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Theorem List for Metamath Proof Explorer - 31901-32000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvhvaddcbv 31901* Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)
 |-  .+  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E )  |->  <.
 ( ( 1st `  f
 )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
 )  .+^  ( 2nd `  g
 ) ) >. )   =>    |-  .+  =  ( h  e.  ( T  X.  E ) ,  i  e.  ( T  X.  E )  |->  <. ( ( 1st `  h )  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h )  .+^  ( 2nd `  i ) ) >. )
 
Theoremdvhvaddval 31902* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)
 |-  .+  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E )  |->  <.
 ( ( 1st `  f
 )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
 )  .+^  ( 2nd `  g
 ) ) >. )   =>    |-  ( ( F  e.  ( T  X.  E )  /\  G  e.  ( T  X.  E ) )  ->  ( F  .+  G )  =  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  (
 ( 2nd `  F )  .+^  ( 2nd `  G ) ) >. )
 
Theoremdvhfvadd 31903* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+b  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E )  |->  <.
 ( ( 1st `  f
 )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
 )  .+^  ( 2nd `  g
 ) ) >. )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  .+  =  .+b  )
 
Theoremdvhvadd 31904 The vector sum operation for the constructed full vector space H. (Contributed by NM, 11-Feb-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .+^  =  (
 +g  `  D )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E ) 
 /\  G  e.  ( T  X.  E ) ) )  ->  ( F  .+  G )  =  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  (
 ( 2nd `  F )  .+^  ( 2nd `  G ) ) >. )
 
Theoremdvhopvadd 31905 The vector sum operation for the constructed full vector space H. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .+^  =  (
 +g  `  D )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E ) )  ->  ( <. F ,  Q >.  .+  <. G ,  R >. )  =  <. ( F  o.  G ) ,  ( Q  .+^  R ) >. )
 
Theoremdvhopvadd2 31906* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 31905 and/or dvhfplusr 31896. (Contributed by NM, 26-Sep-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+b  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  Q  e.  E )  /\  ( G  e.  T  /\  R  e.  E ) )  ->  ( <. F ,  Q >.  .+b  <. G ,  R >. )  =  <. ( F  o.  G ) ,  ( Q  .+  R ) >. )
 
Theoremdvhvaddcl 31907 Closure of the vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E ) 
 /\  G  e.  ( T  X.  E ) ) )  ->  ( F  .+  G )  e.  ( T  X.  E ) )
 
TheoremdvhvaddcomN 31908 Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E ) 
 /\  G  e.  ( T  X.  E ) ) )  ->  ( F  .+  G )  =  ( G  .+  F ) )
 
Theoremdvhvaddass 31909 Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E ) 
 /\  G  e.  ( T  X.  E )  /\  I  e.  ( T  X.  E ) ) ) 
 ->  ( ( F  .+  G )  .+  I )  =  ( F  .+  ( G  .+  I ) ) )
 
Theoremdvhvscacbv 31910* Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)
 |-  .x.  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
 ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
 ) ) >. )   =>    |-  .x.  =  (
 t  e.  E ,  g  e.  ( T  X.  E )  |->  <. ( t `
  ( 1st `  g
 ) ) ,  (
 t  o.  ( 2nd `  g ) ) >. )
 
Theoremdvhvscaval 31911* The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)
 |-  .x.  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
 ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
 ) ) >. )   =>    |-  ( ( U  e.  E  /\  F  e.  ( T  X.  E ) )  ->  ( U 
 .x.  F )  =  <. ( U `  ( 1st `  F ) ) ,  ( U  o.  ( 2nd `  F ) )
 >. )
 
Theoremdvhfvsca 31912* Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  f  e.  ( T  X.  E )  |->  <. ( s `
  ( 1st `  f
 ) ) ,  (
 s  o.  ( 2nd `  f ) ) >. ) )
 
Theoremdvhvsca 31913 Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  F  e.  ( T  X.  E ) ) ) 
 ->  ( R  .x.  F )  =  <. ( R `
  ( 1st `  F ) ) ,  ( R  o.  ( 2nd `  F ) ) >. )
 
Theoremdvhopvsca 31914 Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  F  e.  T  /\  X  e.  E )
 )  ->  ( R  .x.  <. F ,  X >. )  =  <. ( R `
  F ) ,  ( R  o.  X ) >. )
 
Theoremdvhvscacl 31915 Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  E  /\  F  e.  ( T  X.  E ) ) ) 
 ->  ( R  .x.  F )  e.  ( T  X.  E ) )
 
Theoremtendoinvcl 31916* Closure of multiplicative inverse for endomorphism. We use the scalar inverse of the vector space since it is much simpler than the direct inverse of cdleml8 31794. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  N  =  (
 invr `  F )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  S  =/=  O )  ->  ( ( N `  S )  e.  E  /\  ( N `  S )  =/=  O ) )
 
Theoremtendolinv 31917* Left multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  N  =  (
 invr `  F )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  S  =/=  O )  ->  ( ( N `  S )  o.  S )  =  (  _I  |`  T ) )
 
Theoremtendorinv 31918* Right multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  N  =  (
 invr `  F )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  S  =/=  O )  ->  ( S  o.  ( N `  S ) )  =  (  _I  |`  T ) )
 
Theoremdvhgrp 31919 The full vector space  U constructed from a Hilbert lattice 
K (given a fiducial hyperplane 
W) is a group. (Contributed by NM, 19-Oct-2013.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  D )   &    |-  I  =  ( inv
 g `  D )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  U  e.  Grp )
 
Theoremdvhlveclem 31920 Lemma for dvhlvec 31921. TODO: proof substituting inner part first shorter/longer than substituting outer part first? TODO: break up into smaller lemmas? TODO: does  ph  -> method shorten proof? (Contributed by NM, 22-Oct-2013.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  D )   &    |-  I  =  ( inv
 g `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  .x. 
 =  ( .s `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
 
Theoremdvhlvec 31921 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 31922 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 31923* 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 31924 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 31928 and dihpN 32148. (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 31925 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 31926* 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 31927* 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 31928* 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 31929* 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 31930* 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 31931 Extend class notation with subspace orthocomplement for  DVecA partial vector space.
 class  ocA
 
Definitiondf-docaN 31932* 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 31933* 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 31934* 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 31935* 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 31936 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 31937 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 31938 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 31939 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 31940 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 31941* 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 31942* 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 31847 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 31943 Extend class notation with subspace join for  DVecA partial vector space.
 class  vA
 
Definitiondf-djaN 31944* 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 31945* 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 31946* 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 31947 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 31948 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 31949 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 31950 Extend class notation with isomorphism B.
 class  DIsoB
 
Definitiondf-dib 31951* 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 31952* 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 31953* 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 31954* 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 31955* 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 31956* 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 31957* 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 31958* 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 31959* 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 31960* 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 31961 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 31962 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 31963 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 31964* 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 31965 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 31966 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 31967 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 31968* 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 31969* 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 31970 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 31971 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 31972 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 31973 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 31974 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 31975 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 31976 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 31977* 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 31978* 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 31979 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 31980* 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 31981 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 31982 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 31983* 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 31984 Extend class notation with isomorphism C.
 class  DIsoC
 
Definitiondf-dic 31985* 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 31986* 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 31987* 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 31988* 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 31989* 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 31990* 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 31991* 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 31992* 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 31993* 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 31994* 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 31995* 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 31996* 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 31997 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 31998 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 31999* 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 32000 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 )
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