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Theorem List for Metamath Proof Explorer - 22801-22900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremstcltr2i 22801* Property of a strong classical state. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( S  e.  States  /\ 
 A. x  e.  CH  A. y  e.  CH  (
 ( ( S `  x )  =  1  ->  ( S `  y
 )  =  1 ) 
 ->  x  C_  y ) ) )   &    |-  A  e.  CH   =>    |-  ( ph  ->  ( ( S `
  A )  =  1  ->  A  =  ~H ) )
 
Theoremstcltrlem1 22802* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( S  e.  States  /\ 
 A. x  e.  CH  A. y  e.  CH  (
 ( ( S `  x )  =  1  ->  ( S `  y
 )  =  1 ) 
 ->  x  C_  y ) ) )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  (
 ( S `  B )  =  1  ->  ( S `  ( ( _|_ `  A )  vH  ( A  i^i  B ) ) )  =  1 ) )
 
Theoremstcltrlem2 22803* Lemma for strong classical state theorem. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( S  e.  States  /\ 
 A. x  e.  CH  A. y  e.  CH  (
 ( ( S `  x )  =  1  ->  ( S `  y
 )  =  1 ) 
 ->  x  C_  y ) ) )   &    |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( ph  ->  B  C_  ( ( _|_ `  A )  vH  ( A  i^i  B ) ) )
 
Theoremstcltrthi 22804* Theorem for classically strong set of states. If there exists a "classically strong set of states" on lattice  CH (or actually any ortholattice, which would have an identical proof), then any two elements of the lattice commute, i.e., the lattice is distributive. (Proof due to Mladen Pavicic.) Theorem 3.3 of [MegPav2000] p. 2344. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  E. s  e.  States  A. x  e.  CH  A. y  e.  CH  (
 ( ( s `  x )  =  1  ->  ( s `  y
 )  =  1 ) 
 ->  x  C_  y )   =>    |-  B  C_  ( ( _|_ `  A )  vH  ( A  i^i  B ) )
 
15.9.50  Covers relation; modular pairs
 
Definitiondf-cv 22805* Define the covers relation (on the Hilbert lattice). Definition 3.2.18 of [PtakPulmannova] p. 68, whose notation we use. Ptak/Pulmannova's notation  A  <oH  B is read " B covers  A " or " A is covered by  B " , and it means that  B is larger than  A and there is nothing in between. See cvbr 22808 and cvbr2 22809 for membership relations. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  <oH  =  { <. x ,  y >.  |  ( ( x  e.  CH  /\  y  e. 
 CH )  /\  ( x  C.  y  /\  -.  E. z  e.  CH  ( x  C.  z  /\  z  C.  y ) ) ) }
 
Definitiondf-md 22806* Define the modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M for "the ordered pair <x,y> is a modular pair." See mdbr 22820 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
 |-  MH  =  { <. x ,  y >.  |  ( ( x  e.  CH  /\  y  e. 
 CH )  /\  A. z  e.  CH  ( z 
 C_  y  ->  (
 ( z  vH  x )  i^i  y )  =  ( z  vH  ( x  i^i  y ) ) ) ) }
 
Definitiondf-dmd 22807* Define the dual modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M* for "the ordered pair <x,y> is a dual modular pair." See dmdbr 22825 for membership relation. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  MH*  =  { <. x ,  y >.  |  ( ( x  e.  CH  /\  y  e. 
 CH )  /\  A. z  e.  CH  ( y 
 C_  z  ->  (
 ( z  i^i  x )  vH  y )  =  ( z  i^i  ( x  vH  y ) ) ) ) }
 
Theoremcvbr 22808* Binary relation expressing  B covers  A, which means that  B is larger than  A and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( A  C.  B  /\  -.  E. x  e.  CH  ( A  C.  x  /\  x  C.  B ) ) ) )
 
Theoremcvbr2 22809* Binary relation expressing  B covers  A. Definition of covers in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( A  C.  B  /\  A. x  e. 
 CH  ( ( A 
 C.  x  /\  x  C_  B )  ->  x  =  B ) ) ) )
 
