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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempjpjhthi 22001* Projection Theorem: Any Hilbert space vector  A can be decomposed into a member  x of a closed subspace  H and a member  y of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  H  e.  CH   =>    |-  E. x  e.  H  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y )
 
Theorempjop 22002 Orthocomplement projection in terms of projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( ( proj  h `  ( _|_ `  H )
 ) `  A )  =  ( A  -h  (
 ( proj  h `  H ) `  A ) ) )
 
Theorempjpo 22003 Projection in terms of orthocomplement projection. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( ( proj  h `  H ) `  A )  =  ( A  -h  ( ( proj  h `  ( _|_ `  H )
 ) `  A )
 ) )
 
Theorempjopi 22004 Orthocomplement projection in terms of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  (
 ( proj  h `  ( _|_ `  H ) ) `
  A )  =  ( A  -h  (
 ( proj  h `  H ) `  A ) )
 
Theorempjpoi 22005 Projection in terms of orthocomplement projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  (
 ( proj  h `  H ) `  A )  =  ( A  -h  (
 ( proj  h `  ( _|_ `  H ) ) `
  A ) )
 
Theorempjoc1i 22006 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  ( A  e.  H  <->  ( ( proj  h `
  ( _|_ `  H ) ) `  A )  =  0h )
 
Theorempjchi 22007 Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  ( A  e.  H  <->  ( ( proj  h `
  H ) `  A )  =  A )
 
Theorempjoccl 22008 The part of a vector that belongs to the orthocomplemented space. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( A  -h  (
 ( proj  h `  H ) `  A ) )  e.  ( _|_ `  H ) )
 
Theorempjoc1 22009 Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( A  e.  H  <->  ( ( proj  h `  ( _|_ `  H ) ) `
  A )  =  0h ) )
 
Theorempjomli 22010 Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. Derived using projections; compare omlsi 21979. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  SH   =>    |-  (
 ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H )  ->  A  =  B )
 
Theorempjoml 22011 Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. Derived using projections; compare omlsi 21979. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  SH )  /\  ( A  C_  B  /\  ( B  i^i  ( _|_ `  A )
 )  =  0H )
 )  ->  A  =  B )
 
Theorempjococi 22012 Proof of orthocomplement theorem using projections. Compare ococ 21981. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( _|_ `  ( _|_ `  H ) )  =  H
 
Theorempjoc2i 22013 Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  ( A  e.  ( _|_ `  H )  <->  ( ( proj  h `
  H ) `  A )  =  0h )
 
Theorempjoc2 22014 Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( A  e.  ( _|_ `  H )  <->  ( ( proj  h `
  H ) `  A )  =  0h ) )
 
17.5.3  Hilbert lattice operations
 
Theoremsh0le 22015 The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  0H  C_  A )
 
Theoremch0le 22016 The zero subspace is the smallest member of  CH. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  0H  C_  A )
 
Theoremshle0 22017 No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )
 
Theoremchle0 22018 No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  C_  0H  <->  A  =  0H ) )
 
Theoremchnlen0 22019 A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  ( B  e.  CH  ->  ( -.  A  C_  B  ->  -.  A  =  0H )
 )
 
Theoremch0pss 22020 The zero subspace is a proper subset of nonzero Hilbert lattice elements. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( 0H  C.  A  <->  A  =/=  0H )
 )
 
Theoremorthin 22021 The intersection of orthogonal subspaces is the zero subspace. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  C_  ( _|_ `  B )  ->  ( A  i^i  B )  =  0H ) )
 
Theoremssjo 22022 The lattice join of a subset with its orthocomplement is the whole space. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( A  vH  ( _|_ `  A ) )  =  ~H )
 
Theoremshne0i 22023* A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
 
Theoremshs0i 22024 Hilbert subspace sum with the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  ( A  +H  0H )  =  A
 
Theoremshs00i 22025 Two subspaces are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  (
 ( A  =  0H  /\  B  =  0H )  <->  ( A  +H  B )  =  0H )
 
Theoremch0lei 22026 The closed subspace zero is the smallest member of  CH. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |- 
 0H  C_  A
 
Theoremchle0i 22027 No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  C_  0H  <->  A  =  0H )
 
Theoremchne0i 22028* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
 
Theoremchocini 22029 Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  i^i  ( _|_ `  A ) )  =  0H
 
Theoremchj0i 22030 Join with lattice zero in  CH. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  vH  0H )  =  A
 
Theoremchm1i 22031 Meet with lattice one in  CH. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  i^i  ~H )  =  A
 
Theoremchjcli 22032 Closure of  CH join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  B )  e. 
 CH
 
Theoremchsleji 22033 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  +H  B )  C_  ( A  vH  B )
 
Theoremchseli 22034* Membership in subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y ) )
 
Theoremchincli 22035 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  i^i  B )  e. 
 CH
 
Theoremchsscon3i 22036 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  <->  ( _|_ `  B )  C_  ( _|_ `  A ) )
 
Theoremchsscon1i 22037 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( _|_ `  A )  C_  B  <->  ( _|_ `  B )  C_  A )
 
Theoremchsscon2i 22038 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  ( _|_ `  B ) 
 <->  B  C_  ( _|_ `  A ) )
 
Theoremchcon2i 22039 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  =  ( _|_ `  B )  <->  B  =  ( _|_ `  A ) )
 
Theoremchcon1i 22040 Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( _|_ `  A )  =  B  <->  ( _|_ `  B )  =  A )
 
Theoremchcon3i 22041 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  =  B  <->  ( _|_ `  B )  =  ( _|_ `  A ) )
 
