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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembaerlem3lem2 30701 Lemma for baerlem3 30704. (Contributed by NM, 9-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  .+  =  ( +g  `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  R  =  (Scalar `  W )   &    |-  B  =  (
 Base `  R )   &    |-  .+^  =  (
 +g  `  R )   &    |-  L  =  ( -g `  R )   &    |-  Q  =  ( 0g
 `  R )   &    |-  I  =  ( inv g `  R )   =>    |-  ( ph  ->  ( N `  { ( Y 
 .-  Z ) }
 )  =  ( ( ( N `  { Y } )  .(+)  ( N `
  { Z }
 ) )  i^i  (
 ( N `  { ( X  .-  Y ) }
 )  .(+)  ( N `  { ( X  .-  Z ) } )
 ) ) )
 
Theorembaerlem5alem2 30702 Lemma for baerlem5a 30705. (Contributed by NM, 9-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  .+  =  ( +g  `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  R  =  (Scalar `  W )   &    |-  B  =  (
 Base `  R )   &    |-  .+^  =  (
 +g  `  R )   &    |-  L  =  ( -g `  R )   &    |-  Q  =  ( 0g
 `  R )   &    |-  I  =  ( inv g `  R )   =>    |-  ( ph  ->  ( N `  { ( X 
 .-  ( Y  .+  Z ) ) }
 )  =  ( ( ( N `  { ( X  .-  Y ) }
 )  .(+)  ( N `  { Z } ) )  i^i  ( ( N `
  { ( X 
 .-  Z ) }
 )  .(+)  ( N `  { Y } ) ) ) )
 
Theorembaerlem5blem2 30703 Lemma for baerlem5b 30706. (Contributed by NM, 13-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  .+  =  ( +g  `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  R  =  (Scalar `  W )   &    |-  B  =  (
 Base `  R )   &    |-  .+^  =  (
 +g  `  R )   &    |-  L  =  ( -g `  R )   &    |-  Q  =  ( 0g
 `  R )   &    |-  I  =  ( inv g `  R )   =>    |-  ( ph  ->  ( N `  { ( Y 
 .+  Z ) }
 )  =  ( ( ( N `  { Y } )  .(+)  ( N `
  { Z }
 ) )  i^i  (
 ( N `  { ( X  .-  ( Y  .+  Z ) ) }
 )  .(+)  ( N `  { X } ) ) ) )
 
Theorembaerlem3 30704 An equality that holds when  X,  Y,  Z are independent (non-colinear) vectors. Part (3) in [Baer] p. 45. TODO fix ref. (Contributed by NM, 9-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( N ` 
 { ( Y  .-  Z ) } )  =  ( ( ( N `
  { Y }
 )  .(+)  ( N `  { Z } ) )  i^i  ( ( N `
  { ( X 
 .-  Y ) }
 )  .(+)  ( N `  { ( X  .-  Z ) } )
 ) ) )
 
Theorembaerlem5a 30705 An equality that holds when  X,  Y,  Z are independent (non-colinear) vectors. First equation of part (5) in [Baer] p. 46. (Contributed by NM, 10-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ph  ->  ( N `  { ( X 
 .-  ( Y  .+  Z ) ) }
 )  =  ( ( ( N `  { ( X  .-  Y ) }
 )  .(+)  ( N `  { Z } ) )  i^i  ( ( N `
  { ( X 
 .-  Z ) }
 )  .(+)  ( N `  { Y } ) ) ) )
 
Theorembaerlem5b 30706 An equality that holds when  X,  Y,  Z are independent (non-colinear) vectors. Second equation of part (5) in [Baer] p. 46. (Contributed by NM, 13-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ph  ->  ( N `  { ( Y 
 .+  Z ) }
 )  =  ( ( ( N `  { Y } )  .(+)  ( N `
  { Z }
 ) )  i^i  (
 ( N `  { ( X  .-  ( Y  .+  Z ) ) }
 )  .(+)  ( N `  { X } ) ) ) )
 
