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Theorem List for Metamath Proof Explorer - 32601-32700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmapdh8 32601* Part (8) in [Baer] p. 48. Given a reference vector  X, the value of function  I at a vector  T is independent of the choice of auxiliary vectors  Y and  Z. Unlike Baer's, our version does not require  X,  Y, and  Z to be independent, and also is defined for all  Y and  Z that are not colinear with  X or  T. We do this to make the definition of Baer's sigma function more straightforward. (This part eliminates  T  =/=  .0..) (Contributed by NM, 13-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( x  e.  _V  |->  if ( ( 2nd `  x )  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `  { ( 2nd `  x ) }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) }
 ) )  =  ( J `  { (
 ( 2nd `  ( 1st `  x ) ) R h ) } )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z }
 ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { T }
 ) )   &    |-  ( ph  ->  ( N `  { Z } )  =/=  ( N `  { T }
 ) )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  ( I `  <. Y ,  ( I `  <. X ,  F ,  Y >. ) ,  T >. )  =  ( I `  <. Z ,  ( I `  <. X ,  F ,  Z >. ) ,  T >. ) )
 
Theoremmapdh9a 32602* Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 32603? (Contributed by NM, 14-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( x  e.  _V  |->  if ( ( 2nd `  x )  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `  { ( 2nd `  x ) }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) }
 ) )  =  ( J `  { (
 ( 2nd `  ( 1st `  x ) ) R h ) } )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  E! y  e.  D  A. z  e.  V  ( -.  z  e.  (
 ( N `  { X } )  u.  ( N `  { T }
 ) )  ->  y  =  ( I `  <. z ,  ( I `  <. X ,  F ,  z >. ) ,  T >. ) ) )
 
Theoremmapdh9aOLDN 32603* Lemma for part (9) in [Baer] p. 48. (Contributed by NM, 14-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( x  e.  _V  |->  if ( ( 2nd `  x )  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `  { ( 2nd `  x ) }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) }
 ) )  =  ( J `  { (
 ( 2nd `  ( 1st `  x ) ) R h ) } )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  E! y  e.  D  A. z  e.  V  ( -.  z  e.  ( N `  { X ,  T } )  ->  y  =  ( I `  <. z ,  ( I `  <. X ,  F ,  z >. ) ,  T >. ) ) )
 
Syntaxchdma1 32604 Extend class notation with preliminary map from vectors to functionals in the closed kernel dual space.
 class HDMap1
 
Syntaxchdma 32605 Extend class notation with map from vectors to functionals in the closed kernel dual space.
 class HDMap
 
Definitiondf-hdmap1 32606* Define preliminary map from vectors to functionals in the closed kernel dual space. See hdmap1fval 32609 description for more details. (Contributed by NM, 14-May-2015.)
 |- HDMap1  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { a  |  [. ( ( DVecH `  k ) `  w )  /  u ]. [. ( Base `  u )  /  v ]. [. ( LSpan `  u )  /  n ].
 [. ( (LCDual `  k
 ) `  w )  /  c ]. [. ( Base `  c )  /  d ]. [. ( LSpan `  c )  /  j ]. [. ( (mapd `  k ) `  w )  /  m ]. a  e.  ( x  e.  (
 ( v  X.  d
 )  X.  v )  |->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  ( iota_ h  e.  d (
 ( m `  ( n `  { ( 2nd `  x ) } )
 )  =  ( j `
  { h }
 )  /\  ( m `  ( n `  { (
 ( 1st `  ( 1st `  x ) ) (
 -g `  u )
 ( 2nd `  x )
 ) } ) )  =  ( j `  { ( ( 2nd `  ( 1st `  x ) ) ( -g `  c ) h ) } ) ) ) ) ) } )
 )
 
