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Theorem List for Metamath Proof Explorer - 35501-35600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhdmaprnlem10N 35501* Lemma for hdmaprnN 35506. Show  s is in the range of  S. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  s  e.  ( D  \  { Q } ) )   &    |-  ( ph  ->  v  e.  V )   &    |-  ( ph  ->  ( M `  ( N `
  { v }
 ) )  =  ( L `  { s } ) )   &    |-  ( ph  ->  u  e.  V )   &    |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )   &    |-  D  =  ( Base `  C )   &    |-  Q  =  ( 0g `  C )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  .+  =  ( +g  `  U )   =>    |-  ( ph  ->  E. t  e.  V  ( S `  t )  =  s )
 
Theoremhdmaprnlem11N 35502* Lemma for hdmaprnN 35506. Show  s is in the range of  S. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  s  e.  ( D  \  { Q } ) )   &    |-  ( ph  ->  v  e.  V )   &    |-  ( ph  ->  ( M `  ( N `
  { v }
 ) )  =  ( L `  { s } ) )   &    |-  ( ph  ->  u  e.  V )   &    |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )   &    |-  D  =  ( Base `  C )   &    |-  Q  =  ( 0g `  C )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  .+  =  ( +g  `  U )   =>    |-  ( ph  ->  s  e.  ran  S )
 
Theoremhdmaprnlem15N 35503* Lemma for hdmaprnN 35506. Eliminate  u. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .0.  =  ( 0g `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  s  e.  ( D  \  {  .0.  } ) )   &    |-  ( ph  ->  v  e.  V )   &    |-  ( ph  ->  ( M `  ( N `  { v } )
 )  =  ( L `
  { s }
 ) )   =>    |-  ( ph  ->  s  e.  ran  S )
 
Theoremhdmaprnlem16N 35504 Lemma for hdmaprnN 35506. Eliminate  v. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .0.  =  ( 0g `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  s  e.  ( D  \  {  .0.  } ) )   =>    |-  ( ph  ->  s  e.  ran  S )
 
Theoremhdmaprnlem17N 35505 Lemma for hdmaprnN 35506. Include zero. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  .0.  =  ( 0g `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  s  e.  D )   =>    |-  ( ph  ->  s  e.  ran  S )
 
TheoremhdmaprnN 35506 Part of proof of part 12 in [Baer] p. 49 line 21, As=B. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ran  S  =  D )
 
Theoremhdmapf1oN 35507 Part 12 in [Baer] p. 49. The map from vectors to functionals with closed kernels maps one-to-one onto. Combined with hdmapadd 35485, this shows the map is an automorphism from the additive group of vectors to the additive group of functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  ( Base `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  S : V -1-1-onto-> D )
 
Theoremhdmap14lem1a 35508 Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F  e.  B )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  ( ph  ->  F  =/=  .0.  )   =>    |-  ( ph  ->  ( L `  { ( S `  X ) }
 )  =  ( L `
  { ( S `
  ( F  .x.  X ) ) } )
 )
 
Theoremhdmap14lem2a 35509* Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include  F  =  .0. so it can be used in hdmap14lem10 35519. (Contributed by NM, 31-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  L  =  ( LSpan `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  E. g  e.  A  ( S `  ( F  .x.  X ) )  =  ( g 
 .xb  ( S `  X ) ) )
 
Theoremhdmap14lem1 35510 Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  (
 Base `  R )   &    |-  Z  =  ( 0g `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  L  =  (
 LSpan `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  Q  =  ( 0g `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  ( B  \  { Z } ) )   =>    |-  ( ph  ->  ( L ` 
 { ( S `  X ) } )  =  ( L `  { ( S `  ( F  .x.  X ) ) } )
 )
 
Theoremhdmap14lem2N 35511* Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include  F  =  Z so it can be used in hdmap14lem10 35519. (Contributed by NM, 31-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  (
 Base `  R )   &    |-  Z  =  ( 0g `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  L  =  (
 LSpan `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  Q  =  ( 0g `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  ( B  \  { Z } ) )   =>    |-  ( ph  ->  E. g  e.  ( A  \  { Q } ) ( S `
  ( F  .x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
 
Theoremhdmap14lem3 35512* Prior to part 14 in [Baer] p. 49, line 26. (Contributed by NM, 31-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  (
 Base `  R )   &    |-  Z  =  ( 0g `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  L  =  (
 LSpan `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  Q  =  ( 0g `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  ( B  \  { Z } ) )   =>    |-  ( ph  ->  E! g  e.  ( A  \  { Q } ) ( S `
  ( F  .x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
 
