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Theorem List for Metamath Proof Explorer - 23801-23900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorematne0 23801 An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  ->  A  =/=  0H )
 
Theorematss 23802 A lattice element smaller than an atom is either the atom or zero. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  C_  B  ->  ( A  =  B  \/  A  =  0H )
 ) )
 
Theorematsseq 23803 Two atoms in a subset relationship are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms )  ->  ( A  C_  B  <->  A  =  B ) )
 
Theorematcveq0 23804 A Hilbert lattice element covered by an atom must be the zero subspace. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  <oH  B  <->  A  =  0H ) )
 
Theoremh1da 23805 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  ( _|_ `  ( _|_ `  { A }
 ) )  e. HAtoms )
 
Theoremspansna 23806 The span of the singleton of a vector is an atom. (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  ( span `  { A }
 )  e. HAtoms )
 
Theoremsh1dle 23807 A 1-dimensional subspace is less than or equal to any subspace containing its generating vector. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  SH  /\  B  e.  A ) 
 ->  ( _|_ `  ( _|_ `  { B }
 ) )  C_  A )
 
Theoremch1dle 23808 A 1-dimensional subspace is less than or equal to any member of  CH containing its generating vector. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  A ) 
 ->  ( _|_ `  ( _|_ `  { B }
 ) )  C_  A )
 
Theorematom1d 23809* The 1-dimensional subspaces of Hilbert space are its atoms. Part of Remark 10.3.5 of [BeltramettiCassinelli] p. 107. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  ( A  e. HAtoms  <->  E. x  e.  ~H  ( x  =/=  0h  /\  A  =  ( span ` 
 { x } )
 ) )
 
18.8.3  Superposition principle
 
Theoremsuperpos 23810* Superposition Principle. If  A and  B are distinct atoms, there exists a third atom, distinct from  A and  B, that is the superposition of  A and  B. Definition 3.4-3(a) in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms  /\  A  =/=  B )  ->  E. x  e. HAtoms  ( x  =/=  A  /\  x  =/=  B  /\  x  C_  ( A  vH  B ) ) )
 
18.8.4  Atoms, exchange and covering properties, atomicity
 
Theoremchcv1 23811 The Hilbert lattice has the covering property. Proposition 1(ii) of [Kalmbach] p. 140 (and its converse). (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( -.  B  C_  A  <->  A  <oH  ( A 
 vH  B ) ) )
 
Theoremchcv2 23812 The Hilbert lattice has the covering property. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  C.  ( A  vH  B )  <->  A  <oH  ( A 
 vH  B ) ) )
 
Theoremchjatom 23813 The join of a closed subspace and an atom equals their subspace sum. Special case of remark in [Kalmbach] p. 65, stating that if  A or  B is finite dimensional, then this equality holds. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremshatomici 23814* The lattice of Hilbert subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  ( A  =/=  0H  ->  E. x  e. HAtoms  x  C_  A )
 
Theoremhatomici 23815* The Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( A  =/=  0H  ->  E. x  e. HAtoms  x  C_  A )
 
Theoremhatomic 23816* A Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. Also Definition 3.4-2 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  A  =/=  0H )  ->  E. x  e. HAtoms  x  C_  A )
 
Theoremshatomistici 23817* The lattice of Hilbert subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  SH   =>    |-  A  =  ( span ` 
 U. { x  e. HAtoms  |  x  C_  A }
 )
 
Theoremhatomistici 23818*  CH is atomistic, i.e. any element is the supremum of its atoms. Remark in [Kalmbach] p. 140. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  =  (  \/H  ` 
 { x  e. HAtoms  |  x  C_  A } )
 
Theoremchpssati 23819* Two Hilbert lattice elements in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C.  B  ->  E. x  e. HAtoms  ( x  C_  B  /\  -.  x  C_  A ) )
 
Theoremchrelati 23820* The Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C.  B  ->  E. x  e. HAtoms  ( A  C.  ( A  vH  x )  /\  ( A  vH  x ) 
 C_  B ) )
 