Theorembaerlem5amN 30707 An equality that holds when  X,  Y,  Z are independent (non-colinear) vectors. Subtraction version of first equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 30709 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ph  ->  ( N `  { ( X 
 .-  ( Y  .-  Z ) ) }
 )  =  ( ( ( N `  { ( X  .-  Y ) }
 )  .(+)  ( N `  { Z } ) )  i^i  ( ( N `
  { ( X 
 .+  Z ) }
 )  .(+)  ( N `  { Y } ) ) ) )
 
Theorembaerlem5bmN 30708 An equality that holds when  X,  Y,  Z are independent (non-colinear) vectors. Subtraction version of second equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 30709 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ph  ->  ( N `  { ( Y 
 .-  Z ) }
 )  =  ( ( ( N `  { Y } )  .(+)  ( N `
  { Z }
 ) )  i^i  (
 ( N `  { ( X  .-  ( Y  .-  Z ) ) }
 )  .(+)  ( N `  { X } ) ) ) )
 
Theorembaerlem5abmN 30709 An equality that holds when  X,  Y,  Z are independent (non-colinear) vectors. Subtraction versions of first and second equations of part (5) in [Baer] p. 46, conjoined to share commonality in their proofs. TODO: Delete if not be needed. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ph  ->  (
 ( N `  { ( X  .-  ( Y  .-  Z ) ) }
 )  =  ( ( ( N `  { ( X  .-  Y ) }
 )  .(+)  ( N `  { Z } ) )  i^i  ( ( N `
  { ( X 
 .+  Z ) }
 )  .(+)  ( N `  { Y } ) ) )  /\  ( N `
  { ( Y 
 .-  Z ) }
 )  =  ( ( ( N `  { Y } )  .(+)  ( N `
  { Z }
 ) )  i^i  (
 ( N `  { ( X  .-  ( Y  .-  Z ) ) }
 )  .(+)  ( N `  { X } ) ) ) ) )
 
Theoremmapdindp0 30710 Vector independence lemma. (Contributed by NM, 29-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { X ,  Y } ) )   &    |-  ( ph  ->  ( Y  .+  Z )  =/=  .0.  )   =>    |-  ( ph  ->  ( N `  { ( Y  .+  Z ) } )  =  ( N `  { Y } ) )
 
Theoremmapdindp1 30711 Vector independence lemma. (Contributed by NM, 1-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { X ,  Y } ) )   =>    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { ( Y 
 .+  Z ) }
 ) )
 
Theoremmapdindp2 30712 Vector independence lemma. (Contributed by NM, 1-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { X ,  Y } ) )   =>    |-  ( ph  ->  -.  w  e.  ( N `
  { X ,  ( Y  .+  Z ) } ) )
 
Theoremmapdindp3 30713 Vector independence lemma. (Contributed by NM, 29-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { X ,  Y } ) )   =>    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { ( w 
 .+  Y ) }
 ) )
 
Theoremmapdindp4 30714 Vector independence lemma. (Contributed by NM, 29-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { X ,  Y } ) )   =>    |-  ( ph  ->  -.  Z  e.  ( N `
  { X ,  ( w  .+  Y ) } ) )
 
Theoremmapdhval 30715* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  Q  =  ( 0g `  C )   &    |-  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  ->  X  e.  A )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  Y  e.  E )   =>    |-  ( 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 ) }
 ) ) ) ) )
 
Theoremmapdhval0 30716* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  .0.  =  ( 0g `  U )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >. )  =  Q )
 
Theoremmapdhval2 30717* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
 |-  Q  =  ( 0g `  C )   &    |-  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  ->  X  e.  A )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  (
 iota_ h  e.  D ( ( M `  ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( F R h ) }
 ) ) ) )
 
Theoremmapdhcl 30718* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( 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 )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
 