Theoremhdmap14lem4a 35513* Simplify  ( A  \  { Q } ) in hdmap14lem3 35512 to provide a slightly simpler definition later. (Contributed by NM, 31-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  (
 Base `  R )   &    |-  Z  =  ( 0g `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  L  =  (
 LSpan `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  Q  =  ( 0g `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  ( B  \  { Z } ) )   =>    |-  ( ph  ->  ( E! g  e.  ( A  \  { Q } )
 ( S `  ( F  .x.  X ) )  =  ( g  .xb  ( S `  X ) )  <->  E! g  e.  A  ( S `  ( F 
 .x.  X ) )  =  ( g  .xb  ( S `  X ) ) ) )
 
Theoremhdmap14lem4 35514* Simplify  ( A  \  { Q } ) in hdmap14lem3 35512 to provide a slightly simpler definition later. TODO: Use hdmap14lem4a 35513 if that one is also used directly elsewhere. Otherwise, merge hdmap14lem4a 35513 into this one. (Contributed by NM, 31-May-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  (
 Base `  R )   &    |-  Z  =  ( 0g `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  L  =  (
 LSpan `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  Q  =  ( 0g `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  ( B  \  { Z } ) )   =>    |-  ( ph  ->  E! g  e.  A  ( S `  ( F  .x.  X ) )  =  ( g 
 .xb  ( S `  X ) ) )
 
Theoremhdmap14lem6 35515* Case where  F is zero. (Contributed by NM, 1-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  (
 Base `  R )   &    |-  Z  =  ( 0g `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  L  =  (
 LSpan `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  Q  =  ( 0g `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  =  Z )   =>    |-  ( ph  ->  E! g  e.  A  ( S `  ( F 
 .x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
 
Theoremhdmap14lem7 35516* Combine cases of  F. TODO: Can this be done at once in hdmap14lem3 35512, in order to get rid of hdmap14lem6 35515? Perhaps modify lspsneu 18424 to become  E! k  e.  K instead of  E! k  e.  ( K  \  {  .0.  } )? (Contributed by NM, 1-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  (
 Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  E! g  e.  A  ( S `  ( F  .x.  X ) )  =  ( g  .xb  ( S `  X ) ) )
 
Theoremhdmap14lem8 35517 Part of proof of part 14 in [Baer] p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .+b  =  ( +g  `  C )   &    |-  .xb  =  ( .s `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ph  ->  I  e.  A )   &    |-  ( ph  ->  ( S `  ( F 
 .x.  X ) )  =  ( G  .xb  ( S `  X ) ) )   &    |-  ( ph  ->  ( S `  ( F 
 .x.  Y ) )  =  ( I  .xb  ( S `  Y ) ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  J  e.  A )   &    |-  ( ph  ->  ( S `  ( F  .x.  ( X 
 .+  Y ) ) )  =  ( J 
 .xb  ( S `  ( X  .+  Y ) ) ) )   =>    |-  ( ph  ->  ( ( J  .xb  ( S `  X ) ) 
 .+b  ( J  .xb  ( S `  Y ) ) )  =  ( ( G  .xb  ( S `  X ) ) 
 .+b  ( I  .xb  ( S `  Y ) ) ) )
 
Theoremhdmap14lem9 35518 Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 1-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .+b  =  ( +g  `  C )   &    |-  .xb  =  ( .s `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ph  ->  I  e.  A )   &    |-  ( ph  ->  ( S `  ( F 
 .x.  X ) )  =  ( G  .xb  ( S `  X ) ) )   &    |-  ( ph  ->  ( S `  ( F 
 .x.  Y ) )  =  ( I  .xb  ( S `  Y ) ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  J  e.  A )   &    |-  ( ph  ->  ( S `  ( F  .x.  ( X 
 .+  Y ) ) )  =  ( J 
 .xb  ( S `  ( X  .+  Y ) ) ) )   =>    |-  ( ph  ->  G  =  I )
 
Theoremhdmap14lem10 35519 Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 3-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .+b  =  ( +g  `  C )   &    |-  .xb  =  ( .s `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ph  ->  I  e.  A )   &    |-  ( ph  ->  ( S `  ( F 
 .x.  X ) )  =  ( G  .xb  ( S `  X ) ) )   &    |-  ( ph  ->  ( S `  ( F 
 .x.  Y ) )  =  ( I  .xb  ( S `  Y ) ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  G  =  I )
 
