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Theorem List for Metamath Proof Explorer - 30001-30100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremdocafvalN 30001* Subspace orthocomplement for 
DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  N  =  ( ( ocA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  N  =  ( x  e.  ~P T  |->  ( I `  ( ( (  ._|_  `  ( `' I `  |^|
 { z  e.  ran  I  |  x  C_  z } ) )  .\/  (  ._|_  `  W )
 )  ./\  W ) ) ) )
 
TheoremdocavalN 30002* Subspace orthocomplement for 
DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  N  =  ( ( ocA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T )  ->  ( N `  X )  =  ( I `  ( ( (  ._|_  `  ( `' I `  |^|
 { z  e.  ran  I  |  X  C_  z } ) )  .\/  (  ._|_  `  W )
 )  ./\  W ) ) )
 
TheoremdocaclN 30003 Closure of subspace orthocomplement for  DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ._|_  =  ( ( ocA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  T )  ->  (  ._|_  `  X )  e.  ran  I )
 
TheoremdiaocN 30004 Value of partial isomorphism A at lattice orthocomplement (using a Sasaki projection to get orthocomplement relative to the fiducial co-atom  W). (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  N  =  ( ( ocA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I
 )  ->  ( I `  ( ( (  ._|_  `  X )  .\/  (  ._|_  `  W ) ) 
 ./\  W ) )  =  ( N `  ( I `  X ) ) )
 
Theoremdoca2N 30005 Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ._|_  =  ( ( ocA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I
 )  ->  (  ._|_  `  (  ._|_  `  ( I `
  X ) ) )  =  ( I `
  X ) )
 
Theoremdoca3N 30006 Double orthocomplement of partial isomorphism A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ._|_  =  ( ( ocA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I
 )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
 
TheoremdvadiaN 30007 Any closed subspace is a member of the range of partial isomorphism A, showing the isomorphism maps onto the set of closed 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 )   &    |-  ._|_  =  ( ( ocA `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  S  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) 
 ->  X  e.  ran  I
 )
 
TheoremdiarnN 30008* Partial isomorphism A maps onto the set of all closed subspaces of partial vector space A. Part of Lemma M of [Crawley] p. 121 line 12, with closed subspaces rather than subspaces. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ._|_  =  ( ( ocA `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  ran  I  =  { x  e.  S  |  (  ._|_  `  (  ._|_  `  x ) )  =  x } )
 
Theoremdiaf1oN 30009* The partial isomorphism A for a lattice  K is a one-to-one, onto function. Part of Lemma M of [Crawley] p. 121 line 12, with closed subspaces rather than subspaces. See diadm 29914 for the domain. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ._|_  =  ( ( ocA `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  I : dom  I -1-1-onto-> { x  e.  S  |  (  ._|_  `  (  ._|_  `  x ) )  =  x } )
 
SyntaxcdjaN 30010 Extend class notation with subspace join for  DVecA partial vector space.
 class  vA
 
Definitiondf-djaN 30011* Define (closed) subspace join for  DVecA partial vector space. (Contributed by NM, 6-Dec-2013.)
 |-  vA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ~P (
 ( LTrn `  k ) `  w ) ,  y  e.  ~P ( ( LTrn `  k ) `  w )  |->  ( ( ( ocA `  k ) `  w ) `  (
 ( ( ( ocA `  k ) `  w ) `  x )  i^i  ( ( ( ocA `  k ) `  w ) `  y ) ) ) ) ) )
 
TheoremdjaffvalN 30012* Subspace join for  DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  ( vA `  K )  =  ( w  e.  H  |->  ( x  e. 
 ~P ( ( LTrn `  K ) `  w ) ,  y  e.  ~P ( ( LTrn `  K ) `  w )  |->  ( ( ( ocA `  K ) `  w ) `  ( ( ( ( ocA `  K ) `  w ) `  x )  i^i  ( ( ( ocA `  K ) `  w ) `  y
 ) ) ) ) ) )
 
TheoremdjafvalN 30013* Subspace join for  DVecA partial vector space. TODO: take out hypothesis .i, no longer used. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ._|_  =  ( ( ocA `  K ) `  W )   &    |-  J  =  ( ( vA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  J  =  ( x  e.  ~P T ,  y  e.  ~P T  |->  (  ._|_  `  (
 (  ._|_  `  x )  i^i  (  ._|_  `  y ) ) ) ) )
 