Theoremcvcon3 22810 Contraposition law for the covers relation. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  <->  ( _|_ `  B )  <oH  ( _|_ `  A ) ) )
 
Theoremcvpss 22811 The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  A 
 C.  B ) )
 
Theoremcvnbtwn 22812 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  -.  ( A  C.  C  /\  C  C.  B ) ) )
 
Theoremcvnbtwn2 22813 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  ( ( A  C.  C  /\  C  C_  B )  ->  C  =  B ) ) )
 
Theoremcvnbtwn3 22814 The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  ( ( A  C_  C  /\  C  C.  B ) 
 ->  C  =  A ) ) )
 
Theoremcvnbtwn4 22815 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  <oH  B  ->  ( ( A  C_  C  /\  C  C_  B )  ->  ( C  =  A  \/  C  =  B ) ) ) )
 
Theoremcvnsym 22816 The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  <oH  B  ->  -.  B  <oH  A ) )
 
Theoremcvnref 22817 The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  -.  A  <oH  A )
 
Theoremcvntr 22818 The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( ( A  <oH  B 
 /\  B  <oH  C ) 
 ->  -.  A  <oH  C ) )
 
Theoremspansncv2 22819 Hilbert space has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  ~H )  ->  ( -.  ( span ` 
 { B } )  C_  A  ->  A  <oH  ( A  vH  ( span ` 
 { B } )
 ) ) )
 
Theoremmdbr 22820* Binary relation expressing  <. A ,  B >. is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  ( x  C_  B  ->  (
 ( x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
 
Theoremmdi 22821 Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH  B  /\  C  C_  B )
 )  ->  ( ( C  vH  A )  i^i 
 B )  =  ( C  vH  ( A  i^i  B ) ) )
 
Theoremmdbr2 22822* Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 22820. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  ( x  C_  B  ->  (
 ( x  vH  A )  i^i  B )  C_  ( x  vH  ( A  i^i  B ) ) ) ) )
 
Theoremmdbr3 22823* Binary relation expressing the modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
 ( ( x  i^i  B )  vH  A )  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
 
Theoremmdbr4 22824* Binary relation expressing the modular pair property. This version quantifies an ordering instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
 ( ( x  i^i  B )  vH  A )  i^i  B )  C_  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
 
Theoremdmdbr 22825* Binary relation expressing the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  ( B  C_  x  ->  (
 ( x  i^i  A )  vH  B )  =  ( x  i^i  ( A  vH  B ) ) ) ) )
 
Theoremdmdmd 22826 The dual modular pair property expressed in terms of the modular pair property, that hold in Hilbert lattices. Remark 29.6 of [MaedaMaeda] p. 130. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  ( _|_ `  A )  MH  ( _|_ `  B ) ) )
 
Theoremmddmd 22827 The modular pair property expressed in terms of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  ( _|_ `  A )  MH*  ( _|_ `  B ) ) )
 
Theoremdmdi 22828 Consequence of the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH*  B  /\  B  C_  C )
 )  ->  ( ( C  i^i  A )  vH  B )  =  ( C  i^i  ( A  vH  B ) ) )
 
Theoremdmdbr2 22829* Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr 22825. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  ( B  C_  x  ->  ( x  i^i  ( A  vH  B ) )  C_  ( ( x  i^i  A )  vH  B ) ) ) )
 
Theoremdmdi2 22830 Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH*  B  /\  B  C_  C )
 )  ->  ( C  i^i  ( A  vH  B ) )  C_  ( ( C  i^i  A ) 
 vH  B ) )
 
Theoremdmdbr3 22831* Binary relation expressing the dual modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  (
 ( ( x  vH  B )  i^i  A ) 
 vH  B )  =  ( ( x  vH  B )  i^i  ( A 
 vH  B ) ) ) )
 
