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Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsshhococi 23001 The join of two Hilbert space subsets (not necessarily closed subspaces) equals the join of their closures (double orthocomplements). (Contributed by NM, 1-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  C_ 
 ~H   &    |-  B  C_  ~H   =>    |-  ( A  vH  B )  =  ( ( _|_ `  ( _|_ `  A ) )  vH  ( _|_ `  ( _|_ `  B ) ) )
 
Theoremhne0 23002 Hilbert space has a nonzero vector iff it is not trivial. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
 |-  ( ~H  =/=  0H  <->  E. x  e.  ~H  x  =/=  0h )
 
Theoremchsup0 23003 The supremum of the empty set. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  (  \/H  `  (/) )  =  0H
 
Theoremh1deoi 23004 Membership in orthocomplement of 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (New usage is discouraged.)
 |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  { B }
 ) 
 <->  ( A  e.  ~H  /\  ( A  .ih  B )  =  0 )
 )
 
Theoremh1dei 23005* Membership in 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  ( A  e.  ~H 
 /\  A. x  e.  ~H  ( ( B  .ih  x )  =  0  ->  ( A  .ih  x )  =  0 ) ) )
 
Theoremh1did 23006 A generating vector belongs to the 1-dimensional subspace it generates. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  A  e.  ( _|_ `  ( _|_ `  { A }
 ) ) )
 
Theoremh1dn0 23007 A nonzero vector generates a (nonzero) 1-dimensional subspace. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  ( _|_ `  ( _|_ `  { A }
 ) )  =/=  0H )
 
Theoremh1de2i 23008 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 17-Jul-2001.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  ->  ( ( B  .ih  B )  .h  A )  =  ( ( A 
 .ih  B )  .h  B ) )
 
Theoremh1de2bi 23009 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( B  =/=  0h  ->  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  A  =  (
 ( ( A  .ih  B )  /  ( B 
 .ih  B ) )  .h  B ) ) )
 
Theoremh1de2ctlem 23010* Lemma for h1de2ci 23011. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  E. x  e.  CC  A  =  ( x  .h  B ) )
 
Theoremh1de2ci 23011* Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 21-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  E. x  e.  CC  A  =  ( x  .h  B ) )
 
Theoremspansni 23012 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( span `  { A }
 )  =  ( _|_ `  ( _|_ `  { A } ) )
 
Theoremelspansni 23013* Membership in the span of a singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( B  e.  ( span `  { A }
 ) 
 <-> 
 E. x  e.  CC  B  =  ( x  .h  A ) )
 
Theoremspansn 23014 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 span `  { A }
 )  =  ( _|_ `  ( _|_ `  { A } ) ) )
 
Theoremspansnch 23015 The span of a Hilbert space singleton belongs to the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 span `  { A }
 )  e.  CH )
 
Theoremspansnsh 23016 The span of a Hilbert space singleton is a subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 span `  { A }
 )  e.  SH )
 
Theoremspansnchi 23017 The span of a singleton in Hilbert space is a closed subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( span `  { A }
 )  e.  CH
 
Theoremspansnid 23018 A vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  A  e.  ( span `  { A } ) )
 
Theoremspansnmul 23019 A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  CC )  ->  ( B  .h  A )  e.  ( span ` 
 { A } )
 )
 
Theoremelspansncl 23020 A member of a span of a singleton is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ( span ` 
 { A } )
 )  ->  B  e.  ~H )
 
Theoremelspansn 23021* Membership in the span of a singleton. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( B  e.  ( span ` 
 { A } )  <->  E. x  e.  CC  B  =  ( x  .h  A ) ) )
 
Theoremelspansn2 23022 Membership in the span of a singleton. All members are collinear with the generating vector. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  B  =/=  0h )  ->  ( A  e.  ( span `  { B }
 ) 
 <->  A  =  ( ( ( A  .ih  B )  /  ( B  .ih  B ) )  .h  B ) ) )
 
