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Theorem List for Metamath Proof Explorer - 32501-32600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmapdpglem18 32501* Lemma for mapdpg 32518. Baer p. 45, line 7: "Then y =/= 0..." (Contributed by NM, 20-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  .(+)  =  (
 LSSum `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  F  =  ( Base `  C )   &    |-  ( ph  ->  t  e.  (
 ( M `  ( N `  { X }
 ) )  .(+)  ( M `
  ( N `  { Y } ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  (
 Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  R  =  ( -g `  C )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  Q  =  ( 0g `  U )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   &    |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { t }
 ) )   &    |-  .0.  =  ( 0g `  A )   &    |-  ( ph  ->  g  e.  B )   &    |-  ( ph  ->  z  e.  ( M `  ( N `  { Y } ) ) )   &    |-  ( ph  ->  t  =  ( ( g  .x.  G ) R z ) )   &    |-  ( ph  ->  X  =/=  Q )   &    |-  ( ph  ->  Y  =/=  Q )   &    |-  E  =  ( ( ( invr `  A ) `  g )  .x.  z
 )   =>    |-  ( ph  ->  E  =/=  ( 0g `  C ) )
 
Theoremmapdpglem19 32502* Lemma for mapdpg 32518. Baer p. 45, line 8: "...is in (Fy)*..." (Contributed by NM, 20-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  .(+)  =  (
 LSSum `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  F  =  ( Base `  C )   &    |-  ( ph  ->  t  e.  (
 ( M `  ( N `  { X }
 ) )  .(+)  ( M `
  ( N `  { Y } ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  (
 Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  R  =  ( -g `  C )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  Q  =  ( 0g `  U )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   &    |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { t }
 ) )   &    |-  .0.  =  ( 0g `  A )   &    |-  ( ph  ->  g  e.  B )   &    |-  ( ph  ->  z  e.  ( M `  ( N `  { Y } ) ) )   &    |-  ( ph  ->  t  =  ( ( g  .x.  G ) R z ) )   &    |-  ( ph  ->  X  =/=  Q )   &    |-  ( ph  ->  Y  =/=  Q )   &    |-  E  =  ( ( ( invr `  A ) `  g )  .x.  z
 )   =>    |-  ( ph  ->  E  e.  ( M `  ( N `  { Y }
 ) ) )
 
Theoremmapdpglem20 32503* Lemma for mapdpg 32518. Baer p. 45, line 8: "...so that (Fy)*=Gy'." (Contributed by NM, 20-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  .(+)  =  (
 LSSum `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  F  =  ( Base `  C )   &    |-  ( ph  ->  t  e.  (
 ( M `  ( N `  { X }
 ) )  .(+)  ( M `
  ( N `  { Y } ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  (
 Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  R  =  ( -g `  C )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  Q  =  ( 0g `  U )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   &    |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { t }
 ) )   &    |-  .0.  =  ( 0g `  A )   &    |-  ( ph  ->  g  e.  B )   &    |-  ( ph  ->  z  e.  ( M `  ( N `  { Y } ) ) )   &    |-  ( ph  ->  t  =  ( ( g  .x.  G ) R z ) )   &    |-  ( ph  ->  X  =/=  Q )   &    |-  ( ph  ->  Y  =/=  Q )   &    |-  E  =  ( ( ( invr `  A ) `  g )  .x.  z
 )   =>    |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( J `  { E } ) )
 
Theoremmapdpglem21 32504* Lemma for mapdpg 32518. (Contributed by NM, 20-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  .(+)  =  (
 LSSum `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  F  =  ( Base `  C )   &    |-  ( ph  ->  t  e.  (
 ( M `  ( N `  { X }
 ) )  .(+)  ( M `
  ( N `  { Y } ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  (
 Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  R  =  ( -g `  C )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  Q  =  ( 0g `  U )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   &    |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { t }
 ) )   &    |-  .0.  =  ( 0g `  A )   &    |-  ( ph  ->  g  e.  B )   &    |-  ( ph  ->  z  e.  ( M `  ( N `  { Y } ) ) )   &    |-  ( ph  ->  t  =  ( ( g  .x.  G ) R z ) )   &    |-  ( ph  ->  X  =/=  Q )   &    |-  ( ph  ->  Y  =/=  Q )   &    |-  E  =  ( ( ( invr `  A ) `  g )  .x.  z
 )   =>    |-  ( ph  ->  (
 ( ( invr `  A ) `  g )  .x.  t )  =  ( G R E ) )
 
