HomeHome Metamath Proof Explorer
Theorem List (p. 219 of 309)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30843)
 

Theorem List for Metamath Proof Explorer - 21801-21900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempjhthlem2 21801* Lemma for pjhth 21802. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  ( ph  ->  A  e.  ~H )   =>    |-  ( ph  ->  E. x  e.  H  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y
 ) )
 
Theorempjhth 21802 Projection Theorem: Any Hilbert space vector  A can be decomposed uniquely 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, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  ( H  +H  ( _|_ `  H ) )  =  ~H )
 
Theorempjhtheu 21803* Projection Theorem: Any Hilbert space vector  A can be decomposed uniquely 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. See pjhtheu2 21825 for the uniqueness of  y. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  E! x  e.  H  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) )
 
15.9.20  Projectors
 
Definitiondf-pjh 21804* Define the projection function on a Hilbert space, as a mapping from the Hilbert lattice to a function on Hilbert space. Every closed subspace is associated with a unique projection function. Remark in [Kalmbach] p. 66, adopted as a definition.  ( proj  h `  H
) `  A is the projection of vector  A onto closed subspace  H. Note that the range of  proj  h is the set of all projection operators, so  T  e.  ran  proj 
h means that  T is a projection operator. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
 |-  proj  h  =  ( h  e. 
 CH  |->  ( x  e. 
 ~H  |->  ( iota_ z  e.  h E. y  e.  ( _|_ `  h ) x  =  (
 z  +h  y )
 ) ) )
 
Theorempjhfval 21805* The value of the projection map. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  ( proj  h `  H )  =  ( x  e. 
 ~H  |->  ( iota_ z  e.  H E. y  e.  ( _|_ `  H ) x  =  (
 z  +h  y )
 ) ) )
 
Theorempjhval 21806* Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( ( proj  h `  H ) `  A )  =  ( iota_ x  e.  H E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) ) )
 
Theorempjpreeq 21807* Equality with a projection. This version of pjeq 21808 does not assume the Axiom of Choice via pjhth 21802. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ( H  +H  ( _|_ `  H ) ) )  ->  ( ( ( proj  h `
  H ) `  A )  =  B  <->  ( B  e.  H  /\  E. x  e.  ( _|_ `  H ) A  =  ( B  +h  x ) ) ) )
 
Theorempjeq 21808* Equality with a projection. (Contributed by NM, 20-Jan-2007.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( ( ( proj  h `
  H ) `  A )  =  B  <->  ( B  e.  H  /\  E. x  e.  ( _|_ `  H ) A  =  ( B  +h  x ) ) ) )
 
Theoremaxpjcl 21809 Closure of a projection in its subspace. If we consider this together with axpjpj 21829 to be axioms, the need for the ax-hcompl 21611 can often be avoided for the kinds of theorems we are interested in here. An interesting project is to see how far we can go by using them in place of it. In particular, we can prove the orthomodular law pjomli 21844.) (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( ( proj  h `  H ) `  A )  e.  H )
 
Theorempjhcl 21810 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( ( proj  h `  H ) `  A )  e.  ~H )
 
15.9.21  Orthomodular law
 
Theoremomlsilem 21811 Lemma for orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  G  e.  SH   &    |-  H  e.  SH   &    |-  G  C_  H   &    |-  ( H  i^i  ( _|_ `  G )
 )  =  0H   &    |-  A  e.  H   &    |-  B  e.  G   &    |-  C  e.  ( _|_ `  G )   =>    |-  ( A  =  ( B  +h  C ) 
 ->  A  e.  G )
 
Theoremomlsii 21812 Subspace inference form of orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  SH   &    |-  A  C_  B   &    |-  ( B  i^i  ( _|_ `  A )
 )  =  0H   =>    |-  A  =  B
 
Theoremomlsi 21813 Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  SH   =>    |-  (
 ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H )  ->  A  =  B )
 
Theoremococi 21814 Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  ( _|_ `  A ) )  =  A
 
Theoremococ 21815 Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( _|_ `  ( _|_ `  A ) )  =  A )
 
Theoremdfch2 21816 Alternate definition of the Hilbert lattice. (Contributed by NM, 8-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  CH  =  { x  e.  ~P ~H  |  ( _|_ `  ( _|_ `  x ) )  =  x }
 
Theoremococin 21817* The double complement is the smallest closed subspace containing a subset of Hilbert space. Remark 3.12(B) of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( _|_ `  ( _|_ `  A ) )  =  |^| { x  e.  CH  |  A  C_  x } )
 
Theoremhsupval2 21818* Alternate definition of supremum of a subset of the Hilbert lattice. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. We actually define it on any collection of Hilbert space subsets, not just the Hilbert lattice  CH, to allow more general theorems. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  C_  ~P ~H  ->  ( 
 \/H  `  A )  =  |^| { x  e. 
 CH  |  U. A  C_  x } )
 