TheoremdjavalN 30014 Subspace join for  DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ._|_  =  ( ( ocA `  K ) `  W )   &    |-  J  =  ( ( vA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T ) )  ->  ( X J Y )  =  (  ._|_  `  (
 (  ._|_  `  X )  i^i  (  ._|_  `  Y ) ) ) )
 
TheoremdjaclN 30015 Closure of subspace join for 
DVecA partial vector space. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  J  =  ( ( vA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  C_  T  /\  Y  C_  T ) )  ->  ( X J Y )  e.  ran  I )
 
TheoremdjajN 30016 Transfer lattice join to  DVecA partial vector space closed subspace join. Part of Lemma M of [Crawley] p. 120 line 29, with closed subspace join rather than subspace sum. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  J  =  ( ( vA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  dom  I  /\  Y  e.  dom  I ) )  ->  ( I `  ( X  .\/  Y ) )  =  (
 ( I `  X ) J ( I `  Y ) ) )
 
Syntaxcdib 30017 Extend class notation with isomorphism B.
 class  DIsoB
 
Definitiondf-dib 30018* Isomorphism B is isomorphism A extended with an extra dimension set to the zero vector component i.e. the zero endormorphism. Its domain is lattice elements less than or equal to the fiducial co-atom  w. (Contributed by NM, 8-Dec-2013.)
 |-  DIsoB  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  dom  ( ( DIsoA `  k ) `  w )  |->  ( ( ( ( DIsoA `  k
 ) `  w ) `  x )  X.  {
 ( f  e.  (
 ( LTrn `  k ) `  w )  |->  (  _I  |`  ( Base `  k )
 ) ) } )
 ) ) )
 
Theoremdibffval 30019* The partial isomorphism B for a lattice  K. (Contributed by NM, 8-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 DIsoB `  K )  =  ( w  e.  H  |->  ( x  e.  dom  ( ( DIsoA `  K ) `  w )  |->  ( ( ( ( DIsoA `  K ) `  w ) `  x )  X.  { ( f  e.  (
 ( LTrn `  K ) `  w )  |->  (  _I  |`  B ) ) }
 ) ) ) )
 
Theoremdibfval 30020* The partial isomorphism B for a lattice  K. (Contributed by NM, 8-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  J  =  ( (
 DIsoA `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  dom  J  |->  ( ( J `  x )  X.  {  .0.  } ) ) )
 
Theoremdibval 30021* The partial isomorphism B for a lattice  K. (Contributed by NM, 8-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  J  =  ( (
 DIsoA `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  dom  J )  ->  ( I `  X )  =  ( ( J `
  X )  X.  {  .0.  } ) )
 
TheoremdibopelvalN 30022* Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  J  =  ( (
 DIsoA `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  dom  J )  ->  ( <. F ,  S >.  e.  ( I `  X )  <->  ( F  e.  ( J `  X ) 
 /\  S  =  .0.  ) ) )
 
Theoremdibval2 30023* Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  J  =  ( (
 DIsoA `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  =  ( ( J `  X )  X.  {  .0.  } ) )
 
Theoremdibopelval2 30024* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  J  =  ( (
 DIsoA `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( <. F ,  S >.  e.  ( I `  X )  <->  ( F  e.  ( J `  X ) 
 /\  S  =  .0.  ) ) )
 
Theoremdibval3N 30025* Value of the partial isomorphism B for a lattice  K. (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  =  ( {
 f  e.  T  |  ( R `  f ) 
 .<_  X }  X.  {  .0.  } ) )
 
Theoremdibelval3 30026* Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( Y  e.  ( I `  X )  <->  E. f  e.  T  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f )  .<_  X ) ) )
 
Theoremdibopelval3 30027* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( <. F ,  S >.  e.  ( I `  X )  <->  ( ( F  e.  T  /\  ( R `  F )  .<_  X )  /\  S  =  .0.  ) ) )
 
Theoremdibelval1st 30028 Membership in value of the partial isomorphism B for a lattice  K. (Contributed by NM, 13-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  J  =  ( ( DIsoA `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X ) )  ->  ( 1st `  Y )  e.  ( J `  X ) )
 
