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Theorem List for Metamath Proof Explorer - 21701-21800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremshocorth 21701 Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  ->  ( A  .ih  B )  =  0 ) )
 
Theoremococss 21702 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  A  C_  ( _|_ `  ( _|_ `  A ) ) )
 
Theoremshococss 21703 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  A 
 C_  ( _|_ `  ( _|_ `  A ) ) )
 
Theoremshorth 21704 Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
 |-  ( H  e.  SH  ->  ( G  C_  ( _|_ `  H )  ->  (
 ( A  e.  G  /\  B  e.  H ) 
 ->  ( A  .ih  B )  =  0 )
 ) )
 
Theoremocin 21705 Intersection of a Hilbert subspace and its complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A ) )  =  0H )
 
Theoremoccon3 21706 Hilbert lattice contraposition law. (Contributed by Mario Carneiro, 18-May-2014.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( A  C_  ( _|_ `  B )  <->  B  C_  ( _|_ `  A ) ) )
 
Theoremocnel 21707 A nonzero vector in the complement of a subspace does not belong to the subspace. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( H  e.  SH  /\  A  e.  ( _|_ `  H )  /\  A  =/=  0h )  ->  -.  A  e.  H )
 
Theoremchocvali 21708* Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of  A is the set of vectors that are orthogonal to all vectors in  A. (Contributed by NM, 8-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  A )  =  { x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 }
 
Theoremshuni 21709 Two subspaces with trivial intersection have a unique decomposition of the elements of the subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  H  e.  SH )   &    |-  ( ph  ->  K  e.  SH )   &    |-  ( ph  ->  ( H  i^i  K )  =  0H )   &    |-  ( ph  ->  A  e.  H )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  C  e.  H )   &    |-  ( ph  ->  D  e.  K )   &    |-  ( ph  ->  ( A  +h  B )  =  ( C  +h  D ) )   =>    |-  ( ph  ->  ( A  =  C  /\  B  =  D )
 )
 
Theoremchocunii 21710 Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of [Beran] p. 102 (uniqueness part). (Contributed by NM, 23-Oct-1999.) (Proof shortened by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  /\  ( C  e.  H  /\  D  e.  ( _|_ `  H ) ) )  ->  ( ( R  =  ( A  +h  B ) 
 /\  R  =  ( C  +h  D ) )  ->  ( A  =  C  /\  B  =  D ) ) )
 
Theorempjhthmo 21711* Projection Theorem, uniqueness part. Any two disjoint subspaces yield a unique decomposition of vectors into each subspace. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  SH  /\  ( A  i^i  B )  =  0H )  ->  E* x ( x  e.  A  /\  E. y  e.  B  C  =  ( x  +h  y ) ) )
 
Theoremoccllem 21712 Lemma for occl 21713. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  C_  ~H )   &    |-  ( ph  ->  F  e.  Cauchy )   &    |-  ( ph  ->  F : NN
 --> ( _|_ `  A ) )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  ( (  ~~>v  `  F )  .ih  B )  =  0 )
 
Theoremoccl 21713 Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( A  C_  ~H  ->  ( _|_ `  A )  e. 
 CH )
 
Theoremshoccl 21714 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  SH  ->  ( _|_ `  A )  e.  CH )
 
Theoremchoccl 21715 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( _|_ `  A )  e. 
 CH )
 
Theoremchoccli 21716 Closure of  CH orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  A )  e.  CH
 
15.9.18  Subspace sum, span, lattice join, lattice supremum
 
Definitiondf-shs 21717* Define subspace sum in  SH. See shsval 21721, shsval2i 21796, and shsval3i 21797 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 21718* Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 21742 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 21719* Define Hilbert lattice join. See chjval 21761 for its value and chjcl 21766 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 21764. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
 |-  vH  =  ( x  e.  ~P ~H ,  y  e.  ~P ~H  |->  ( _|_ `  ( _|_ `  ( x  u.  y ) ) ) )
 
Definitiondf-chsup 21720 Define the supremum of a set of Hilbert lattice elements. See chsupval2 21819 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 21748. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)
 |-  \/H  =  ( x  e.  ~P ~P ~H  |->  ( _|_ `  ( _|_ `  U. x ) ) )
 
Theoremshsval 21721 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 21722 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 21723* 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 21724* 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 21725* 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 21726 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 21727 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 21728 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 21729 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 21730 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 21731 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 21732 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 21733 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 21734 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 21735 The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  ( _|_ `  0H )  =  ~H
 
Theoremchoc1 21736 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 21737 Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
 |-  ( _|_ `  (/) )  =  ~H
 
Theoremshintcli 21738 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 21739 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 21740 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 21741 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 21742* 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 21743 Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 21818. (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 21744 The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 21819. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  C_  CH  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ ` 
 U. A ) ) )
 
Theoremspancl 21745 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 21746 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 21747 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 21748 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 21749 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 21750 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 21751 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 21752 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 21753 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 21754 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 21755 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 21756 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 21757 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 21758 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 21759 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 21760 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 21761 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 21762 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 21763 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 21764 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 21765 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 21766 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 21767 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 21768 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 21769 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 21770 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 21771 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 21772 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 21773 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 21774 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 21775 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 21776 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 21777 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 21778 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 21779 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 21780 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 21781 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 21782 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 21783 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 21784 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 21785  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 21786 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 21787 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 21788 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 21789 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 21790 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 21791 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 21792 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 21793 Idempotent law for Hilbert subspace sum. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  ( A  +H  A )  =  A
 
Theoremshslubi 21794 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 21795 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 21796* 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 21797 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 21798 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 21799 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 ) ) )
 
15.9.19  Projection theorem
 
Theorempjhthlem1 21800* 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  ->  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 )
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