HomeHome Metamath Proof Explorer
Theorem List (p. 301 of 310)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30955)
 

Theorem List for Metamath Proof Explorer - 30001-30100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvafmulr 30001* Ring multiplication 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 )   &    |-  F  =  (Scalar `  U )   &    |-  .x.  =  ( .r `  F )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  t  e.  E  |->  ( s  o.  t ) ) )
 
Theoremdvamulr 30002 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 30003 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 30004* 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 30005 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 30006* 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 30007 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 30008 Identity property of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  ( F  e.  T  ->  ( (  _I  |`  T ) `
  F )  =  F )
 
Theoremtendospcl 30009 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 30010 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 30011 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 30012 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 30013* 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 30014* 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 30015 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 30016 Lemma for dvalvec 30017. (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 30017 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 30018 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 30019 Extend class notation with partial isomorphism A.
 class  DIsoA
 
Definitiondf-disoa 30020* 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 30021* 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 30022* 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 30023* 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 30024 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 30025* 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 30026* 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 30027 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 30028 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 30029 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 30030 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 30031 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 30032 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 30033 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 30034 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 30035 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 30036 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 30037 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 30038 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 30039 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 30040 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 30041 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 30042 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 30043 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 30044 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 30045 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 30046* 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 30047 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 30048 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 30049 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 30050 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 30051 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 30052* 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 30053 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 30054 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 30055 Lemma for dia2dim 30068. 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 30056 Lemma for dia2dim 30068. 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 30057 Lemma for dia2dim 30068. 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 30058 Lemma for dia2dim 30068. 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 30059 Lemma for dia2dim 30068. 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 30060 Lemma for dia2dim 30068. 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 30061 Lemma for dia2dim 30068. 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 30062 Lemma for dia2dim 30068. 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 30063 Lemma for dia2dim 30068. 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 30064 Lemma for dia2dim 30068. 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 30065 Lemma for dia2dim 30068. 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 30066 Lemma for dia2dim 30068. 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 30067 Lemma for dia2dim 30068. 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 30068 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 30069 Extend class notation with constructed full vector space H.
 class  DVecH
 
Definitiondf-dvech 30070* 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 30071* 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 30072* 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 30073 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 30074 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 30075* 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 30076* 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 30077 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 30078 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 30079 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 30080* 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 30081* 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 30082* 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 30083 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 30084 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 30085* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 30084 and/or dvhfplusr 30075. (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 30086 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 30087 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 30088 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 30089* 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 30090* 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 30091* 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 30092 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 30093 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 30094 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 30095* 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 29973. (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 30096* 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 30097* 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 30098 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 30099 Lemma for dvhlvec 30100. 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 30100 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 )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-30955
  Copyright terms: Public domain < Previous  Next >