Theoremhdmap14lem11 35520 Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 3-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .+b  =  ( +g  `  C )   &    |-  .xb  =  ( .s `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ph  ->  I  e.  A )   &    |-  ( ph  ->  ( S `  ( F 
 .x.  X ) )  =  ( G  .xb  ( S `  X ) ) )   &    |-  ( ph  ->  ( S `  ( F 
 .x.  Y ) )  =  ( I  .xb  ( S `  Y ) ) )   =>    |-  ( ph  ->  G  =  I )
 
Theoremhdmap14lem12 35521* Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  B )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  U )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  G  e.  A )   =>    |-  ( ph  ->  (
 ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X ) )  <->  A. y  e.  ( V  \  {  .0.  }
 ) ( S `  ( F  .x.  y ) )  =  ( G 
 .xb  ( S `  y ) ) ) )
 
Theoremhdmap14lem13 35522* Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  B )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  U )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  G  e.  A )   =>    |-  ( ph  ->  (
 ( S `  ( F  .x.  X ) )  =  ( G  .xb  ( S `  X ) )  <->  A. y  e.  V  ( S `  ( F 
 .x.  y ) )  =  ( G  .xb  ( S `  y ) ) ) )
 
Theoremhdmap14lem14 35523* Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 6-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  B )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   =>    |-  ( ph  ->  E! g  e.  A  A. x  e.  V  ( S `  ( F  .x.  x ) )  =  ( g 
 .xb  ( S `  x ) ) )
 
Theoremhdmap14lem15 35524* Part of proof of part 14 in [Baer] p. 50 line 3. Convert scalar base of dual to scalar base of vector space. (Contributed by NM, 6-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  E! g  e.  B  A. x  e.  V  ( S `  ( F  .x.  x ) )  =  ( g  .xb  ( S `  x ) ) )
 
Syntaxchg 35525 Extend class notation with g-map.
 class HGMap
 
Definitiondf-hgmap 35526* Define map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
 |- HGMap  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { a  |  [. ( ( DVecH `  k ) `  w )  /  u ]. [. ( Base `  (Scalar `  u ) )  /  b ]. [. ( (HDMap `  k ) `  w )  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b  A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u ) v ) )  =  ( y ( .s `  ( (LCDual `  k ) `  w ) ) ( m `
  v ) ) ) ) } )
 )
 
Theoremhgmapffval 35527* Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
 |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  X  ->  (HGMap `  K )  =  ( w  e.  H  |->  { a  |  [. (
 ( DVecH `  K ) `  w )  /  u ].
 [. ( Base `  (Scalar `  u ) )  /  b ]. [. ( (HDMap `  K ) `  w )  /  m ]. a  e.  ( x  e.  b  |->  ( iota_ y  e.  b  A. v  e.  ( Base `  u ) ( m `  ( x ( .s `  u ) v ) )  =  ( y ( .s `  ( (LCDual `  K ) `  w ) ) ( m `
  v ) ) ) ) } )
 )
 
Theoremhgmapfval 35528* Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  M  =  ( (HDMap `  K ) `  W )   &    |-  I  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  Y  /\  W  e.  H ) )   =>    |-  ( ph  ->  I  =  ( x  e.  B  |->  ( iota_ y  e.  B  A. v  e.  V  ( M `  ( x 
 .x.  v ) )  =  ( y  .xb  ( M `  v ) ) ) ) )
 
Theoremhgmapval 35529* Value of map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. Function sigma of scalar f in part 14 of [Baer] p. 50 line 4. TODO: variable names are inherited from older version. Maybe make more consistent with hdmap14lem15 35524. (Contributed by NM, 25-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  M  =  ( (HDMap `  K ) `  W )   &    |-  I  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  Y  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( I `  X )  =  ( iota_ y  e.  B  A. v  e.  V  ( M `  ( X  .x.  v ) )  =  ( y 
 .xb  ( M `  v ) ) ) )
 
TheoremhgmapfnN 35530 Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  G  Fn  B )
 
Theoremhgmapcl 35531 Closure of scalar sigma map i.e. the map from the vector space scalar base to the dual space scalar base. (Contributed by NM, 6-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( G `  F )  e.  B )
 
Theoremhgmapdcl 35532 Closure of the vector space to dual space scalar map, in the scalar sigma map. (Contributed by NM, 6-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  Q  =  (Scalar `  C )   &    |-  A  =  ( Base `  Q )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( G `  F )  e.  A )
 
