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Theorem List for Metamath Proof Explorer - 30101-30200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvhlmod 30101 The full vector space  U constructed from a Hilbert lattice 
K (given a fiducial hyperplane 
W) is a left module. (Contributed by NM, 23-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  U  e.  LMod )
 
Theoremdvh0g 30102* The zero vector of vector space H has the zero translation as its first member and the zero trace-preserving endomorphism as the second. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
 
Theoremdvheveccl 30103 Properties of a unit vector that we will use later as a convenient reference vector. This vector is called "e" in the remark after Lemma M of [Crawley] p. 121. line 17. See also dvhopN 30107 and dihpN 30327. (Contributed by NM, 27-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  B  =  ( Base `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  E  =  <. (  _I  |`  B ) ,  (  _I  |`  T ) >.   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  E  e.  ( V  \  {  .0.  } ) )
 
TheoremdvhopclN 30104 Closure of a  DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( F  e.  T  /\  U  e.  E ) 
 ->  <. F ,  U >.  e.  ( T  X.  E ) )
 
TheoremdvhopaddN 30105* Sum of  DVecH vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
 |-  A  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E )  |->  <.
 ( ( 1st `  f
 )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
 ) P ( 2nd `  g ) ) >. )   =>    |-  ( ( ( F  e.  T  /\  U  e.  E )  /\  ( G  e.  T  /\  V  e.  E )
 )  ->  ( <. F ,  U >. A <. G ,  V >. )  = 
 <. ( F  o.  G ) ,  ( U P V ) >. )
 
TheoremdvhopspN 30106* Scalar product of  DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
 |-  S  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
 ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
 ) ) >. )   =>    |-  ( ( R  e.  E  /\  ( F  e.  T  /\  U  e.  E )
 )  ->  ( R S <. F ,  U >. )  =  <. ( R `
  F ) ,  ( R  o.  U ) >. )
 
TheoremdvhopN 30107* Decompose a  DVecH vector expressed as an ordered pair into the sum of two components, the first from the translation group vector base of  DVecA and the other from the one-dimensional vector subspace  E. Part of Lemma M of [Crawley] p. 121, line 18. We represent their e, sigma, f by 
<. (  _I  |`  B ) ,  (  _I  |`  T )
>.,  U,  <. F ,  O >.. We swapped the order of vector sum (their juxtaposition i.e. composition) to show  <. F ,  O >. first. Note that  O and  (  _I  |`  T ) are the zero and one of the division ring  E, and  (  _I  |`  B ) is the zero of the translation group.  S is the scalar product. (Contributed by NM, 21-Nov-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( c  e.  T  |->  ( ( a `  c )  o.  (
 b `  c )
 ) ) )   &    |-  A  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E )  |->  <.
 ( ( 1st `  f
 )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
 ) P ( 2nd `  g ) ) >. )   &    |-  S  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <. ( s `  ( 1st `  f )
 ) ,  ( s  o.  ( 2nd `  f
 ) ) >. )   &    |-  O  =  ( c  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  U  e.  E )
 )  ->  <. F ,  U >.  =  ( <. F ,  O >. A ( U S <. (  _I  |`  B ) ,  (  _I  |`  T ) >. ) ) )
 
Theoremdvhopellsm 30108* Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  S  /\  Y  e.  S )  ->  ( <. F ,  T >.  e.  ( X  .(+)  Y )  <->  E. x E. y E. z E. w ( ( <. x ,  y >.  e.  X  /\  <. z ,  w >.  e.  Y )  /\  <. F ,  T >.  =  ( <. x ,  y >.  .+  <. z ,  w >. ) ) ) )
 
Theoremcdlemm10N 30109* The image of the map  G is the entire one-dimensional subspace  ( I `  V ). Remark after Lemma M of [Crawley] p. 121 line 23. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  C  =  { r  e.  A  |  ( r  .<_  ( P 
 .\/  V )  /\  -.  r  .<_  W ) }   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  s )   &    |-  G  =  ( q  e.  C  |->  ( iota_ f  e.  T ( f `  P )  =  q )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) ) 
 ->  ran  G  =  ( I `  V ) )
 
SyntaxcocaN 30110 Extend class notation with subspace orthocomplement for  DVecA partial vector space.
 class  ocA
 
Definitiondf-docaN 30111* Define subspace orthocomplement for  DVecA partial vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 6-Dec-2013.)
 |-  ocA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ~P (
 ( LTrn `  k ) `  w )  |->  ( ( ( DIsoA `  k ) `  w ) `  (
 ( ( ( oc
 `  k ) `  ( `' ( ( DIsoA `  k
 ) `  w ) `  |^| { z  e. 
 ran  ( ( DIsoA `  k ) `  w )  |  x  C_  z } ) ) (
 join `  k ) ( ( oc `  k
 ) `  w )
 ) ( meet `  k
 ) w ) ) ) ) )
 
TheoremdocaffvalN 30112* 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 )   =>    |-  ( K  e.  V  ->  ( ocA `  K )  =  ( w  e.  H  |->  ( x  e. 
 ~P ( ( LTrn `  K ) `  w )  |->  ( ( (
 DIsoA `  K ) `  w ) `  (
 ( (  ._|_  `  ( `' ( ( DIsoA `  K ) `  w ) `  |^|
 { z  e.  ran  ( ( DIsoA `  K ) `  w )  |  x  C_  z }
 ) )  .\/  (  ._|_  `  w ) ) 
 ./\  w ) ) ) ) )
 
TheoremdocafvalN 30113* 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 30114* 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 30115 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 30116 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 30117 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 30118 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 30119 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 30120* 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 30121* 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 30026 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 30122 Extend class notation with subspace join for  DVecA partial vector space.
 class  vA
 
Definitiondf-djaN 30123* 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 30124* 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 30125* 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 30126 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 30127 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 30128 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 30129 Extend class notation with isomorphism B.
 class  DIsoB
 
Definitiondf-dib 30130* 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 30131* 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 30132* 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 30133* 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 30134* 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 30135* 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 30136* 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 30137* 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 30138* 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 30139* 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 30140 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 30141 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 30142 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 30143* 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 30144 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 30145 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 30146 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 30147* 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 30148* 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 30149 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 30150 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 30151 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 30152 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 30153 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 30154 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 30155 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 30156* 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 30157* 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 30158 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 30159* 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 30160 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 30161 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 30162* 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 30163 Extend class notation with isomorphism C.
 class  DIsoC
 
Definitiondf-dic 30164* 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 30165* 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 30166* 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 30167* 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 30168* 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 30169* 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 30170* 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 30171* 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 30172* 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 30173* 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 30174* 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 30175* 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 30176 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 30177 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 30178* 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 30179 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 30180 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 30181 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 30182 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 30183 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 30184 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 30185* 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 30186* 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 30187* 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 30188* 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 30189* 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 30190* 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 30191* 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 30192 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 30193* 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 30194* 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 30195* 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 30196* 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 30197 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 30198* 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 30199* 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 30200* 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 ) )
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