Theoremdmdbr4 22832* Binary relation expressing the dual modular pair property. This version quantifies an ordering instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  (
 ( x  vH  B )  i^i  ( A  vH  B ) )  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B ) ) )
 
Theoremdmdi4 22833 Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  MH*  B  ->  ( ( C  vH  B )  i^i  ( A  vH  B ) )  C_  ( ( ( C 
 vH  B )  i^i 
 A )  vH  B ) ) )
 
Theoremdmdbr5 22834* Binary relation expressing the dual modular pair property. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  A. x  e.  CH  ( x  C_  ( A  vH  B )  ->  x  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B ) ) ) )
 
Theoremmddmd2 22835* Relationship between modular pairs and dual-modular pairs. Lemma 1.2 of [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A. x  e.  CH  A  MH  x  <->  A. x  e.  CH  A  MH*  x ) )
 
Theoremmdsl0 22836 A sublattice condition that transfers the modular pair property. Exercise 12 of [Kalmbach] p. 103. Also Lemma 1.5.3 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH )  /\  ( C  e.  CH 
 /\  D  e.  CH ) )  ->  ( ( ( ( C  C_  A  /\  D  C_  B )  /\  ( A  i^i  B )  =  0H )  /\  A  MH  B ) 
 ->  C  MH  D ) )
 
Theoremssmd1 22837 Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  A  MH  B )
 
Theoremssmd2 22838 Ordering implies the modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  B  MH  A )
 
Theoremssdmd1 22839 Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  A  MH*  B )
 
Theoremssdmd2 22840 Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  ( _|_ `  B )  MH  ( _|_ `  A ) )
 
Theoremdmdsl3 22841 Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  MH*  A  /\  A  C_  C  /\  C  C_  ( A  vH  B ) ) ) 
 ->  ( ( C  i^i  B )  vH  A )  =  C )
 
Theoremmdsl3 22842 Sublattice mapping for a modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH  B  /\  ( A  i^i  B )  C_  C  /\  C  C_  B ) )  ->  ( ( C  vH  A )  i^i  B )  =  C )
 
Theoremmdslle1i 22843 Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( B  MH*  A  /\  A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) )  ->  ( C  C_  D  <->  ( C  i^i  B )  C_  ( D  i^i  B ) ) )
 
Theoremmdslle2i 22844 Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( A  MH  B  /\  ( A  i^i  B )  C_  ( C  i^i  D )  /\  ( C 
 vH  D )  C_  B )  ->  ( C 
 C_  D  <->  ( C  vH  A )  C_  ( D 
 vH  A ) ) )
 
Theoremmdslj1i 22845 Join preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D ) 
 /\  ( C  vH  D )  C_  ( A 
 vH  B ) ) )  ->  ( ( C  vH  D )  i^i 
 B )  =  ( ( C  i^i  B )  vH  ( D  i^i  B ) ) )
 
Theoremmdslj2i 22846 Meet preservation of the reverse mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  ( C  i^i  D ) 
 /\  ( C  vH  D )  C_  B ) )  ->  ( ( C  i^i  D )  vH  A )  =  (
 ( C  vH  A )  i^i  ( D  vH  A ) ) )
 
Theoremmdsl1i 22847* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A. x  e.  CH  (
 ( ( A  i^i  B )  C_  x  /\  x  C_  ( A  vH  B ) )  ->  ( x  C_  B  ->  ( ( x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) )  <->  A  MH  B )
 
Theoremmdsl2i 22848* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 28-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH  B  <->  A. x  e.  CH  ( ( ( A  i^i  B )  C_  x  /\  x  C_  B )  ->  ( ( x 
 vH  A )  i^i 
 B )  C_  ( x  vH  ( A  i^i  B ) ) ) )
 
Theoremmdsl2bi 22849* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH  B  <->  A. x  e.  CH  ( ( ( A  i^i  B )  C_  x  /\  x  C_  B )  ->  ( ( x 
 vH  A )  i^i 
 B )  =  ( x  vH  ( A  i^i  B ) ) ) )
 
Theoremcvmdi 22850 The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  i^i  B )  <oH  B  ->  A  MH  B )
 