Theoremhgmapvs 35533 Part 15 of [Baer] p. 50 line 6. Also line 15 in [Holland95] p. 14. (Contributed by NM, 6-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  .xb  =  ( .s `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( S `  ( F  .x.  X ) )  =  ( ( G `  F )  .xb  ( S `  X ) ) )
 
Theoremhgmapval0 35534 Value of the scalar sigma map at zero. (Contributed by NM, 12-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  .0.  =  ( 0g `  R )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ( G `  .0.  )  =  .0.  )
 
Theoremhgmapval1 35535 Value of the scalar sigma map at one. (Contributed by NM, 12-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  .1.  =  ( 1r `  R )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ( G `  .1.  )  =  .1.  )
 
Theoremhgmapadd 35536 Part 15 of [Baer] p. 50 line 13. (Contributed by NM, 6-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( G `  ( X  .+  Y ) )  =  ( ( G `  X )  .+  ( G `
  Y ) ) )
 
Theoremhgmapmul 35537 Part 15 of [Baer] p. 50 line 16. The multiplication is reversed after converting to the dual space scalar to the vector space scalar. (Contributed by NM, 7-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( G `  ( X  .x.  Y ) )  =  ( ( G `  Y )  .x.  ( G `  X ) ) )
 
Theoremhgmaprnlem1N 35538 Lemma for hgmaprnN 35543. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  .xb  =  ( .s `  C )   &    |-  Q  =  ( 0g `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  z  e.  A )   &    |-  ( ph  ->  t  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  s  e.  V )   &    |-  ( ph  ->  ( S `  s )  =  ( z  .xb  ( S `  t ) ) )   &    |-  ( ph  ->  k  e.  B )   &    |-  ( ph  ->  s  =  ( k  .x.  t )
 )   =>    |-  ( ph  ->  z  e.  ran  G )
 
Theoremhgmaprnlem2N 35539 Lemma for hgmaprnN 35543. Part 15 of [Baer] p. 50 line 20. We only require a subset relation, rather than equality, so that the case of zero  z is taken care of automatically. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  .xb  =  ( .s `  C )   &    |-  Q  =  ( 0g `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  z  e.  A )   &    |-  ( ph  ->  t  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  s  e.  V )   &    |-  ( ph  ->  ( S `  s )  =  ( z  .xb  ( S `  t ) ) )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   &    |-  L  =  ( LSpan `  C )   =>    |-  ( ph  ->  ( N `  { s } )  C_  ( N `  { t } ) )
 
Theoremhgmaprnlem3N 35540* Lemma for hgmaprnN 35543. Eliminate  k. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  .xb  =  ( .s `  C )   &    |-  Q  =  ( 0g `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  z  e.  A )   &    |-  ( ph  ->  t  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  s  e.  V )   &    |-  ( ph  ->  ( S `  s )  =  ( z  .xb  ( S `  t ) ) )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   &    |-  L  =  ( LSpan `  C )   =>    |-  ( ph  ->  z  e.  ran  G )
 
Theoremhgmaprnlem4N 35541* Lemma for hgmaprnN 35543. Eliminate  s. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  .xb  =  ( .s `  C )   &    |-  Q  =  ( 0g `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  z  e.  A )   &    |-  ( ph  ->  t  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  z  e.  ran 
 G )
 
Theoremhgmaprnlem5N 35542 Lemma for hgmaprnN 35543. Eliminate  t. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  D  =  (
 Base `  C )   &    |-  P  =  (Scalar `  C )   &    |-  A  =  ( Base `  P )   &    |-  .xb  =  ( .s `  C )   &    |-  Q  =  ( 0g `  C )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  z  e.  A )   =>    |-  ( ph  ->  z  e.  ran  G )
 
TheoremhgmaprnN 35543 Part of proof of part 16 in [Baer] p. 50 line 23, Fs=G, except that we use the original vector space scalars for the range. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ran  G  =  B )
 
Theoremhgmap11 35544 The scalar sigma map is one-to-one. (Contributed by NM, 7-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( G `  X )  =  ( G `  Y )  <->  X  =  Y ) )
 
Theoremhgmapf1oN 35545 The scalar sigma map is a one-to-one onto function. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  G : B -1-1-onto-> B )
 
Theoremhgmapeq0 35546 The scalar sigma map is zero iff its argument is zero. (Contributed by NM, 12-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (
 ( G `  X )  =  .0.  <->  X  =  .0.  ) )
 
