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Theorem List for Metamath Proof Explorer - 16801-16900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremznrrg 16801 The regular elements of ℤ/nℤ are exactly the units. (This theorem fails for  N  =  0, where all nonzero integers are regular, but only  pm 1 are units.) (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  U  =  (Unit `  Y )   &    |-  E  =  (RLReg `  Y )   =>    |-  ( N  e.  NN  ->  E  =  U )
 
Theoremcygznlem1 16802* Lemma for cygzn 16806. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  ( n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   =>    |-  ( ( ph  /\  ( K  e.  ZZ  /\  M  e.  ZZ )
 )  ->  ( ( L `  K )  =  ( L `  M ) 
 <->  ( K  .x.  X )  =  ( M  .x.  X ) ) )
 
Theoremcygznlem2a 16803* Lemma for cygzn 16806. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  ( n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   &    |-  F  =  ran  ( m  e. 
 ZZ  |->  <. ( L `  m ) ,  ( m  .x.  X ) >. )   =>    |-  ( ph  ->  F :
 ( Base `  Y ) --> B )
 
Theoremcygznlem2 16804* Lemma for cygzn 16806. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by Mario Carneiro, 23-Dec-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  ( n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   &    |-  F  =  ran  ( m  e. 
 ZZ  |->  <. ( L `  m ) ,  ( m  .x.  X ) >. )   =>    |-  ( ( ph  /\  M  e.  ZZ )  ->  ( F `  ( L `  M ) )  =  ( M  .x.  X ) )
 
Theoremcygznlem3 16805* A cyclic group with  n elements is isomorphic to  ZZ  /  n ZZ. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  ( n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   &    |-  F  =  ran  ( m  e. 
 ZZ  |->  <. ( L `  m ) ,  ( m  .x.  X ) >. )   =>    |-  ( ph  ->  G  ~=ph𝑔  Y )
 
Theoremcygzn 16806 A cyclic group with  n elements is isomorphic to  ZZ  /  n ZZ, and an infinite cyclic group is isomorphic to  ZZ 
/  0 ZZ  ~~  ZZ. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( G  e. CycGrp  ->  G 
 ~=ph𝑔  Y )
 
Theoremcygth 16807* The "fundamental theorem of cyclic groups". Cyclic groups are exactly the additive groups  ZZ  /  n ZZ, for 
0  <_  n (where  n  =  0 is the infinite cyclic group 
ZZ), up to isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( G  e. CycGrp  <->  E. n  e.  NN0  G 
 ~=ph𝑔  (ℤ/n `  n ) )
 
Theoremcyggic 16808 Cyclic groups are isomorphic precisely when they have the same order. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  (
 Base `  H )   =>    |-  ( ( G  e. CycGrp  /\  H  e. CycGrp )  ->  ( G  ~=ph𝑔 
 H 
 <->  B  ~~  C ) )
 
Theoremfrgpcyg 16809 A free group is cyclic iff it has zero or one generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (freeGrp `  I
 )   =>    |-  ( I  ~<_  1o  <->  G  e. CycGrp )
 
10.12  Hilbert spaces
 
10.12.1  Definition and basic properties
 
Syntaxcphl 16810 Extend class notation with class all pre-Hilbert spaces.
 class  PreHil
 
Syntaxcipf 16811 Extend class notation with inner product function.
 class  .i f
 
Definitiondf-phl 16812* Define class all generalized pre-Hilbert (inner product) spaces. (Contributed by NM, 22-Sep-2011.)
 |-  PreHil  =  { g  e. 
 LVec  |  [. ( Base `  g )  /  v ]. [. ( .i `  g )  /  h ].
 [. (Scalar `  g )  /  f ]. ( f  e.  *Ring  /\  A. x  e.  v  ( ( y  e.  v  |->  ( y h x ) )  e.  ( g LMHom  (ringLMod `  f ) )  /\  ( ( x h x )  =  ( 0g `  f ) 
 ->  x  =  ( 0g `  g ) ) 
 /\  A. y  e.  v  ( ( * r `
  f ) `  ( x h y ) )  =  ( y h x ) ) ) }
 
Definitiondf-ipf 16813* Define group addition function. Usually we will use  +g directly instead of  + f, and they have the same behavior in most cases. The main advantage of  + f is that it is a guaranteed function (mndplusf 14661), while  +g only has closure (mndcl 14650). (Contributed by Mario Carneiro, 12-Aug-2015.)
 |- 
 .i f  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( .i
 `  g ) y ) ) )
 