Theoremdibelval1st1 30029 Membership in value of the partial isomorphism B for a lattice  K. (Contributed by NM, 13-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X ) )  ->  ( 1st `  Y )  e.  T )
 
Theoremdibelval1st2N 30030 Membership in value of the partial isomorphism B for a lattice  K. (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X ) )  ->  ( R `  ( 1st `  Y ) )  .<_  X )
 
Theoremdibelval2nd 30031* Membership in value of the partial isomorphism B for a lattice  K. (Contributed by NM, 13-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X ) )  ->  ( 2nd `  Y )  =  .0.  )
 
Theoremdibn0 30032 The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  =/=  (/) )
 
Theoremdibfna 30033 Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  J  =  ( ( DIsoA `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  dom  J )
 
Theoremdibdiadm 30034 Domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  J  =  ( ( DIsoA `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  dom  J )
 
TheoremdibfnN 30035* Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
 
TheoremdibdmN 30036* Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  { x  e.  B  |  x  .<_  W }
 )
 
TheoremdibeldmN 30037 Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom 
 I 
 <->  ( X  e.  B  /\  X  .<_  W ) ) )
 
Theoremdibord 30038 The isomorphism B for a lattice  K is order-preserving in the region under co-atom  W. (Contributed by NM, 24-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  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 ) )
 
Theoremdib11N 30039 The isomorphism B for a lattice  K is one-to-one in the region under co-atom  W. (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( ( I `  X )  =  ( I `  Y )  <->  X  =  Y ) )
 
Theoremdibf11N 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  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
 
TheoremdibclN 30041 Closure of partial isomorphism B for a lattice  K. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I
 )  ->  ( I `  X )  e.  ran  I )
 
Theoremdibvalrel 30042 The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )
 
Theoremdib0 30043 The value of partial isomorphism B at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 27-Mar-2014.)
 |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  O  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .0.  )  =  { O } )
 
Theoremdib1dim 30044* Two expressions for the 1-dimensional subspaces of vector space H. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  ->  ( I `  ( R `  F ) )  =  { g  e.  ( T  X.  E )  | 
 E. s  e.  E  g  =  <. ( s `
  F ) ,  O >. } )
 
TheoremdibglbN 30045* Partial isomorphism B of a lattice glb. (Contributed by NM, 9-Mar-2014.) (New usage is discouraged.)
 |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  dom  I  /\  S  =/=  (/) ) )  ->  ( I `  ( G `
  S ) )  =  |^|_ x  e.  S  ( I `  x ) )
 
TheoremdibintclN 30046 The intersection of partial isomorphism B closed subspaces is a closed subspace. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) ) 
 ->  |^| S  e.  ran  I )
 
Theoremdib1dim2 30047* Two expressions for a 1-dimensional subspace of vector space H (when  F is a nonzero vector i.e. non-identity translation). (Contributed by NM, 24-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( I `  ( R `  F ) )  =  ( N `  { <. F ,  O >. } ) )
 
Theoremdibss 30048 The partial isomorphism B maps to a set of vectors in full vector space H. (Contributed by NM, 1-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  C_  V )
 
Theoremdiblss 30049 The value of partial isomorphism B is a subspace of partial vector space H. TODO: use dib* specific theorems instead of dia* ones to shorten proof? (Contributed by NM, 11-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  S  =  (
 LSubSp `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  e.  S )
 
Theoremdiblsmopel 30050* Membership in subspace sum for partial isomorphism B. (Contributed by NM, 21-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  V  =  ( (
 DVecA `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  ( LSSum `  V )   &    |-  .+b  =  ( LSSum `  U )   &    |-  J  =  ( ( DIsoA `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( X  e.  B  /\  X  .<_  W ) )   &    |-  ( ph  ->  ( Y  e.  B  /\  Y  .<_  W ) )   =>    |-  ( ph  ->  ( <. F ,  S >.  e.  ( ( I `  X )  .+b  ( I `
  Y ) )  <-> 
 ( F  e.  (
 ( J `  X )  .(+)  ( J `  Y ) )  /\  S  =  O )
 ) )
 
