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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdleml7 30301* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( oc `  K ) `  W )   &    |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( ( h `  Q )  .\/  ( R `
  ( b  o.  `' ( s `  h ) ) ) ) )   &    |-  Y  =  ( ( Q  .\/  ( R `  g ) ) 
 ./\  ( Z  .\/  ( R `  ( g  o.  `' b ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  Q )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if (
 ( s `  h )  =  h ,  g ,  X )
 )   &    |-  E  =  ( (
 TEndo `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( ( U  o.  s ) `  h )  =  ( (  _I  |`  T ) `  h ) )
 
Theoremcdleml8 30302* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( oc `  K ) `  W )   &    |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( ( h `  Q )  .\/  ( R `
  ( b  o.  `' ( s `  h ) ) ) ) )   &    |-  Y  =  ( ( Q  .\/  ( R `  g ) ) 
 ./\  ( Z  .\/  ( R `  ( g  o.  `' b ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  Q )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if (
 ( s `  h )  =  h ,  g ,  X )
 )   &    |-  E  =  ( (
 TEndo `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  ( U  o.  s
 )  =  (  _I  |`  T ) )
 
Theoremcdleml9 30303* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( oc `  K ) `  W )   &    |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( ( h `  Q )  .\/  ( R `
  ( b  o.  `' ( s `  h ) ) ) ) )   &    |-  Y  =  ( ( Q  .\/  ( R `  g ) ) 
 ./\  ( Z  .\/  ( R `  ( g  o.  `' b ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  Q )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if (
 ( s `  h )  =  h ,  g ,  X )
 )   &    |-  E  =  ( (
 TEndo `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) )  /\  ( s  e.  E  /\  s  =/=  .0.  ) )  ->  U  =/=  .0.  )
 
Theoremdva1dim 30304* Two expressions for the 1-dimensional subspaces of partial vector space A. Remark in [Crawley] p. 120 line 21, but using a non-identity translation (nonzero vector) 
F whose trace is  P rather than  P itself;  F exists by cdlemf 29882. 
E is the division ring base by erngdv 30312, and  s `  F is the scalar product by dvavsca 30336. 
F must be a non-identity translation for the expression to be a 1-dimensional subspace, although the theorem doesn't require it. (Contributed by NM, 14-Oct-2013.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  { g  |  E. s  e.  E  g  =  ( s `  F ) }  =  {
 g  e.  T  |  ( R `  g ) 
 .<_  ( R `  F ) } )
 
Theoremdvhb1dimN 30305* Two expressions for the 1-dimensional subspaces of vector space H, in the isomorphism B case where the 2nd vector component is zero. (Contributed by NM, 23-Feb-2014.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .0.  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  ->  { g  e.  ( T  X.  E )  |  E. s  e.  E  g  =  <. ( s `  F ) ,  .0.  >. }  =  { g  e.  ( T  X.  E )  |  ( ( R `  ( 1st `  g )
 )  .<_  ( R `  F )  /\  ( 2nd `  g )  =  .0.  ) } )
 
Theoremerng1lem 30306 Value of the endomorphism division ring unit. (Contributed by NM, 12-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  D  e.  Ring )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  ( 1r `  D )  =  (  _I  |`  T ) )
 
Theoremerngdvlem1 30307* Lemma for erngrng 30311. (Contributed by NM, 4-Aug-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  B  =  ( Base `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  (
 b `  f )
 ) ) )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
 
Theoremerngdvlem2N 30308* Lemma for erngrng 30311. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  B  =  ( Base `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  (
 b `  f )
 ) ) )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Abel )
 
Theoremerngdvlem3 30309* Lemma for erngrng 30311. (Contributed by NM, 6-Aug-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  B  =  ( Base `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  (
 b `  f )
 ) ) )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   &    |-  .+  =  (
 a  e.  E ,  b  e.  E  |->  ( a  o.  b ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Ring )
 
