ) `  w ) >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  (
 ( TEndo `  k ) `  w ) ,  f  e.  ( ( ( LTrn `  k ) `  w )  X.  ( ( TEndo `  k ) `  w ) )  |->  <. ( s `
  ( 1st `  f
 ) ) ,  (
 s  o.  ( 2nd `  f ) ) >. )
 >. } ) ) )
 
Theoremdvhfset 31270* The constructed full vector space H for a lattice  K. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 DVecH `  K )  =  ( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( (
 ( LTrn `  K ) `  w )  X.  (
 ( TEndo `  K ) `  w ) ) >. , 
 <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  K ) `  w )  X.  ( ( TEndo `  K ) `  w ) ) ,  g  e.  ( ( ( LTrn `  K ) `  w )  X.  ( ( TEndo `  K ) `  w ) )  |->  <. ( ( 1st `  f )  o.  ( 1st `  g
 ) ) ,  ( h  e.  ( ( LTrn `  K ) `  w )  |->  ( ( ( 2nd `  f
 ) `  h )  o.  ( ( 2nd `  g
 ) `  h )
 ) ) >. ) >. , 
 <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. }  u.  { <. ( .s
 `  ndx ) ,  (
 s  e.  ( (
 TEndo `  K ) `  w ) ,  f  e.  ( ( ( LTrn `  K ) `  w )  X.  ( ( TEndo `  K ) `  w ) )  |->  <. ( s `
  ( 1st `  f
 ) ) ,  (
 s  o.  ( 2nd `  f ) ) >. )
 >. } ) ) )
 
Theoremdvhset 31271* The constructed full vector space H for a lattice  K. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   =>    |-  ( ( K  e.  X  /\  W  e.  H )  ->  U  =  ( { <. ( Base ` 
 ndx ) ,  ( T  X.  E ) >. , 
 <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
 |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `  h )  o.  (
 ( 2nd `  g ) `  h ) ) )
 >. ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s
 `  ndx ) ,  (
 s  e.  E ,  f  e.  ( T  X.  E )  |->  <. ( s `
  ( 1st `  f
 ) ) ,  (
 s  o.  ( 2nd `  f ) ) >. )
 >. } ) )
 
Theoremdvhsca 31272 The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (Scalar `  U )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  F  =  D )
 
Theoremdvhbase 31273 The ring base set of the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  C  =  ( Base `  F )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  C  =  E )
 
Theoremdvhfplusr 31274* Ring addition operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .+  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  .+b  =  ( +g  `  F )   =>    |-  (
 ( K  e.  V  /\  W  e.  H ) 
 ->  .+b  =  .+  )
 
Theoremdvhfmulr 31275* Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .x.  =  ( .r `  F )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  t  e.  E  |->  ( s  o.  t ) ) )
 
Theoremdvhmulr 31276 Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .x.  =  ( .r `  F )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  S  e.  E )
 )  ->  ( R  .x.  S )  =  ( R  o.  S ) )
 
Theoremdvhvbase 31277 The vectors (vector base set) of the constructed full vector space H are all translations (for a fiducial co-atom  W). (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  V  =  ( T  X.  E ) )
 
Theoremdvhelvbasei 31278 Vector membership in the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( ( K  e.  X  /\  W  e.  H )  /\  ( F  e.  T  /\  S  e.  E ) )  ->  <. F ,  S >.  e.  V )
 
Theoremdvhvaddcbv 31279* Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)
 |-  .+  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E )  |->  <.
 ( ( 1st `  f
 )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
 )  .+^  ( 2nd `  g
 ) ) >. )   =>    |-  .+  =  ( h  e.  ( T  X.  E ) ,  i  e.  ( T  X.  E )  |->  <. ( ( 1st `  h )  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h )  .+^  ( 2nd `  i ) ) >. )
 
Theoremdvhvaddval 31280* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)
 |-  .+  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E )  |->  <.
 ( ( 1st `  f
 )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
 )  .+^  ( 2nd `  g
 ) ) >. )   =>    |-  ( ( F  e.  ( T  X.  E )  /\  G  e.  ( T  X.  E ) )  ->  ( F  .+  G )  =  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  (
 ( 2nd `  F )  .+^  ( 2nd `  G ) ) >. )
 
Theoremdvhfvadd 31281* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+b  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E )  |->  <.
 ( ( 1st `  f
 )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
 )  .+^  ( 2nd `  g
 ) ) >. )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  .+  =  .+b  )
 
Theoremdvhvadd 31282 The vector sum operation for the constructed full vector space H. (Contributed by NM, 11-Feb-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .+^  =  (
 +g  `  D )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E ) 
 /\  G  e.  ( T  X.  E ) ) )  ->  ( F  .+  G )  =  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  (
 ( 2nd `  F )  .+^  ( 2nd `  G ) ) >. )
 