Theoremhdmapipcl 35547 The inner product (Hermitian form)  ( X ,  Y
) will be defined as  ( ( S `  Y ) `  X ). Show closure. (Contributed by NM, 7-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( S `  Y ) `  X )  e.  B )
 
Theoremhdmapln1 35548 Linearity property that will be used for inner product. TODO: try to combine hypotheses in hdmap*ln* series. (Contributed by NM, 7-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .+^  =  ( +g  `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  ( ( S `  Z ) `  ( ( A 
 .x.  X )  .+  Y ) )  =  (
 ( A  .X.  (
 ( S `  Z ) `  X ) )  .+^  ( ( S `  Z ) `  Y ) ) )
 
Theoremhdmaplna1 35549 Additive property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  R  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   =>    |-  ( ph  ->  (
 ( S `  Z ) `  ( X  .+  Y ) )  =  ( ( ( S `
  Z ) `  X )  .+^  ( ( S `  Z ) `
  Y ) ) )
 
Theoremhdmaplns1 35550 Subtraction property of first (inner product) argument. (Contributed by NM, 12-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .-  =  ( -g `  U )   &    |-  R  =  (Scalar `  U )   &    |-  N  =  ( -g `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   =>    |-  ( ph  ->  (
 ( S `  Z ) `  ( X  .-  Y ) )  =  ( ( ( S `
  Z ) `  X ) N ( ( S `  Z ) `  Y ) ) )
 
Theoremhdmaplnm1 35551 Multiplicative property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  (
 ( S `  Y ) `  ( A  .x.  X ) )  =  ( A  .X.  ( ( S `  Y ) `  X ) ) )
 
Theoremhdmaplna2 35552 Additive property of second (inner product) argument. (Contributed by NM, 10-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  R  =  (Scalar `  U )   &    |-  .+^  =  (
 +g  `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   =>    |-  ( ph  ->  (
 ( S `  ( Y  .+  Z ) ) `
  X )  =  ( ( ( S `
  Y ) `  X )  .+^  ( ( S `  Z ) `
  X ) ) )
 
Theoremhdmapglnm2 35553 g-linear property of second (inner product) argument. Line 19 in [Holland95] p. 14. (Contributed by NM, 10-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  (
 ( S `  ( A  .x.  Y ) ) `
  X )  =  ( ( ( S `
  Y ) `  X )  .X.  ( G `
  A ) ) )
 
Theoremhdmapgln2 35554 g-linear property that will be used for inner product. (Contributed by NM, 14-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .+^  =  ( +g  `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  ( ( S `  (
 ( A  .x.  Y )  .+  Z ) ) `
  X )  =  ( ( ( ( S `  Y ) `
  X )  .X.  ( G `  A ) )  .+^  ( ( S `  Z ) `  X ) ) )
 
Theoremhdmaplkr 35555 Kernel of the vector to dual map. Line 16 in [Holland95] p. 14. TODO: eliminate  F hypothesis. (Contributed by NM, 9-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  Y  =  (LKer `  U )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( Y `  ( S `
  X ) )  =  ( O `  { X } ) )
 
Theoremhdmapellkr 35556 Membership in the kernel (as shown by hdmaplkr 35555) of the vector to dual map. Line 17 in [Holland95] p. 14. (Contributed by NM, 16-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( ( S `  X ) `  Y )  =  .0.  <->  Y  e.  ( O `  { X }
 ) ) )
 
Theoremhdmapip0 35557 Zero property that will be used for inner product. (Contributed by NM, 9-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  R  =  (Scalar `  U )   &    |-  Z  =  ( 0g
 `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( ( ( S `  X ) `  X )  =  Z  <->  X  =  .0.  ) )
 
Theoremhdmapip1 35558 Construct a proportional vector  Y whose inner product with the original  X equals one. (Contributed by NM, 13-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  R  =  (Scalar `  U )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( invr `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  Y  =  ( ( N `  ( ( S `  X ) `  X ) )  .x.  X )   =>    |-  ( ph  ->  ( ( S `  X ) `  Y )  =  .1.  )
 
Theoremhdmapip0com 35559 Commutation property of Baer's sigma map (Holland's A map). Line 20 of [Holland95] p. 14. Also part of Lemma 1 of [Baer] p. 110 line 7. (Contributed by NM, 9-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( ( S `  X ) `  Y )  =  .0.  <->  ( ( S `
  Y ) `  X )  =  .0.  ) )
 
