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Theorem List for Metamath Proof Explorer - 22701-22800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcvnref 22701 The covers relation is not reflexive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  -.  A  <oH  A )
 
Theoremcvntr 22702 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 22703 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 22704* 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 22705 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 22706* Binary relation expressing the modular pair property. This version has a weaker constraint than mdbr 22704. (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 22707* 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 22708* 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 22709* 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 22710 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 22711 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 22712 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 22713* Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr 22709. (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 22714 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 22715* 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 22716* 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 22717 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 22718* 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 22719* 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 22720 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 22721 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 22722 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 22723 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 22724 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 22725 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 22726 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 22727 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 22728 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 22729 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 22730 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 22731* 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 22732* 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 22733* 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 22734 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 22735 Lemma for mdslmd1i 22739. (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 22736 Lemma for mdslmd1i 22739. (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 22737* Lemma for mdslmd1i 22739. (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 22738* Lemma for mdslmd1i 22739. (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 22739 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 22740 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 22741 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 22742 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 22743 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 22744* 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 22745 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 22746 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 22747 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 22748 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 22749 and elat2 22750 for membership relations. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |- HAtoms  =  { x  e.  CH  |  0H  <oH  x }
 
Theoremela 22749 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 22750* 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 22751 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 22752 An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  ->  0H  <oH  A )
 
Theorematssch 22753 Atoms are a subset of the Hilbert lattice. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |- HAtoms  C_  CH
 
Theorematelch 22754 An atom is a Hilbert lattice element. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  ->  A  e.  CH )
 
Theorematne0 22755 An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  ->  A  =/=  0H )
 
Theorematss 22756 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 22757 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 22758 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 22759 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 22760 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 22761 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 22762 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 22763* 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 22764* 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 22765 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 22766 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 22767 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 22768* 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 22769* 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 22770* 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 22771* 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 22772*  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 22773* 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 22774* 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 22775* 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 22776* 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 22777* 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 22778 Lemma for cvexchi 22779. (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 22779 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 22780* 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 22781* 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 22782* 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 22783* 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 22784 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 ) ) )
 
Theoremcvp 22785 The Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  (
 ( A  i^i  B )  =  0H  <->  A  <oH  ( A 
 vH  B ) ) )
 
Theorematnssm0 22786 The meet of a Hilbert lattice element and an incomparable atom is the zero subspace. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( -.  B  C_  A  <->  ( A  i^i  B )  =  0H )
 )
 
Theorematnemeq0 22787 The meet of distinct atoms is the zero subspace. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms )  ->  ( A  =/=  B  <->  ( A  i^i  B )  =  0H )
 )
 
Theorematssma 22788 The meet with an atom's superset is the atom. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( A  i^i  B )  e. HAtoms ) )
 
Theorematcv0eq 22789 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms )  ->  ( 0H  <oH  ( A  vH  B )  <->  A  =  B )
 )
 
Theorematcv1 22790 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e. HAtoms  /\  C  e. HAtoms )  /\  A  <oH  ( B  vH  C ) )  ->  ( A  =  0H  <->  B  =  C ) )
 
Theorematexch 22791 The Hilbert lattice satisfies the atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem related to vector analysis was originally proved by Hermann Grassmann in 1862. Also Definition 3.4-3(b) in [MegPav2000] p. 2345 (PDF p. 8) (use atnemeq0 22787 to obtain atom inequality). (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms ) 
 ->  ( ( B  C_  ( A  vH  C ) 
 /\  ( A  i^i  B )  =  0H )  ->  C  C_  ( A  vH  B ) ) )
 
Theorematomli 22792 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition 3.2.17 of [PtakPulmannova] p. 66. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( B  e. HAtoms  ->  ( ( A  vH  B )  i^i  ( _|_ `  A ) )  e.  (HAtoms  u. 
 { 0H } )
 )
 
Theorematoml2i 22793 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition P8(ii) of [BeltramettiCassinelli1] p. 400. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\ 
 -.  B  C_  A )  ->  ( ( A 
 vH  B )  i^i  ( _|_ `  A ) )  e. HAtoms )
 
Theorematordi 22794 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  A  C_H  B ) 
 ->  ( B  C_  A  \/  B  C_  ( _|_ `  A ) ) )
 
Theorematcvatlem 22795 Lemma for atcvati 22796. (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( ( B  e. HAtoms  /\  C  e. HAtoms )  /\  ( A  =/=  0H  /\  A  C.  ( B 
 vH  C ) ) )  ->  ( -.  B  C_  A  ->  A  e. HAtoms ) )
 
Theorematcvati 22796 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (Contributed by NM, 28-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  C  e. HAtoms )  ->  (
 ( A  =/=  0H  /\  A  C.  ( B 
 vH  C ) ) 
 ->  A  e. HAtoms ) )
 
Theorematcvat2i 22797 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  C  e. HAtoms )  ->  (
 ( -.  B  =  C  /\  A  <oH  ( B 
 vH  C ) ) 
 ->  A  e. HAtoms ) )
 
Theorematord 22798 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms  /\  A  C_H  B )  ->  ( B  C_  A  \/  B  C_  ( _|_ `  A )
 ) )
 
Theorematcvat2 22799 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 29-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms ) 
 ->  ( ( -.  B  =  C  /\  A  <oH  ( B  vH  C ) )  ->  A  e. HAtoms ) )
 
15.9.54  Irreducibility
 
Theoremchirredlem1 22800* Lemma for chirredi 22804. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( ( p  e. HAtoms  /\  ( q  e. 
 CH  /\  q  C_  ( _|_ `  A ) ) )  /\  ( ( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q ) ) )  ->  ( p  i^i  ( _|_ `  r
 ) )  =  0H )
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