Theoremdvhopvadd 31283 The vector sum operation for the constructed full vector space H. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .+^  =  (
 +g  `  D )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E ) )  ->  ( <. F ,  Q >.  .+  <. G ,  R >. )  =  <. ( F  o.  G ) ,  ( Q  .+^  R ) >. )
 
Theoremdvhopvadd2 31284* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 31283 and/or dvhfplusr 31274. (Contributed by NM, 26-Sep-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+b  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E ) )  ->  ( <. F ,  Q >.  .+b  <. G ,  R >. )  =  <. ( F  o.  G ) ,  ( Q  .+  R ) >. )
 
Theoremdvhvaddcl 31285 Closure of the vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E ) 
 /\  G  e.  ( T  X.  E ) ) )  ->  ( F  .+  G )  e.  ( T  X.  E ) )
 
TheoremdvhvaddcomN 31286 Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E ) 
 /\  G  e.  ( T  X.  E ) ) )  ->  ( F  .+  G )  =  ( G  .+  F ) )
 
Theoremdvhvaddass 31287 Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E ) 
 /\  G  e.  ( T  X.  E )  /\  I  e.  ( T  X.  E ) ) ) 
 ->  ( ( F  .+  G )  .+  I )  =  ( F  .+  ( G  .+  I ) ) )
 
Theoremdvhvscacbv 31288* Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)
 |-  .x.  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
 ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
 ) ) >. )   =>    |-  .x.  =  (
 t  e.  E ,  g  e.  ( T  X.  E )  |->  <. ( t `
  ( 1st `  g
 ) ) ,  (
 t  o.  ( 2nd `  g ) ) >. )
 
Theoremdvhvscaval 31289* The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)
 |-  .x.  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
 ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
 ) ) >. )   =>    |-  ( ( U  e.  E  /\  F  e.  ( T  X.  E ) )  ->  ( U 
 .x.  F )  =  <. ( U `  ( 1st `  F ) ) ,  ( U  o.  ( 2nd `  F ) )
 >. )
 
Theoremdvhfvsca 31290* Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  f  e.  ( T  X.  E )  |->  <. ( s `
  ( 1st `  f
 ) ) ,  (
 s  o.  ( 2nd `  f ) ) >. ) )
 
Theoremdvhvsca 31291 Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  F  e.  ( T  X.  E ) ) ) 
 ->  ( R  .x.  F )  =  <. ( R `
  ( 1st `  F ) ) ,  ( R  o.  ( 2nd `  F ) ) >. )
 
Theoremdvhopvsca 31292 Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  F  e.  T  /\  X  e.  E )
 )  ->  ( R  .x.  <. F ,  X >. )  =  <. ( R `
  F ) ,  ( R  o.  X ) >. )
 
Theoremdvhvscacl 31293 Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  E  /\  F  e.  ( T  X.  E ) ) ) 
 ->  ( R  .x.  F )  e.  ( T  X.  E ) )
 
Theoremtendoinvcl 31294* Closure of multiplicative inverse for endomorphism. We use the scalar inverse of the vector space since it is much simpler than the direct inverse of cdleml8 31172. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  N  =  (
 invr `  F )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  S  =/=  O )  ->  ( ( N `  S )  e.  E  /\  ( N `  S )  =/=  O ) )
 
Theoremtendolinv 31295* Left multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  N  =  (
 invr `  F )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  S  =/=  O )  ->  ( ( N `  S )  o.  S )  =  (  _I  |`  T ) )
 
Theoremtendorinv 31296* Right multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  N  =  (
 invr `  F )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  S  =/=  O )  ->  ( S  o.  ( N `  S ) )  =  (  _I  |`  T ) )
 
Theoremdvhgrp 31297 The full vector space  U constructed from a Hilbert lattice 
K (given a fiducial hyperplane 
W) is a group. (Contributed by NM, 19-Oct-2013.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  D )   &    |-  I  =  ( inv
 g `  D )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  U  e.  Grp )
 
Theoremdvhlveclem 31298 Lemma for dvhlvec 31299. TODO: proof substituting inner part first shorter/longer than substituting outer part first? TODO: break up into smaller lemmas? TODO: does  ph  -> method shorten proof? (Contributed by NM, 22-Oct-2013.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  D )   &    |-  I  =  ( inv
 g `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  .x. 
 =  ( .s `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
 
Theoremdvhlvec 31299 The full vector space  U constructed from a Hilbert lattice 
K (given a fiducial hyperplane 
W) is a left module. (Contributed by NM, 23-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  U  e.  LVec )
 
Theoremdvhlmod 31300 The full vector space  U constructed from a Hilbert lattice 
K (given a fiducial hyperplane 
W) is a left module. (Contributed by NM, 23-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  U  e.  LMod )
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