Theoremmapdpglem22 32505* Lemma for mapdpg 32518. Baer p. 45, line 9: "(F(x-y))* = ... = G(x'-y')." (Contributed by NM, 20-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  .(+)  =  (
 LSSum `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  F  =  ( Base `  C )   &    |-  ( ph  ->  t  e.  (
 ( M `  ( N `  { X }
 ) )  .(+)  ( M `
  ( N `  { Y } ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  (
 Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  R  =  ( -g `  C )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  Q  =  ( 0g `  U )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   &    |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { t }
 ) )   &    |-  .0.  =  ( 0g `  A )   &    |-  ( ph  ->  g  e.  B )   &    |-  ( ph  ->  z  e.  ( M `  ( N `  { Y } ) ) )   &    |-  ( ph  ->  t  =  ( ( g  .x.  G ) R z ) )   &    |-  ( ph  ->  X  =/=  Q )   &    |-  ( ph  ->  Y  =/=  Q )   &    |-  E  =  ( ( ( invr `  A ) `  g )  .x.  z
 )   =>    |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R E ) }
 ) )
 
Theoremmapdpglem23 32506* Lemma for mapdpg 32518. Baer p. 45, line 10: "and so y' meets all our requirements." Our  h is Baer's y'. (Contributed by NM, 20-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  .(+)  =  (
 LSSum `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  F  =  ( Base `  C )   &    |-  ( ph  ->  t  e.  (
 ( M `  ( N `  { X }
 ) )  .(+)  ( M `
  ( N `  { Y } ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  (
 Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  R  =  ( -g `  C )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  Q  =  ( 0g `  U )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   &    |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { t }
 ) )   &    |-  .0.  =  ( 0g `  A )   &    |-  ( ph  ->  g  e.  B )   &    |-  ( ph  ->  z  e.  ( M `  ( N `  { Y } ) ) )   &    |-  ( ph  ->  t  =  ( ( g  .x.  G ) R z ) )   &    |-  ( ph  ->  X  =/=  Q )   &    |-  ( ph  ->  Y  =/=  Q )   &    |-  E  =  ( ( ( invr `  A ) `  g )  .x.  z
 )   =>    |-  ( ph  ->  E. h  e.  F  ( ( M `
  ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `
  { ( X 
 .-  Y ) }
 ) )  =  ( J `  { ( G R h ) }
 ) ) )
 
Theoremmapdpglem30a 32507 Lemma for mapdpg 32518. (Contributed by NM, 22-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   =>    |-  ( ph  ->  G  =/=  ( 0g `  C ) )
 
Theoremmapdpglem30b 32508 Lemma for mapdpg 32518. (Contributed by NM, 22-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R h ) }
 ) ) ) )   &    |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { i } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R i ) }
 ) ) ) )   =>    |-  ( ph  ->  i  =/=  ( 0g `  C ) )
 
Theoremmapdpglem25 32509 Lemma for mapdpg 32518. Baer p. 45 line 12: "Then we have Gy' = Gy'' and G(x'-y') = G(x'-y'')." (Contributed by NM, 21-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R h ) }
 ) ) ) )   &    |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { i } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R i ) }
 ) ) ) )   =>    |-  ( ph  ->  ( ( J `  { h }
 )  =  ( J `
  { i }
 )  /\  ( J ` 
 { ( G R h ) } )  =  ( J `  { ( G R i ) }
 ) ) )
 
Theoremmapdpglem26 32510* Lemma for mapdpg 32518. Baer p. 45 line 14: "Consequently there exist numbers u,v in G neither of which is 0 such that y = uy'' and..." (We scope $d  u ph locally to avoid clashes with later substitutions into 
ph.) (Contributed by NM, 22-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R h ) }
 ) ) ) )   &    |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { i } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R i ) }
 ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  ( Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  O  =  ( 0g `  A )   =>    |-  ( ph  ->  E. u  e.  ( B  \  { O } ) h  =  ( u  .x.  i
 ) )
 