Theoremisphl 16814* The predicate "is a generalized pre-Hilbert (inner product) space". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .,  =  ( .i `  W )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .*  =  ( * r `  F )   &    |-  Z  =  ( 0g `  F )   =>    |-  ( W  e.  PreHil  <->  ( W  e.  LVec  /\  F  e.  *Ring  /\  A. x  e.  V  ( ( y  e.  V  |->  ( y 
 .,  x ) )  e.  ( W LMHom  (ringLMod `  F ) )  /\  ( ( x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x  .,  y ) )  =  ( y  .,  x ) ) ) )
 
Theoremphllvec 16815 A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  PreHil  ->  W  e.  LVec )
 
Theoremphllmod 16816 A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  PreHil  ->  W  e.  LMod )
 
Theoremphlsrng 16817 The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  PreHil  ->  F  e.  *Ring )
 
Theoremphllmhm 16818* The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  G  =  ( x  e.  V  |->  ( x  .,  A ) )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  G  e.  ( W LMHom 
 (ringLMod `  F ) ) )
 
Theoremipcl 16819 Closure of the inner product operation in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  ( Base `  F )   =>    |-  (
 ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .,  B )  e.  K )
 
Theoremipcj 16820 Conjugate of an inner product in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .*  =  ( * r `  F )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (  .*  `  ( A  .,  B ) )  =  ( B  .,  A ) )
 
Theoremiporthcom 16821 Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .,  B )  =  Z  <->  ( B  .,  A )  =  Z ) )
 
Theoremip0l 16822 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  (  .0.  .,  A )  =  Z )
 
Theoremip0r 16823 Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  ( A  .,  .0.  )  =  Z )
 
Theoremipeq0 16824 The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  ( ( A  .,  A )  =  Z  <->  A  =  .0.  ) )
 
Theoremipdir 16825 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .+^  =  (
 +g  `  F )   =>    |-  (
 ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .+  B )  .,  C )  =  (
 ( A  .,  C )  .+^  ( B  .,  C ) ) )
 
Theoremipdi 16826 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .+^  =  (
 +g  `  F )   =>    |-  (
 ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( A  .,  ( B  .+  C ) )  =  (
 ( A  .,  B )  .+^  ( A  .,  C ) ) )
 
Theoremip2di 16827 Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .+^  =  (
 +g  `  F )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  (
 ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( ( A 
 .,  C )  .+^  ( B  .,  D ) )  .+^  ( ( A  .,  D )  .+^  ( B  .,  C ) ) ) )
 
Theoremipsubdir 16828 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  S  =  ( -g `  F )   =>    |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .-  B )  .,  C )  =  (
 ( A  .,  C ) S ( B  .,  C ) ) )
 
Theoremipsubdi 16829 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  S  =  ( -g `  F )   =>    |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( A  .,  ( B  .-  C ) )  =  (
 ( A  .,  B ) S ( A  .,  C ) ) )
 
Theoremip2subdi 16830 Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  S  =  ( -g `  F )   &    |-  .+  =  ( +g  `  F )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  (
 ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( ( A 
 .,  C )  .+  ( B  .,  D ) ) S ( ( A  .,  D ) 
 .+  ( B  .,  C ) ) ) )
 
Theoremipass 16831 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  F )   =>    |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .x.  B )  .,  C )  =  ( A  .X.  ( B  .,  C ) ) )
 
Theoremipassr 16832 "Associative" law for second argument of inner product (compare ipass 16831). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  F )   &    |-  .*  =  ( * r `  F )   =>    |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
 )  ->  ( A  .,  ( C  .x.  B ) )  =  (
 ( A  .,  B )  .X.  (  .*  `  C ) ) )
 
Theoremipassr2 16833 "Associative" law for inner product. Conjugate version of ipassr 16832. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  F )   &    |-  .*  =  ( * r `  F )   =>    |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
 )  ->  ( ( A  .,  B )  .X.  C )  =  ( A 
 .,  ( (  .*  `  C )  .x.  B ) ) )
 
Theoremipffval 16834* The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  .x.  =  ( .i f `  W )   =>    |- 
 .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
 
Theoremipfval 16835 The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  .x.  =  ( .i f `  W )   =>    |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( X  .x.  Y )  =  ( X 
 .,  Y ) )
 
Theoremipfeq 16836 If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  .x.  =  ( .i f `  W )   =>    |-  (  .,  Fn  ( V  X.  V )  ->  .x.  =  .,  )
 