Theoremchunssji 22042 Union is smaller than  CH join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  u.  B )  C_  ( A  vH  B )
 
Theoremchjcomi 22043 Commutative law for join in  CH. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  B )  =  ( B  vH  A )
 
Theoremchub1i 22044  CH join is an upper bound of two elements. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  A  C_  ( A  vH  B )
 
Theoremchub2i 22045  CH join is an upper bound of two elements. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  A  C_  ( B  vH  A )
 
Theoremchlubi 22046 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  vH  B )  C_  C )
 
Theoremchlubii 22047 Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 22046). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  C_  C  /\  B  C_  C )  ->  ( A  vH  B )  C_  C )
 
Theoremchlej1i 22048 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( A  C_  B  ->  ( A  vH  C )  C_  ( B  vH  C ) )
 
Theoremchlej2i 22049 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( A  C_  B  ->  ( C  vH  A )  C_  ( C  vH  B ) )
 
Theoremchlej12i 22050 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( A  C_  B  /\  C  C_  D )  ->  ( A  vH  C )  C_  ( B  vH  D ) )
 
Theoremchlejb1i 22051 Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  <->  ( A  vH  B )  =  B )
 
Theoremchdmm1i 22052 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 ) )
 
Theoremchdmm2i 22053 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 ) )
 
Theoremchdmm3i 22054 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 )
 
Theoremchdmm4i 22055 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 )
 
Theoremchdmj1i 22056 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 ) )
 
Theoremchdmj2i 22057 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 ) )
 
Theoremchdmj3i 22058 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 )
 
Theoremchdmj4i 22059 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 )
 
Theoremchnlei 22060 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( -.  B  C_  A  <->  A  C.  ( A 
 vH  B ) )
 
Theoremchjassi 22061 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 ) )
 
Theoremchj00i 22062 Two Hilbert lattice elements are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  =  0H  /\  B  =  0H )  <->  ( A  vH  B )  =  0H )
 
Theoremchjoi 22063 The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  vH  ( _|_ `  A ) )  =  ~H
 
Theoremchj1i 22064 Join with Hilbert lattice unit. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  vH  ~H )  =  ~H
 
Theoremchm0i 22065 Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  i^i  0H )  =  0H
 
Theoremchm0 22066 Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  i^i  0H )  =  0H )
 
Theoremshjshsi 22067 Hilbert lattice join equals the double orthocomplement of subspace sum. (Contributed by NM, 27-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  +H  B ) ) )
 
Theoremshjshseli 22068 A closed subspace sum equals Hilbert lattice join. Part of Lemma 31.1.5 of [MaedaMaeda] p. 136. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  (
 ( A  +H  B )  e.  CH  <->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremchne0 22069* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h ) )
 
Theoremchocin 22070 Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  i^i  ( _|_ `  A ) )  =  0H )
 
Theoremchssoc 22071 A closed subspace less than its orthocomplement is zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  C_  ( _|_ `  A ) 
 <->  A  =  0H )
 )
 
Theoremchj0 22072 Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  vH  0H )  =  A )
 
Theoremchslej 22073 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  +H  B )  C_  ( A  vH  B ) )
 
Theoremchincl 22074 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  B )  e.  CH )
 
Theoremchsscon3 22075 Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( _|_ `  B )  C_  ( _|_ `  A ) ) )
 
Theoremchsscon1 22076 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( ( _|_ `  A )  C_  B  <->  ( _|_ `  B )  C_  A ) )
 
Theoremchsscon2 22077 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  ( _|_ `  B )  <->  B  C_  ( _|_ `  A ) ) )
 
Theoremchpsscon3 22078 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C.  B  <->  ( _|_ `  B )  C.  ( _|_ `  A ) ) )
 
Theoremchpsscon1 22079 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( ( _|_ `  A )  C.  B  <->  ( _|_ `  B )  C.  A ) )
 
Theoremchpsscon2 22080 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C.  ( _|_ `  B )  <->  B  C.  ( _|_ `  A ) ) )
 
Theoremchjcom 22081 Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B )  =  ( B  vH  A ) )
 
Theoremchub1 22082 Hilbert lattice join is greater than or equal to its first argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  A  C_  ( A  vH  B ) )
 
Theoremchub2 22083 Hilbert lattice join is greater than or equal to its second argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  A  C_  ( B  vH  A ) )
 
Theoremchlub 22084 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  vH  B )  C_  C ) )
 
Theoremchlej1 22085 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  A  C_  B )  ->  ( A  vH  C )  C_  ( B  vH  C ) )
 
Theoremchlej2 22086 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  A  C_  B )  ->  ( C  vH  A )  C_  ( C  vH  B ) )
 
Theoremchlejb1 22087 Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( A  vH  B )  =  B ) )
 
Theoremchlejb2 22088 Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( B  vH  A )  =  B ) )
 
Theoremchnle 22089 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( -.  B  C_  A 
 <->  A  C.  ( A 
 vH  B ) ) )
 
Theoremchjo 22090 The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  vH  ( _|_ `  A ) )  =  ~H )
 
Theoremchabs1 22091 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  ( A  i^i  B ) )  =  A )
 
Theoremchabs2 22092 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  ( A  vH  B ) )  =  A )
 
Theoremchabs1i 22093 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  ( A  i^i  B ) )  =  A
 
Theoremchabs2i 22094 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  i^i  ( A  vH  B ) )  =  A
 
Theoremchjidm 22095 Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  vH  A )  =  A )
 
Theoremchjidmi 22096 Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  vH  A )  =  A
 
Theoremchj12i 22097 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 22098 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 22099 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 22100 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 ) ) )
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