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Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremspansnsh 22101 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 22102 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 22103 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 22104 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 22105 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 22106* 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 22107 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 22108 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 22109 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 22110 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 22111 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 22112 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 22113 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 22114 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 22115 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 22116 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 22117 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 22118 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 22119 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 22120 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 22121 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 22122 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 )
 ) )
 
17.5.5  Commutes relation for Hilbert lattice elements
 
Definitiondf-cm 22123* 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 22130 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 22124 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 22125 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 22126 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 22127 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 22128 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 22129* 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 22130 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 22131 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 22132 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 22133 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 22134 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 22135 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 22136 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 22137 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 22138 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 22139 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 22140 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 22141 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 22142 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 22143 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 22144 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 22145 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 22146 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 22147 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 22148 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 22149 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 22150 The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  C_H  A
 
Theorempjoml2 22151 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 22152 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 22153 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 22154 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 22155 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 22156 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 22157 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 )
 
17.5.6  Foulis-Holland theorem
 
Theoremfh1 22158 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 22159 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 22160 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 22161 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 22162 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 22163 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 22164 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 22165 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 22166 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 )
 
17.5.7  Quantum Logic Explorer axioms
 
Theoremqlax1i 22167 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 22168 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 22169 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 22170 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 22171 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 22172 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 22173 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 22174 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 22175 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 22176 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
 
17.5.8  Orthogonal subspaces
 
Theoremchscllem1 22177* Lemma for chscl 22181. (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 22178* Lemma for chscl 22181. (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 22179* Lemma for chscl 22181. (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 22180* Lemma for chscl 22181. (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 22181 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 22182 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 21933, although "the hard part" of this proof, chscl 22181, 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 22183 Corollary of osumi 22182. (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 22184 Corollary of osumi 22182, 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 22185 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 ) )
 
Theoremspansnji 22186 The subspace sum of a closed subspace and a one-dimensional subspace equals their join. (Proof suggested by Eric Schechter 1-Jun-2004.) (Contributed by NM, 1-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  ~H   =>    |-  ( A  +H  ( span `  { B } ) )  =  ( A  vH  ( span `  { B }
 ) )
 
Theoremspansnj 22187 The subspace sum of a closed subspace and a one-dimensional subspace equals their join. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  ~H )  ->  ( A  +H  ( span `  { B }
 ) )  =  ( A  vH  ( span ` 
 { B } )
 ) )
 
Theoremspansnscl 22188 The subspace sum of a closed subspace and a one-dimensional subspace is closed. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  ~H )  ->  ( A  +H  ( span `  { B }
 ) )  e.  CH )
 
Theoremsumspansn 22189 The sum of two vectors belong to the span of one of them iff the other vector also belongs. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  e.  ( span `  { A }
 ) 
 <->  B  e.  ( span ` 
 { A } )
 ) )
 
Theoremspansnm0i 22190 The meet of different one-dimensional subspaces is the zero subspace. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( -.  A  e.  ( span ` 
 { B } )  ->  ( ( span `  { A } )  i^i  ( span ` 
 { B } )
 )  =  0H )
 
Theoremnonbooli 22191 A Hilbert lattice with two or more dimensions fails the distributive law and therefore cannot be a Boolean algebra. This counterexample demonstrates a condition where 
( ( H  i^i  F )  vH  ( H  i^i  G ) )  =  0H but  ( H  i^i  ( F  vH  G ) )  =/=  0H. The antecedent specifies that the vectors  A and  B are nonzero and non-colinear. The last three hypotheses assign one-dimensional subspaces to  F,  G, and  H. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  F  =  ( span `  { A }
 )   &    |-  G  =  ( span ` 
 { B } )   &    |-  H  =  ( span `  { ( A  +h  B ) }
 )   =>    |-  ( -.  ( A  e.  G  \/  B  e.  F )  ->  ( H  i^i  ( F  vH  G ) )  =/=  ( ( H  i^i  F )  vH  ( H  i^i  G ) ) )
 