Theoremipffn 16837 The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i f `  W )   =>    |-  ., 
 Fn  ( V  X.  V )
 
Theoremphlipf 16838 The inner product operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i f `  W )   &    |-  S  =  (Scalar `  W )   &    |-  K  =  ( Base `  S )   =>    |-  ( W  e.  PreHil  ->  .,  : ( V  X.  V ) --> K )
 
Theoremip2eq 16839* Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   =>    |-  (
 ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  =  B  <->  A. x  e.  V  ( x  .,  A )  =  ( x  .,  B ) ) )
 
Theoremisphld 16840* Properties that determine a pre-Hilbert (inner product) space. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  ( ph  ->  V  =  ( Base `  W )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  W )
 )   &    |-  ( ph  ->  .x.  =  ( .s `  W ) )   &    |-  ( ph  ->  I  =  ( .i `  W ) )   &    |-  ( ph  ->  .0.  =  ( 0g `  W ) )   &    |-  ( ph  ->  F  =  (Scalar `  W ) )   &    |-  ( ph  ->  K  =  ( Base `  F )
 )   &    |-  ( ph  ->  .+^  =  (
 +g  `  F )
 )   &    |-  ( ph  ->  .X.  =  ( .r `  F ) )   &    |-  ( ph  ->  .*  =  ( * r `
  F ) )   &    |-  ( ph  ->  O  =  ( 0g `  F ) )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  F  e.  *Ring )   &    |-  ( ( ph  /\  x  e.  V  /\  y  e.  V )  ->  ( x I y )  e.  K )   &    |-  ( ( ph  /\  q  e.  K  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  ( ( ( q 
 .x.  x )  .+  y ) I z )  =  ( ( q  .X.  ( x I z ) )  .+^  ( y I z ) ) )   &    |-  (
 ( ph  /\  x  e.  V  /\  ( x I x )  =  O )  ->  x  =  .0.  )   &    |-  ( ( ph  /\  x  e.  V  /\  y  e.  V )  ->  (  .*  `  ( x I y ) )  =  ( y I x ) )   =>    |-  ( ph  ->  W  e.  PreHil )
 
Theoremphlpropd 16841* If two structures have the same components (properties), one is a pre-Hilbert space iff the other one is. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ph  ->  F  =  (Scalar `  K ) )   &    |-  ( ph  ->  F  =  (Scalar `  L ) )   &    |-  P  =  ( Base `  F )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  =  ( x ( .s
 `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( .i `  K ) y )  =  ( x ( .i
 `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  PreHil  <->  L  e.  PreHil ) )
 
10.12.2  Orthocomplements and closed subspaces
 
Syntaxcocv 16842 Extend class notation with orthocomplement of a subspace.
 class  ocv
 
Syntaxccss 16843 Extend class notation with set of closed subspaces.
 class  CSubSp
 
Syntaxcthl 16844 Extend class notation with the Hilbert lattice.
 class toHL
 
Definitiondf-ocv 16845* Define orthocomplement of a subspace. (Contributed by NM, 7-Oct-2011.)
 |- 
 ocv  =  ( h  e.  _V  |->  ( s  e. 
 ~P ( Base `  h )  |->  { x  e.  ( Base `  h )  | 
 A. y  e.  s  ( x ( .i `  h ) y )  =  ( 0g `  (Scalar `  h ) ) } ) )
 
Definitiondf-css 16846* Define set of closed subspaces. (Contributed by NM, 7-Oct-2011.)
 |-  CSubSp  =  ( h  e. 
 _V  |->  { s  |  s  =  ( ( ocv `  h ) `  (
 ( ocv `  h ) `  s ) ) }
 )
 
Definitiondf-thl 16847 Define the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |- toHL  =  ( h  e.  _V  |->  ( (toInc `  ( CSubSp `  h ) ) sSet  <. ( oc
 `  ndx ) ,  ( ocv `  h ) >. ) )
 
Theoremocvfval 16848* The orthocomplement operation. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  F )   &    |- 
 ._|_  =  ( ocv `  W )   =>    |-  ( W  e.  X  -> 
 ._|_  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0.  } ) )
 
Theoremocvval 16849* Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  F )   &    |- 
 ._|_  =  ( ocv `  W )   =>    |-  ( S  C_  V  ->  (  ._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
 
