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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremchsscon3i 22001 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  <->  ( _|_ `  B )  C_  ( _|_ `  A ) )
 
Theoremchsscon1i 22002 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( _|_ `  A )  C_  B  <->  ( _|_ `  B )  C_  A )
 
Theoremchsscon2i 22003 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  ( _|_ `  B ) 
 <->  B  C_  ( _|_ `  A ) )
 
Theoremchcon2i 22004 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  =  ( _|_ `  B )  <->  B  =  ( _|_ `  A ) )
 
Theoremchcon1i 22005 Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( _|_ `  A )  =  B  <->  ( _|_ `  B )  =  A )
 
Theoremchcon3i 22006 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  =  B  <->  ( _|_ `  B )  =  ( _|_ `  A ) )
 
Theoremchunssji 22007 Union is smaller than  CH join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  u.  B )  C_  ( A  vH  B )
 
Theoremchjcomi 22008 Commutative law for join in  CH. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  B )  =  ( B  vH  A )
 
Theoremchub1i 22009  CH join is an upper bound of two elements. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  A  C_  ( A  vH  B )
 
Theoremchub2i 22010  CH join is an upper bound of two elements. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  A  C_  ( B  vH  A )
 
Theoremchlubi 22011 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  vH  B )  C_  C )
 
Theoremchlubii 22012 Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 22011). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  C_  C  /\  B  C_  C )  ->  ( A  vH  B )  C_  C )
 
Theoremchlej1i 22013 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( A  C_  B  ->  ( A  vH  C )  C_  ( B  vH  C ) )
 
Theoremchlej2i 22014 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( A  C_  B  ->  ( C  vH  A )  C_  ( C  vH  B ) )
 
Theoremchlej12i 22015 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( A  C_  B  /\  C  C_  D )  ->  ( A  vH  C )  C_  ( B  vH  D ) )
 
Theoremchlejb1i 22016 Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  <->  ( A  vH  B )  =  B )
 
Theoremchdmm1i 22017 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( _|_ `  ( A  i^i  B ) )  =  ( ( _|_ `  A )  vH  ( _|_ `  B ) )
 
Theoremchdmm2i 22018 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( _|_ `  ( ( _|_ `  A )  i^i  B ) )  =  ( A  vH  ( _|_ `  B ) )
 
Theoremchdmm3i 22019 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( _|_ `  ( A  i^i  ( _|_ `  B )
 ) )  =  ( ( _|_ `  A )  vH  B )
 
Theoremchdmm4i 22020 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( _|_ `  ( ( _|_ `  A )  i^i  ( _|_ `  B ) ) )  =  ( A 
 vH  B )
 
Theoremchdmj1i 22021 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( _|_ `  ( A  vH  B ) )  =  ( ( _|_ `  A )  i^i  ( _|_ `  B ) )
 
Theoremchdmj2i 22022 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( _|_ `  ( ( _|_ `  A )  vH  B ) )  =  ( A  i^i  ( _|_ `  B ) )
 
Theoremchdmj3i 22023 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( _|_ `  ( A  vH  ( _|_ `  B )
 ) )  =  ( ( _|_ `  A )  i^i  B )
 
Theoremchdmj4i 22024 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( _|_ `  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) )  =  ( A  i^i  B )
 
Theoremchnlei 22025 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( -.  B  C_  A  <->  A  C.  ( A 
 vH  B ) )
 
Theoremchjassi 22026 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  vH  B )  vH  C )  =  ( A  vH  ( B  vH  C ) )
 
Theoremchj00i 22027 Two Hilbert lattice elements are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  =  0H  /\  B  =  0H )  <->  ( A  vH  B )  =  0H )
 
Theoremchjoi 22028 The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  vH  ( _|_ `  A ) )  =  ~H
 
Theoremchj1i 22029 Join with Hilbert lattice unit. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  vH  ~H )  =  ~H
 
Theoremchm0i 22030 Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  i^i  0H )  =  0H
 
Theoremchm0 22031 Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  i^i  0H )  =  0H )
 
Theoremshjshsi 22032 Hilbert lattice join equals the double orthocomplement of subspace sum. (Contributed by NM, 27-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( A  vH  B )  =  ( _|_ `  ( _|_ `  ( A  +H  B ) ) )
 
Theoremshjshseli 22033 A closed subspace sum equals Hilbert lattice join. Part of Lemma 31.1.5 of [MaedaMaeda] p. 136. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  (
 ( A  +H  B )  e.  CH  <->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremchne0 22034* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h ) )
 
Theoremchocin 22035 Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  i^i  ( _|_ `  A ) )  =  0H )
 
Theoremchssoc 22036 A closed subspace less than its orthocomplement is zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  C_  ( _|_ `  A ) 
 <->  A  =  0H )
 )
 
Theoremchj0 22037 Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  vH  0H )  =  A )
 