Theoremerngdvlem4 30310* Lemma for erngdv 30312. (Contributed by NM, 11-Aug-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  B  =  ( Base `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  (
 b `  f )
 ) ) )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   &    |-  .+  =  (
 a  e.  E ,  b  e.  E  |->  ( a  o.  b ) )   &    |-  .\/ 
 =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( oc `  K ) `  W )   &    |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( ( h `  Q )  .\/  ( R `  ( b  o.  `' ( s `
  h ) ) ) ) )   &    |-  Y  =  ( ( Q  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  Q )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if (
 ( s `  h )  =  h ,  g ,  X )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  D  e. 
 DivRing )
 
Theoremerngrng 30311 An endomorphism ring is a ring. Todo: fix comment. (Contributed by NM, 4-Aug-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Ring )
 
Theoremerngdv 30312 An endomorphism ring is a division ring. Todo: fix comment. (Contributed by NM, 11-Aug-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e. 
 DivRing )
 
Theoremerng0g 30313* The division ring zero of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |- 
 .0.  =  ( 0g `  D )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  O )
 
Theoremerng1r 30314 The division ring unit of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  .1.  =  ( 1r `  D )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .1.  =  (  _I  |`  T ) )
 
Theoremerngdvlem1-rN 30315* Lemma for erngrng 30311. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   &    |-  B  =  ( Base `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  ( b `  f ) ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
 
Theoremerngdvlem2-rN 30316* Lemma for erngrng 30311. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   &    |-  B  =  ( Base `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  ( b `  f ) ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Abel )
 
Theoremerngdvlem3-rN 30317* Lemma for erngrng 30311. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   &    |-  B  =  ( Base `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  ( b `  f ) ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   &    |-  M  =  ( a  e.  E ,  b  e.  E  |->  ( b  o.  a ) )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Ring )
 
Theoremerngdvlem4-rN 30318* Lemma for erngdv 30312. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   &    |-  B  =  ( Base `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  P  =  ( a  e.  E ,  b  e.  E  |->  ( f  e.  T  |->  ( ( a `  f )  o.  ( b `  f ) ) ) )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( a  e.  E  |->  ( f  e.  T  |->  `' (
 a `  f )
 ) )   &    |-  M  =  ( a  e.  E ,  b  e.  E  |->  ( b  o.  a ) )   &    |-  .\/ 
 =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( oc `  K ) `  W )   &    |-  Z  =  ( ( Q  .\/  ( R `  b ) )  ./\  ( ( h `  Q )  .\/  ( R `  ( b  o.  `' ( s `
  h ) ) ) ) )   &    |-  Y  =  ( ( Q  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
 g  o.  `' b
 ) ) ) )   &    |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B ) 
 /\  ( R `  b )  =/=  ( R `  ( s `  h ) )  /\  ( R `  b )  =/=  ( R `  g ) )  ->  ( z `  Q )  =  Y )
 )   &    |-  U  =  ( g  e.  T  |->  if (
 ( s `  h )  =  h ,  g ,  X )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  T  /\  h  =/=  (  _I  |`  B ) ) )  ->  D  e. 
 DivRing )
 
Theoremerngrng-rN 30319 An endomorphism ring is a ring. Todo: fix comment. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Ring )
 
Theoremerngdv-rN 30320 An endomorphism ring is a division ring. Todo: fix comment. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing R `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
 
Syntaxcdveca 30321 Extend class notation with constructed vector space A.
 class  DVecA
 
Definitiondf-dveca 30322* Define constructed partial vector space A. (Contributed by NM, 8-Oct-2013.)
 |-  DVecA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  k
 ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  k
 ) `  w ) ,  g  e.  (
 ( LTrn `  k ) `  w )  |->  ( f  o.  g ) )
 >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  k
 ) `  w ) >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  (
 ( TEndo `  k ) `  w ) ,  f  e.  ( ( LTrn `  k
 ) `  w )  |->  ( s `  f
 ) ) >. } )
 ) )
 