Theoremmdslmd1lem1 22851 Lemma for mdslmd1i 22855. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   &    |-  R  e.  CH   =>    |-  ( ( ( A  MH  B  /\  B  MH* 
 A )  /\  (
 ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) ) )  ->  ( (
 ( R  vH  A )  C_  D  ->  (
 ( ( R  vH  A )  vH  C )  i^i  D )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) ) )  ->  ( (
 ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) 
 C_  R  /\  R  C_  ( D  i^i  B ) )  ->  ( ( R  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  ( R  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) )
 
Theoremmdslmd1lem2 22852 Lemma for mdslmd1i 22855. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   &    |-  R  e.  CH   =>    |-  ( ( ( A  MH  B  /\  B  MH* 
 A )  /\  (
 ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) ) )  ->  ( (
 ( R  i^i  B )  C_  ( D  i^i  B )  ->  ( (
 ( R  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
 C_  ( ( R  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) )  ->  (
 ( ( C  i^i  D )  C_  R  /\  R  C_  D )  ->  ( ( R  vH  C )  i^i  D ) 
 C_  ( R  vH  ( C  i^i  D ) ) ) ) )
 
Theoremmdslmd1lem3 22853* Lemma for mdslmd1i 22855. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( x  e.  CH  /\  ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
 C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) ) ) ) 
 ->  ( ( ( x 
 vH  A )  C_  D  ->  ( ( ( x  vH  A ) 
 vH  C )  i^i 
 D )  C_  (
 ( x  vH  A )  vH  ( C  i^i  D ) ) )  ->  ( ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  x  /\  x  C_  ( D  i^i  B ) ) 
 ->  ( ( x  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
 C_  ( x  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) )
 
Theoremmdslmd1lem4 22854* Lemma for mdslmd1i 22855. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( x  e.  CH  /\  ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
 C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) ) ) ) 
 ->  ( ( ( x  i^i  B )  C_  ( D  i^i  B ) 
 ->  ( ( ( x  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
 C_  ( ( x  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) )  ->  (
 ( ( C  i^i  D )  C_  x  /\  x  C_  D )  ->  ( ( x  vH  C )  i^i  D ) 
 C_  ( x  vH  ( C  i^i  D ) ) ) ) )
 
Theoremmdslmd1i 22855 Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (meet version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D ) 
 /\  ( C  vH  D )  C_  ( A 
 vH  B ) ) )  ->  ( C  MH  D  <->  ( C  i^i  B )  MH  ( D  i^i  B ) ) )
 
Theoremmdslmd2i 22856 Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (join version). (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A  i^i  B )  C_  ( C  i^i  D ) 
 /\  ( C  vH  D )  C_  B ) )  ->  ( C  MH  D  <->  ( C  vH  A )  MH  ( D  vH  A ) ) )
 
Theoremmdsldmd1i 22857 Preservation of the dual modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D ) 
 /\  ( C  vH  D )  C_  ( A 
 vH  B ) ) )  ->  ( C  MH* 
 D 
 <->  ( C  i^i  B )  MH*  ( D  i^i  B ) ) )
 
Theoremmdslmd3i 22858 Modular pair conditions that imply the modular pair property in a sublattice. Lemma 1.5.1 of [MaedaMaeda] p. 2. (Contributed by NM, 23-Dec-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( ( A  MH  B  /\  ( A  i^i  B )  MH  C ) 
 /\  ( ( A  i^i  C )  C_  D  /\  D  C_  A ) )  ->  D  MH  ( B  i^i  C ) )
 
Theoremmdslmd4i 22859 Modular pair condition that implies the modular pair property in a sublattice. Lemma 1.5.2 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( A  MH  B  /\  ( ( A  i^i  B )  C_  C  /\  C  C_  A )  /\  ( ( A  i^i  B )  C_  D  /\  D  C_  B ) ) 
 ->  C  MH  D )
 