Syntaxcdic 30051 Extend class notation with isomorphism C.
 class  DIsoC
 
Definitiondf-dic 30052* Isomorphism C has domain of lattice atoms that are not less than or equal to the fiducial co-atom 
w. The value is a one-dimensional subspace generated by the pair consisting of the  iota_ vector below and the endomorphism ring unit. Definition of phi(q) in [Crawley] p. 121. Note that we use the fixed atom  ( ( oc k )  w ) to represent the p in their "Choose an atom p..." on line 21. (Contributed by NM, 15-Dec-2013.)
 |-  DIsoC  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( q  e.  { r  e.  ( Atoms `  k )  |  -.  r ( le `  k ) w }  |->  { <. f ,  s >.  |  ( f  =  ( s `  ( iota_
 g  e.  ( (
 LTrn `  k ) `  w ) ( g `
  ( ( oc
 `  k ) `  w ) )  =  q ) )  /\  s  e.  ( ( TEndo `  k ) `  w ) ) }
 ) ) )
 
Theoremdicffval 30053* The partial isomorphism C for a lattice  K. (Contributed by NM, 15-Dec-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( K  e.  V  ->  ( DIsoC `  K )  =  ( w  e.  H  |->  ( q  e. 
 { r  e.  A  |  -.  r  .<_  w }  |->  { <. f ,  s >.  |  ( f  =  ( s `  ( iota_
 g  e.  ( (
 LTrn `  K ) `  w ) ( g `
  ( ( oc
 `  K ) `  w ) )  =  q ) )  /\  s  e.  ( ( TEndo `  K ) `  w ) ) }
 ) ) )
 
Theoremdicfval 30054* The partial isomorphism C for a lattice  K. (Contributed by NM, 15-Dec-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
  ( iota_ g  e.  T ( g `  P )  =  q
 ) )  /\  s  e.  E ) } )
 )
 
Theoremdicval 30055* The partial isomorphism C for a lattice  K. (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  ( I `  Q )  =  { <. f ,  s >.  |  (
 f  =  ( s `
  ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) } )
 
Theoremdicopelval 30056* Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 15-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  F  e.  _V   &    |-  S  e.  _V   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  ( <. F ,  S >.  e.  ( I `  Q )  <->  ( F  =  ( S `  ( iota_ g  e.  T ( g `
  P )  =  Q ) )  /\  S  e.  E )
 ) )
 
TheoremdicelvalN 30057* Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  ( Y  e.  ( I `  Q )  <->  ( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y )  =  ( ( 2nd `  Y ) `  ( iota_ g  e.  T ( g `  P )  =  Q )
 )  /\  ( 2nd `  Y )  e.  E ) ) ) )
 
Theoremdicval2 30058* The partial isomorphism C for a lattice  K. (Contributed by NM, 20-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  G  =  (
 iota_ g  e.  T ( g `  P )  =  Q )   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  =  { <. f ,  s >.  |  ( f  =  ( s `  G )  /\  s  e.  E ) } )
 
Theoremdicelval3 30059* Member of the partial isomorphism C. (Contributed by NM, 26-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  G  =  (
 iota_ g  e.  T ( g `  P )  =  Q )   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( Y  e.  ( I `  Q )  <->  E. s  e.  E  Y  =  <. ( s `
  G ) ,  s >. ) )
 
Theoremdicopelval2 30060* Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 20-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  G  =  (
 iota_ g  e.  T ( g `  P )  =  Q )   &    |-  F  e.  _V   &    |-  S  e.  _V   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  Q ) 
 <->  ( F  =  ( S `  G ) 
 /\  S  e.  E ) ) )
 
Theoremdicelval2N 30061* Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  G  =  (
 iota_ g  e.  T ( g `  P )  =  Q )   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( Y  e.  ( I `  Q )  <->  ( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y )  =  ( ( 2nd `  Y ) `  G )  /\  ( 2nd `  Y )  e.  E ) ) ) )
 
TheoremdicfnN 30062* Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { p  e.  A  |  -.  p  .<_  W }
 )
 
TheoremdicdmN 30063* Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  { p  e.  A  |  -.  p  .<_  W }
 )
 
TheoremdicvalrelN 30064 The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )
 
Theoremdicssdvh 30065 The partial isomorphism C maps to a set of vectors in full vector space H. (Contributed by NM, 19-Jan-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  C_  V )
 