Definitiondf-hdmap 32607* Define map from vectors to functionals in the closed kernel dual space. See hdmapfval 32642 description for more details. (Contributed by NM, 15-May-2015.)
 |- HDMap  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { a  |  [. <. (  _I  |`  ( Base `  k ) ) ,  (  _I  |`  ( (
 LTrn `  k ) `  w ) ) >.  /  e ]. [. (
 ( DVecH `  k ) `  w )  /  u ].
 [. ( Base `  u )  /  v ]. [. (
 (HDMap1 `  k ) `  w )  /  i ]. a  e.  (
 t  e.  v  |->  (
 iota_ y  e.  ( Base `  ( (LCDual `  k
 ) `  w )
 ) A. z  e.  v  ( -.  z  e.  (
 ( ( LSpan `  u ) `  { e }
 )  u.  ( (
 LSpan `  u ) `  { t } )
 )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  k ) `  w ) `  e ) ,  z >. ) ,  t >. ) ) ) ) } ) )
 
Theoremhdmap1ffval 32608* Preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 14-May-2015.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  X  ->  (HDMap1 `  K )  =  ( w  e.  H  |->  { a  |  [. (
 ( DVecH `  K ) `  w )  /  u ].
 [. ( Base `  u )  /  v ]. [. ( LSpan `  u )  /  n ]. [. ( (LCDual `  K ) `  w )  /  c ]. [. ( Base `  c )  /  d ]. [. ( LSpan `  c )  /  j ]. [. ( (mapd `  K ) `  w )  /  m ]. a  e.  ( x  e.  (
 ( v  X.  d
 )  X.  v )  |->  if ( ( 2nd `  x )  =  ( 0g `  u ) ,  ( 0g `  c ) ,  ( iota_ h  e.  d (
 ( m `  ( n `  { ( 2nd `  x ) } )
 )  =  ( j `
  { h }
 )  /\  ( m `  ( n `  { (
 ( 1st `  ( 1st `  x ) ) (
 -g `  u )
 ( 2nd `  x )
 ) } ) )  =  ( j `  { ( ( 2nd `  ( 1st `  x ) ) ( -g `  c ) h ) } ) ) ) ) ) } )
 )
 
Theoremhdmap1fval 32609* Preliminary map from vectors to functionals in the closed kernel dual space. TODO: change span  J to the convention  L for this section. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  A  /\  W  e.  H ) )   =>    |-  ( ph  ->  I  =  ( x  e.  (
 ( V  X.  D )  X.  V )  |->  if ( ( 2nd `  x )  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `  { ( 2nd `  x ) }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) }
 ) )  =  ( J `  { (
 ( 2nd `  ( 1st `  x ) ) R h ) } )
 ) ) ) ) )
 
Theoremhdmap1vallem 32610* Value of preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  A  /\  W  e.  H ) )   &    |-  ( ph  ->  T  e.  ( ( V  X.  D )  X.  V ) )   =>    |-  ( ph  ->  ( I `  T )  =  if ( ( 2nd `  T )  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `  { ( 2nd `  T ) }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( ( 1st `  ( 1st `  T ) )  .-  ( 2nd `  T ) ) }
 ) )  =  ( J `  { (
 ( 2nd `  ( 1st `  T ) ) R h ) } )
 ) ) ) )
 
Theoremhdmap1val 32611* Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 32536.) TODO: change  I  =  ( x  e.  _V  |->... to  ( ph  ->  ( I `  <. X ,  F ,  Y  >  )  =... in e.g. mapdh8 32601 to shorten proofs with no $d on  x. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  A  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  if ( Y  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( F R h ) }
 ) ) ) ) )
 
Theoremhdmap1val0 32612 Value of preliminary map from vectors to functionals at zero. (Restated mapdhval0 32537.) (Contributed by NM, 17-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  Q  =  ( 0g `  C )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >. )  =  Q )
 
Theoremhdmap1val2 32613* Value of preliminary map from vectors to functionals in the closed kernel dual space, for nonzero  Y. (Contributed by NM, 16-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  (
 iota_ h  e.  D ( ( M `  ( N `  { Y } ) )  =  ( L `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( L `
  { ( F R h ) }
 ) ) ) )
 