Theoremcsmdsymi 22860* Cross-symmetry implies M-symmetry. Theorem 1.9.1 of [MaedaMaeda] p. 3. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A. c  e.  CH  ( c  MH  B  ->  B  MH*  c )  /\  A  MH  B ) 
 ->  B  MH  A )
 
Theoremmdexchi 22861 An exchange lemma for modular pairs. Lemma 1.6 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  MH  B  /\  C  MH  ( A  vH  B )  /\  ( C  i^i  ( A 
 vH  B ) ) 
 C_  A )  ->  ( ( C  vH  A )  MH  B  /\  ( ( C  vH  A )  i^i  B )  =  ( A  i^i  B ) ) )
 
Theoremcvmd 22862 The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  ( A  i^i  B ) 
 <oH  B )  ->  A  MH  B )
 
Theoremcvdmd 22863 The covering property implies the dual modular pair property. Lemma 7.5.2 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  B  <oH  ( A  vH  B ) )  ->  A  MH*  B )
 
15.9.51  Atoms
 
Definitiondf-at 22864 Define the set of atoms in a Hilbert lattice. An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is a smallest nonzero element of the lattice. Definition of atom in [Kalmbach] p. 15. See ela 22865 and elat2 22866 for membership relations. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |- HAtoms  =  { x  e.  CH  |  0H  <oH  x }
 
Theoremela 22865 Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  <->  ( A  e.  CH 
 /\  0H  <oH  A ) )
 
Theoremelat2 22866* Expanded membership relation for the set of atoms, i.e. the predicate "is an atom (of the Hilbert lattice)." An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is a smallest nonzero element of the lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  <->  ( A  e.  CH 
 /\  ( A  =/=  0H 
 /\  A. x  e.  CH  ( x  C_  A  ->  ( x  =  A  \/  x  =  0H )
 ) ) ) )
 
Theoremelatcv0 22867 A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  e. HAtoms  <->  0H  <oH  A ) )
 
Theorematcv0 22868 An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  ->  0H  <oH  A )
 
Theorematssch 22869 Atoms are a subset of the Hilbert lattice. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |- HAtoms  C_  CH
 
Theorematelch 22870 An atom is a Hilbert lattice element. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  ->  A  e.  CH )
 
Theorematne0 22871 An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  ->  A  =/=  0H )
 
Theorematss 22872 A lattice element smaller than an atom is either the atom or zero. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  C_  B  ->  ( A  =  B  \/  A  =  0H )
 ) )
 
Theorematsseq 22873 Two atoms in a subset relationship are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms )  ->  ( A  C_  B  <->  A  =  B ) )
 
Theorematcveq0 22874 A Hilbert lattice element covered by an atom must be the zero subspace. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  <oH  B  <->  A  =  0H ) )
 
Theoremh1da 22875 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  ( _|_ `  ( _|_ `  { A }
 ) )  e. HAtoms )
 
Theoremspansna 22876 The span of the singleton of a vector is an atom. (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  ( span `  { A }
 )  e. HAtoms )
 
Theoremsh1dle 22877 A 1-dimensional subspace is less than or equal to any subspace containing its generating vector. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  A ) 
 ->  ( _|_ `  ( _|_ `  { B }
 ) )  C_  A )
 
Theoremch1dle 22878 A 1-dimensional subspace is less than or equal to any member of  CH containing its generating vector. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  A ) 
 ->  ( _|_ `  ( _|_ `  { B }
 ) )  C_  A )
 
Theorematom1d 22879* The 1-dimensional subspaces of Hilbert space are its atoms. Part of Remark 10.3.5 of [BeltramettiCassinelli] p. 107. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  <->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( span ` 
 { x } )
 ) )
 
15.9.52  Superposition principle
 
Theoremsuperpos 22880* Superposition Principle. If  A and  B are distinct atoms, there exists a third atom, distinct from  A and  B, that is the superposition of  A and  B. Definition 3.4-3(a) in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms  /\  A  =/=  B )  ->  E. x  e. HAtoms  ( x  =/=  A  /\  x  =/=  B  /\  x  C_  ( A  vH  B ) ) )
 