Theoremspansncvi 22192 Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of [Kalmbach] p. 153. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  ~H   =>    |-  ( ( A  C.  B  /\  B  C_  ( A  vH  ( span `  { C } ) ) ) 
 ->  B  =  ( A 
 vH  ( span `  { C } ) ) )
 
Theoremspansncv 22193 Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of [Kalmbach] p. 153. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  ~H )  ->  ( ( A  C.  B  /\  B  C_  ( A  vH  ( span `  { C } ) ) ) 
 ->  B  =  ( A 
 vH  ( span `  { C } ) ) ) )
 
17.5.9  Orthoarguesian laws 5OA and 3OA
 
Theorem5oalem1 22194 Lemma for orthoarguesian law 5OA. (Contributed by NM, 1-Apr-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  R  e.  SH   =>    |-  (
 ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y
 ) )  /\  (
 z  e.  C  /\  ( x  -h  z
 )  e.  R ) )  ->  v  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) ) )
 
Theorem5oalem2 22195 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   =>    |-  (
 ( ( ( x  e.  A  /\  y  e.  B )  /\  (
 z  e.  C  /\  w  e.  D )
 )  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( x  -h  z )  e.  (
 ( A  +H  C )  i^i  ( B  +H  D ) ) )
 
Theorem5oalem3 22196 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   &    |-  F  e.  SH   &    |-  G  e.  SH   =>    |-  (
 ( ( ( ( x  e.  A  /\  y  e.  B )  /\  ( z  e.  C  /\  w  e.  D ) )  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  (
 f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) ) 
 ->  ( x  -h  z
 )  e.  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G ) ) ) )
 
Theorem5oalem4 22197 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   &    |-  F  e.  SH   &    |-  G  e.  SH   =>    |-  (
 ( ( ( ( x  e.  A  /\  y  e.  B )  /\  ( z  e.  C  /\  w  e.  D ) )  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  (
 f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) ) 
 ->  ( x  -h  z
 )  e.  ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  (
 ( C  +H  F )  i^i  ( D  +H  G ) ) ) ) )
 
Theorem5oalem5 22198 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-May-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   &    |-  F  e.  SH   &    |-  G  e.  SH   &    |-  R  e.  SH   &    |-  S  e.  SH   =>    |-  (
 ( ( ( ( x  e.  A  /\  y  e.  B )  /\  ( z  e.  C  /\  w  e.  D ) )  /\  ( ( f  e.  F  /\  g  e.  G )  /\  ( v  e.  R  /\  u  e.  S ) ) )  /\  ( ( ( x  +h  y )  =  ( v  +h  u )  /\  ( z  +h  w )  =  (
 v  +h  u )
 )  /\  ( f  +h  g )  =  ( v  +h  u ) ) )  ->  ( x  -h  z )  e.  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  (
 ( C  +H  R )  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
 ( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) )
 
Theorem5oalem6 22199 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   &    |-  F  e.  SH   &    |-  G  e.  SH   &    |-  R  e.  SH   &    |-  S  e.  SH   =>    |-  (
 ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) ) 
 /\  ( ( z  e.  C  /\  w  e.  D )  /\  h  =  ( z  +h  w ) ) )  /\  ( ( ( f  e.  F  /\  g  e.  G )  /\  h  =  ( f  +h  g
 ) )  /\  (
 ( v  e.  R  /\  u  e.  S )  /\  h  =  ( v  +h  u ) ) ) )  ->  h  e.  ( B  +H  ( A  i^i  ( C  +H  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  (
 ( C  +H  R )  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
 ( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) ) ) ) )
 
Theorem5oalem7 22200 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   &    |-  F  e.  SH   &    |-  G  e.  SH   &    |-  R  e.  SH   &    |-  S  e.  SH   =>    |-  (
 ( ( A  +H  B )  i^i  ( C  +H  D ) )  i^i  ( ( F  +H  G )  i^i  ( R  +H  S ) ) )  C_  ( B  +H  ( A  i^i  ( C  +H  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  (
 ( C  +H  R )  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
 ( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) ) ) )
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