Theoremchsupval2 21819* The value of the supremum of a set of closed subspaces of Hilbert space. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  C_  CH  ->  (  \/H  `  A )  =  |^| { x  e.  CH  |  U. A  C_  x }
 )
 
Theoremsshjval2 21820* Value of join in the set of closed subspaces of Hilbert space  CH. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( A  vH  B )  =  |^| { x  e. 
 CH  |  ( A  u.  B )  C_  x } )
 
Theoremchsupid 21821* A subspace is the supremum of all smaller subspaces. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  (  \/H  `  { x  e.  CH  |  x  C_  A }
 )  =  A )
 
Theoremchsupsn 21822 Value of supremum of subset of 
CH on a singleton. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  (  \/H  `  { A } )  =  A )
 
Theoremshlub 21823 Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH  /\  C  e.  CH )  ->  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  vH  B )  C_  C ) )
 
Theoremshlubi 21824 Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  CH   =>    |-  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  vH  B )  C_  C )
 
15.9.22  Projectors (cont.)
 
Theorempjhtheu2 21825* Uniqueness of  y for the projection theorem. (Contributed by NM, 6-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  E! y  e.  ( _|_ `  H ) E. x  e.  H  A  =  ( x  +h  y
 ) )
 
Theorempjcli 21826 Closure of a projection in its subspace. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( ( proj  h `  H ) `  A )  e.  H )
 
Theorempjhcli 21827 Closure of a projection in Hilbert space. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( ( proj  h `  H ) `  A )  e.  ~H )
 
Theorempjpjpre 21828 Decomposition of a vector into projections. This formulation of axpjpj 21829 avoids pjhth 21802. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  H  e.  CH )   &    |-  ( ph  ->  A  e.  ( H  +H  ( _|_ `  H ) ) )   =>    |-  ( ph  ->  A  =  ( ( ( proj  h `
  H ) `  A )  +h  (
 ( proj  h `  ( _|_ `  H ) ) `
  A ) ) )
 
Theoremaxpjpj 21829 Decomposition of a vector into projections. See comment in axpjcl 21809. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  A  =  ( ( ( proj  h `  H ) `  A )  +h  ( ( proj  h `  ( _|_ `  H )
 ) `  A )
 ) )
 
Theorempjclii 21830 Closure of a projection in its subspace. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  (
 ( proj  h `  H ) `  A )  e.  H
 
Theorempjhclii 21831 Closure of a projection in Hilbert space. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  (
 ( proj  h `  H ) `  A )  e. 
 ~H
 
Theorempjpj0i 21832 Decomposition of a vector into projections. (Contributed by NM, 26-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  A  =  ( ( ( proj  h `
  H ) `  A )  +h  (
 ( proj  h `  ( _|_ `  H ) ) `
  A ) )
 
Theorempjpji 21833 Decomposition of a vector into projections. (Contributed by NM, 6-Nov-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  A  =  ( ( ( proj  h `
  H ) `  A )  +h  (
 ( proj  h `  ( _|_ `  H ) ) `
  A ) )
 
Theorempjpjhth 21834* 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.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  E. x  e.  H  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) )
 
Theorempjpjhthi 21835* 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 21836 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 21837 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 21838 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 21839 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 21840 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 21841 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 21842 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 21843 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 21844 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 21813. (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 21845 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 21813. (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 21846 Proof of orthocomplement theorem using projections. Compare ococ 21815. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( _|_ `  ( _|_ `  H ) )  =  H
 
Theorempjoc2i 21847 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 21848 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 ) )
 
15.9.23  Hilbert lattice operations
 
Theoremsh0le 21849 The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  0H  C_  A )
 
Theoremch0le 21850 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 21851 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 21852 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 21853 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 21854 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 21855 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 21856 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 21857* 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 21858 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 21859 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 21860 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 21861 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 21862* 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 21863 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 21864 Join with lattice zero in  CH. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  vH  0H )  =  A
 
Theoremchm1i 21865 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 21866 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 21867 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 21868* 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 21869 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 21870 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 21871 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 21872 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 21873 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 21874 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 21875 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 21876 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 21877 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 21878  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 21879  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 21880 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 21881 Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 21880). (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 21882 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 21883 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 21884 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 21885 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 21886 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 21887 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 21888 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 21889 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 21890 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 21891 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 21892 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 21893 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 21894 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 21895 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 21896 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 21897 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 21898 Join with Hilbert lattice unit. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  vH  ~H )  =  ~H
 
Theoremchm0i 21899 Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  i^i  0H )  =  0H
 
Theoremchm0 21900 Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  i^i  0H )  =  0H )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30843
  Copyright terms: Public domain < Previous  Next >