Theoremelocv 16850* Elementhood in the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  F )   &    |- 
 ._|_  =  ( ocv `  W )   =>    |-  ( A  e.  (  ._|_  `  S )  <->  ( S  C_  V  /\  A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
 
Theoremocvi 16851 Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  F )   &    |- 
 ._|_  =  ( ocv `  W )   =>    |-  ( ( A  e.  (  ._|_  `  S )  /\  B  e.  S ) 
 ->  ( A  .,  B )  =  .0.  )
 
Theoremocvss 16852 The orthocomplement of a subset is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  (  ._|_  `  S )  C_  V
 
Theoremocvocv 16853 A set is contained in its double orthocomplement. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  ( ( W  e.  PreHil  /\  S  C_  V )  ->  S  C_  (  ._|_  `  (  ._|_  `  S ) ) )
 
Theoremocvlss 16854 The orthocomplement of a subset is a linear subspace of the pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  ._|_  =  ( ocv `  W )   &    |-  L  =  ( LSubSp `  W )   =>    |-  (
 ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  e.  L )
 
Theoremocv2ss 16855 Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   =>    |-  ( T  C_  S  ->  (  ._|_  `  S ) 
 C_  (  ._|_  `  T ) )
 
Theoremocvin 16856 An orthocomplement has trivial intersection with the original subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   &    |-  L  =  (
 LSubSp `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  PreHil  /\  S  e.  L ) 
 ->  ( S  i^i  (  ._|_  `  S ) )  =  {  .0.  }
 )
 
Theoremocvsscon 16857 Two ways to say that  S and  T are orthogonal subspaces. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  ( ( W  e.  PreHil  /\  S  C_  V  /\  T  C_  V )  ->  ( S  C_  (  ._|_  `  T )  <->  T  C_  (  ._|_  `  S ) ) )
 
Theoremocvlsp 16858 The orthocomplement of a linear span. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  ._|_  =  ( ocv `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  ( N `
  S ) )  =  (  ._|_  `  S ) )
 
Theoremocv0 16859 The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  (  ._|_  `  (/) )  =  V
 
Theoremocvz 16860 The orthocomplement of the zero subspace. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  ._|_  =  ( ocv `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( W  e.  PreHil  ->  (  ._|_  `  {  .0.  }
 )  =  V )
 
Theoremocv1 16861 The orthocomplement of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  ._|_  =  ( ocv `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( W  e.  PreHil  ->  (  ._|_  `  V )  =  {  .0.  } )
 
Theoremunocv 16862 The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   =>    |-  (  ._|_  `  ( A  u.  B ) )  =  ( (  ._|_  `  A )  i^i  (  ._|_  `  B ) )
 
Theoremiunocv 16863* The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   &    |-  V  =  (
 Base `  W )   =>    |-  (  ._|_  `  U_ x  e.  A  B )  =  ( V  i^i  |^|_ x  e.  A  (  ._|_  `  B ) )
 
Theoremcssval 16864* The set of closed subspaces of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   &    |-  C  =  (
 CSubSp `  W )   =>    |-  ( W  e.  X  ->  C  =  {
 s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) }
 )
 
Theoremiscss 16865 The predicate "is a closed subspace" (of a pre-Hilbert space). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   &    |-  C  =  (
 CSubSp `  W )   =>    |-  ( W  e.  X  ->  ( S  e.  C 
 <->  S  =  (  ._|_  `  (  ._|_  `  S ) ) ) )
 
Theoremcssi 16866 Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   &    |-  C  =  (
 CSubSp `  W )   =>    |-  ( S  e.  C  ->  S  =  ( 
 ._|_  `  (  ._|_  `  S ) ) )
 
Theoremcssss 16867 A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  C  =  (
 CSubSp `  W )   =>    |-  ( S  e.  C  ->  S  C_  V )
 
Theoremiscss2 16868 It is sufficient to prove that the double orthocomplement is a subset of the target set to show that the set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  C  =  (
 CSubSp `  W )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  (
 ( W  e.  PreHil  /\  S  C_  V )  ->  ( S  e.  C  <->  ( 
 ._|_  `  (  ._|_  `  S ) )  C_  S ) )
 
Theoremocvcss 16869 The orthocomplement of any set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  C  =  (
 CSubSp `  W )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  (
 ( W  e.  PreHil  /\  S  C_  V )  ->  (  ._|_  `  S )  e.  C )
 
Theoremcssincl 16870 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  C  =  ( CSubSp `  W )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  C  /\  B  e.  C )  ->  ( A  i^i  B )  e.  C )
 
