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Theorem List for Metamath Proof Explorer - 16401-16500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremocvi 16401 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 16402 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 16403 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 16404 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 16405 Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   =>    |-  ( T  C_  S  ->  (  ._|_  `  S ) 
 C_  (  ._|_  `  T ) )
 
Theoremocvin 16406 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 16407 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 16408 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 16409 The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  (  ._|_  `  (/) )  =  V
 
Theoremocvz 16410 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 16411 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 16412 The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   =>    |-  (  ._|_  `  ( A  u.  B ) )  =  ( (  ._|_  `  A )  i^i  (  ._|_  `  B ) )
 
Theoremiunocv 16413* 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 16414* 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 16415 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 16416 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 16417 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 16418 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 16419 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 16420 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 16421 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 16422 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 16423 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 16424 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 16425 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 13363: 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 13401. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  C  =  (
 CSubSp `  W )   =>    |-  ( W  e.  PreHil  ->  C  e.  (Moore `  V ) )
 
Theoremmrccss 16426 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 16427 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 16428 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 16429 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 16430 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 16431 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 16432 Extend class notation with orthogonal projection function.
 class  proj
 
Syntaxchs 16433 Extend class notation with class of all Hilbert spaces.
 class  Hil
 
Syntaxcobs 16434 Extend class notation with the set of orthonormal bases.
 class OBasis
 
Definitiondf-pj 16435* Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 14783, 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 16436 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 16437* 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 16438* 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 16439 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 16440 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 16441* 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 16442 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 16443 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 16444 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 16445 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 16446 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 16447 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 16448 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 16449 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 16450 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 ) )
 
Theoremishil2 16451* The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)
 |-  V  =  ( Base `  H )   &    |-  .(+)  =  ( LSSum `  H )   &    |-  ._|_  =  ( ocv `  H )   &    |-  C  =  ( CSubSp `  H )   =>    |-  ( H  e.  Hil  <->  ( H  e.  PreHil  /\ 
 A. s  e.  C  ( s  .(+)  (  ._|_  `  s ) )  =  V ) )
 
Theoremisobs 16452* The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .1.  =  ( 1r `  F )   &    |- 
 .0.  =  ( 0g `  F )   &    |-  ._|_  =  ( ocv `  W )   &    |-  Y  =  ( 0g `  W )   =>    |-  ( B  e.  (OBasis `  W )  <->  ( W  e.  PreHil  /\  B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } )
 ) )
 
Theoremobsip 16453 The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .1.  =  ( 1r `  F )   &    |- 
 .0.  =  ( 0g `  F )   =>    |-  ( ( B  e.  (OBasis `  W )  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .,  Q )  =  if ( P  =  Q ,  .1.  ,  .0.  ) )
 
Theoremobsipid 16454 A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |- 
 .,  =  ( .i
 `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .1.  =  ( 1r `  F )   =>    |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B )  ->  ( A  .,  A )  =  .1.  )
 
Theoremobsrcl 16455 Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  ( B  e.  (OBasis `  W )  ->  W  e.  PreHil )
 
Theoremobsss 16456 An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  V  =  ( Base `  W )   =>    |-  ( B  e.  (OBasis `  W )  ->  B  C_  V )
 
Theoremobsne0 16457 A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |- 
 .0.  =  ( 0g `  W )   =>    |-  ( ( B  e.  (OBasis `  W )  /\  A  e.  B )  ->  A  =/=  .0.  )
 
Theoremobsocv 16458 An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  ._|_  =  ( ocv `  W )   =>    |-  ( B  e.  (OBasis `  W )  ->  (  ._|_  `  B )  =  {  .0.  } )
 
Theoremobs2ocv 16459 The double orthocomplement (closure) of an orthonormal basis is the whole space. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   &    |-  V  =  (
 Base `  W )   =>    |-  ( B  e.  (OBasis `  W )  ->  (  ._|_  `  (  ._|_  `  B ) )  =  V )
 