Theoremdvafset 30323* The constructed partial vector space A for a lattice  K. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 DVecA `  K )  =  ( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. ,  <. (
 +g  `  ndx ) ,  ( f  e.  (
 ( LTrn `  K ) `  w ) ,  g  e.  ( ( LTrn `  K ) `  w )  |->  ( f  o.  g ) ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `
  K ) `  w ) >. }  u.  {
 <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w ) ,  f  e.  ( (
 LTrn `  K ) `  w )  |->  ( s `
  f ) )
 >. } ) ) )
 
Theoremdvaset 30324* The constructed partial vector space A for a lattice  K. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   =>    |-  ( ( K  e.  X  /\  W  e.  H )  ->  U  =  ( { <. ( Base ` 
 ndx ) ,  T >. ,  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) >. , 
 <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f ) ) >. } ) )
 
Theoremdvasca 30325 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom  W). (Contributed by NM, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( EDRing `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  F  =  D )
 
Theoremdvabase 30326 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom  W). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  C  =  ( Base `  F )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  C  =  E )
 
Theoremdvafplusg 30327* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .+  =  ( +g  `  F )   =>    |-  (
 ( K  e.  V  /\  W  e.  H ) 
 ->  .+  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `
  f )  o.  ( t `  f
 ) ) ) ) )
 
Theoremdvaplusg 30328* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .+  =  ( +g  `  F )   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  S  e.  E ) )  ->  ( R 
 .+  S )  =  ( f  e.  T  |->  ( ( R `  f )  o.  ( S `  f ) ) ) )
 
Theoremdvaplusgv 30329 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .+  =  ( +g  `  F )   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  S  e.  E  /\  G  e.  T ) )  ->  ( ( R  .+  S ) `  G )  =  (
 ( R `  G )  o.  ( S `  G ) ) )
 
Theoremdvafmulr 30330* Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .x.  =  ( .r `  F )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  t  e.  E  |->  ( s  o.  t ) ) )
 
Theoremdvamulr 30331 Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  F  =  (Scalar `  U )   &    |-  .x.  =  ( .r `  F )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  S  e.  E )
 )  ->  ( R  .x.  S )  =  ( R  o.  S ) )
 
Theoremdvavbase 30332 The vectors (vector base set) of the constructed partial vector space A are all translations (for a fiducial co-atom  W). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  V  =  T )
 
Theoremdvafvadd 30333* The vector sum operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( K  e.  X  /\  W  e.  H ) 
 ->  .+  =  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) )
 
Theoremdvavadd 30334 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   =>    |-  (
 ( ( K  e.  V  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) )  ->  ( F 
 .+  G )  =  ( F  o.  G ) )
 
Theoremdvafvsca 30335* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  (
 s  e.  E ,  f  e.  T  |->  ( s `
  f ) ) )
 
Theoremdvavsca 30336 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .x.  =  ( .s `  U )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( R  e.  E  /\  F  e.  T )
 )  ->  ( R  .x.  F )  =  ( R `  F ) )
 
Theoremtendospid 30337 Identity property of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  ( F  e.  T  ->  ( (  _I  |`  T ) `
  F )  =  F )
 
Theoremtendospcl 30338 Closure of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T )  ->  ( U `  F )  e.  T )
 
Theoremtendospass 30339 Associative law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  X  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  F  e.  T )
 )  ->  ( ( U  o.  V ) `  F )  =  ( U `  ( V `  F ) ) )
 
Theoremtendospdi1 30340 Forward distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T )
 )  ->  ( U `  ( F  o.  G ) )  =  (
 ( U `  F )  o.  ( U `  G ) ) )
 
Theoremtendocnv 30341 Converse of a trace-preserving endomorphism value. (Contributed by NM, 7-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  F  e.  T )  ->  `' ( S `  F )  =  ( S `  `' F ) )
 
