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Theorem List for Metamath Proof Explorer - 30801-30843   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhdmapinvlem2 30801 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 30802 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 30803 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 30718. 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 30804 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 30805 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 30806 Lemma for hgmapvv 30808. 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 30807 Lemma for hgmapvv 30808. 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 30808 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 30809* Lemma for hdmapg 30812. (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 30810 Lemma for hdmapg 30812. (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 30811 Lemma for hdmapg 30812. 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 30812 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 30813* 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 30814 Extend class notation with the final constructed Hilbert space.
 class HLHil
 
Definitiondf-hlhil 30815* 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 30816* 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 30817 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 30818 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 30819 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 30820 Lemma for hlhilsbase2 30824. (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 30821 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 30822 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 30823 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 30824 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 30825 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 30826 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 30827 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 30828 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 30829 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 30830* 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 30831 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 30832 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 30833 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 30834 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 30835 Lemma for hlhilsrng 30836. (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 30836 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 30837 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 30838 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 30839 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 30840 The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 30818 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 30841* Lemma for hlhil 18639. (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 30842* Lemma for hlhil 18639. (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 30843 Construction of a Hilbert space (df-hil 16436)  U from a Hilbert lattice (df-hlat 28230) 
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 29988) 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 16436. It can be eliminated if we just want to 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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