Theoremdicelval1sta 30066* Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 16-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q ) )  ->  ( 1st `  Y )  =  ( ( 2nd `  Y ) `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) )
 
Theoremdicelval1stN 30067 Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 16-Feb-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  Y  e.  ( I `  Q ) ) 
 ->  ( 1st `  Y )  e.  T )
 
Theoremdicelval2nd 30068 Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 16-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  Y  e.  ( I `  Q ) ) 
 ->  ( 2nd `  Y )  e.  E )
 
Theoremdicvaddcl 30069 Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q ) ) )  ->  ( X  .+  Y )  e.  ( I `  Q ) )
 
Theoremdicvscacl 30070 Membership in value of the partial isomorphism C is closed under scalar product. (Contributed by NM, 16-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  E  /\  Y  e.  ( I `
  Q ) ) )  ->  ( X  .x.  Y )  e.  ( I `  Q ) )
 
Theoremdicn0 30071 The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  =/=  (/) )
 
Theoremdiclss 30072 The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  e.  S )
 
Theoremdiclspsn 30073* The value of isomorphism C is spanned by vector  F. Part of proof of Lemma N of [Crawley] p. 121 line 29. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  =  ( N `  { <. F ,  (  _I  |`  T )
 >. } ) )
 
Theoremcdlemn2 30074* Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  F  =  ( iota_ h  e.  T ( h `  Q )  =  S )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  S  .<_  ( Q 
 .\/  X ) )  ->  ( R `  F ) 
 .<_  X )
 
Theoremcdlemn2a 30075* Part of proof of Lemma N of [Crawley] p. 121. (Contributed by NM, 24-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoB `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  Q )  =  S )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  S  .<_  ( Q 
 .\/  X ) )  ->  ( N `  { <. F ,  O >. } )  C_  ( I `  X ) )
 
Theoremcdlemn3 30076* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   &    |-  J  =  ( iota_ h  e.  T ( h `  Q )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( J  o.  F )  =  G )
 
Theoremcdlemn4 30077* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  F  =  (
 iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   &    |-  J  =  ( iota_ h  e.  T ( h `  Q )  =  R )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 ->  <. G ,  (  _I  |`  T ) >.  =  ( <. F ,  (  _I  |`  T ) >.  .+ 
 <. J ,  O >. ) )
 
Theoremcdlemn4a 30078* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 24-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  F  =  (
 iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   &    |-  J  =  ( iota_ h  e.  T ( h `  Q )  =  R )   &    |-  N  =  ( LSpan `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( N `  { <. G ,  (  _I  |`  T ) >. } )  C_  (
 ( N `  { <. F ,  (  _I  |`  T )
 >. } )  .(+)  ( N `
  { <. J ,  O >. } ) ) )
 
Theoremcdlemn5pre 30079* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  N  =  ( LSpan `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   &    |-  M  =  ( iota_ h  e.  T ( h `  Q )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  R  .<_  ( Q 
 .\/  X ) )  ->  ( J `  R ) 
 C_  ( ( J `
  Q )  .(+)  ( I `  X ) ) )
 
Theoremcdlemn5 30080 Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  R  .<_  ( Q 
 .\/  X ) )  ->  ( J `  R ) 
 C_  ( ( J `
  Q )  .(+)  ( I `  X ) ) )
 
Theoremcdlemn6 30081* Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T ) )  ->  ( <. ( s `  F ) ,  s >.  .+  <. g ,  O >. )  =  <. ( ( s `  F )  o.  g ) ,  s >. )
 
Theoremcdlemn7 30082* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. G ,  (  _I  |`  T ) >.  =  ( <. ( s `
  F ) ,  s >.  .+  <. g ,  O >. ) ) ) 
 ->  ( G  =  ( ( s `  F )  o.  g )  /\  (  _I  |`  T )  =  s ) )
 
Theoremcdlemn8 30083* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. G ,  (  _I  |`  T ) >.  =  ( <. ( s `
  F ) ,  s >.  .+  <. g ,  O >. ) ) ) 
 ->  g  =  ( G  o.  `' F ) )
 