Theoremmapdheq 30719* Lemmma for ~? mapdh . The defining equation for h(x,x',y)=y' in part (2) in [Baer] p. 45 line 24. (Contributed by NM, 4-Apr-2015.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( 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  ->  G  e.  D )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  (
 ( I `  <. X ,  F ,  Y >. )  =  G  <->  ( ( M `
  ( N `  { Y } ) )  =  ( J `  { G } )  /\  ( M `  ( N `
  { ( X 
 .-  Y ) }
 ) )  =  ( J `  { ( F R G ) }
 ) ) ) )
 
Theoremmapdheq2 30720* Lemmma for ~? mapdh . One direction of part (2) in [Baer] p. 45. (Contributed by NM, 4-Apr-2015.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( 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  ->  G  e.  D )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  (
 ( I `  <. X ,  F ,  Y >. )  =  G  ->  ( I `  <. Y ,  G ,  X >. )  =  F ) )
 
Theoremmapdheq2biN 30721* Lemmma for ~? mapdh . Part (2) in [Baer] p. 45. The bidirectional version of mapdheq2 30720 seems to require an additional hypothesis not mentioned in Baer. TODO fix ref. TODO: We probably don't need this; delete if never used. (Contributed by NM, 4-Apr-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( 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  ->  G  e.  D )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  ( M `  ( N `
  { Y }
 ) )  =  ( J `  { G } ) )   =>    |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <->  ( I `  <. Y ,  G ,  X >. )  =  F ) )
 
Theoremmapdheq4lem 30722* Lemma for mapdheq4 30723. Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( 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  ->  -.  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 ) } )
 )  =  ( J `
  { ( G R E ) }
 ) )
 
Theoremmapdheq4 30723* Lemma for ~? mapdh . Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( 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  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( N `  { Y } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )   =>    |-  ( ph  ->  ( I `  <. Y ,  G ,  Z >. )  =  E )
 
Theoremmapdh6lem1N 30724* Lemma for mapdh6N 30738. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 ) ) } ) )  =  ( J `  { ( F R ( G  .+b  E ) ) } ) )
 
Theoremmapdh6lem2N 30725* Lemma for mapdh6N 30738. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 ) } )
 )  =  ( J `
  { ( G 
 .+b  E ) } )
 )
 
Theoremmapdh6aN 30726* Lemma for mapdh6N 30738. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 >. ) ) )
 
Theoremmapdh6b0N 30727* Lemmma for mapdh6N 30738. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( ( N `  { X } )  i^i  ( N `
  { Y ,  Z } ) )  =  {  .0.  } )   =>    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
 
Theoremmapdh6bN 30728* Lemmma for mapdh6N 30738. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 >. ) ) )
 
Theoremmapdh6cN 30729* Lemmma for mapdh6N 30738. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 >. ) ) )
 
Theoremmapdh6dN 30730* Lemmma for mapdh6N 30738. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 ) >. ) ) )
 
Theoremmapdh6eN 30731* Lemmma for mapdh6N 30738. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 >. ) ) )
 
Theoremmapdh6fN 30732* Lemmma for mapdh6N 30738. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 >. ) ) )
 
Theoremmapdh6gN 30733* Lemmma for mapdh6N 30738. Part (6) of [Baer] p. 47 line 39. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 >. ) ) )
 
Theoremmapdh6hN 30734* Lemmma for mapdh6N 30738. Part (6) of [Baer] p. 48 line 2. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 >. ) ) )
 
Theoremmapdh6iN 30735* Lemmma for mapdh6N 30738. Eliminate auxiliary vector  w. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 >. ) ) )
 
Theoremmapdh6jN 30736* Lemmma for mapdh6N 30738. Eliminate  ( N { Y } ) = ( N  { Z } ) hypothesis. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 >. ) ) )
 
Theoremmapdh6kN 30737* Lemmma for mapdh6N 30738. Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
 |-  Q  =  ( 0g `  C )   &    |-  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 ) } )
 ) ) ) )   &    |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  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 )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  F  e.  D )   &    |-  ( ph  ->  ( M `  ( N `
  { X }
 ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  .+  =  ( +g  `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( 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 >. ) ) )
 