Theoremtendospdi2 30342* Reverse distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
 |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  P  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   =>    |-  ( ( U  e.  E  /\  V  e.  E  /\  F  e.  T )  ->  ( ( U P V ) `
  F )  =  ( ( U `  F )  o.  ( V `  F ) ) )
 
TheoremtendospcanN 30343* Cancellation law for trace-perserving endomorphism values (used as scalar product). (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( f  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  E  /\  S  =/=  O )  /\  ( F  e.  T  /\  G  e.  T ) )  ->  ( ( S `  F )  =  ( S `  G ) 
 <->  F  =  G ) )
 
Theoremdvaabl 30344 The constructed partial vector space A for a lattice  K is an abelian group. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Abel )
 
Theoremdvalveclem 30345 Lemma for dvalvec 30346. (Contributed by NM, 11-Oct-2013.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  D  =  (Scalar `  U )   &    |-  B  =  ( Base `  K )   &    |-  .+^  =  (
 +g  `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  .x. 
 =  ( .s `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
 
Theoremdvalvec 30346 The constructed partial vector space A for a lattice  K is a left vector space. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
 
Theoremdva0g 30347 The zero vector of partial vector space A. (Contributed by NM, 9-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  (  _I  |`  B ) )
 
Syntaxcdia 30348 Extend class notation with partial isomorphism A.
 class  DIsoA
 
Definitiondf-disoa 30349* Define partial isomorphism A. (Contributed by NM, 15-Oct-2013.)
 |-  DIsoA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  { y  e.  ( Base `  k )  |  y ( le `  k
 ) w }  |->  { f  e.  ( (
 LTrn `  k ) `  w )  |  (
 ( ( trL `  k
 ) `  w ) `  f ) ( le `  k ) x }
 ) ) )
 
Theoremdiaffval 30350* The partial isomorphism A for a lattice  K. (Contributed by NM, 15-Oct-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  ( DIsoA `  K )  =  ( w  e.  H  |->  ( x  e.  { y  e.  B  |  y  .<_  w }  |->  { f  e.  (
 ( LTrn `  K ) `  w )  |  ( ( ( trL `  K ) `  w ) `  f )  .<_  x }
 ) ) )
 
Theoremdiafval 30351* The partial isomorphism A for a lattice  K. (Contributed by NM, 15-Oct-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  {
 y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
 .<_  x } ) )
 
Theoremdiaval 30352* The partial isomorphism A for a lattice  K. Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  =  { f  e.  T  |  ( R `
  f )  .<_  X } )
 
Theoremdiaelval 30353 Member of the partial isomorphism A for a lattice  K. (Contributed by NM, 3-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( F  e.  ( I `  X )  <->  ( F  e.  T  /\  ( R `  F )  .<_  X ) ) )
 
Theoremdiafn 30354* Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
 
Theoremdiadm 30355* Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  { x  e.  B  |  x  .<_  W }
 )
 
Theoremdiaeldm 30356 Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom 
 I 
 <->  ( X  e.  B  /\  X  .<_  W ) ) )
 
TheoremdiadmclN 30357 A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  dom  I
 )  ->  X  e.  B )
 
TheoremdiadmleN 30358 A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X  .<_  W )
 
Theoremdian0 30359 The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  =/=  (/) )
 
Theoremdia0eldmN 30360 The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  dom  I )
 
Theoremdia1eldmN 30361 The fiducial hyperplane (largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  dom  I )
 
Theoremdiass 30362 The value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  C_  T )
 
Theoremdiael 30363 A member of the value of the partial isomorphism A is a translation i.e. a vector. (Contributed by NM, 17-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  F  e.  ( I `  X ) )  ->  F  e.  T )
 
Theoremdiatrl 30364 Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  F  e.  ( I `  X ) )  ->  ( R `  F ) 
 .<_  X )
 
TheoremdiaelrnN 30365 Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  S  e.  ran  I
 )  ->  S  C_  T )
 