Theoremobselocv 16460 A basis element is in the orthocomplement of a subset of the basis iff it is not in the subset. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |- 
 ._|_  =  ( ocv `  W )   =>    |-  ( ( B  e.  (OBasis `  W )  /\  C  C_  B  /\  A  e.  B )  ->  ( A  e.  (  ._|_  `  C )  <->  -.  A  e.  C ) )
 
Theoremobs2ss 16461 A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.)
 |-  ( ( B  e.  (OBasis `  W )  /\  C  e.  (OBasis `  W )  /\  C  C_  B )  ->  C  =  B )
 
Theoremobslbs 16462 An orthogonal basis is a linear basis iff the span of the basis elements is closed (which is usually not true). (Contributed by Mario Carneiro, 29-Oct-2015.)
 |-  J  =  (LBasis `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  C  =  (
 CSubSp `  W )   =>    |-  ( B  e.  (OBasis `  W )  ->  ( B  e.  J  <->  ( N `  B )  e.  C ) )
 
PART 11  BASIC TOPOLOGY
 
11.1  Topology
 
11.1.1  Topological spaces
 
Syntaxctop 16463 Extend class notation with the class of all topologies.
 class  Top
 
Syntaxctopon 16464 The class function of all topologies over a base set.
 class TopOn
 
SyntaxctpsOLD 16465 Extend class notation with the class of all topological spaces.
 class  TopSp OLD
 
Syntaxctps 16466 Extend class notation with the class of all topological spaces.
 class  TopSp
 
Syntaxctb 16467 Extend class notation with the class of all topological bases.
 class  TopBases
 
Definitiondf-top 16468* Define the (proper) class of all topologies. See istop2g 16474 for an alternate way to express finite intersection and istps5OLD 16494 for a standard definition in terms of both members of a topological space. (Contributed by NM, 3-Mar-2006.)
 |- 
 Top  =  { x  |  ( A. y  e. 
 ~P  x U. y  e.  x  /\  A. y  e.  x  A. z  e.  x  ( y  i^i  z )  e.  x ) }
 
Definitiondf-topspOLD 16469* Define the class of all topological spaces, each of which is an ordered pair the second of which is a topology on the first. See istps5OLD 16494 for a standard way to express a topological space. (Contributed by NM, 8-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 TopSp OLD  =  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y ) }
 
Definitiondf-bases 16470* Define the class of all topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 16518). Note that "bases" is the plural of "basis." (Contributed by NM, 17-Jul-2006.)
 |-  TopBases 
 =  { x  |  A. y  e.  x  A. z  e.  x  ( y  i^i  z ) 
 C_  U. ( x  i^i  ~P ( y  i^i  z
 ) ) }
 
Definitiondf-topon 16471* Define the set of topologies with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- TopOn  =  ( b  e.  _V  |->  { j  e.  Top  |  b  =  U. j }
 )
 
Definitiondf-topsp 16472 Define the class of all topological spaces (structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
 |- 
 TopSp  =  { f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) ) }
 
Theoremistopg 16473* Express the predicate " J is a topology." Note: In the literature, a topology is often represented by a script letter T, which resembles the letter J. This confusion may have led to J being used by some authors - e.g. K. D. Joshi, Introduction to General Topology (1983), p. 114 - and it is convenient for us since we later use  T to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  ( J  e.  A  ->  ( J  e.  Top  <->  ( A. x ( x  C_  J  ->  U. x  e.  J )  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J ) ) )
 
Theoremistop2g 16474* Express the predicate " J is a topology," using "the intersection of the elements of any finite subcollection" instead of the intersection of any two elements. (Contributed by NM, 19-Jul-2006.)
 |-  ( J  e.  A  ->  ( J  e.  Top  <->  ( A. x ( x  C_  J  ->  U. x  e.  J )  /\  A. x ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
 
Theoremuniopn 16475 The union of a subset of a topology is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
 |-  ( ( J  e.  Top  /\  A  C_  J )  ->  U. A  e.  J )
 