Theoremmapdh6N 30738* 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 15714. TODO: If $ds with  I and  ph becomes a problem later, cbv's on  I variables here to get rid of them. . Maybe reorder hypotheses in lemmas to the more consistent order of this theorem so they can be shared with this theorem. (Contributed by NM, 1-May-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 )   &    |-  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  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y ,  Z } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )   =>    |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `  <. X ,  F ,  Y >. )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )
 
Theoremmapdh7eN 30739* Part (7) of [Baer] p. 48 line 10 (5 of 6 cases). (Note: 1 of 6 and 2 of 6 are hypotheses a and b.) (Contributed by NM, 2-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 `  { u } ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  u  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  v  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N `  { u } )  =/=  ( N `  { v }
 ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { u ,  v } ) )   &    |-  ( ph  ->  ( I `  <. u ,  F ,  w >. )  =  E )   =>    |-  ( ph  ->  ( I `  <. w ,  E ,  u >. )  =  F )
 
Theoremmapdh7cN 30740* Part (7) of [Baer] p. 48 line 10 (3 of 6 cases). (Contributed by NM, 2-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 `  { u } ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  u  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  v  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N `  { u } )  =/=  ( N `  { v }
 ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { u ,  v } ) )   &    |-  ( ph  ->  ( I `  <. u ,  F ,  v >. )  =  G )   =>    |-  ( ph  ->  ( I `  <. v ,  G ,  u >. )  =  F )
 
Theoremmapdh7dN 30741* Part (7) of [Baer] p. 48 line 10 (4 of 6 cases). (Contributed by NM, 2-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 `  { u } ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  u  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  v  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N `  { u } )  =/=  ( N `  { v }
 ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { u ,  v } ) )   &    |-  ( ph  ->  ( I `  <. u ,  F ,  v >. )  =  G )   &    |-  ( ph  ->  ( I `  <. u ,  F ,  w >. )  =  E )   =>    |-  ( ph  ->  ( I `  <. v ,  G ,  w >. )  =  E )
 
Theoremmapdh7fN 30742* Part (7) of [Baer] p. 48 line 10 (6 of 6 cases). (Contributed by NM, 2-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 `  { u } ) )  =  ( J `  { F } ) )   &    |-  ( ph  ->  u  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  v  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  w  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( N `  { u } )  =/=  ( N `  { v }
 ) )   &    |-  ( ph  ->  -.  w  e.  ( N `
  { u ,  v } ) )   &    |-  ( ph  ->  ( I `  <. u ,  F ,  v >. )  =  G )   &    |-  ( ph  ->  ( I `  <. u ,  F ,  w >. )  =  E )   =>    |-  ( ph  ->  ( I `  <. w ,  E ,  v >. )  =  G )
 
Theoremmapdh75e 30743* Part (7) of [Baer] p. 48 line 10 (5 of 6 cases).  X,  Y,  Z are Baer's u, v, w. (Note: Cases 1 of 6 and 2 of 6 are hypotheses mapdh75b here and mapdh75a in mapdh75cN 30744.) (Contributed by NM, 2-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  ->  ( I `  <. X ,  F ,  Z >. )  =  E )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Z } )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( I `  <. Z ,  E ,  X >. )  =  F )
 
Theoremmapdh75cN 30744* Part (7) of [Baer] p. 48 line 10 (3 of 6 cases). (Contributed by NM, 2-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  ->  ( I `  <. X ,  F ,  Y >. )  =  G )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( I `  <. Y ,  G ,  X >. )  =  F )
 
Theoremmapdh75d 30745* Part (7) of [Baer] p. 48 line 10 (4 of 6 cases). (Contributed by NM, 2-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  ->  ( I `  <. X ,  F ,  Y >. )  =  G )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )   &    |-  ( ph  ->  ( N `  { Y }
 )  =/=  ( N ` 
 { Z } )
 )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( I `  <. Y ,  G ,  Z >. )  =  E )
 