Theoremhdmapinvlem1 35560 Line 27 in [Baer] p. 110. We use  C for Baer's u. Our unit vector  E has the required properties for his w by hdmapevec2 35478. Our  ( ( S `  E ) `  C ) means the inner product  <. C ,  E >. i.e. his f(u,w) (note argument reversal). (Contributed by NM, 11-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  C  e.  ( O `  { E } ) )   =>    |-  ( ph  ->  ( ( S `  E ) `  C )  =  .0.  )
 
Theoremhdmapinvlem2 35561 Line 28 in [Baer] p. 110, 0 = f(w,u). (Contributed by NM, 11-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  C  e.  ( O `  { E } ) )   =>    |-  ( ph  ->  ( ( S `  C ) `  E )  =  .0.  )
 
Theoremhdmapinvlem3 35562 Line 30 in [Baer] p. 110, f(sw + u, tw - v) = 0. (Contributed by NM, 12-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .x. 
 =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  (
 Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  C  e.  ( O `  { E } ) )   &    |-  ( ph  ->  D  e.  ( O `  { E }
 ) )   &    |-  ( ph  ->  I  e.  B )   &    |-  ( ph  ->  J  e.  B )   &    |-  ( ph  ->  ( I  .X.  ( G `  J ) )  =  ( ( S `  D ) `  C ) )   =>    |-  ( ph  ->  (
 ( S `  (
 ( J  .x.  E )  .-  D ) ) `
  ( ( I 
 .x.  E )  .+  C ) )  =  .0.  )
 
Theoremhdmapinvlem4 35563 Part 1.1 of Proposition 1 of [Baer] p. 110. We use  C,  D,  I, and  J for Baer's u, v, s, and t. Our unit vector  E has the required properties for his w by hdmapevec2 35478. Our  ( ( S `  D ) `  C ) means his f(u,v) (note argument reversal). (Contributed by NM, 12-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .x. 
 =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  (
 Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  C  e.  ( O `  { E } ) )   &    |-  ( ph  ->  D  e.  ( O `  { E }
 ) )   &    |-  ( ph  ->  I  e.  B )   &    |-  ( ph  ->  J  e.  B )   &    |-  ( ph  ->  ( I  .X.  ( G `  J ) )  =  ( ( S `  D ) `  C ) )   =>    |-  ( ph  ->  ( J  .X.  ( G `  I ) )  =  ( ( S `  C ) `  D ) )
 
Theoremhdmapglem5 35564 Part 1.2 in [Baer] p. 110 line 34, f(u,v) alpha = f(v,u). (Contributed by NM, 12-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .-  =  ( -g `  U )   &    |-  .x. 
 =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  (
 Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  C  e.  ( O `  { E } ) )   &    |-  ( ph  ->  D  e.  ( O `  { E }
 ) )   &    |-  ( ph  ->  I  e.  B )   &    |-  ( ph  ->  J  e.  B )   =>    |-  ( ph  ->  ( G `  ( ( S `
  D ) `  C ) )  =  ( ( S `  C ) `  D ) )
 
Theoremhgmapvvlem1 35565 Involution property of scalar sigma map. Line 10 in [Baer] p. 111, t sigma squared = t. Our  E,  C,  D,  Y,  X correspond to Baer's w, h, k, s, t. (Contributed by NM, 13-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  N  =  (
 invr `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( B  \  {  .0.  } ) )   &    |-  ( ph  ->  C  e.  ( O `  { E } ) )   &    |-  ( ph  ->  D  e.  ( O `  { E }
 ) )   &    |-  ( ph  ->  ( ( S `  D ) `  C )  =  .1.  )   &    |-  ( ph  ->  Y  e.  ( B  \  {  .0.  } ) )   &    |-  ( ph  ->  ( Y  .X.  ( G `  X ) )  =  .1.  )   =>    |-  ( ph  ->  ( G `  ( G `  X ) )  =  X )
 
Theoremhgmapvvlem2 35566 Lemma for hgmapvv 35568. Eliminate  Y (Baer's s). (Contributed by NM, 13-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  N  =  (
 invr `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( B  \  {  .0.  } ) )   &    |-  ( ph  ->  C  e.  ( O `  { E } ) )   &    |-  ( ph  ->  D  e.  ( O `  { E }
 ) )   &    |-  ( ph  ->  ( ( S `  D ) `  C )  =  .1.  )   =>    |-  ( ph  ->  ( G `  ( G `  X ) )  =  X )
 