Theoremhdmap1eq 32614 The defining equation for h(x,x',y)=y' in part (2) in [Baer] p. 45 line 24. (Contributed by NM, 16-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  G  e.  D )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )   =>    |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <->  ( ( M `  ( N `  { Y }
 ) )  =  ( L `  { G } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( L `
  { ( F R G ) }
 ) ) ) )
 
Theoremhdmap1cbv 32615* Frequently used lemma to change bound variables in  L hypothesis. (Contributed by NM, 15-May-2015.)
 |-  L  =  ( x  e.  _V  |->  if ( ( 2nd `  x )  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `  { ( 2nd `  x ) }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) }
 ) )  =  ( J `  { (
 ( 2nd `  ( 1st `  x ) ) R h ) } )
 ) ) ) )   =>    |-  L  =  ( y  e.  _V  |->  if ( ( 2nd `  y )  =  .0. 
 ,  Q ,  ( iota_
 i  e.  D ( ( M `  ( N `  { ( 2nd `  y ) } )
 )  =  ( J `
  { i }
 )  /\  ( M `  ( N `  { (
 ( 1st `  ( 1st `  y ) )  .-  ( 2nd `  y )
 ) } ) )  =  ( J `  { ( ( 2nd `  ( 1st `  y
 ) ) R i ) } ) ) ) ) )
 
Theoremhdmap1valc 32616* Connect the value of the preliminary map from vectors to functionals  I to the hypothesis  L used by earlier theorems. Note: the  X  e.  ( V  \  {  .0.  } ) hypothesis could be the more general  X  e.  V but the former will be easier to use. TODO: use the  I function directly in those theorems, so this theorem becomes unnecessary? TODO: The hdmap1cbv 32615 is probably unnecessary, but it would mean different $d's later on. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  Y  e.  V )   &    |-  L  =  ( x  e.  _V  |->  if ( ( 2nd `  x )  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `  { ( 2nd `  x ) }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) }
 ) )  =  ( J `  { (
 ( 2nd `  ( 1st `  x ) ) R h ) } )
 ) ) ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  ( L `  <. X ,  F ,  Y >. ) )
 
Theoremhdmap1cl 32617 Convert closure theorem mapdhcl 32539 to use HDMap1 function. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
 
Theoremhdmap1eq2 32618 Convert mapdheq2 32541 to use HDMap1 function. (Contributed by NM, 16-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )   =>    |-  ( ph  ->  ( I `  <. Y ,  G ,  X >. )  =  F )
 
Theoremhdmap1eq4N 32619 Convert mapdheq4 32544 to use HDMap1 function. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z }
 ) )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  B )   =>    |-  ( ph  ->  ( I `  <. Y ,  G ,  Z >. )  =  B )
 
Theoremhdmap1l6lem1 32620 Lemma for hdmap1l6 32634. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N ` 
 { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )   =>    |-  ( ph  ->  ( M `  ( N `  { ( X  .-  ( Y  .+  Z ) ) } ) )  =  ( L `  { ( F R ( G  .+b  E ) ) } ) )
 
Theoremhdmap1l6lem2 32621 Lemma for hdmap1l6 32634. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N ` 
 { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )   =>    |-  ( ph  ->  ( M `  ( N `  { ( Y  .+  Z ) } )
 )  =  ( L `
  { ( G 
 .+b  E ) } )
 )
 
Theoremhdmap1l6a 32622 Lemma for hdmap1l6 32634. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N ` 
 { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `  <. X ,  F ,  Y >. ) 
 .+b  ( I `  <. X ,  F ,  Z >. ) ) )
 
Theoremhdmap1l6b0N 32623 Lemmma for hdmap1l6 32634. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( ( N `  { X } )  i^i  ( N `
  { Y ,  Z } ) )  =  {  .0.  } )   =>    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
 