Theoremmapdpglem27 32511* Lemma for mapdpg 32518. Baer p. 45 line 16: "v(x'-y'') = x'-y'" (with equality swapped). (Contributed by NM, 22-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R h ) }
 ) ) ) )   &    |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { i } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R i ) }
 ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  ( Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  O  =  ( 0g `  A )   =>    |-  ( ph  ->  E. v  e.  ( B  \  { O } ) ( G R h )  =  ( v  .x.  ( G R i ) ) )
 
Theoremmapdpglem29 32512* Lemma for mapdpg 32518. Baer p. 45 line 16: "But Gx' and Gy'' are distinct points and so x' and y'' are independent elements in B. (Contributed by NM, 22-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R h ) }
 ) ) ) )   &    |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { i } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R i ) }
 ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  ( Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  O  =  ( 0g `  A )   &    |-  ( ph  ->  v  e.  B )   &    |-  ( ph  ->  h  =  ( u  .x.  i ) )   &    |-  ( ph  ->  ( G R h )  =  (
 v  .x.  ( G R i ) ) )   =>    |-  ( ph  ->  ( J `  { G }
 )  =/=  ( J ` 
 { i } )
 )
 
Theoremmapdpglem28 32513* Lemma for mapdpg 32518. Baer p. 45 line 18: "vx'-vy'' = x'-uy''". (Contributed by NM, 22-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R h ) }
 ) ) ) )   &    |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { i } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R i ) }
 ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  ( Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  O  =  ( 0g `  A )   &    |-  ( ph  ->  v  e.  B )   &    |-  ( ph  ->  h  =  ( u  .x.  i ) )   &    |-  ( ph  ->  ( G R h )  =  (
 v  .x.  ( G R i ) ) )   =>    |-  ( ph  ->  (
 ( v  .x.  G ) R ( v  .x.  i ) )  =  ( G R ( u  .x.  i )
 ) )
 
Theoremmapdpglem30 32514* Lemma for mapdpg 32518. Baer p. 45 line 18: "Hence we deduce (from mapdpglem28 32513, using lvecindp2 15908) that v = 1 and v = u...". TODO: would it be shorter to have only the  v  =  ( 1r `  A ) part and use mapdpglem28.u2 in mapdpglem31 32515? (Contributed by NM, 22-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R h ) }
 ) ) ) )   &    |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { i } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R i ) }
 ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  ( Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  O  =  ( 0g `  A )   &    |-  ( ph  ->  v  e.  B )   &    |-  ( ph  ->  h  =  ( u  .x.  i ) )   &    |-  ( ph  ->  ( G R h )  =  (
 v  .x.  ( G R i ) ) )   &    |-  ( ph  ->  u  e.  B )   =>    |-  ( ph  ->  ( v  =  ( 1r
 `  A )  /\  v  =  u )
 )
 
Theoremmapdpglem31 32515* Lemma for mapdpg 32518. Baer p. 45 line 19: "...and we have consequently that y' = y'', as we claimed." (Contributed by NM, 23-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   &    |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R h ) }
 ) ) ) )   &    |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { i } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R i ) }
 ) ) ) )   &    |-  A  =  (Scalar `  U )   &    |-  B  =  ( Base `  A )   &    |-  .x.  =  ( .s `  C )   &    |-  O  =  ( 0g `  A )   &    |-  ( ph  ->  v  e.  B )   &    |-  ( ph  ->  h  =  ( u  .x.  i ) )   &    |-  ( ph  ->  ( G R h )  =  (
 v  .x.  ( G R i ) ) )   &    |-  ( ph  ->  u  e.  B )   =>    |-  ( ph  ->  h  =  i )
 
Theoremmapdpglem24 32516* Lemma for mapdpg 32518. Existence part - consolidate hypotheses in mapdpglem23 32506. (Contributed by NM, 21-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   =>    |-  ( ph  ->  E. h  e.  F  ( ( M `  ( N `  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R h ) }
 ) ) )
 
Theoremmapdpglem32 32517* Lemma for mapdpg 32518. Uniqueness part - consolidate hypotheses in mapdpglem31 32515. (Contributed by NM, 23-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   =>    |-  ( ( ph  /\  ( h  e.  F  /\  i  e.  F )  /\  ( ( ( M `  ( N `
  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R h ) }
 ) )  /\  (
 ( M `  ( N `  { Y }
 ) )  =  ( J `  { i } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R i ) }
 ) ) ) ) 
 ->  h  =  i
 )
 