Theoremcss0 16871 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  C  =  ( CSubSp `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( W  e.  PreHil  ->  {  .0.  }  e.  C )
 
Theoremcss1 16872 The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  C  =  (
 CSubSp `  W )   =>    |-  ( W  e.  PreHil  ->  V  e.  C )
 
Theoremcsslss 16873 A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  C  =  ( CSubSp `  W )   &    |-  L  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  PreHil  /\  S  e.  C )  ->  S  e.  L )
 
Theoremlsmcss 16874 A subset of a pre-Hilbert space whose double orthocomplement has a projection decomposition is a closed subspace. This is the core of the proof that a topologically closed subspace is algebraically closed in a Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  C  =  ( CSubSp `  W )   &    |-  V  =  (
 Base `  W )   &    |-  ._|_  =  ( ocv `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  S  C_  V )   &    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  S ) )  C_  ( S 
 .(+)  (  ._|_  `  S ) ) )   =>    |-  ( ph  ->  S  e.  C )
 
Theoremcssmre 16875 The closed subspaces of a pre-Hilbert space are a Moore system. Unlike many of our other examples of closure systems, this one is not usually an algebraic closure system df-acs 13769: consider the Hilbert space of sequences  NN --> RR with convergent sum; the subspace of all sequences with finite support is the classic example of a non-closed subspace, but for every finite set of sequences of finite support, there is a finite-dimensional (and hence closed) subspace containing all of the sequences, so if closed subspaces were an algebraic closure system this would violate acsfiel 13834. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  C  =  (
 CSubSp `  W )   =>    |-  ( W  e.  PreHil  ->  C  e.  (Moore `  V ) )
 
Theoremmrccss 16876 The Moore closure corresponding to the system of closed subspaces is the double orthocomplement operation. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  ._|_  =  ( ocv `  W )   &    |-  C  =  ( CSubSp `  W )   &    |-  F  =  (mrCls `  C )   =>    |-  (
 ( W  e.  PreHil  /\  S  C_  V )  ->  ( F `  S )  =  (  ._|_  `  (  ._|_  `  S ) ) )
 
Theoremthlval 16877 Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  K  =  (toHL `  W )   &    |-  C  =  (
 CSubSp `  W )   &    |-  I  =  (toInc `  C )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  ( W  e.  V  ->  K  =  ( I sSet  <. ( oc `  ndx ) ,  ._|_  >. ) )
 
Theoremthlbas 16878 Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  K  =  (toHL `  W )   &    |-  C  =  (
 CSubSp `  W )   =>    |-  C  =  (
 Base `  K )
 
Theoremthlle 16879 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  K  =  (toHL `  W )   &    |-  C  =  (
 CSubSp `  W )   &    |-  I  =  (toInc `  C )   &    |-  .<_  =  ( le `  I
 )   =>    |- 
 .<_  =  ( le `  K )
 
Theoremthlleval 16880 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  K  =  (toHL `  W )   &    |-  C  =  (
 CSubSp `  W )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( S  e.  C  /\  T  e.  C )  ->  ( S  .<_  T  <->  S  C_  T ) )
 
Theoremthloc 16881 Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  K  =  (toHL `  W )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  ._|_  =  ( oc `  K )
 
10.12.3  Orthogonal projection and orthonormal bases
 
Syntaxcpj 16882 Extend class notation with orthogonal projection function.
 class  proj
 
Syntaxchs 16883 Extend class notation with class of all Hilbert spaces.
 class  Hil
 
Syntaxcobs 16884 Extend class notation with the set of orthonormal bases.
 class OBasis
 
Definitiondf-pj 16885* Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 15226, but we restrict the domain of this function to only total projection functions. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 proj  =  ( h  e.  _V  |->  ( ( x  e.  ( LSubSp `  h )  |->  ( x (
 proj 1 `  h ) ( ( ocv `  h ) `  x ) ) )  i^i  ( _V 
 X.  ( ( Base `  h )  ^m  ( Base `  h ) ) ) ) )
 
Definitiondf-hil 16886 Define class of all Hilbert spaces. Based on Proposition 4.5, p. 176, Gudrun Kalmbach, Quantum Measures and Spaces, Kluwer, Dordrecht, 1998. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 16-Oct-2015.)
 |- 
 Hil  =  { h  e.  PreHil  |  dom  ( proj `  h )  =  ( CSubSp `  h ) }
 