Theoremhdmap1l6b 32624 Lemmma for hdmap1l6 32634. (Contributed by NM, 24-Apr-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  Y  =  .0.  )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `  <. X ,  F ,  Y >. ) 
 .+b  ( I `  <. X ,  F ,  Z >. ) ) )
 
Theoremhdmap1l6c 32625 Lemmma for hdmap1l6 32634. (Contributed by NM, 24-Apr-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  =  .0.  )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `  <. X ,  F ,  Y >. )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )
 
Theoremhdmap1l6d 32626 Lemmma for hdmap1l6 32634. (Contributed by NM, 1-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N ` 
 { Y } )  =  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { X ,  Y } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y  .+  Z ) ) >. )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. ) ) )
 
Theoremhdmap1l6e 32627 Lemmma for hdmap1l6 32634. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N ` 
 { Y } )  =  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { X ,  Y } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( ( w  .+  Y )  .+  Z ) >. )  =  ( ( I `  <. X ,  F ,  ( w  .+  Y )
 >. )  .+b  ( I `
  <. X ,  F ,  Z >. ) ) )
 
Theoremhdmap1l6f 32628 Lemmma for hdmap1l6 32634. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N ` 
 { Y } )  =  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { X ,  Y } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( w  .+  Y ) >. )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  Y >. ) ) )
 
Theoremhdmap1l6g 32629 Lemmma for hdmap1l6 32634. Part (6) of [Baer] p. 47 line 39. (Contributed by NM, 1-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N ` 
 { Y } )  =  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { X ,  Y } ) )   =>    |-  ( ph  ->  ( ( I `  <. X ,  F ,  w >. ) 
 .+b  ( I `  <. X ,  F ,  ( Y  .+  Z )
 >. ) )  =  ( ( ( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  Y >. ) )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )
 
Theoremhdmap1l6h 32630 Lemmma for hdmap1l6 32634. Part (6) of [Baer] p. 48 line 2. (Contributed by NM, 1-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N ` 
 { Y } )  =  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { X ,  Y } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `  <. X ,  F ,  Y >. )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )
 
Theoremhdmap1l6i 32631 Lemmma for hdmap1l6 32634. Eliminate auxiliary vector  w. (Contributed by NM, 1-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N ` 
 { Y } )  =  ( N `  { Z } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `  <. X ,  F ,  Y >. )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )
 
Theoremhdmap1l6j 32632 Lemmma for hdmap1l6 32634. Eliminate  ( N { Y } ) = ( N  { Z } ) hypothesis. (Contributed by NM, 1-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `  <. X ,  F ,  Y >. ) 
 .+b  ( I `  <. X ,  F ,  Z >. ) ) )
 
Theoremhdmap1l6k 32633 Lemmma for hdmap1l6 32634. Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `  <. X ,  F ,  Y >. )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )
 
Theoremhdmap1l6 32634 Part (6) of [Baer] p. 47 line 6. Note that we use  -.  X  e.  ( N `  { Y ,  Z }
) which is equivalent to Baer's "Fx  i^i (Fy + Fz)" by lspdisjb 15895. (Contributed by NM, 17-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `  <. X ,  F ,  Y >. )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )
 
Theoremhdmap1p6N 32635 (Convert mapdh6N 32559 to use HDMap1 function.) Part (6) of [Baer] p. 47 line 6. Note that we use  -.  X  e.  ( N `  { Y ,  Z } ) which is equivalent to Baer's "Fx  i^i (Fy + Fz)" by lspdisjb 15895. TODO: No longer used and should be deleted. Use hdmap1l6 32634 instead. Also delete unused mapdh6N 32559 etc. leading up to this. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  .+b  =  ( +g  `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `  <. X ,  F ,  Y >. )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )
 