Theoremmapdpg 32518* Part 1 of proof of the first fundamental theorem of projective geometry. Part (1) in [Baer] p. 44. Our notation corresponds to Baer's as follows:  M for *,  N `  { } for F(),  J `  { } for G(),  X for x,  G for x',  Y for y,  h for y'. TODO: Rename variables per mapdhval 32536. (Contributed by NM, 22-Mar-2015.)
 |-  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 )   &    |-  F  =  ( Base `  C )   &    |-  R  =  ( -g `  C )   &    |-  J  =  ( LSpan `  C )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )   =>    |-  ( ph  ->  E! h  e.  F  ( ( M `  ( N `  { Y }
 ) )  =  ( J `  { h } )  /\  ( M `
  ( N `  { ( X  .-  Y ) } )
 )  =  ( J `
  { ( G R h ) }
 ) ) )
 
Theorembaerlem3lem1 32519 Lemma for baerlem3 32525. (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  ->  a  e.  B )   &    |-  ( ph  ->  b  e.  B )   &    |-  ( ph  ->  d  e.  B )   &    |-  ( ph  ->  e  e.  B )   &    |-  ( ph  ->  j  =  ( ( a  .x.  Y )  .+  ( b  .x.  Z ) ) )   &    |-  ( ph  ->  j  =  ( ( d  .x.  ( X  .-  Y ) ) 
 .+  ( e  .x.  ( X  .-  Z ) ) ) )   =>    |-  ( ph  ->  j  =  ( a  .x.  ( Y  .-  Z ) ) )
 
Theorembaerlem5alem1 32520 Lemma for baerlem5a 32526. (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  ->  a  e.  B )   &    |-  ( ph  ->  b  e.  B )   &    |-  ( ph  ->  d  e.  B )   &    |-  ( ph  ->  e  e.  B )   &    |-  ( ph  ->  j  =  ( ( a  .x.  ( X  .-  Y ) ) 
 .+  ( b  .x.  Z ) ) )   &    |-  ( ph  ->  j  =  ( ( d  .x.  ( X  .-  Z ) ) 
 .+  ( e  .x.  Y ) ) )   =>    |-  ( ph  ->  j  =  ( a  .x.  ( X  .-  ( Y 
 .+  Z ) ) ) )
 
Theorembaerlem5blem1 32521 Lemma for baerlem5b 32527. (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  ->  a  e.  B )   &    |-  ( ph  ->  b  e.  B )   &    |-  ( ph  ->  d  e.  B )   &    |-  ( ph  ->  e  e.  B )   &    |-  ( ph  ->  j  =  ( ( a  .x.  Y )  .+  ( b  .x.  Z ) ) )   &    |-  ( ph  ->  j  =  ( ( d  .x.  ( X  .-  ( Y  .+  Z ) ) ) 
 .+  ( e  .x.  X ) ) )   =>    |-  ( ph  ->  j  =  ( ( I `
  d )  .x.  ( Y  .+  Z ) ) )
 
Theorembaerlem3lem2 32522 Lemma for baerlem3 32525. (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 32523 Lemma for baerlem5a 32526. (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 32524 Lemma for baerlem5b 32527. (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 32525 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 32526 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 32527 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 32528 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 32530 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 32529 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 32530 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 32530 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 32531 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 32532 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 32533 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 32534 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 32535 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 32536* 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 32537* 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 32538* 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 32539* 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 32540* 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 32541* 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 32542* Lemmma for ~? mapdh . Part (2) in [Baer] p. 45. The bidirectional version of mapdheq2 32541 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 32543* Lemma for mapdheq4 32544. 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 32544* 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 32545* Lemma for mapdh6N 32559. 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 32546* Lemma for mapdh6N 32559. 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 32547* Lemma for mapdh6N 32559. 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 32548* Lemmma for mapdh6N 32559. (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 32549* Lemmma for mapdh6N 32559. (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 32550* Lemmma for mapdh6N 32559. (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 32551* Lemmma for mapdh6N 32559. (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 )