Theoremspansncol 23023 The singletons of collinear vectors have the same span. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( span `  { ( B  .h  A ) }
 )  =  ( span ` 
 { A } )
 )
 
Theoremspansneleqi 23024 Membership relation implied by equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( ( span `  { A }
 )  =  ( span ` 
 { B } )  ->  A  e.  ( span ` 
 { B } )
 ) )
 
Theoremspansneleq 23025 Membership relation that implies equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( B  e.  ~H  /\  A  =/=  0h )  ->  ( A  e.  ( span `  { B }
 )  ->  ( span ` 
 { A } )  =  ( span `  { B }
 ) ) )
 
Theoremspansnss 23026 The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  A ) 
 ->  ( span `  { B }
 )  C_  A )
 
Theoremelspansn3 23027 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  A  /\  C  e.  ( span ` 
 { B } )
 )  ->  C  e.  A )
 
Theoremelspansn4 23028 A span membership condition implying two vectors belong to the same subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  SH  /\  B  e.  ~H )  /\  ( C  e.  ( span `  { B }
 )  /\  C  =/=  0h ) )  ->  ( B  e.  A  <->  C  e.  A ) )
 
Theoremelspansn5 23029 A vector belonging to both a subspace and the span of the singleton of a vector not in it must be zero. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( ( ( B  e.  ~H 
 /\  -.  B  e.  A )  /\  ( C  e.  ( span `  { B } )  /\  C  e.  A ) )  ->  C  =  0h )
 )
 
Theoremspansnss2 23030 The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  ~H )  ->  ( B  e.  A  <->  (
 span `  { B }
 )  C_  A )
 )
 
Theoremnormcan 23031 Cancellation-type law that "extracts" a vector  A from its inner product with a proportional vector  B. (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( B  e.  ~H  /\  B  =/=  0h  /\  A  e.  ( span ` 
 { B } )
 )  ->  ( (
 ( A  .ih  B )  /  ( ( normh `  B ) ^ 2
 ) )  .h  B )  =  A )
 
Theorempjspansn 23032 A projection on the span of a singleton. (The proof ws shortened by Mario Carneiro, 15-Dec-2013.) (Contributed by NM, 28-May-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  ->  ( ( proj  h `  ( span `  { A }
 ) ) `  B )  =  ( (
 ( B  .ih  A )  /  ( ( normh `  A ) ^ 2
 ) )  .h  A ) )
 
Theoremspansnpji 23033 A subset of Hilbert space is orthogonal to the span of the singleton of a projection onto its orthocomplement. (Contributed by NM, 4-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  C_ 
 ~H   &    |-  B  e.  ~H   =>    |-  A  C_  ( _|_ `  ( span `  { (
 ( proj  h `  ( _|_ `  A ) ) `
  B ) }
 ) )
 
Theoremspanunsni 23034 The span of the union of a closed subspace with a singleton equals the span of its union with an orthogonal singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  ~H   =>    |-  ( span `  ( A  u.  { B } ) )  =  ( span `  ( A  u.  { ( (
 proj  h `  ( _|_ `  A ) ) `  B ) } )
 )
 
Theoremspanpr 23035 The span of a pair of vectors. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( span `  { ( A  +h  B ) }
 )  C_  ( span ` 
 { A ,  B } ) )
 
Theoremh1datomi 23036 A 1-dimensional subspace is an atom. (Contributed by NM, 20-Jul-2001.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  ~H   =>    |-  ( A  C_  ( _|_ `  ( _|_ `  { B }
 ) )  ->  ( A  =  ( _|_ `  ( _|_ `  { B } ) )  \/  A  =  0H )
 )
 
Theoremh1datom 23037 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  ~H )  ->  ( A  C_  ( _|_ `  ( _|_ `  { B } ) )  ->  ( A  =  ( _|_ `  ( _|_ `  { B } ) )  \/  A  =  0H )
 ) )
 