Theoremhdmap1eulem 32636* Lemma for hdmap1eu 32638. TODO: combine with hdmap1eu 32638 or at least share some hypotheses. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  T  e.  V )   &    |-  L  =  ( x  e.  _V  |->  if (
 ( 2nd `  x )  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `  { ( 2nd `  x ) }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) }
 ) )  =  ( J `  { (
 ( 2nd `  ( 1st `  x ) ) R h ) } )
 ) ) ) )   =>    |-  ( ph  ->  E! y  e.  D  A. z  e.  V  ( -.  z  e.  ( ( N `  { X } )  u.  ( N `  { T } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. X ,  F ,  z >. ) ,  T >. ) ) )
 
Theoremhdmap1eulemOLDN 32637* Lemma for hdmap1euOLDN 32639. TODO: combine with hdmap1euOLDN 32639 or at least share some hypotheses. (Contributed by NM, 15-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  Q  =  ( 0g
 `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  T  e.  V )   &    |-  L  =  ( x  e.  _V  |->  if (
 ( 2nd `  x )  =  .0.  ,  Q ,  ( iota_ h  e.  D ( ( M `  ( N `  { ( 2nd `  x ) }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( ( 1st `  ( 1st `  x ) )  .-  ( 2nd `  x ) ) }
 ) )  =  ( J `  { (
 ( 2nd `  ( 1st `  x ) ) R h ) } )
 ) ) ) )   =>    |-  ( ph  ->  E! y  e.  D  A. z  e.  V  ( -.  z  e.  ( N `  { X ,  T } )  ->  y  =  ( I `  <. z ,  ( I `  <. X ,  F ,  z >. ) ,  T >. ) ) )
 
Theoremhdmap1eu 32638* Convert mapdh9a 32602 to use the HDMap1 notation. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  E! y  e.  D  A. z  e.  V  ( -.  z  e.  ( ( N `  { X } )  u.  ( N `  { T } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. X ,  F ,  z >. ) ,  T >. ) ) )
 
Theoremhdmap1euOLDN 32639* Convert mapdh9aOLDN 32603 to use the HDMap1 notation. (Contributed by NM, 15-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  E! y  e.  D  A. z  e.  V  ( -.  z  e.  ( N `  { X ,  T } )  ->  y  =  ( I `  <. z ,  ( I `  <. X ,  F ,  z >. ) ,  T >. ) ) )
 
Theoremhdmap1neglem1N 32640 Lemma for hdmapneg 32661. TODO: Not used; delete. (Contributed by NM, 23-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  ( inv g `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  S  =  ( inv g `  C )   &    |-  L  =  (
 LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )   =>    |-  ( ph  ->  ( I `  <. ( R `  X ) ,  ( S `  F ) ,  ( R `  Y ) >. )  =  ( S `  G ) )
 
Theoremhdmapffval 32641* Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  X  ->  (HDMap `  K )  =  ( w  e.  H  |->  { a  |  [. <. (  _I  |`  ( Base `  K )
 ) ,  (  _I  |`  ( ( LTrn `  K ) `  w ) )
 >.  /  e ]. [. (
 ( DVecH `  K ) `  w )  /  u ].
 [. ( Base `  u )  /  v ]. [. (
 (HDMap1 `  K ) `  w )  /  i ]. a  e.  (
 t  e.  v  |->  (
 iota_ y  e.  ( Base `  ( (LCDual `  K ) `  w ) )
 A. z  e.  v  ( -.  z  e.  (
 ( ( LSpan `  u ) `  { e }
 )  u.  ( (
 LSpan `  u ) `  { t } )
 )  ->  y  =  ( i `  <. z ,  ( i `  <. e ,  ( ( (HVMap `  K ) `  w ) `  e ) ,  z >. ) ,  t >. ) ) ) ) } ) )
 
Theoremhdmapfval 32642* Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  A  /\  W  e.  H ) )   =>    |-  ( ph  ->  S  =  ( t  e.  V  |->  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  (
 ( N `  { E } )  u.  ( N `  { t }
 ) )  ->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `  E ) ,  z >. ) ,  t >. ) ) ) ) )
 
