HomeHome Metamath Proof Explorer
Theorem List (p. 355 of 402)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26575)
  Hilbert Space Explorer  Hilbert Space Explorer
(26576-28098)
  Users' Mathboxes  Users' Mathboxes
(28099-40159)
 

Theorem List for Metamath Proof Explorer - 35401-35500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhdmaprnlem7N 35401 Part of proof of part 12 in [Baer] p. 49 line 19, s-St  e. G(u'+s) = P*. (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  s  e.  ( D  \  { Q } ) )   &    |-  ( ph  ->  v  e.  V )   &    |-  ( ph  ->  ( M `  ( N `
  { v }
 ) )  =  ( L `  { s } ) )   &    |-  ( ph  ->  u  e.  V )   &    |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )   &    |-  D  =  ( Base `  C )   &    |-  Q  =  ( 0g `  C )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( ph  ->  t  e.  (
 ( N `  { v } )  \  {  .0.  } ) )   &    |-  .+  =  ( +g  `  U )   &    |-  ( ph  ->  ( L `  { ( ( S `
  u )  .+b  s ) } )  =  ( M `  ( N `  { ( u 
 .+  t ) }
 ) ) )   =>    |-  ( ph  ->  ( s ( -g `  C ) ( S `  t ) )  e.  ( L `  { (
 ( S `  u )  .+b  s ) }
 ) )
 
Theoremhdmaprnlem8N 35402 Part of proof of part 12 in [Baer] p. 49 line 19, s-St  e. (Ft)* = T*. (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  s  e.  ( D  \  { Q } ) )   &    |-  ( ph  ->  v  e.  V )   &    |-  ( ph  ->  ( M `  ( N `
  { v }
 ) )  =  ( L `  { s } ) )   &    |-  ( ph  ->  u  e.  V )   &    |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )   &    |-  D  =  ( Base `  C )   &    |-  Q  =  ( 0g `  C )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( ph  ->  t  e.  (
 ( N `  { v } )  \  {  .0.  } ) )   &    |-  .+  =  ( +g  `  U )   &    |-  ( ph  ->  ( L `  { ( ( S `
  u )  .+b  s ) } )  =  ( M `  ( N `  { ( u 
 .+  t ) }
 ) ) )   =>    |-  ( ph  ->  ( s ( -g `  C ) ( S `  t ) )  e.  ( M `  ( N `  { t }
 ) ) )
 
Theoremhdmaprnlem9N 35403 Part of proof of part 12 in [Baer] p. 49 line 21, s=S(t). TODO: we seem to be going back and forth with mapd11 35182 and mapdcnv11N 35202. Use better hypotheses and/or theorems? (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  C  =  ( (LCDual `  K ) `  W )   &    |-  L  =  ( LSpan `  C )   &    |-  M  =  ( (mapd `  K ) `  W )   &    |-  S  =  ( (HDMap `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  s  e.  ( D  \  { Q } ) )   &    |-  ( ph  ->  v  e.  V )   &    |-  ( ph  ->  ( M `  ( N `
  { v }
 ) )  =  ( L `  { s } ) )   &    |-  ( ph  ->  u  e.  V )   &    |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )   &    |-  D  =  ( Base `  C )   &    |-  Q  =  ( 0g `  C )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .+b  =  ( +g  `  C )   &    |-  ( ph  ->  t  e.  (
 ( N `  { v } )  \  {  .0.  } ) )   &    |-  .+  =  ( +g  `  U )   &    |-  ( ph  ->  ( L `  { ( ( S `
  u )  .+b  s ) } )  =  ( M `  ( N `  { ( u 
 .+  t ) }
 ) ) )   =>    |-  ( ph  ->  s  =  ( S `  t ) )
 
Theoremhdmaprnlem3eN 35404* Lemma for hdmaprnN 35410. (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.  (
 ( N `  { v } )  \  {  .0.  } ) ( L `  { ( ( S `
  u )  .+b  s ) } )  =  ( M `  ( N `  { ( u 
 .+  t ) }
 ) ) )
 
Theoremhdmaprnlem10N 35405* Lemma for hdmaprnN 35410. 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 35406* Lemma for hdmaprnN 35410. 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 35407* Lemma for hdmaprnN 35410. 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 35408 Lemma for hdmaprnN 35410. 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 35409 Lemma for hdmaprnN 35410. 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 35410 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 35411 Part 12 in [Baer] p. 49. The map from vectors to functionals with closed kernels maps one-to-one onto. Combined with hdmapadd 35389, 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 35412 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 35413* Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include  F  =  .0. so it can be used in hdmap14lem10 35423. (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 35414 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 35415* Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include  F  =  Z so it can be used in hdmap14lem10 35423. (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 35416* 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 35417* Simplify  ( A  \  { Q } ) in hdmap14lem3 35416 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 35418* Simplify  ( A  \  { Q } ) in hdmap14lem3 35416 to provide a slightly simpler definition later. TODO: Use hdmap14lem4a 35417 if that one is also used directly elsewhere. Otherwise, merge hdmap14lem4a 35417 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 35419* 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 35420* Combine cases of  F. TODO: Can this be done at once in hdmap14lem3 35416, in order to get rid of hdmap14lem6 35419? Perhaps modify lspsneu 18351 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 35421 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 35422 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 35423 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 35424 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 35425* 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 35426* 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 35427* 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 35428* 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 35429 Extend class notation with g-map.
 class HGMap
 
Definitiondf-hgmap 35430* 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 35431* 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 35432* 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 35433* 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 35428. (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 35434 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 35435 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 35436 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 35437 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 35438 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 35439 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 35440 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 35441 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 35442 Lemma for hgmaprnN 35447. (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 35443 Lemma for hgmaprnN 35447. 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 35444* Lemma for hgmaprnN 35447. 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 35445* Lemma for hgmaprnN 35447. 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 35446 Lemma for hgmaprnN 35447. 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 35447 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 35448 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 35449 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 35450 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 35451 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 35452 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 35453 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 35454 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 35455 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 35456 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 35457 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 35458 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 35459 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 35460 Membership in the kernel (as shown by hdmaplkr 35459) 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 35461 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 35462 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 35463 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 35464 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 35382. 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 35465 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 35466 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 35467 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 35382. 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 35468 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 35469 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 35470 Lemma for hgmapvv 35472. 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 35471 Lemma for hgmapvv 35472. 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 35472 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 35473* Lemma for hdmapg 35476. (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 35474 Lemma for hdmapg 35476. (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 35475 Lemma for hdmapg 35476. 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 35476 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 35477* 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 35478 Extend class notation with the final constructed Hilbert space.
 class HLHil
 
Definitiondf-hlhil 35479* 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 35480* 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 35481 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 35482 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 35483 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 35484 Lemma for hlhilsbase2 35488. (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 35485 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 35486 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 35487 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 35488 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 35489 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 35490 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 35491 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 35492 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 35493 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 35494* 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 35495 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 35496 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 35497 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 35498 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 )
 </