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Theorem List for Metamath Proof Explorer - 34601-34700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvhlveclem 34601 Lemma for dvhlvec 34602. TODO: proof substituting inner part first shorter/longer than substituting outer part first? TODO: break up into smaller lemmas? TODO: does  ph  -> method shorten proof? (Contributed by NM, 22-Oct-2013.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  D )   &    |-  I  =  ( invg `  D )   &    |-  .X. 
 =  ( .r `  D )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
 
Theoremdvhlvec 34602 The full vector space  U constructed from a Hilbert lattice 
K (given a fiducial hyperplane 
W) is a left module. (Contributed by NM, 23-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  U  e.  LVec )
 
Theoremdvhlmod 34603 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 34604* 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 34605 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 34609 and dihpN 34829. (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 34606 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 34607* 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 34608* 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 34609* 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 34610* 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 34611* 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 34612 Extend class notation with subspace orthocomplement for  DVecA partial vector space.
 class  ocA
 
Definitiondf-docaN 34613* 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 34614* 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 34615* 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 34616* 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 34617 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 34618 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 34619 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 34620 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 34621 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 34622* 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 34623* 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 34528 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 34624 Extend class notation with subspace join for  DVecA partial vector space.
 class  vA
 
Definitiondf-djaN 34625* 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 34626* 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 34627* 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 34628 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 34629 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 34630 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 34631 Extend class notation with isomorphism B.
 class  DIsoB
 
Definitiondf-dib 34632* 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 34633* 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 34634* 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 34635* 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 34636* 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 34637* 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 34638* 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 34639* 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 34640* 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 34641* 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 34642 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 34643 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 34644 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 34645* 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 34646 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 34647 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 34648 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 34649* 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 34650* 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 34651 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 34652 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 34653 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 34654 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 34655 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 34656 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 34657 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 34658* 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 34659* 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 34660 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 34661* 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 34662 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 34663 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 34664* 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 34665 Extend class notation with isomorphism C.
 class  DIsoC
 
Definitiondf-dic 34666* 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 34667* 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 34668* 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 34669* 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 34670* 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 34671* 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 34672* 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 34673* 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 34674* 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 34675* 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 34676* 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 34677* 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 34678 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 34679 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 34680* 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 34681 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 34682 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 34683 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 34684 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 34685 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 34686 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 34687* 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 34688* 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 34689* 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 34690* 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 34691* 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 34692* 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 34693* 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 34694 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 34695* 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 34696* 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 34697* 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 34698* 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 34699 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 34700* 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 ) )
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