Theoremhdmapval 32643* Value of map from vectors to functionals in the closed kernel dual space. This is the function sigma on line 27 above part 9 in [Baer] p. 48. We select a convenient fixed reference vector  E to be  <. 0 ,  1 >. (corresponding to vector u on p. 48 line 7) whose span is the lattice isomorphism map of the fiducial atom  P  =  ( ( oc `  K
) `  W ) (see dvheveccl 31924). 
( J `  E
) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 32581 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our  z that the  A. z  e.  V ranges over. The middle term  ( I `  <. E ,  ( J `
 E ) ,  z >. ) provides isolation to allow  E and  T to assume the same value without conflict. Closure is shown by hdmapcl 32645. If a separate auxiliary vector is known, hdmapval2 32647 provides a version without quantification. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  A  /\  W  e.  H ) )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  ( S `  T )  =  ( iota_ y  e.  D A. z  e.  V  ( -.  z  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  ->  y  =  ( I `  <. z ,  ( I `  <. E ,  ( J `  E ) ,  z >. ) ,  T >. ) ) ) )
 
TheoremhdmapfnN 32644 Functionality of map from vectors to functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  S  Fn  V )
 
Theoremhdmapcl 32645 Closure of map from vectors to functionals with closed kernels. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  ( S `  T )  e.  D )
 
Theoremhdmapval2lem 32646* Lemma for hdmapval2 32647. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  T  e.  V )   &    |-  ( ph  ->  F  e.  D )   =>    |-  ( ph  ->  (
 ( S `  T )  =  F  <->  A. z  e.  V  ( -.  z  e.  (
 ( N `  { E } )  u.  ( N `  { T }
 ) )  ->  F  =  ( I `  <. z ,  ( I `  <. E ,  ( J `  E ) ,  z >. ) ,  T >. ) ) ) )
 
Theoremhdmapval2 32647 Value of map from vectors to functionals with a specific auxiliary vector. TODO: Would shorter proofs result if the .ne hypothesis were changed to two  =/= hypothesis? Consider hdmaplem1 32583 through hdmaplem4 32586, which would become obsolete. (Contributed by NM, 15-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  T  e.  V )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( ( N `  { E } )  u.  ( N `  { T } ) ) )   =>    |-  ( ph  ->  ( S `  T )  =  ( I `  <. X ,  ( I `  <. E ,  ( J `  E ) ,  X >. ) ,  T >. ) )
 
Theoremhdmapval0 32648 Value of map from vectors to functionals at zero. Note: we use dvh3dim 32258 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 32659 could be derived from the other to shorten proof. (Contributed by NM, 17-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  Q  =  ( 0g
 `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ( S `  .0.  )  =  Q )
 
Theoremhdmapeveclem 32649 Lemma for hdmapevec 32650. TODO: combine with hdmapevec 32650 if it shortens overall. (Contributed by NM, 16-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( ( N `  { E } )  u.  ( N `  { E } ) ) )   =>    |-  ( ph  ->  ( S `  E )  =  ( J `  E ) )
 
Theoremhdmapevec 32650 Value of map from vectors to functionals at the reference vector  E. (Contributed by NM, 16-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ( S `  E )  =  ( J `  E ) )
 
Theoremhdmapevec2 32651 The inner product of the reference vector  E with itself is nonzero. This shows the inner product condition in the proof of Theorem 3.6 of [Holland95] p. 14 line 32,  [ e , e  ]  =/=  0 is satisfied. TODO: remove redundant hypothesis hdmapevec.j. (Contributed by NM, 1-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ph  ->  ( ( S `  E ) `  E )  =  .1.  )
 