18.5.5  Commutes relation for Hilbert lattice elements
 
Definitiondf-cm 23038* Define the commutes relation (on the Hilbert lattice). Definition of commutes in [Kalmbach] p. 20, who uses the notation xCy for "x commutes with y." See cmbri 23045 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
 |-  C_H  =  { <. x ,  y >.  |  ( ( x  e.  CH  /\  y  e. 
 CH )  /\  x  =  ( ( x  i^i  y )  vH  ( x  i^i  ( _|_ `  y
 ) ) ) ) }
 
Theoremcmbr 23039 Binary relation expressing  A commutes with  B. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B ) ) ) ) )
 
Theorempjoml2i 23040 Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  ->  ( A  vH  ( ( _|_ `  A )  i^i  B ) )  =  B )
 
Theorempjoml3i 23041 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( B  C_  A  ->  ( A  i^i  ( ( _|_ `  A )  vH  B ) )  =  B )
 
Theorempjoml4i 23042 Variation of orthomodular law. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  ( B  i^i  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) ) )  =  ( A  vH  B )
 
Theorempjoml5i 23043 The orthomodular law. Remark in [Kalmbach] p. 22. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  ( ( _|_ `  A )  i^i  ( A  vH  B ) ) )  =  ( A 
 vH  B )
 
Theorempjoml6i 23044* An equivalent of the orthomodular law. Theorem 29.13(e) of [MaedaMaeda] p. 132. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  ->  E. x  e.  CH  ( A  C_  ( _|_ `  x )  /\  ( A  vH  x )  =  B )
 )
 
Theoremcmbri 23045 Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  A  =  (
 ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B )
 ) ) )
 
Theoremcmcmlem 23046 Commutation is symmetric. Theorem 3.4 of [Beran] p. 45. (Contributed by NM, 3-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  ->  B  C_H  A )
 
Theoremcmcmi 23047 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  B  C_H  A )
 
Theoremcmcm2i 23048 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  A  C_H  ( _|_ `  B ) )
 
Theoremcmcm3i 23049 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  ( _|_ `  A )  C_H  B )
 
Theoremcmcm4i 23050 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  ( _|_ `  A )  C_H  ( _|_ `  B ) )
 
Theoremcmbr2i 23051 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  A  =  (
 ( A  vH  B )  i^i  ( A  vH  ( _|_ `  B )
 ) ) )
 
Theoremcmcmii 23052 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  C_H  B   =>    |-  B  C_H  A
 
Theoremcmcm2ii 23053 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  C_H  B   =>    |-  A  C_H  ( _|_ `  B )
 
Theoremcmcm3ii 23054 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  C_H  B   =>    |-  ( _|_ `  A )  C_H  B
 
Theoremcmbr3i 23055 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  ( A  i^i  ( ( _|_ `  A )  vH  B ) )  =  ( A  i^i  B ) )
 
Theoremcmbr4i 23056 Alternate definition for the commutes relation. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  ( A  i^i  ( ( _|_ `  A )  vH  B ) ) 
 C_  B )
 
Theoremlecmi 23057 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  ->  A  C_H  B )
 
Theoremlecmii 23058 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  C_  B   =>    |-  A  C_H  B
 
Theoremcmj1i 23059 A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  A  C_H  ( A  vH  B )
 
Theoremcmj2i 23060 A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  B  C_H  ( A  vH  B )
 
Theoremcmm1i 23061 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  A  C_H  ( A  i^i  B )
 
Theoremcmm2i 23062 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  B  C_H  ( A  i^i  B )
 
Theoremcmbr3 23063 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  ( A  i^i  ( ( _|_ `  A )  vH  B ) )  =  ( A  i^i  B ) ) )
 
Theoremcm0 23064 The zero Hilbert lattice element commutes with every element. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  0H  C_H 
 A )
 
Theoremcmidi 23065 The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  C_H  A
 
Theorempjoml2 23066 Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  ( A  vH  ( ( _|_ `  A )  i^i  B ) )  =  B )
 