Theoremcdlemn9 30084* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. G ,  (  _I  |`  T ) >.  =  ( <. ( s `
  F ) ,  s >.  .+  <. g ,  O >. ) ) ) 
 ->  ( g `  Q )  =  R )
 
Theoremcdlemn10 30085 Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( g  e.  T  /\  ( g `
  Q )  =  S  /\  ( R `
  g )  .<_  X ) )  ->  S  .<_  ( Q  .\/  X ) )
 
Theoremcdlemn11a 30086* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( J `  N )  C_  ( ( J `  Q ) 
 .(+)  ( I `  X ) ) )  ->  <. G ,  (  _I  |`  T ) >.  e.  ( J `  N ) )
 
Theoremcdlemn11b 30087* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( J `  N )  C_  ( ( J `  Q ) 
 .(+)  ( I `  X ) ) )  ->  <. G ,  (  _I  |`  T ) >.  e.  (
 ( J `  Q )  .(+)  ( I `  X ) ) )
 
Theoremcdlemn11c 30088* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( J `  N )  C_  ( ( J `  Q ) 
 .(+)  ( I `  X ) ) )  ->  E. y  e.  ( J `  Q ) E. z  e.  ( I `  X ) <. G ,  (  _I  |`  T ) >.  =  ( y  .+  z ) )
 
Theoremcdlemn11pre 30089* Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 30086, cdlemn11b 30087, cdlemn11c 30088, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( J `  N )  C_  ( ( J `  Q ) 
 .(+)  ( I `  X ) ) )  ->  N  .<_  ( Q  .\/  X ) )
 
Theoremcdlemn11 30090 Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( J `  R )  C_  ( ( J `  Q ) 
 .(+)  ( I `  X ) ) )  ->  R  .<_  ( Q  .\/  X ) )
 
Theoremcdlemn 30091 Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) )  ->  ( R  .<_  ( Q  .\/  X ) 
 <->  ( J `  R )  C_  ( ( J `
  Q )  .(+)  ( I `  X ) ) ) )
 
Theoremdihordlem6 30092* Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T ) )  ->  ( <. ( s `  G ) ,  s >.  .+  <. g ,  O >. )  =  <. ( ( s `  G )  o.  g ) ,  s >. )
 
Theoremdihordlem7 30093* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G ) ,  s >.  .+ 
 <. g ,  O >. ) ) )  ->  (
 f  =  ( ( s `  G )  o.  g )  /\  O  =  s )
 )
 
Theoremdihordlem7b 30094* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G ) ,  s >.  .+ 
 <. g ,  O >. ) ) )  ->  (
 f  =  g  /\  O  =  s )
 )
 
Theoremdihjustlem 30095 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B )  /\  ( Q  .\/  ( X  ./\  W ) )  =  ( R  .\/  ( X  ./\  W ) ) )  ->  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  (
 ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) )
 
Theoremdihjust 30096 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B )  /\  ( Q  .\/  ( X  ./\  W ) )  =  ( R  .\/  ( X  ./\  W ) ) )  ->  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  =  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) )
 
Theoremdihord1 30097 Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change  ( Q  .\/  ( X  ./\  W ) )  =  X to  Q  .<_  X using lhpmcvr3 28903, here and all theorems below. (Contributed by NM, 2-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( Q 
 .\/  ( X  ./\  W ) )  =  X  /\  ( R  .\/  ( Y  ./\  W ) )  =  Y  /\  X  .<_  Y ) )  ->  ( ( J `  Q )  .(+)  ( I `
  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `
  ( Y  ./\  W ) ) ) )
 
Theoremdihord2a 30098 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( Q 
 .\/  ( X  ./\  W ) )  =  X  /\  ( R  .\/  ( Y  ./\  W ) )  =  Y  /\  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  (
 ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) ) ) 
 ->  Q  .<_  ( R  .\/  ( Y  ./\  W ) ) )
 
Theoremdihord2b 30099 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( J `
  Q )  .(+)  ( I `  ( X 
 ./\  W ) ) ) 
 C_  ( ( J `
  R )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) )  ->  ( I `  ( X  ./\  W ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) )
 
Theoremdihord2cN 30100* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f )  .<_  ( X  ./\  W )
 ) )  ->  <. f ,  O >.  e.  ( I `  ( X  ./\  W ) ) )
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