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Theorem List for Metamath Proof Explorer - 21901-22000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremshsva 21901 Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( ( C  e.  A  /\  D  e.  B )  ->  ( C  +h  D )  e.  ( A  +H  B ) ) )
 
Theoremshsel1 21902 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  A  ->  C  e.  ( A  +H  B ) ) )
 
Theoremshsel2 21903 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  B  ->  C  e.  ( A  +H  B ) ) )
 
Theoremshsvs 21904 Vector subtraction belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( ( C  e.  A  /\  D  e.  B )  ->  ( C  -h  D )  e.  ( A  +H  B ) ) )
 
Theoremshsub1 21905 Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  A  C_  ( A  +H  B ) )
 
Theoremshsub2 21906 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 ) )
 
Theoremchoc0 21907 The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  ( _|_ `  0H )  =  ~H
 
Theoremchoc1 21908 The orthocomplement of the unit subspace is the zero subspace. Does not require Axiom of Choice. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( _|_ `  ~H )  =  0H
 
Theoremchocnul 21909 Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
 |-  ( _|_ `  (/) )  =  ~H
 
Theoremshintcli 21910 Closure of intersection of a non-empty subset of  SH. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  ( A  C_  SH  /\  A  =/= 
 (/) )   =>    |- 
 |^| A  e.  SH
 
Theoremshintcl 21911 The intersection of a non-empty set of subspaces is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  C_  SH  /\  A  =/=  (/) )  ->  |^| A  e.  SH )
 
Theoremchintcli 21912 The intersection of a non-empty set of closed subspaces is a closed subspace. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  ( A  C_  CH  /\  A  =/=  (/) )   =>    |- 
 |^| A  e.  CH
 
Theoremchintcl 21913 The intersection (infimum) of a non-empty subset of  CH belongs to  CH. Part of Theorem 3.13 of [Beran] p. 108. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  C_  CH  /\  A  =/=  (/) )  ->  |^| A  e.  CH )
 
Theoremspanval 21914* Value of the linear span of a subset of Hilbert space. The span is the intersection of all subspaces constraining the subset. Definition of span in [Schechter] p. 276. (Contributed by NM, 2-Jun-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( span `  A )  = 
 |^| { x  e.  SH  |  A  C_  x }
 )
 
Theoremhsupval 21915 Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 21990. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( A  C_  ~P ~H  ->  ( 
 \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )
 
Theoremchsupval 21916 The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 21991. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  C_  CH  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ ` 
 U. A ) ) )
 
Theoremspancl 21917 The span of a subset of Hilbert space is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( span `  A )  e. 
 SH )
 
Theoremelspancl 21918 A member of a span is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  e.  ( span `  A ) )  ->  B  e.  ~H )
 
Theoremshsupcl 21919 Closure of the subspace supremum of set of subsets of Hilbert space. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
 |-  ( A  C_  ~P ~H  ->  (
 span `  U. A )  e.  SH )
 
Theoremhsupcl 21920 Closure of supremum of set of subsets of Hilbert space. Note that the supremum belongs to  CH even if the subsets do not. (Contributed by NM, 10-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  ( A  C_  ~P ~H  ->  ( 
 \/H  `  A )  e.  CH )
 
Theoremchsupcl 21921 Closure of supremum of subset of 
CH. Definition of supremum in Proposition 1 of [Kalmbach] p. 65. Shows that  CH is a complete lattice. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 10-Nov-1999.) (New usage is discouraged.)
 |-  ( A  C_  CH  ->  (  \/H  `  A )  e.  CH )
 
Theoremhsupss 21922 Subset relation for supremum of Hilbert space subsets. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~P ~H  /\  B  C_  ~P ~H )  ->  ( A  C_  B  ->  (  \/H  `  A ) 
 C_  (  \/H  `  B ) ) )
 
Theoremchsupss 21923 Subset relation for supremum of subset of  CH. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  (
 ( A  C_  CH  /\  B  C_  CH )  ->  ( A  C_  B  ->  ( 
 \/H  `  A )  C_  (  \/H  `  B ) ) )
 
Theoremhsupunss 21924 The union of a set of Hilbert space subsets is smaller than its supremum. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  ( A  C_  ~P ~H  ->  U. A  C_  (  \/H  `  A ) )
 
Theoremchsupunss 21925 The union of a set of closed subspaces is smaller than its supremum. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |-  ( A  C_  CH  ->  U. A  C_  (  \/H  `  A ) )
 
Theoremspanss2 21926 A subset of Hilbert space is included in its span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  A  C_  ( span `  A )
 )
 
Theoremshsupunss 21927 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 21928 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 21929 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 21930 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 21931 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 21932 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 21933 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 21934 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 21935 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 21936 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 21937 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 21938 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 21939 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 21940 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 21941 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 21942 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 21943 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 21944 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 21945 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 21946 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 21947 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 21948 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 21949 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 21950 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 21951 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 21952 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 21953 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 21954 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 21955 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 21956 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 21957  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 21958 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 21959 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 21960 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 21961 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 21962 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 21963 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 21964 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 21965 Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  ( A  +H  A )  =  A
 
Theoremshslubi 21966 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 21967 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 21968* 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 21969 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 21970 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 21971 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 ) ) )
 
17.4.5  Projection theorem
 
Theorempjhthlem1 21972* Lemma for pjhth 21974. (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 21973* Lemma for pjhth 21974. (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 21974 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 21975* 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 21997 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 ) )
 
17.4.6  Projectors
 
Definitiondf-pjh 21976* 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 21977* 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 21978* 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 21979* Equality with a projection. This version of pjeq 21980 does not assume the Axiom of Choice via pjhth 21974. (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 21980* 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 21981 Closure of a projection in its subspace. If we consider this together with axpjpj 22001 to be axioms, the need for the ax-hcompl 21783 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 22016.) (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 21982 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 )
 
17.5  Properties of Hilbert subspaces
 
17.5.1  Orthomodular law
 
Theoremomlsilem 21983 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 21984 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 21985 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 21986 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 21987 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 21988 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 21989* 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 21990* 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 21991* 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 21992* 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 21993* 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 21994 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 21995 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 21996 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 )
 
17.5.2  Projectors (cont.)
 
Theorempjhtheu2 21997* 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 21998 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 21999 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 22000 Decomposition of a vector into projections. This formulation of axpjpj 22001 avoids pjhth 21974. (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 ) ) )
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