Theoremmapdh75fN 30746* Part (7) of [Baer] p. 48 line 10 (6 of 6 cases). (Contributed by NM, 2-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  ->  ( I `  <. X ,  F ,  Y >. )  =  G )   &    |-  ( ph  ->  ( I `  <. X ,  F ,  Z >. )  =  E )   &    |-  ( ph  ->  ( N `  { Y }
 )  =/=  ( N ` 
 { Z } )
 )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Z  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( I `  <. Z ,  E ,  Y >. )  =  G )
 
Syntaxchvm 30747 Extend class notation with vector to dual map.
 class HVMap
 
Definitiondf-hvmap 30748* Extend class notation with a map from each nonzero vector  x to a unique nonzero functional in the closed kernel dual space. (We could extend it to include the zero vector, but that is unnecessary for our purposes.) TODO: This pattern is used several times earlier e.g. lcf1o 30542, dochfl1 30467- should we update those to use this definition? (Contributed by NM, 23-Mar-2015.)
 |- HVMap  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ( ( Base `  ( ( DVecH `  k
 ) `  w )
 )  \  { ( 0g `  ( ( DVecH `  k ) `  w ) ) } )  |->  ( v  e.  ( Base `  ( ( DVecH `  k ) `  w ) )  |->  ( iota_ j  e.  ( Base `  (Scalar `  ( ( DVecH `  k
 ) `  w )
 ) ) E. t  e.  ( ( ( ocH `  k ) `  w ) `  { x }
 ) v  =  ( t ( +g  `  (
 ( DVecH `  k ) `  w ) ) ( j ( .s `  ( ( DVecH `  k
 ) `  w )
 ) x ) ) ) ) ) ) )
 
Theoremhvmapffval 30749* Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  X  ->  (HVMap `  K )  =  ( w  e.  H  |->  ( x  e.  ( (
 Base `  ( ( DVecH `  K ) `  w ) )  \  { ( 0g `  ( ( DVecH `  K ) `  w ) ) } )  |->  ( v  e.  ( Base `  ( ( DVecH `  K ) `  w ) )  |->  ( iota_ j  e.  ( Base `  (Scalar `  ( ( DVecH `  K ) `  w ) ) ) E. t  e.  ( ( ( ocH `  K ) `  w ) `  { x }
 ) v  =  ( t ( +g  `  (
 ( DVecH `  K ) `  w ) ) ( j ( .s `  ( ( DVecH `  K ) `  w ) ) x ) ) ) ) ) ) )
 
Theoremhvmapfval 30750* Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  (
 Base `  S )   &    |-  M  =  ( (HVMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  A  /\  W  e.  H ) )   =>    |-  ( ph  ->  M  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ j  e.  R E. t  e.  ( O `  { x } ) v  =  ( t  .+  (
 j  .x.  x )
 ) ) ) ) )
 
Theoremhvmapval 30751* Value of map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  (
 Base `  S )   &    |-  M  =  ( (HVMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  A  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( M `  X )  =  ( v  e.  V  |->  (
 iota_ j  e.  R E. t  e.  ( O `  { X }
 ) v  =  ( t  .+  ( j 
 .x.  X ) ) ) ) )
 
TheoremhvmapvalvalN 30752* Value of value of map (i.e. functional value) from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  (
 Base `  S )   &    |-  M  =  ( (HVMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  A  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( M `  X ) `  Y )  =  ( iota_ j  e.  R E. t  e.  ( O `  { X }
 ) Y  =  ( t  .+  ( j 
 .x.  X ) ) ) )
 
TheoremhvmapidN 30753 The value of the vector to functional map, at the vector, is one. (Contributed by NM, 23-Mar-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .1.  =  ( 1r `  S )   &    |-  M  =  ( (HVMap `  K ) `  W )   &    |-  (