Theoremdialss 30366 The value of partial isomorphism A is a subspace of partial vector space A. Part of Lemma M of [Crawley] p. 120 line 26. (Contributed by NM, 17-Jan-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  S  =  (
 LSubSp `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  e.  S )
 
Theoremdiaord 30367 The partial isomorphism A for a lattice  K is order-preserving in the region under co-atom  W. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( ( I `  X )  C_  ( I `
  Y )  <->  X  .<_  Y ) )
 
Theoremdia11N 30368 The partial isomorphism A for a lattice  K is one-to-one in the region under co-atom  W. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( ( I `  X )  =  ( I `  Y )  <->  X  =  Y ) )
 
Theoremdiaf11N 30369 The partial isomorphism A for a lattice  K is a one-to-one function. . Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
 
TheoremdiaclN 30370 Closure of partial isomorphism A for a lattice  K. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I
 )  ->  ( I `  X )  e.  ran  I )
 
TheoremdiacnvclN 30371 Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I
 )  ->  ( `' I `  X )  e. 
 dom  I )
 
Theoremdia0 30372 The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .0.  )  =  { (  _I  |`  B ) }
 )
 
Theoremdia1N 30373 The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  W )  =  T )
 
Theoremdia1elN 30374 The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  T  e.  ran  I )
 
TheoremdiaglbN 30375* Partial isomorphism A of a lattice glb. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
 |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  dom  I  /\  S  =/=  (/) ) )  ->  ( I `  ( G `
  S ) )  =  |^|_ x  e.  S  ( I `  x ) )
 
TheoremdiameetN 30376 Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
 |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  dom  I 
 /\  Y  e.  dom  I ) )  ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `  X )  i^i  ( I `  Y ) ) )
 
TheoremdiainN 30377 Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  ran  I 
 /\  Y  e.  ran  I ) )  ->  ( X  i^i  Y )  =  ( I `  (
 ( `' I `  X )  ./\  ( `' I `  Y ) ) ) )
 
TheoremdiaintclN 30378 The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  ran  I  /\  S  =/=  (/) ) ) 
 ->  |^| S  e.  ran  I )
 
TheoremdiasslssN 30379 The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  ran  I  C_  S )
 
TheoremdiassdvaN 30380 The partial isomorphism A maps to a set of vectors in partial vector space A. (Contributed by NM, 1-Jan-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  V  =  (
 Base `  U )   =>    |-  ( ( ( K  e.  Y  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  C_  V )
 
Theoremdia1dim 30381* Two expressions for the 1-dimensional subspaces of partial vector space A (when  F is a nonzero vector i.e. non-identity translation). Remark after Lemma L in [Crawley] p. 120 line 21. (Contributed by NM, 15-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( I `  ( R `  F ) )  =  { g  | 
 E. s  e.  E  g  =  ( s `  F ) } )
 
Theoremdia1dim2 30382 Two expressions for a 1-dimensional subspace of partial vector space A (when  F is a nonzero vector i.e. non-identity translation). (Contributed by NM, 15-Jan-2014.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  U  =  ( ( DVecA `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  N  =  ( LSpan `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  ->  ( I `  ( R `  F ) )  =  ( N `
  { F }
 ) )
 
Theoremdia1dimid 30383 A vector (translation) belongs to the 1-dim subspace it generates. (Contributed by NM, 8-Sep-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  F  e.  ( I `
  ( R `  F ) ) )
 
Theoremdia2dimlem1 30384 Lemma for dia2dim 30397. Show properties of the auxiliary atom  Q. Part of proof of Lemma M in [Crawley] p. 121 line 3. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( F  e.  T  /\  ( F `
  P )  =/= 
 P ) )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/= 
 U )   =>    |-  ( ph  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
 
Theoremdia2dimlem2 30385 Lemma for dia2dim 30397. Define a translation  G whose trace is atom  U. Part of proof of Lemma M in [Crawley] p. 121 line 4. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( F  e.  T  /\  ( F `
  P )  =/= 
 P ) )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   &    |-  ( ph  ->  G  e.  T )   &    |-  ( ph  ->  ( G `  P )  =  Q )   =>    |-  ( ph  ->  ( R `  G )  =  U )
 