Definitiondf-obs 16887* Define the set of all orthonormal bases for a pre-Hilbert space. An orthonormal basis is a set of mutually orthogonal vectors with norm 1 and such that the linear span is dense in the whole space. (As this is an "algebraic" definition, before we have topology available, we express this denseness by saying that the double orthocomplement is the whole space, or equivalently, the single orthocomplement is trivial.) (Contributed by Mario Carneiro, 23-Oct-2015.)
 |- OBasis  =  ( h  e.  PreHil  |->  { b  e.  ~P ( Base `  h )  |  ( A. x  e.  b  A. y  e.  b  ( x ( .i `  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) ) 
 /\  ( ( ocv `  h ) `  b
 )  =  { ( 0g `  h ) }
 ) } )
 
Theorempjfval 16888* The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  L  =  (
 LSubSp `  W )   &    |-  ._|_  =  ( ocv `  W )   &    |-  P  =  ( proj 1 `  W )   &    |-  K  =  ( proj `  W )   =>    |-  K  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
 
Theorempjdm 16889 A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  L  =  (
 LSubSp `  W )   &    |-  ._|_  =  ( ocv `  W )   &    |-  P  =  ( proj 1 `  W )   &    |-  K  =  ( proj `  W )   =>    |-  ( T  e.  dom  K  <-> 
 ( T  e.  L  /\  ( T P ( 
 ._|_  `  T ) ) : V --> V ) )
 
Theorempjpm 16890 The projection map is a partial function from subspaces of the pre-Hilbert space to total operators. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  L  =  (
 LSubSp `  W )   &    |-  K  =  ( proj `  W )   =>    |-  K  e.  ( ( V  ^m  V )  ^pm  L )
 
Theorempjfval2 16891* Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   &    |-  P  =  (
 proj 1 `  W )   &    |-  K  =  ( proj `  W )   =>    |-  K  =  ( x  e.  dom  K  |->  ( x P (  ._|_  `  x ) ) )
 
Theorempjval 16892 Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   &    |-  P  =  (
 proj 1 `  W )   &    |-  K  =  ( proj `  W )   =>    |-  ( T  e.  dom  K 
 ->  ( K `  T )  =  ( T P (  ._|_  `  T ) ) )
 
Theorempjdm2 16893 A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  L  =  (
 LSubSp `  W )   &    |-  ._|_  =  ( ocv `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  K  =  ( proj `  W )   =>    |-  ( W  e.  PreHil  ->  ( T  e.  dom  K  <->  ( T  e.  L  /\  ( T  .(+)  ( 
 ._|_  `  T ) )  =  V ) ) )
 
Theorempjff 16894 A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  K  =  ( proj `  W )   =>    |-  ( W  e.  PreHil  ->  K : dom  K --> ( W LMHom  W ) )
 
Theorempjf 16895 A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  K  =  ( proj `  W )   &    |-  V  =  (
 Base `  W )   =>    |-  ( T  e.  dom 
 K  ->  ( K `  T ) : V --> V )
 
Theorempjf2 16896 A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  K  =  ( proj `  W )   &    |-  V  =  (
 Base `  W )   =>    |-  ( ( W  e.  PreHil  /\  T  e.  dom 
 K )  ->  ( K `  T ) : V --> T )
 
Theorempjfo 16897 A projection is a surjection onto the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  K  =  ( proj `  W )   &    |-  V  =  (
 Base `  W )   =>    |-  ( ( W  e.  PreHil  /\  T  e.  dom 
 K )  ->  ( K `  T ) : V -onto-> T )
 
Theorempjcss 16898 A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  K  =  ( proj `  W )   &    |-  C  =  (
 CSubSp `  W )   =>    |-  ( W  e.  PreHil  ->  dom  K  C_  C )
 
Theoremocvpj 16899 The orthocomplement of a projection subspace is a projection subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  K  =  ( proj `  W )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  ( ( W  e.  PreHil  /\  T  e.  dom 
 K )  ->  (  ._|_  `  T )  e. 
 dom  K )
 
Theoremishil 16900 The predicate "is a Hilbert space" (over a *-division ring). A Hilbert space is a pre-Hilbert space such that all closed subspaces have a projection decomposition. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  K  =  ( proj `  H )   &    |-  C  =  (
 CSubSp `  H )   =>    |-  ( H  e.  Hil  <->  ( H  e.  PreHil  /\  dom  K  =  C ) )
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