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Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremchj12i 22101 A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( A  vH  ( B  vH  C ) )  =  ( B  vH  ( A  vH  C ) )
 
Theoremchj4i 22102 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( A  vH  B )  vH  ( C  vH  D ) )  =  ( ( A  vH  C )  vH  ( B 
 vH  D ) )
 
Theoremchjjdiri 22103 Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  vH  B )  vH  C )  =  ( ( A 
 vH  C )  vH  ( B  vH  C ) )
 
Theoremchdmm1 22104 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  i^i  B ) )  =  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) )
 
Theoremchdmm2 22105 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  i^i  B ) )  =  ( A  vH  ( _|_ `  B ) ) )
 
Theoremchdmm3 22106 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  i^i  ( _|_ `  B ) ) )  =  ( ( _|_ `  A )  vH  B ) )
 
Theoremchdmm4 22107 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  i^i  ( _|_ `  B ) ) )  =  ( A  vH  B ) )
 
Theoremchdmj1 22108 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  vH  B ) )  =  ( ( _|_ `  A )  i^i  ( _|_ `  B ) ) )
 
Theoremchdmj2 22109 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  vH  B ) )  =  ( A  i^i  ( _|_ `  B ) ) )
 
Theoremchdmj3 22110 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  vH  ( _|_ `  B ) ) )  =  ( ( _|_ `  A )  i^i  B ) )
 
Theoremchdmj4 22111 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  vH  ( _|_ `  B ) ) )  =  ( A  i^i  B ) )
 
Theoremchjass 22112 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( ( A  vH  B )  vH  C )  =  ( A  vH  ( B  vH  C ) ) )
 
Theoremchj12 22113 A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  vH  ( B 
 vH  C ) )  =  ( B  vH  ( A  vH  C ) ) )
 
Theoremchj4 22114 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH )  /\  ( C  e.  CH 
 /\  D  e.  CH ) )  ->  ( ( A  vH  B ) 
 vH  ( C  vH  D ) )  =  ( ( A  vH  C )  vH  ( B 
 vH  D ) ) )
 
Theoremledii 22115 An ortholattice is distributive in one ordering direction. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  i^i  B )  vH  ( A  i^i  C ) ) 
 C_  ( A  i^i  ( B  vH  C ) )
 
Theoremlediri 22116 An ortholattice is distributive in one ordering direction. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  i^i  C )  vH  ( B  i^i  C ) ) 
 C_  ( ( A 
 vH  B )  i^i 
 C )
 
Theoremlejdii 22117 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( A  vH  ( B  i^i  C ) ) 
 C_  ( ( A 
 vH  B )  i^i  ( A  vH  C ) )
 
Theoremlejdiri 22118 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  i^i  B )  vH  C ) 
 C_  ( ( A 
 vH  C )  i^i  ( B  vH  C ) )
 
Theoremledi 22119 An ortholattice is distributive in one ordering direction. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) ) 
 C_  ( A  i^i  ( B  vH  C ) ) )
 
17.5.4  Span (cont.) and one-dimensional subspaces
 
Theoremspansn0 22120 The span of the singleton of the zero vector is the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
 |-  ( span `  { 0h }
 )  =  0H
 
Theoremspan0 22121 The span of the empty set is the zero subspace. Remark 11.6.e of [Schechter] p. 276. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  ( span `  (/) )  =  0H
 
Theoremelspani 22122* Membership in the span of a subset of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  B  e.  _V   =>    |-  ( A  C_  ~H  ->  ( B  e.  ( span `  A )  <->  A. x  e.  SH  ( A  C_  x  ->  B  e.  x )
 ) )
 
Theoremspanuni 22123 The span of a union is the subspace sum of spans. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  A  C_ 
 ~H   &    |-  B  C_  ~H   =>    |-  ( span `  ( A  u.  B ) )  =  ( ( span `  A )  +H  ( span `  B ) )
 
Theoremspanun 22124 The span of a union is the subspace sum of spans. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( span `  ( A  u.  B ) )  =  ( ( span `  A )  +H  ( span `  B ) ) )
 
Theoremsshhococi 22125 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 22126 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 22127 The supremum of the empty set. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  (  \/H  `  (/) )  =  0H
 
Theoremh1deoi 22128 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 22129* 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 22130 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 22131 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 22132 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 22133 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 22134* Lemma for h1de2ci 22135. (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 22135* 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 22136 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 22137* 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 22138 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 22139 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 22140 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 22141 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 22142 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 22143 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 22144 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 22145* 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 22146 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 22147 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 22148 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 22149 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 22150 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 22151 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 22152 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 22153 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 22154 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 22155 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 22156 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 22157 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 22158 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 22159 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 22160 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 22161 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 22162* 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 22169 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 22163 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 22164 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 22165 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 22166 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 22167 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 22168* 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 22169 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 22170 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 22171 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 22172 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 22173 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 22174 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 22175 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 22176 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 22177 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 22178 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 22179 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 22180 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 22181 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 22182 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 22183 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 22184 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 22185 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 22186 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 22187 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 22188 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 22189 The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  C_H  A
 
Theorempjoml2 22190 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 22191 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 22192 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 22193 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 22194 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 22195 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 22196 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 22197 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 22198 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 22199 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 22200 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 ) )
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