Theorempjoml3 23067 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( B  C_  A  ->  ( A  i^i  (
 ( _|_ `  A )  vH  B ) )  =  B ) )
 
Theorempjoml5 23068 The orthomodular law. Remark in [Kalmbach] p. 22. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  (
 ( _|_ `  A )  i^i  ( A  vH  B ) ) )  =  ( A  vH  B ) )
 
Theoremcmcm 23069 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  B 
 C_H  A ) )
 
Theoremcmcm3 23070 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  ( _|_ `  A )  C_H  B ) )
 
Theoremcmcm2 23071 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A 
 C_H  ( _|_ `  B ) ) )
 
Theoremlecm 23072 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  A  C_H  B )
 
18.5.6  Foulis-Holland theorem
 
Theoremfh1 23073 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. First of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( A  i^i  ( B  vH  C ) )  =  (
 ( A  i^i  B )  vH  ( A  i^i  C ) ) )
 
Theoremfh2 23074 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Second of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( A  i^i  ( B  vH  C ) )  =  (
 ( A  i^i  B )  vH  ( A  i^i  C ) ) )
 
Theoremcm2j 23075 A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  A  C_H  ( B  vH  C ) )
 
Theoremfh1i 23076 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. First of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  ( A  i^i  ( B  vH  C ) )  =  ( ( A  i^i  B )  vH  ( A  i^i  C ) )
 
Theoremfh2i 23077 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Second of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  ( B  i^i  ( A  vH  C ) )  =  ( ( B  i^i  A )  vH  ( B  i^i  C ) )
 
Theoremfh3i 23078 Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  ( A  vH  ( B  i^i  C ) )  =  ( ( A 
 vH  B )  i^i  ( A  vH  C ) )
 
Theoremfh4i 23079 Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  ( B  vH  ( A  i^i  C ) )  =  ( ( B 
 vH  A )  i^i  ( B  vH  C ) )
 
Theoremcm2ji 23080 A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 11-May-2009.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  A  C_H  ( B 
 vH  C )
 
Theoremcm2mi 23081 A lattice element that commutes with two others also commutes with their meet. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 11-May-2009.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  A  C_H  ( B  i^i  C )
 
18.5.7  Quantum Logic Explorer axioms
 
Theoremqlax1i 23082 One of the equations showing  CH is an ortholattice. (This corresponds to axiom "ax-1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  =  ( _|_ `  ( _|_ `  A ) )
 
Theoremqlax2i 23083 One of the equations showing  CH is an ortholattice. (This corresponds to axiom "ax-2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  B )  =  ( B  vH  A )
 
Theoremqlax3i 23084 One of the equations showing  CH is an ortholattice. (This corresponds to axiom "ax-3" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  vH  B )  vH  C )  =  ( A  vH  ( B  vH  C ) )
 
Theoremqlax4i 23085 One of the equations showing  CH is an ortholattice. (This corresponds to axiom "ax-4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  ( B  vH  ( _|_ `  B )
 ) )  =  ( B  vH  ( _|_ `  B ) )
 
Theoremqlax5i 23086 One of the equations showing  CH is an ortholattice. (This corresponds to axiom "ax-5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  ( _|_ `  (
 ( _|_ `  A )  vH  B ) ) )  =  A
 
Theoremqlaxr1i 23087 One of the conditions showing 
CH is an ortholattice. (This corresponds to axiom "ax-r1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  =  B   =>    |-  B  =  A
 
Theoremqlaxr2i 23088 One of the conditions showing 
CH is an ortholattice. (This corresponds to axiom "ax-r2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  =  B   &    |-  B  =  C   =>    |-  A  =  C
 
Theoremqlaxr4i 23089 One of the conditions showing 
CH is an ortholattice. (This corresponds to axiom "ax-r4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  =  B   =>    |-  ( _|_ `  A )  =  ( _|_ `  B )
 
Theoremqlaxr5i 23090 One of the conditions showing 
CH is an ortholattice. (This corresponds to axiom "ax-r5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  =  B   =>    |-  ( A  vH  C )  =  ( B  vH  C )
 