Theoremdia2dimlem3 30386 Lemma for dia2dim 30397. Define a translation  D whose trace is atom  V. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( F  e.  T  /\  ( F `
  P )  =/= 
 P ) )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/= 
 U )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   &    |-  ( ph  ->  D  e.  T )   &    |-  ( ph  ->  ( D `  Q )  =  ( F `  P ) )   =>    |-  ( ph  ->  ( R `  D )  =  V )
 
Theoremdia2dimlem4 30387 Lemma for dia2dim 30397. Show that the composition (sum) of translations (vectors)  G and  D equals  F. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  G  e.  T )   &    |-  ( ph  ->  ( G `  P )  =  Q )   &    |-  ( ph  ->  D  e.  T )   &    |-  ( ph  ->  ( D `  Q )  =  ( F `  P ) )   =>    |-  ( ph  ->  ( D  o.  G )  =  F )
 
Theoremdia2dimlem5 30388 Lemma for dia2dim 30397. The sum of vectors  G and  D belongs to the sum of the subspaces generated by them. Thus  F  =  ( G  o.  D ) belongs to the subspace sum. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( F  e.  T  /\  ( F `
  P )  =/= 
 P ) )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/= 
 U )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   &    |-  ( ph  ->  G  e.  T )   &    |-  ( ph  ->  ( G `  P )  =  Q )   &    |-  ( ph  ->  D  e.  T )   &    |-  ( ph  ->  ( D `  Q )  =  ( F `  P ) )   =>    |-  ( ph  ->  F  e.  ( ( I `  U )  .(+)  ( I `
  V ) ) )
 
Theoremdia2dimlem6 30389 Lemma for dia2dim 30397. Eliminate auxiliary translations  G and  D. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  ( F  e.  T  /\  ( F `
  P )  =/= 
 P ) )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/= 
 U )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   =>    |-  ( ph  ->  F  e.  ( ( I `
  U )  .(+)  ( I `  V ) ) )
 
Theoremdia2dimlem7 30390 Lemma for dia2dim 30397. Eliminate  ( F `  P )  =/=  P condition. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  ( R `  F ) 
 .<_  ( U  .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/=  U )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   =>    |-  ( ph  ->  F  e.  ( ( I `  U )  .(+)  ( I `
  V ) ) )
 
Theoremdia2dimlem8 30391 Lemma for dia2dim 30397. Eliminate no-longer used auxiliary atoms  P and  Q. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  ( R `  F )  =/= 
 U )   &    |-  ( ph  ->  ( R `  F )  =/=  V )   =>    |-  ( ph  ->  F  e.  ( ( I `
  U )  .(+)  ( I `  V ) ) )
 
Theoremdia2dimlem9 30392 Lemma for dia2dim 30397. Eliminate  ( R `  F )  =/=  U,  V conditions. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  ( R `  F )  .<_  ( U 
 .\/  V ) )   &    |-  ( ph  ->  U  =/=  V )   =>    |-  ( ph  ->  F  e.  ( ( I `  U )  .(+)  ( I `
  V ) ) )
 
Theoremdia2dimlem10 30393 Lemma for dia2dim 30397. Convert membership in closed subspace  ( I `  ( U  .\/  V ) ) to a lattice ordering. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  F  e.  ( I `  ( U  .\/  V ) ) )   =>    |-  ( ph  ->  ( R `  F ) 
 .<_  ( U  .\/  V ) )
 
Theoremdia2dimlem11 30394 Lemma for dia2dim 30397. Convert ordering hypothesis on  R `  F to subspace membership  F  e.  ( I `
 ( U  .\/  V ) ). (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  F  e.  T )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  F  e.  ( I `  ( U  .\/  V ) ) )   =>    |-  ( ph  ->  F  e.  ( ( I `  U )  .(+)  ( I `
  V ) ) )
 