Theoremhdmapval3lemN 32652 Value of map from vectors to functionals at arguments not colinear with the reference vector 
E. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( N `  { T } )  =/=  ( N `  { E }
 ) )   &    |-  ( ph  ->  T  e.  ( V  \  { ( 0g `  U ) } )
 )   &    |-  ( ph  ->  x  e.  V )   &    |-  ( ph  ->  -.  x  e.  ( N `
  { E ,  T } ) )   =>    |-  ( ph  ->  ( S `  T )  =  ( I `  <. E ,  ( J `
  E ) ,  T >. ) )
 
Theoremhdmapval3N 32653 Value of map from vectors to functionals at arguments not colinear with the reference vector  E. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( N `  { T } )  =/=  ( N `  { E }
 ) )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  ( S `  T )  =  ( I `  <. E ,  ( J `
  E ) ,  T >. ) )
 
Theoremhdmap10lem 32654 Lemma for hdmap10 32655. (Contributed by NM, 17-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  .0.  =  ( 0g `  U )   &    |-  D  =  ( Base `  C )   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  T  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( M `  ( N `  { T } ) )  =  ( L `  { ( S `  T ) }
 ) )
 
Theoremhdmap10 32655 Part 10 in [Baer] p. 48 line 33, (Ft)* = G(tS) in their notation (S = sigma). (Contributed by NM, 17-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  ( M `  ( N `
  { T }
 ) )  =  ( L `  { ( S `  T ) }
 ) )
 
Theoremhdmap11lem1 32656 Lemma for hdmapadd 32658. (Contributed by NM, 26-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .+b  =  ( +g  `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  E  =  <. (  _I  |`  ( Base `  K )
 ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) )
 >.   &    |- 
 .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  D  =  ( Base `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   &    |-  ( ph  ->  z  e.  V )   &    |-  ( ph  ->  -.  z  e.  ( N `  { X ,  Y } ) )   &    |-  ( ph  ->  ( N ` 
 { z } )  =/=  ( N `  { E } ) )   =>    |-  ( ph  ->  ( S `  ( X 
 .+  Y ) )  =  ( ( S `
  X )  .+b  ( S `  Y ) ) )
 
Theoremhdmap11lem2 32657 Lemma for hdmapadd 32658. (Contributed by NM, 26-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .+b  =  ( +g  `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  E  =  <. (  _I  |`  ( Base `  K )
 ) ,  (  _I  |`  ( ( LTrn `  K ) `  W ) )
 >.   &    |- 
 .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  D  =  ( Base `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  J  =  ( (HVMap `  K ) `  W )   &    |-  I  =  ( (HDMap1 `  K ) `  W )   =>    |-  ( ph  ->  ( S `  ( X 
 .+  Y ) )  =  ( ( S `
  X )  .+b  ( S `  Y ) ) )
 
Theoremhdmapadd 32658 Part 11 in [Baer] p. 48 line 35, (a+b)S = aS+bS in their notation (S = sigma). (Contributed by NM, 22-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .+b  =  ( +g  `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( S `  ( X  .+  Y ) )  =  ( ( S `  X )  .+b  ( S `
  Y ) ) )
 
Theoremhdmapeq0 32659 Part of proof of part 12 in [Baer] p. 49 line 3. (Contributed by NM, 22-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  Q  =  ( 0g
 `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  ( ( S `  T )  =  Q  <->  T  =  .0.  ) )
 
Theoremhdmapnzcl 32660 Nonzero vector closure of map from vectors to functionals with closed kernels. (Contributed by NM, 27-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  Q  =  ( 0g
 `  C )   &    |-  D  =  ( Base `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  T  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( S `  T )  e.  ( D  \  { Q }
 ) )
 
Theoremhdmapneg 32661 Part of proof of part 12 in [Baer] p. 49 line 4. The sigma map of a negative is the negative of the sigma map. (Contributed by NM, 24-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  M  =  ( inv g `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  I  =  ( inv g `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  ( S `  ( M `  T ) )  =  ( I `  ( S `  T ) ) )
 
