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Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremchoccli 25901 Closure of  CH orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  A )  e.  CH
 
20.4.4  Subspace sum, span, lattice join, lattice supremum
 
Definitiondf-shs 25902* Define subspace sum in  SH. See shsval 25906, shsval2i 25981, and shsval3i 25982 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
 |-  +H  =  ( x  e.  SH ,  y  e.  SH  |->  (  +h  " ( x  X.  y ) ) )
 
Definitiondf-span 25903* Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 25927 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  span  =  ( x  e.  ~P ~H  |->  |^| { y  e. 
 SH  |  x  C_  y } )
 
Definitiondf-chj 25904* Define Hilbert lattice join. See chjval 25946 for its value and chjcl 25951 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to  CH; see sshjcl 25949. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
 |-  vH  =  ( x  e.  ~P ~H ,  y  e.  ~P ~H  |->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) ) )
 
Definitiondf-chsup 25905 Define the supremum of a set of Hilbert lattice elements. See chsupval2 26004 for its value. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice  CH, to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl 25933. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)
 |-  \/H  =  ( x  e.  ~P ~P ~H  |->  ( _|_ `  ( _|_ `  U. x ) ) )
 
Theoremshsval 25906 Value of subspace sum of two Hilbert space subspaces. Definition of subspace sum in [Kalmbach] p. 65. (Contributed by NM, 16-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  =  (  +h  " ( A  X.  B ) ) )
 
Theoremshsss 25907 The subspace sum is a subset of Hilbert space. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  C_  ~H )
 
Theoremshsel 25908* Membership in the subspace sum of two Hilbert subspaces. (Contributed by NM, 14-Dec-2004.) (Revised by Mario Carneiro, 29-Jan-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y ) ) )
 
Theoremshsel3 25909* Membership in the subspace sum of two Hilbert subspaces, using vector subtraction. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  -h  y ) ) )
 
Theoremshseli 25910* Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( C  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x  +h  y ) )
 
Theoremshscli 25911 Closure of subspace sum. (Contributed by NM, 15-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  +H  B )  e. 
 SH
 
Theoremshscl 25912 Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  e.  SH )
 
Theoremshscom 25913 Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B )  =  ( B  +H  A ) )
 
Theoremshsva 25914 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 25915 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 25916 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 25917 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 25918 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 25919 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 25920 The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  ( _|_ `  0H )  =  ~H
 
Theoremchoc1 25921 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 25922 Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
 |-  ( _|_ `  (/) )  =  ~H
 
Theoremshintcli 25923 Closure of intersection of a nonempty subset of  SH. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  ( A  C_  SH  /\  A  =/= 
 (/) )   =>    |- 
 |^| A  e.  SH
 
Theoremshintcl 25924 The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  C_  SH  /\  A  =/=  (/) )  ->  |^| A  e.  SH )
 
Theoremchintcli 25925 The intersection of a nonempty 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 25926 The intersection (infimum) of a nonempty 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 25927* 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 25928 Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 26003. (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 25929 The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 26004. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  C_  CH  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ ` 
 U. A ) ) )
 
Theoremspancl 25930 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 25931 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 25932 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 25933 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 25934 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 25935 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 25936 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 25937 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 25938 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 25939 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 25940 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 25941 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 25942 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 25943 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 25944 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 25945 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 25946 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 25947 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 25948 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 25949 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 25950 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 25951 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 25952 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 25953 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 25954 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 25955 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 25956 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 25957 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 25958 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 25959 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 25960 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 25961 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 25962 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 25963 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 25964 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 25965 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 25966 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 25967 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 25968 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 25969 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 25970  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 25971 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 25972 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 25973 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 25974 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 25975 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 25976 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 25977 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 25978 Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  ( A  +H  A )  =  A
 
Theoremshslubi 25979 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 25980 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 25981* 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 25982 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 25983 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 25984 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 ) ) )
 
20.4.5  Projection theorem
 
Theorempjhthlem1 25985* Lemma for pjhth 25987. (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 25986* Lemma for pjhth 25987. (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 25987 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 25988* 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 26010 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 ) )
 
20.4.6  Projectors
 
Definitiondf-pjh 25989* 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 25990* 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 25991* 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 25992* Equality with a projection. This version of pjeq 25993 does not assume the Axiom of Choice via pjhth 25987. (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 25993* 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 25994 Closure of a projection in its subspace. If we consider this together with axpjpj 26014 to be axioms, the need for the ax-hcompl 25795 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 26029.) (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 25995 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 )
 
20.5  Properties of Hilbert subspaces
 
20.5.1  Orthomodular law
 
Theoremomlsilem 25996 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 25997 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 25998 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 25999 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 26000 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 )
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