Theoremhgmapvvlem3 35567 Lemma for hgmapvv 35568. Eliminate  ( ( S `  D
) `  C )  =  .1. (Baer's f(h,k)=1). (Contributed by NM, 13-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  N  =  (
 invr `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( B  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( G `  ( G `  X ) )  =  X )
 
Theoremhgmapvv 35568 Value of a double involution. Part 1.2 of [Baer] p. 110 line 37. (Contributed by NM, 13-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( G `  ( G `
  X ) )  =  X )
 
Theoremhdmapglem7a 35569* Lemma for hdmapg 35572. (Contributed by NM, 14-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  E. u  e.  ( O `
  { E }
 ) E. k  e.  B  X  =  ( ( k  .x.  E )  .+  u ) )
 
Theoremhdmapglem7b 35570 Lemma for hdmapg 35572. (Contributed by NM, 14-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  .X.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .+b  =  ( +g  `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  x  e.  ( O `  { E }
 ) )   &    |-  ( ph  ->  y  e.  ( O `  { E } ) )   &    |-  ( ph  ->  m  e.  B )   &    |-  ( ph  ->  n  e.  B )   =>    |-  ( ph  ->  ( ( S `  (
 ( m  .x.  E )  .+  x ) ) `
  ( ( n 
 .x.  E )  .+  y
 ) )  =  ( ( n  .X.  ( G `  m ) ) 
 .+b  ( ( S `
  x ) `  y ) ) )
 
Theoremhdmapglem7 35571 Lemma for hdmapg 35572. Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). In the proof, our  E,  ( O `  { E } )  X,  Y,  k,  u,  l,  v correspond to Baer's w, H, x, y, x', x'', y' , y'', and our  ( ( S `
 Y ) `  X ) corresponds to Baer's f(x,y). (Contributed by NM, 14-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  <. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  ( (
 LTrn `  K ) `  W ) ) >.   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  .X.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .+b  =  ( +g  `  R )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( G `  ( ( S `
  Y ) `  X ) )  =  ( ( S `  X ) `  Y ) )
 
Theoremhdmapg 35572 Apply the scalar sigma function (involution)  G to an inner product reverses the arguments. The inner product of  X and  Y is represented by  ( ( S `  Y ) `  X
). Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). (Contributed by NM, 14-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( G `  ( ( S `
  Y ) `  X ) )  =  ( ( S `  X ) `  Y ) )
 
Theoremhdmapoc 35573* Express our constructed orthocomplement (polarity) in terms of the Hilbert space definition of orthocomplement. Lines 24 and 25 in [Holland95] p. 14. (Contributed by NM, 17-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  R  =  (Scalar `  U )   &    |-  .0.  =  ( 0g `  R )   &    |-  O  =  ( ( ocH `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  C_  V )   =>    |-  ( ph  ->  ( O `  X )  =  { y  e.  V  |  A. z  e.  X  ( ( S `  z ) `  y
 )  =  .0.  }
 )
 
Syntaxchlh 35574 Extend class notation with the final constructed Hilbert space.
 class HLHil
 
Definitiondf-hlhil 35575* Define our final Hilbert space constructed from a Hilbert lattice. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |- HLHil  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  [_ (
 ( DVecH `  k ) `  w )  /  u ]_
 [_ ( Base `  u )  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
 <. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  (
 ( ( EDRing `  k
 ) `  w ) sSet  <.
 ( *r `  ndx ) ,  ( (HGMap `  k ) `  w ) >. ) >. }  u.  {
 <. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y ) `  x ) ) >. } ) ) )
 
Theoremhlhilset 35576* The final Hilbert space constructed from a Hilbert lattice  K and an arbitrary hyperplane  W in  K. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( (HLHil `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  E  =  ( ( EDRing `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  R  =  ( E sSet  <. ( *r `  ndx ) ,  G >. )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  .,  =  ( x  e.  V ,  y  e.  V  |->  ( ( S `  y ) `  x ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   =>    |-  ( ph  ->  L  =  ( { <. ( Base ` 
 ndx ) ,  V >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i
 `  ndx ) ,  .,  >. } ) )
 
Theoremhlhilsca 35577 The scalar of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  E  =  ( ( EDRing `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  R  =  ( E sSet  <. ( *r `
  ndx ) ,  G >. )   =>    |-  ( ph  ->  R  =  (Scalar `  U )
 )
 
Theoremhlhilbase 35578 The base set of the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  M  =  (
 Base `  L )   =>    |-  ( ph  ->  M  =  ( Base `  U ) )
 
Theoremhlhilplus 35579 The vector addition for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  L )   =>    |-  ( ph  ->  .+  =  ( +g  `  U ) )
 
