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Theorem List for Metamath Proof Explorer - 22801-22900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremshsupunss 22801 The union of a set of subspaces is smaller than its supremum. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
 |-  ( A  C_  SH  ->  U. A  C_  ( span `  U. A ) )
 
Theoremspanid 22802 A subspace of Hilbert space is its own span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  (
 span `  A )  =  A )
 
Theoremspanss 22803 Ordering relationship for the spans of subsets of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( B  C_  ~H  /\  A  C_  B )  ->  ( span `  A )  C_  ( span `  B )
 )
 
Theoremspanssoc 22804 The span of a subset of Hilbert space is less than or equal to its closure (double orthogonal complement). (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( span `  A )  C_  ( _|_ `  ( _|_ `  A ) ) )
 
Theoremsshjval 22805 Value of join for subsets of Hilbert space. (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 )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
 
Theoremshjval 22806 Value of join in  SH. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
 
Theoremchjval 22807 Value of join in  CH. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) ) )
 
Theoremchjvali 22808 Value of join in  CH. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  u.  B ) ) )
 
Theoremsshjval3 22809 Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice  CH. (Contributed by NM, 2-Mar-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( A  vH  B )  =  (  \/H  `  { A ,  B } ) )
 
Theoremsshjcl 22810 Closure of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( A  vH  B )  e.  CH )
 
Theoremshjcl 22811 Closure of join in  SH. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  vH  B )  e.  CH )
 
Theoremchjcl 22812 Closure of join in  CH. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B )  e.  CH )
 
Theoremshjcom 22813 Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  vH  B )  =  ( B  vH  A ) )
 
Theoremshless 22814 Subset implies subset of subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( A  +H  C )  C_  ( B  +H  C ) )
 
Theoremshlej1 22815 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( A  vH  C )  C_  ( B  vH  C ) )
 
Theoremshlej2 22816 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  SH  /\  B  e.  SH  /\  C  e.  SH )  /\  A  C_  B )  ->  ( C  vH  A )  C_  ( C  vH  B ) )
 
Theoremshincli 22817 Closure of intersection of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  i^i  B )  e. 
 SH
 
Theoremshscomi 22818 Commutative law for subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  +H  B )  =  ( B  +H  A )
 
Theoremshsvai 22819 Vector sum belongs to subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  (
 ( C  e.  A  /\  D  e.  B ) 
 ->  ( C  +h  D )  e.  ( A  +H  B ) )
 
Theoremshsel1i 22820 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( C  e.  A  ->  C  e.  ( A  +H  B ) )
 
Theoremshsel2i 22821 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( C  e.  B  ->  C  e.  ( A  +H  B ) )
 
Theoremshsvsi 22822 Vector subtraction belongs to subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  (
 ( C  e.  A  /\  D  e.  B ) 
 ->  ( C  -h  D )  e.  ( A  +H  B ) )
 
Theoremshunssi 22823 Union is smaller than subspace sum. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  u.  B )  C_  ( A  +H  B )
 
Theoremshunssji 22824 Union is smaller than Hilbert lattice join. (Contributed by NM, 11-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  u.  B )  C_  ( A  vH  B )
 
Theoremshsleji 22825 Subspace sum is smaller than Hilbert lattice join. Remark in [Kalmbach] p. 65. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  +H  B )  C_  ( A  vH  B )
 
Theoremshjcomi 22826 Commutative law for join in  SH. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  vH  B )  =  ( B  vH  A )
 
Theoremshsub1i 22827 Subspace sum is an upper bound of its arguments. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  A  C_  ( A  +H  B )
 
Theoremshsub2i 22828 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  A  C_  ( B  +H  A )
 
Theoremshub1i 22829 Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  A  C_  ( A  vH  B )
 
Theoremshjcli 22830 Closure of  CH join. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  vH  B )  e. 
 CH
 
Theoremshjshcli 22831  SH closure of join. (Contributed by NM, 5-May-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  vH  B )  e. 
 SH
 
Theoremshlessi 22832 Subset implies subset of subspace sum. (Contributed by NM, 18-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   =>    |-  ( A  C_  B  ->  ( A  +H  C )  C_  ( B  +H  C ) )
 
Theoremshlej1i 22833 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   =>    |-  ( A  C_  B  ->  ( A  vH  C )  C_  ( B  vH  C ) )
 
