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Theorem List for Metamath Proof Explorer - 32701-32776   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhgmapfval 32701* 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 32702* 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 32697. (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 32703 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 32704 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 32705 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 32706 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 32707 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 32708 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 32709 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 32710 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 32711 Lemma for hgmaprnN 32716. (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 32712 Lemma for hgmaprnN 32716. 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 32713* Lemma for hgmaprnN 32716. 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 32714* Lemma for hgmaprnN 32716. 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 32715 Lemma for hgmaprnN 32716. 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 32716 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 32717 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 32718 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 32719 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 32720 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 32721 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 32722 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 32723 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 32724 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 32725 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 32726 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 32727 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 32728 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 32729 Membership in the kernel (as shown by hdmaplkr 32728) 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 32730 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 32731 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 32732 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 32733 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 32651. 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 32734 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 32735 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 32736 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 32651. 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 32737 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 32738 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 32739 Lemma for hgmapvv 32741. 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 32740 Lemma for hgmapvv 32741. 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 32741 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 32742* Lemma for hdmapg 32745. (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 32743 Lemma for hdmapg 32745. (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 32744 Lemma for hdmapg 32745. 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 32745 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 32746* 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 32747 Extend class notation with the final constructed Hilbert space.
 class HLHil
 
Definitiondf-hlhil 32748* 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 32749* 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 32750 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 32751 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 32752 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 32753 Lemma for hlhilsbase2 32757. (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 32754 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 32755 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 32756 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 32757 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 32758 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 32759 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 32760 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 32761 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 32762 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 32763* 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 32764 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 32765 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 32766 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 32767 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 32768 Lemma for hlhilsrng 32769. (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 32769 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 32770 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 32771 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 32772 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 32773 The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 32751 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 )
 
Theoremhlhilphllem 32774* Lemma for hlhil 18823. (Contributed by NM, 23-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  F  =  (Scalar `  U )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  L )   &    |-  .+  =  ( +g  `  L )   &    |-  .x.  =  ( .s `  L )   &    |-  R  =  (Scalar `  L )   &    |-  B  =  ( Base `  R )   &    |-  .+^  =  ( +g  `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  Q  =  ( 0g `  R )   &    |- 
 .0.  =  ( 0g `  L )   &    |-  .,  =  ( .i `  U )   &    |-  J  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  E  =  ( x  e.  V ,  y  e.  V  |->  ( ( J `  y ) `  x ) )   =>    |-  ( ph  ->  U  e.  PreHil )
 
Theoremhlhilhillem 32775* Lemma for hlhil 18823. (Contributed by NM, 23-Jun-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  F  =  (Scalar `  U )   &    |-  L  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  L )   &    |-  .+  =  ( +g  `  L )   &    |-  .x.  =  ( .s `  L )   &    |-  R  =  (Scalar `  L )   &    |-  B  =  ( Base `  R )   &    |-  .+^  =  ( +g  `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  Q  =  ( 0g `  R )   &    |- 
 .0.  =  ( 0g `  L )   &    |-  .,  =  ( .i `  U )   &    |-  J  =  ( (HDMap `  K ) `  W )   &    |-  G  =  ( (HGMap `  K ) `  W )   &    |-  E  =  ( x  e.  V ,  y  e.  V  |->  ( ( J `  y ) `  x ) )   &    |-  O  =  ( ocv `  U )   &    |-  C  =  ( CSubSp `  U )   =>    |-  ( ph  ->  U  e.  Hil )
 
Theoremhlathil 32776 Construction of a Hilbert space (df-hil 16620)  U from a Hilbert lattice (df-hlat 30163) 
K, where  W is a fixed but arbitrary hyperplane (co-atom) in  K.

The Hilbert space  U is identical to the vector space  ( ( DVecH `  K ) `  W ) (see dvhlvec 31921) except that it is extended with involution and inner product components. The construction of these two components is provided by Theorem 3.6 in [Holland95] p. 13, whose proof we follow loosely.

An example of involution is the complex conjugate when the division ring is the field of complex numbers. The nature of the division ring we constructed is indeterminate, however, until we specialize the initial Hilbert lattice with additional conditions found by Maria Solèr in 1995 and refined by René Mayet in 1998 that result in a division ring isomorphic to 
CC. See additional discussion at http://us.metamath.org/qlegif/mmql.html#what.

 W corresponds to the w in the proof of Theorem 13.4 of [Crawley] p. 111. Such a  W always exists since  HL has lattice rank of at least 4 by df-hil 16620. It can be eliminated if we just want to show the existence of a Hilbert space, as is done in the literature. (Contributed by NM, 23-Jun-2015.)

 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( (HLHil `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  U  e.  Hil )
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