Theoremiunopn 16476* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
 |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  J )  -> 
 U_ x  e.  A  B  e.  J )
 
Theoreminopn 16477 The intersection of two open sets of a topology is also an open set. (Contributed by NM, 17-Jul-2006.)
 |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  i^i  B )  e.  J )
 
Theoremfitop 16478 A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)
 |-  ( J  e.  Top  ->  ( fi `  J )  =  J )
 
Theoremfiinopn 16479 The intersection of a non-empty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
 |-  ( J  e.  Top  ->  ( ( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  |^| A  e.  J ) )
 
Theoremiinopn 16480* The intersection of a non-empty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  |^|_
 x  e.  A  B  e.  J )
 
Theoremunopn 16481 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  u.  B )  e.  J )
 
Theorem0opn 16482 The empty set is an open subset of a topology. (Contributed by Stefan Allan, 27-Feb-2006.)
 |-  ( J  e.  Top  ->  (/) 
 e.  J )
 
Theorem0ntop 16483 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
 |- 
 -.  (/)  e.  Top
 
Theoremtopopn 16484 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  X  e.  J )
 
Theoremeltopss 16485 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  J ) 
 ->  A  C_  X )
 
Theoremriinopn 16486* A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  J )
 
Theoremrintopn 16487 A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  J  /\  A  e.  Fin )  ->  ( X  i^i  |^| A )  e.  J )
 
TheoremeltopspOLD 16488 Construct a topological space from a topology and vice-versa. We say that  A is a topology on  U. A. (This could be proved more efficiently from istpsOLD 16490, but the proof here does not require the Axiom of Regularity.) (Contributed by NM, 8-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. U. J ,  J >.  e.  TopSp OLD  <->  J  e.  Top )
 
TheoremtpsexOLD 16489 Existence implied by membership in a topological space. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  ->  ( A  e.  _V  /\  J  e.  _V ) )
 
TheoremistpsOLD 16490 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  <->  ( J  e.  Top  /\  A  =  U. J ) )
 
Theoremistps2OLD 16491 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  <->  ( ( J  e.  Top  /\  J  C_  ~P A )  /\  ( (/) 
 e.  J  /\  A  e.  J ) ) )
 
Theoremistps3OLD 16492* A standard textbook definition of a topological space. (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  <->  ( ( J 
 C_  ~P A  /\  (/)  e.  J  /\  A  e.  J ) 
 /\  ( A. x ( x  C_  J  ->  U. x  e.  J ) 
 /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J ) ) )
 
Theoremistps4OLD 16493* A standard textbook definition of a topological space. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  <->  ( ( J 
 C_  ~P A  /\  (/)  e.  J  /\  A  e.  J ) 
 /\  ( A. x ( x  C_  J  ->  U. x  e.  J ) 
 /\  A. x ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
 
Theoremistps5OLD 16494* A standard textbook definition of a topological space  <. A ,  J >.: a topology on  A is a collection  J of subsets of  A such that  (/) and  A are in  J and that is closed under union and finite intersection. Definition of topological space in [Munkres] p. 76. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSp OLD  <->  ( ( A. x  e.  J  x  C_  A  /\  (/)  e.  J  /\  A  e.  J ) 
 /\  ( A. x ( x  C_  J  ->  U. x  e.  J ) 
 /\  A. x ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
 
Theoremistopon 16495 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  <->  ( J  e.  Top  /\  B  =  U. J ) )
 
Theoremtopontop 16496 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  J  e.  Top )
 
Theoremtoponuni 16497 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  B  =  U. J )
 
Theoremtoponmax 16498 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  B  e.  J )
 
Theoremtoponss 16499 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  J )  ->  A  C_  X )
 
Theoremtoponcom 16500 If  K is a topology on the base set of topology  J, then  J is a topology on the base of  K. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( ( J  e.  Top  /\  K  e.  (TopOn `  U. J ) )  ->  J  e.  (TopOn `  U. K ) )
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