Theoremshlej2i 22834 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   =>    |-  ( A  C_  B  ->  ( C  vH  A )  C_  ( C  vH  B ) )
 
Theoremshslej 22835 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  C_  ( A  vH  B ) )
 
Theoremshincl 22836 Closure of intersection of two subspaces. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  i^i  B )  e.  SH )
 
Theoremshub1 22837 Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  A  C_  ( A  vH  B ) )
 
Theoremshub2 22838 A subspace is a subset of its Hilbert lattice join with another. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  A  C_  ( B  vH  A ) )
 
Theoremshsidmi 22839 Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  ( A  +H  A )  =  A
 
Theoremshslubi 22840 The least upper bound law for Hilbert subspace sum. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   =>    |-  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  +H  B )  C_  C )
 
Theoremshlesb1i 22841 Hilbert lattice ordering in terms of subspace sum. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  C_  B  <->  ( A  +H  B )  =  B )
 
Theoremshsval2i 22842* An alternate way to express subspace sum. (Contributed by NM, 25-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  +H  B )  = 
 |^| { x  e.  SH  |  ( A  u.  B )  C_  x }
 
Theoremshsval3i 22843 An alternate way to express subspace sum. (Contributed by NM, 25-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  +H  B )  =  ( span `  ( A  u.  B ) )
 
Theoremshmodsi 22844 The modular law holds for subspace sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   =>    |-  ( A  C_  C  ->  ( ( A  +H  B )  i^i  C ) 
 C_  ( A  +H  ( B  i^i  C ) ) )
 
Theoremshmodi 22845 The modular law is implied by the closure of subspace sum. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 23-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   =>    |-  ( ( ( A  +H  B )  =  ( A  vH  B )  /\  A  C_  C )  ->  ( ( A 
 vH  B )  i^i 
 C )  C_  ( A  vH  ( B  i^i  C ) ) )
 
18.4.5  Projection theorem
 
Theorempjhthlem1 22846* Lemma for pjhth 22848. (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  ->  B  e.  H )   &    |-  ( ph  ->  C  e.  H )   &    |-  ( ph  ->  A. x  e.  H  (
 normh `  ( A  -h  B ) )  <_  ( normh `  ( A  -h  x ) ) )   &    |-  T  =  ( (
 ( A  -h  B )  .ih  C )  /  ( ( C  .ih  C )  +  1 ) )   =>    |-  ( ph  ->  (
 ( A  -h  B )  .ih  C )  =  0 )
 
Theorempjhthlem2 22847* Lemma for pjhth 22848. (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 22848 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 22849* 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 22871 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 ) )
 
18.4.6  Projectors
 
Definitiondf-pjh 22850* 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 22851* 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 22852* 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 22853* Equality with a projection. This version of pjeq 22854 does not assume the Axiom of Choice via pjhth 22848. (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 22854* 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 22855 Closure of a projection in its subspace. If we consider this together with axpjpj 22875 to be axioms, the need for the ax-hcompl 22657 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 22890.) (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 22856 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 )
 
18.5  Properties of Hilbert subspaces
 
18.5.1  Orthomodular law
 
Theoremomlsilem 22857 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 22858 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 22859 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 22860 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 22861 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 22862 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 22863* 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 22864* 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 22865* 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 22866* 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 22867* 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 22868 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 22869 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 22870 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 )
 
18.5.2  Projectors (cont.)
 
Theorempjhtheu2 22871* 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 22872 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 22873 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 22874 Decomposition of a vector into projections. This formulation of axpjpj 22875 avoids pjhth 22848. (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 22875 Decomposition of a vector into projections. See comment in axpjcl 22855. (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 22876 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 22877 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 22878 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 22879 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 22880* 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 22881* 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 22882 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 22883 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 22884 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 22885 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 22886 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 22887 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 22888 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 22889 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 22890 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 22859. (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 22891 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 22859. (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 22892 Proof of orthocomplement theorem using projections. Compare ococ 22861. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( _|_ `  ( _|_ `  H ) )  =  H
 
Theorempjoc2i 22893 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 22894 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 ) )
 
18.5.3  Hilbert lattice operations
 
Theoremsh0le 22895 The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  0H  C_  A )
 
Theoremch0le 22896 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 22897 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 22898 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 22899 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 22900 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 )
 )
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