15.9.53  Atoms, exchange and covering properties, atomicity
 
Theoremchcv1 22881 The Hilbert lattice has the covering property. Proposition 1(ii) of [Kalmbach] p. 140 (and its converse). (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( -.  B  C_  A  <->  A  <oH  ( A 
 vH  B ) ) )
 
Theoremchcv2 22882 The Hilbert lattice has the covering property. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  C.  ( A  vH  B )  <->  A  <oH  ( A 
 vH  B ) ) )
 
Theoremchjatom 22883 The join of a closed subspace and an atom equals their subspace sum. Special case of remark in [Kalmbach] p. 65, stating that if  A or  B is finite dimensional, then this equality holds. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremshatomici 22884* The lattice of Hilbert subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  ( A  =/=  0H  ->  E. x  e. HAtoms  x  C_  A )
 
Theoremhatomici 22885* The Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  =/=  0H  ->  E. x  e. HAtoms  x  C_  A )
 
Theoremhatomic 22886* A Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. Also Definition 3.4-2 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  A  =/=  0H )  ->  E. x  e. HAtoms  x  C_  A )
 
Theoremshatomistici 22887* The lattice of Hilbert subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  A  =  ( span ` 
 U. { x  e. HAtoms  |  x  C_  A }
 )
 
Theoremhatomistici 22888*  CH is atomistic, i.e. any element is the supremum of its atoms. Remark in [Kalmbach] p. 140. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  =  (  \/H  ` 
 { x  e. HAtoms  |  x  C_  A } )
 
Theoremchpssati 22889* Two Hilbert lattice elements in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C.  B  ->  E. x  e. HAtoms  ( x  C_  B  /\  -.  x  C_  A ) )
 
Theoremchrelati 22890* The Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C.  B  ->  E. x  e. HAtoms  ( A  C.  ( A  vH  x )  /\  ( A  vH  x ) 
 C_  B ) )
 
Theoremchrelat2i 22891* A consequence of relative atomicity. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( -.  A  C_  B  <->  E. x  e. HAtoms  ( x  C_  A  /\  -.  x  C_  B ) )
 
Theoremcvati 22892* If a Hilbert lattice element covers another, it equals the other joined with some atom. This is a consequence of the relative atomicity of Hilbert space. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  <oH  B  ->  E. x  e. HAtoms  ( A  vH  x )  =  B )
 
Theoremcvbr4i 22893* An alternate way to express the covering property. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  <oH  B  <->  ( A  C.  B  /\  E. x  e. HAtoms  ( A  vH  x )  =  B ) )
 
Theoremcvexchlem 22894 Lemma for cvexchi 22895. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  i^i  B )  <oH  B  ->  A  <oH  ( A  vH  B ) )
 
Theoremcvexchi 22895 The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  i^i  B )  <oH  B  <->  A  <oH  ( A 
 vH  B ) )
 
Theoremchrelat2 22896* A consequence of relative atomicity. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( -.  A  C_  B 
 <-> 
 E. x  e. HAtoms  ( x  C_  A  /\  -.  x  C_  B ) ) )
 
Theoremchrelat3 22897* A consequence of relative atomicity. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  B  <->  A. x  e. HAtoms  ( x  C_  A  ->  x  C_  B ) ) )
 
Theoremchrelat3i 22898* A consequence of the relative atomicity of Hilbert space: the ordering of Hilbert lattice elements is completely determined by the atoms they majorize. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  <->  A. x  e. HAtoms  ( x  C_  A  ->  x  C_  B ) )
 
Theoremchrelat4i 22899* A consequence of relative atomicity. Extensionality principle: two lattice elements are equal iff they majorize the same atoms. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  =  B  <->  A. x  e. HAtoms  ( x  C_  A  <->  x  C_  B ) )
 
Theoremcvexch 22900 The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( ( A  i^i  B )  <oH  B  <->  A  <oH  ( A 
 vH  B ) ) )
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