Theoremqlaxr3i 23091 A variation of the orthomodular law, showing  CH is an orthomodular lattice. (This corresponds to axiom "ax-r3" in the Quantum Logic Explorer.) (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  ( C  vH  ( _|_ `  C )
 )  =  ( ( _|_ `  ( ( _|_ `  A )  vH  ( _|_ `  B )
 ) )  vH  ( _|_ `  ( A  vH  B ) ) )   =>    |-  A  =  B
 
18.5.8  Orthogonal subspaces
 
Theoremchscllem1 23092* Lemma for chscl 23096. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  CH )   &    |-  ( ph  ->  B  e.  CH )   &    |-  ( ph  ->  B 
 C_  ( _|_ `  A ) )   &    |-  ( ph  ->  H : NN --> ( A  +H  B ) )   &    |-  ( ph  ->  H  ~~>v  u )   &    |-  F  =  ( n  e.  NN  |->  ( ( proj  h `
  A ) `  ( H `  n ) ) )   =>    |-  ( ph  ->  F : NN --> A )
 
Theoremchscllem2 23093* Lemma for chscl 23096. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  CH )   &    |-  ( ph  ->  B  e.  CH )   &    |-  ( ph  ->  B 
 C_  ( _|_ `  A ) )   &    |-  ( ph  ->  H : NN --> ( A  +H  B ) )   &    |-  ( ph  ->  H  ~~>v  u )   &    |-  F  =  ( n  e.  NN  |->  ( ( proj  h `
  A ) `  ( H `  n ) ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>v  )
 
Theoremchscllem3 23094* Lemma for chscl 23096. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  CH )   &    |-  ( ph  ->  B  e.  CH )   &    |-  ( ph  ->  B 
 C_  ( _|_ `  A ) )   &    |-  ( ph  ->  H : NN --> ( A  +H  B ) )   &    |-  ( ph  ->  H  ~~>v  u )   &    |-  F  =  ( n  e.  NN  |->  ( ( proj  h `
  A ) `  ( H `  n ) ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  D  e.  B )   &    |-  ( ph  ->  ( H `  N )  =  ( C  +h  D ) )   =>    |-  ( ph  ->  C  =  ( F `  N ) )
 
Theoremchscllem4 23095* Lemma for chscl 23096. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  CH )   &    |-  ( ph  ->  B  e.  CH )   &    |-  ( ph  ->  B 
 C_  ( _|_ `  A ) )   &    |-  ( ph  ->  H : NN --> ( A  +H  B ) )   &    |-  ( ph  ->  H  ~~>v  u )   &    |-  F  =  ( n  e.  NN  |->  ( ( proj  h `
  A ) `  ( H `  n ) ) )   &    |-  G  =  ( n  e.  NN  |->  ( ( proj  h `  B ) `  ( H `  n ) ) )   =>    |-  ( ph  ->  u  e.  ( A  +H  B ) )
 
Theoremchscl 23096 The subspace sum of two closed orthogonal spaces is closed. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  CH )   &    |-  ( ph  ->  B  e.  CH )   &    |-  ( ph  ->  B 
 C_  ( _|_ `  A ) )   =>    |-  ( ph  ->  ( A  +H  B )  e. 
 CH )
 
Theoremosumi 23097 If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67. Note that the (countable) Axiom of Choice is used for this proof via pjhth 22848, although "the hard part" of this proof, chscl 23096, requires no choice. (Contributed by NM, 28-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  ( _|_ `  B )  ->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremosumcori 23098 Corollary of osumi 23097. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  i^i  B )  +H  ( A  i^i  ( _|_ `  B )
 ) )  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B )
 ) )
 
Theoremosumcor2i 23099 Corollary of osumi 23097, showing it holds under the weaker hypothesis that  A and  B commute. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  ->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremosum 23100 If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  ( _|_ `  B ) )  ->  ( A  +H  B )  =  ( A  vH  B ) )
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