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Theorem List for Metamath Proof Explorer - 16501-16600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremznzrhfo 16501 The  ZZ ring homomorphism is a surjection onto 
ZZ  /  n ZZ. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  B  =  ( Base `  Y )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( N  e.  NN0  ->  L : ZZ -onto-> B )
 
Theoremzncyg 16502 The group  ZZ  /  n ZZ is cyclic for all  n (including  n  =  0). (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  Y  e. CycGrp )
 
Theoremzndvds 16503 Express equality of equivalence classes in  ZZ 
/  n ZZ in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( L `  A )  =  ( L `  B )  <->  N  ||  ( A  -  B ) ) )
 
Theoremzndvds0 16504 Special case of zndvds 16503 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   &    |-  .0.  =  ( 0g `  Y )   =>    |-  ( ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( ( L `  A )  =  .0.  <->  N  ||  A ) )
 
Theoremznf1o 16505 The function  F enumerates all equivalence classes in ℤ/nℤ for each  n. When  n  = 
0,  ZZ  /  0 ZZ  =  ZZ  /  {
0 }  ~~  ZZ so we let  W  =  ZZ; otherwise  W  =  { 0 , 
... ,  n  - 
1 } enumerates all the equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.)
 |-  Z  =  (flds  ZZ )   &    |-  Y  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Y )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   =>    |-  ( N  e.  NN0  ->  F : W -1-1-onto-> B )
 
Theoremzzngim 16506 The  ZZ ring homomorphism is an isomorphism for 
N  =  0. (We only show group isomorphism here, but ring isomorphism follows, since it is a bijective ring homomorphism.) (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Y  =  (ℤ/n `  0
 )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  L  e.  ( (flds  ZZ ) GrpIso  Y )
 
Theoremznle2 16507 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   =>    |-  ( N  e.  NN0  ->  .<_  =  ( ( F  o.  <_  )  o.  `' F ) )
 
Theoremznleval 16508 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   &    |-  X  =  ( Base `  Y )   =>    |-  ( N  e.  NN0  ->  ( A  .<_  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( `' F `  A )  <_  ( `' F `  B ) ) ) )
 
Theoremznleval2 16509 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   &    |-  X  =  ( Base `  Y )   =>    |-  ( ( N  e.  NN0  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .<_  B  <->  ( `' F `  A )  <_  ( `' F `  B ) ) )
 
Theoremzntoslem 16510 Lemma for zntos 16511. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   &    |-  X  =  ( Base `  Y )   =>    |-  ( N  e.  NN0  ->  Y  e. Toset )
 
Theoremzntos 16511 The ℤ/nℤ structure is a totally ordered set. (The order is not respected by the operations, except in the case  N  =  0 when it coincides with the ordering on  ZZ.) (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  Y  e. Toset )
 
Theoremznhash 16512 The ℤ/nℤ structure has  n elements. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  B  =  ( Base `  Y )   =>    |-  ( N  e.  NN  ->  ( # `  B )  =  N )
 
Theoremznfi 16513 The ℤ/nℤ structure is a finite ring. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  B  =  ( Base `  Y )   =>    |-  ( N  e.  NN  ->  B  e.  Fin )
 
Theoremznfld 16514 The ℤ/nℤ structure is a finite field when  n is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  Prime  ->  Y  e. Field )
 
Theoremznidomb 16515 The ℤ/nℤ structure is a domain (and hence a field) precisely when  n is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN  ->  ( Y  e. IDomn  <->  N  e.  Prime ) )
 
Theoremznchr 16516 Cyclic rings are defined by their characteristic. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  (chr `  Y )  =  N )
 
Theoremznunit 16517 The units of ℤ/nℤ are the integers coprime to the base. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  U  =  (Unit `  Y )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  (
 ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( ( L `  A )  e.  U  <->  ( A  gcd  N )  =  1 ) )
 
Theoremznunithash 16518 The size of the unit group of ℤ/nℤ. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  U  =  (Unit `  Y )   =>    |-  ( N  e.  NN  ->  ( # `  U )  =  ( phi `  N ) )
 
Theoremznrrg 16519 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 16520* Lemma for cygzn 16524. (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 16521* Lemma for cygzn 16524. (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 16522* Lemma for cygzn 16524. (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 16523* 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 16524 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 16525* 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 16526 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 16527 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 16528 Extend class notation with class all pre-Hilbert spaces.
 class  PreHil
 
Syntaxcipf 16529 Extend class notation with inner product function.
 class  .i f
 
Definitiondf-phl 16530* 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 16531* 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 14383), while  +g only has closure (mndcl 14372). (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 16532* 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 16533 A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  PreHil  ->  W  e.  LVec )
 
Theoremphllmod 16534 A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  PreHil  ->  W  e.  LMod )
 
Theoremphlsrng 16535 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 16536* 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 16537 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 16538 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 16539 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 16540 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 16541 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 16542 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 16543 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 16544 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 16545 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 16546 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 16547 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 16548 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 16549 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 16550 "Associative" law for second argument of inner product (compare ipass 16549). (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 16551 "Associative" law for inner product. Conjugate version of ipassr 16550. (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 16552* 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 16553 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 16554 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 16555 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 16556 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 16557* 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 16558* 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 16559* 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 16560 Extend class notation with orthocomplement of a subspace.
 class  ocv
 
Syntaxccss 16561 Extend class notation with set of closed subspaces.
 class  CSubSp
 
Syntaxcthl 16562 Extend class notation with the Hilbert lattice.
 class toHL
 
Definitiondf-ocv 16563* 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 16564* Define set of closed subspaces. (Contributed by NM, 7-Oct-2011.)
 |-  CSubSp  =  ( h  e. 
 _V  |->  { s  |  s  =  ( ( ocv `  h ) `  (
 ( ocv `  h ) `  s ) ) }
 )
 
Definitiondf-thl 16565 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 16566* 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 16567* 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 16568* 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 16569 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 16570 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 16571 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 16572 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 16573 Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   =>    |-  ( T  C_  S  ->  (  ._|_  `  S ) 
 C_  (  ._|_  `  T ) )
 
Theoremocvin 16574 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 16575 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 16576 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 16577 The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  (  ._|_  `  (/) )  =  V
 
Theoremocvz 16578 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 16579 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 16580 The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   =>    |-  (  ._|_  `  ( A  u.  B ) )  =  ( (  ._|_  `  A )  i^i  (  ._|_  `  B ) )
 
Theoremiunocv 16581* 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 16582* 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 16583 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 16584 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 16585 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 16586 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 16587 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 16588 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 16589 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 16590 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 16591 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 16592 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 16593 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 13491: 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 13556. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  C  =  (
 CSubSp `  W )   =>    |-  ( W  e.  PreHil  ->  C  e.  (Moore `  V ) )
 
Theoremmrccss 16594 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 16595 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 16596 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 16597 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 16598 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 16599 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 16600 Extend class notation with orthogonal projection function.
 class  proj
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