Theoremdia2dimlem12 30395 Lemma for dia2dim 30397. Obtain subset relation. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   &    |-  ( ph  ->  U  =/=  V )   =>    |-  ( ph  ->  ( I `  ( U 
 .\/  V ) )  C_  ( ( I `  U )  .(+)  ( I `
  V ) ) )
 
Theoremdia2dimlem13 30396 Lemma for dia2dim 30397. Eliminate  U  =/=  V condition. (Contributed by NM, 8-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  S  =  ( LSubSp `  Y )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  N  =  ( LSpan `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( U  .\/  V ) )  C_  (
 ( I `  U )  .(+)  ( I `  V ) ) )
 
Theoremdia2dim 30397 A two-dimensional subspace of partial vector space A is closed, or equivalently, the isomorphism of a join of two atoms is a subset of the subspace sum of the isomorphisms of each atom (and thus they are equal, as shown later for the full vector space H). (Contributed by NM, 9-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  Y  =  ( ( DVecA `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  Y )   &    |-  I  =  ( ( DIsoA `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )   &    |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( U  .\/  V ) )  C_  (
 ( I `  U )  .(+)  ( I `  V ) ) )
 
Syntaxcdvh 30398 Extend class notation with constructed full vector space H.
 class  DVecH
 
Definitiondf-dvech 30399* Define constructed full vector space H. (Contributed by NM, 17-Oct-2013.)
 |-  DVecH  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( { <. ( Base `  ndx ) ,  ( ( ( LTrn `  k ) `  w )  X.  ( ( TEndo `  k ) `  w ) ) >. ,  <. (
 +g  `  ndx ) ,  ( f  e.  (
 ( ( LTrn `  k
 ) `  w )  X.  ( ( TEndo `  k
 ) `  w )
 ) ,  g  e.  ( ( ( LTrn `  k ) `  w )  X.  ( ( TEndo `  k ) `  w ) )  |->  <. ( ( 1st `  f )  o.  ( 1st `  g
 ) ) ,  ( h  e.  ( ( LTrn `  k ) `  w )  |->  ( ( ( 2nd `  f
 ) `  h )  o.  ( ( 2nd `  g
 ) `  h )
 ) ) >. ) >. , 
 <. (Scalar `  ndx ) ,  ( ( EDRing `  k
 ) `  w ) >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  (
 ( TEndo `  k ) `  w ) ,  f  e.  ( ( ( LTrn `  k ) `  w )  X.  ( ( TEndo `  k ) `  w ) )  |->  <. ( s `
  ( 1st `  f
 ) ) ,  (
 s  o.  ( 2nd `  f ) ) >. )
 >. } ) ) )
 
Theoremdvhfset 30400* The constructed full vector space H for a lattice  K. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 DVecH `  K )  =  ( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( (
 ( LTrn `  K ) `  w )  X.  (
 ( TEndo `  K ) `  w ) ) >. , 
 <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  K ) `  w )  X.  ( ( TEndo `  K ) `  w ) ) ,  g  e.  ( ( ( LTrn `  K ) `  w )  X.  ( ( TEndo `  K ) `  w ) )  |->  <. ( ( 1st `  f )  o.  ( 1st `  g
 ) ) ,  ( h  e.  ( ( LTrn `  K ) `  w )  |->  ( ( ( 2nd `  f
 ) `  h )  o.  ( ( 2nd `  g
 ) `  h )
 ) ) >. ) >. , 
 <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. }  u.  { <. ( .s
 `  ndx ) ,  (
 s  e.  ( (
 TEndo `  K ) `  w ) ,  f  e.  ( ( ( LTrn `  K ) `  w )  X.  ( ( TEndo `  K ) `  w ) )  |->  <. ( s `
  ( 1st `  f
 ) ) ,  (
 s  o.  ( 2nd `  f ) ) >. )
 >. } ) ) )
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