Theoremchslej 22038 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  +H  B )  C_  ( A  vH  B ) )
 
Theoremchincl 22039 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  B )  e.  CH )
 
Theoremchsscon3 22040 Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( _|_ `  B )  C_  ( _|_ `  A ) ) )
 
Theoremchsscon1 22041 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( ( _|_ `  A )  C_  B  <->  ( _|_ `  B )  C_  A ) )
 
Theoremchsscon2 22042 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  ( _|_ `  B )  <->  B  C_  ( _|_ `  A ) ) )
 
Theoremchpsscon3 22043 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C.  B  <->  ( _|_ `  B )  C.  ( _|_ `  A ) ) )
 
Theoremchpsscon1 22044 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( ( _|_ `  A )  C.  B  <->  ( _|_ `  B )  C.  A ) )
 
Theoremchpsscon2 22045 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C.  ( _|_ `  B )  <->  B  C.  ( _|_ `  A ) ) )
 
Theoremchjcom 22046 Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B )  =  ( B  vH  A ) )
 
Theoremchub1 22047 Hilbert lattice join is greater than or equal to its first argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  A  C_  ( A  vH  B ) )
 
Theoremchub2 22048 Hilbert lattice join is greater than or equal to its second argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  A  C_  ( B  vH  A ) )
 
Theoremchlub 22049 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  vH  B )  C_  C ) )
 
Theoremchlej1 22050 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  A  C_  B )  ->  ( A  vH  C )  C_  ( B  vH  C ) )
 
Theoremchlej2 22051 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  A  C_  B )  ->  ( C  vH  A )  C_  ( C  vH  B ) )
 
Theoremchlejb1 22052 Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( A  vH  B )  =  B ) )
 
Theoremchlejb2 22053 Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( B  vH  A )  =  B ) )
 
Theoremchnle 22054 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( -.  B  C_  A 
 <->  A  C.  ( A 
 vH  B ) ) )
 
Theoremchjo 22055 The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  vH  ( _|_ `  A ) )  =  ~H )
 
Theoremchabs1 22056 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  ( A  i^i  B ) )  =  A )
 
Theoremchabs2 22057 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  ( A  vH  B ) )  =  A )
 
Theoremchabs1i 22058 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  ( A  i^i  B ) )  =  A
 
Theoremchabs2i 22059 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  i^i  ( A  vH  B ) )  =  A
 
Theoremchjidm 22060 Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  ( A  vH  A )  =  A )
 
Theoremchjidmi 22061 Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  vH  A )  =  A
 
Theoremchj12i 22062 A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( A  vH  ( B  vH  C ) )  =  ( B  vH  ( A  vH  C ) )
 
Theoremchj4i 22063 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   =>    |-  (
 ( A  vH  B )  vH  ( C  vH  D ) )  =  ( ( A  vH  C )  vH  ( B 
 vH  D ) )
 
Theoremchjjdiri 22064 Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  vH  B )  vH  C )  =  ( ( A 
 vH  C )  vH  ( B  vH  C ) )
 
Theoremchdmm1 22065 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  i^i  B ) )  =  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) )
 
Theoremchdmm2 22066 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  i^i  B ) )  =  ( A  vH  ( _|_ `  B ) ) )
 
Theoremchdmm3 22067 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  i^i  ( _|_ `  B ) ) )  =  ( ( _|_ `  A )  vH  B ) )
 
Theoremchdmm4 22068 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  i^i  ( _|_ `  B ) ) )  =  ( A  vH  B ) )
 
Theoremchdmj1 22069 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  vH  B ) )  =  ( ( _|_ `  A )  i^i  ( _|_ `  B ) ) )
 
Theoremchdmj2 22070 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  vH  B ) )  =  ( A  i^i  ( _|_ `  B ) ) )
 
Theoremchdmj3 22071 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  vH  ( _|_ `  B ) ) )  =  ( ( _|_ `  A )  i^i  B ) )
 
Theoremchdmj4 22072 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  (
 ( _|_ `  A )  vH  ( _|_ `  B ) ) )  =  ( A  i^i  B ) )
 
Theoremchjass 22073 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( ( A  vH  B )  vH  C )  =  ( A  vH  ( B  vH  C ) ) )
 
Theoremchj12 22074 A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  vH  ( B 
 vH  C ) )  =  ( B  vH  ( A  vH  C ) ) )
 
Theoremchj4 22075 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH )  /\  ( C  e.  CH 
 /\  D  e.  CH ) )  ->  ( ( A  vH  B ) 
 vH  ( C  vH  D ) )  =  ( ( A  vH  C )  vH  ( B 
 vH  D ) ) )
 
Theoremledii 22076 An ortholattice is distributive in one ordering direction. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  i^i  B )  vH  ( A  i^i  C ) ) 
 C_  ( A  i^i  ( B  vH  C ) )
 