Theoremhdmapsub 32662 Part of proof of part 12 in [Baer] p. 49 line 5, (a-b)S = aS-bS in their notation (S = sigma). (Contributed by NM, 26-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  N  =  ( -g `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( S `  ( X 
 .-  Y ) )  =  ( ( S `
  X ) N ( S `  Y ) ) )
 
Theoremhdmap11 32663 Part of proof of part 12 in [Baer] p. 49 line 4, aS=bS iff a=b in their notation (S = sigma). The sigma map is one-to-one. (Contributed by NM, 26-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( S `  X )  =  ( S `  Y )  <->  X  =  Y ) )
 
Theoremhdmaprnlem1N 32664 Part of proof of part 12 in [Baer] p. 49 line 10, Gu'  =/= Gs. Our  ( N `  { v } ) is Baer's T. (Contributed by NM, 26-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  s  e.  ( D  \  { Q } ) )   &    |-  ( ph  ->  v  e.  V )   &    |-  ( ph  ->  ( M `  ( N `
  { v }
 ) )  =  ( L `  { s } ) )   &    |-  ( ph  ->  u  e.  V )   &    |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )   =>    |-  ( ph  ->  ( L `  { ( S `  u ) }
 )  =/=  ( L ` 
 { s } )
 )
 
Theoremhdmaprnlem3N 32665 Part of proof of part 12 in [Baer] p. 49 line 15, T  =/= P. Our  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) ) is Baer's P, where P* = G(u'+s). (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  s  e.  ( D  \  { Q } ) )   &    |-  ( ph  ->  v  e.  V )   &    |-  ( ph  ->  ( M `  ( N `
  { v }
 ) )  =  ( L `  { s } ) )   &    |-  ( ph  ->  u  e.  V )   &    |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )   &    |-  D  =  ( Base `  C )   &    |-  Q  =  ( 0g `  C )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .+b  =  ( +g  `  C )   =>    |-  ( ph  ->  ( N `  { v } )  =/=  ( `' M `  ( L `
  { ( ( S `  u ) 
 .+b  s ) }
 ) ) )
 
Theoremhdmaprnlem3uN 32666 Part of proof of part 12 in [Baer] p. 49. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  s  e.  ( D  \  { Q } ) )   &    |-  ( ph  ->  v  e.  V )   &    |-  ( ph  ->  ( M `  ( N `
  { v }
 ) )  =  ( L `  { s } ) )   &    |-  ( ph  ->  u  e.  V )   &    |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )   &    |-  D  =  ( Base `  C )   &    |-  Q  =  ( 0g `  C )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .+b  =  ( +g  `  C )   =>    |-  ( ph  ->  ( N `  { u } )  =/=  ( `' M `  ( L `
  { ( ( S `  u ) 
 .+b  s ) }
 ) ) )
 
Theoremhdmaprnlem4tN 32667 Lemma for hdmaprnN 32679. TODO: This lemma doesn't quite pay for itself even though used 4 times. Maybe prove this directly instead. (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  s  e.  ( D  \  { Q } ) )   &    |-  ( ph  ->  v  e.  V )   &    |-  ( ph  ->  ( M `  ( N `
  { v }
 ) )  =  ( L `  { s } ) )   &    |-  ( ph  ->  u  e.  V )   &    |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )   &    |-  D  =  ( Base `  C )   &    |-  Q  =  ( 0g `  C )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( ph  ->  t  e.  (
 ( N `  { v } )  \  {  .0.  } ) )   =>    |-  ( ph  ->  t  e.  V )
 
Theoremhdmaprnlem4N 32668 Part of proof of part 12 in [Baer] p. 49 line 19. (T* =) (Ft)* = Gs. (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  s  e.  ( D  \  { Q } ) )   &    |-  ( ph  ->  v  e.  V )   &    |-  ( ph  ->  ( M `  ( N `
  { v }
 ) )  =  ( L `  { s } ) )   &    |-  ( ph  ->  u  e.