Theoremhlhilslem 35580 Lemma for hlhilsbase2 35584. (Contributed by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  ( ( EDRing `  K ) `  W )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  F  = Slot  N   &    |-  N  e.  NN   &    |-  N  <  4   &    |-  C  =  ( F `  E )   =>    |-  ( ph  ->  C  =  ( F `  R ) )
 
Theoremhlhilsbase 35581 The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  ( ( EDRing `  K ) `  W )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  C  =  (
 Base `  E )   =>    |-  ( ph  ->  C  =  ( Base `  R ) )
 
Theoremhlhilsplus 35582 Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  ( ( EDRing `  K ) `  W )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  .+  =  ( +g  `  E )   =>    |-  ( ph  ->  .+  =  ( +g  `  R ) )
 
Theoremhlhilsmul 35583 Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  E  =  ( ( EDRing `  K ) `  W )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  .x.  =  ( .r `  E )   =>    |-  ( ph  ->  .x. 
 =  ( .r `  R ) )
 
Theoremhlhilsbase2 35584 The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  (Scalar `  L )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  C  =  (
 Base `  S )   =>    |-  ( ph  ->  C  =  ( Base `  R ) )
 
Theoremhlhilsplus2 35585 Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  (Scalar `  L )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  .+  =  ( +g  `  S )   =>    |-  ( ph  ->  .+  =  ( +g  `  R ) )
 
Theoremhlhilsmul2 35586 Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  (Scalar `  L )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  .x.  =  ( .r `  S )   =>    |-  ( ph  ->  .x. 
 =  ( .r `  R ) )
 
Theoremhlhils0 35587 The scalar ring zero for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  (Scalar `  L )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ph  ->  .0.  =  ( 0g `  R ) )
 
Theoremhlhils1N 35588 The scalar ring unity for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  (Scalar `  L )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  .1.  =  ( 1r `  S )   =>    |-  ( ph  ->  .1.  =  ( 1r `  R ) )
 
Theoremhlhilvsca 35589 The scalar product for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  .x.  =  ( .s `  L )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  .x. 
 =  ( .s `  U ) )
 
Theoremhlhilip 35590* Inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  L )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  .,  =  ( x  e.  V ,  y  e.  V  |->  ( ( S `  y ) `
  x ) )   =>    |-  ( ph  ->  .,  =  ( .i `  U ) )
 
Theoremhlhilipval 35591 Value of inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  L )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  .,  =  ( .i `  U )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( X  .,  Y )  =  ( ( S `  Y ) `  X ) )
 
Theoremhlhilnvl 35592 The involution operation of the star division ring for the final constructed Hilbert space. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  R  =  (Scalar `  U )   &    |-  .*  =  ( (HGMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  .*  =  ( *r `  R ) )
 
Theoremhlhillvec 35593 The final constructed Hilbert space is a vector space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  U  e.  LVec )
 
Theoremhlhildrng 35594 The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  R  =  (Scalar `  U )   =>    |-  ( ph  ->  R  e. 
 DivRing )
 
Theoremhlhilsrnglem 35595 Lemma for hlhilsrng 35596. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  R  =  (Scalar `  U )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  (Scalar `  L )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  G  =  ( (HGMap `  K ) `  W )   =>    |-  ( ph  ->  R  e.  *Ring )
 
Theoremhlhilsrng 35596 The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 21-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  R  =  (Scalar `  U )   =>    |-  ( ph  ->  R  e.  *Ring )
 
Theoremhlhil0 35597 The zero vector for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  .0.  =  ( 0g `  L )   =>    |-  ( ph  ->  .0.  =  ( 0g `  U ) )
 
Theoremhlhillsm 35598 The vector sum operation for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  .(+)  =  ( LSSum `  L )   =>    |-  ( ph  ->  .(+)  =  (
 LSSum `  U ) )
 
Theoremhlhilocv 35599 The orthocomplement for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  V  =  (
 Base `  L )   &    |-  N  =  ( ( ocH `  K ) `  W )   &    |-  O  =  ( ocv `  U )   &    |-  ( ph  ->  X  C_  V )   =>    |-  ( ph  ->  ( O `  X )  =  ( N `  X ) )
 
Theoremhlhillcs 35600 The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 35578 is applied over and over to conclusion rather than applied once to antecedent - would compressed proof be shorter if applied once to antecedent? (Contributed by NM, 23-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  C  =  ( CSubSp `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  C  =  ran  I )
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