Theoremlediri 22077 An ortholattice is distributive in one ordering direction. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  i^i  C )  vH  ( B  i^i  C ) ) 
 C_  ( ( A 
 vH  B )  i^i 
 C )
 
Theoremlejdii 22078 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( A  vH  ( B  i^i  C ) ) 
 C_  ( ( A 
 vH  B )  i^i  ( A  vH  C ) )
 
Theoremlejdiri 22079 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  i^i  B )  vH  C ) 
 C_  ( ( A 
 vH  C )  i^i  ( B  vH  C ) )
 
Theoremledi 22080 An ortholattice is distributive in one ordering direction. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) ) 
 C_  ( A  i^i  ( B  vH  C ) ) )
 
17.5.4  Span (cont.) and one-dimensional subspaces
 
Theoremspansn0 22081 The span of the singleton of the zero vector is the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
 |-  ( span `  { 0h }
 )  =  0H
 
Theoremspan0 22082 The span of the empty set is the zero subspace. Remark 11.6.e of [Schechter] p. 276. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  ( span `  (/) )  =  0H
 
Theoremelspani 22083* Membership in the span of a subset of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  B  e.  _V   =>    |-  ( A  C_  ~H  ->  ( B  e.  ( span `  A )  <->  A. x  e.  SH  ( A  C_  x  ->  B  e.  x )
 ) )
 
Theoremspanuni 22084 The span of a union is the subspace sum of spans. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
 |-  A  C_ 
 ~H   &    |-  B  C_  ~H   =>    |-  ( span `  ( A  u.  B ) )  =  ( ( span `  A )  +H  ( span `  B ) )
 
Theoremspanun 22085 The span of a union is the subspace sum of spans. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  C_  ~H  /\  B  C_  ~H )  ->  ( span `  ( A  u.  B ) )  =  ( ( span `  A )  +H  ( span `  B ) ) )
 
Theoremsshhococi 22086 The join of two Hilbert space subsets (not necessarily closed subspaces) equals the join of their closures (double orthocomplements). (Contributed by NM, 1-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  C_ 
 ~H   &    |-  B  C_  ~H   =>    |-  ( A  vH  B )  =  ( ( _|_ `  ( _|_ `  A ) )  vH  ( _|_ `  ( _|_ `  B ) ) )
 
Theoremhne0 22087 Hilbert space has a nonzero vector iff it is not trivial. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
 |-  ( ~H  =/=  0H  <->  E. x  e.  ~H  x  =/=  0h )
 
Theoremchsup0 22088 The supremum of the empty set. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  (  \/H  `  (/) )  =  0H
 
Theoremh1deoi 22089 Membership in orthocomplement of 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (New usage is discouraged.)
 |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  { B }
 ) 
 <->  ( A  e.  ~H  /\  ( A  .ih  B )  =  0 )
 )
 
Theoremh1dei 22090* Membership in 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  ( A  e.  ~H 
 /\  A. x  e.  ~H  ( ( B  .ih  x )  =  0  ->  ( A  .ih  x )  =  0 ) ) )
 
Theoremh1did 22091 A generating vector belongs to the 1-dimensional subspace it generates. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  A  e.  ( _|_ `  ( _|_ `  { A }
 ) ) )
 
Theoremh1dn0 22092 A nonzero vector generates a (nonzero) 1-dimensional subspace. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  ( _|_ `  ( _|_ `  { A }
 ) )  =/=  0H )
 
Theoremh1de2i 22093 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 17-Jul-2001.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  ->  ( ( B  .ih  B )  .h  A )  =  ( ( A 
 .ih  B )  .h  B ) )
 
Theoremh1de2bi 22094 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( B  =/=  0h  ->  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  A  =  (
 ( ( A  .ih  B )  /  ( B 
 .ih  B ) )  .h  B ) ) )
 
Theoremh1de2ctlem 22095* Lemma for h1de2ci 22096. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  E. x  e.  CC  A  =  ( x  .h  B ) )
 
Theoremh1de2ci 22096* Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 21-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  B  e.  ~H   =>    |-  ( A  e.  ( _|_ `  ( _|_ `  { B } ) )  <->  E. x  e.  CC  A  =  ( x  .h  B ) )
 
Theoremspansni 22097 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( span `  { A }
 )  =  ( _|_ `  ( _|_ `  { A } ) )
 
Theoremelspansni 22098* Membership in the span of a singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( B  e.  ( span `  { A }
 ) 
 <-> 
 E. x  e.  CC  B  =  ( x  .h  A ) )
 
Theoremspansn 22099 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 span `  { A }
 )  =  ( _|_ `  ( _|_ `  { A } ) ) )
 
Theoremspansnch 22100 The span of a Hilbert space singleton belongs to the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 span `  { A }
 )  e.  CH )
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