Theoremchrelat2i 23821* A consequence of relative atomicity. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( -.  A  C_  B  <->  E. x  e. HAtoms  ( x  C_  A  /\  -.  x  C_  B ) )
 
Theoremcvati 23822* If a Hilbert lattice element covers another, it equals the other joined with some atom. This is a consequence of the relative atomicity of Hilbert space. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  <oH  B  ->  E. x  e. HAtoms  ( A  vH  x )  =  B )
 
Theoremcvbr4i 23823* An alternate way to express the covering property. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  <oH  B  <->  ( A  C.  B  /\  E. x  e. HAtoms  ( A  vH  x )  =  B ) )
 
Theoremcvexchlem 23824 Lemma for cvexchi 23825. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  i^i  B )  <oH  B  ->  A  <oH  ( A  vH  B ) )
 
Theoremcvexchi 23825 The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  i^i  B )  <oH  B  <->  A  <oH  ( A 
 vH  B ) )
 
Theoremchrelat2 23826* A consequence of relative atomicity. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( -.  A  C_  B 
 <-> 
 E. x  e. HAtoms  ( x  C_  A  /\  -.  x  C_  B ) ) )
 
Theoremchrelat3 23827* A consequence of relative atomicity. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  B  <->  A. x  e. HAtoms  ( x  C_  A  ->  x  C_  B ) ) )
 
Theoremchrelat3i 23828* A consequence of the relative atomicity of Hilbert space: the ordering of Hilbert lattice elements is completely determined by the atoms they majorize. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  <->  A. x  e. HAtoms  ( x  C_  A  ->  x  C_  B ) )
 
Theoremchrelat4i 23829* A consequence of relative atomicity. Extensionality principle: two lattice elements are equal iff they majorize the same atoms. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  =  B  <->  A. x  e. HAtoms  ( x  C_  A  <->  x  C_  B ) )
 
Theoremcvexch 23830 The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( ( A  i^i  B )  <oH  B  <->  A  <oH  ( A 
 vH  B ) ) )
 
Theoremcvp 23831 The Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  (
 ( A  i^i  B )  =  0H  <->  A  <oH  ( A 
 vH  B ) ) )
 
Theorematnssm0 23832 The meet of a Hilbert lattice element and an incomparable atom is the zero subspace. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  ( -.  B  C_  A  <->  ( A  i^i  B )  =  0H )
 )
 
Theorematnemeq0 23833 The meet of distinct atoms is the zero subspace. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms )  ->  ( A  =/=  B  <->  ( A  i^i  B )  =  0H )
 )
 
Theorematssma 23834 The meet with an atom's superset is the atom. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e.  CH )  ->  ( A  C_  B  <->  ( A  i^i  B )  e. HAtoms ) )
 
Theorematcv0eq 23835 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e. HAtoms )  ->  ( 0H  <oH  ( A  vH  B )  <->  A  =  B )
 )
 
Theorematcv1 23836 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e. HAtoms  /\  C  e. HAtoms )  /\  A  <oH  ( B  vH  C ) )  ->  ( A  =  0H  <->  B  =  C ) )
 
Theorematexch 23837 The Hilbert lattice satisfies the atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem related to vector analysis was originally proved by Hermann Grassmann in 1862. Also Definition 3.4-3(b) in [MegPav2000] p. 2345 (PDF p. 8) (use atnemeq0 23833 to obtain atom inequality). (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms ) 
 ->  ( ( B  C_  ( A  vH  C ) 
 /\  ( A  i^i  B )  =  0H )  ->  C  C_  ( A  vH  B ) ) )
 
Theorematomli 23838 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition 3.2.17 of [PtakPulmannova] p. 66. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( B  e. HAtoms  ->  ( ( A  vH  B )  i^i  ( _|_ `  A ) )  e.  (HAtoms  u. 
 { 0H } )
 )
 
Theorematoml2i 23839 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition P8(ii) of [BeltramettiCassinelli1] p. 400. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\ 
 -.  B  C_  A )  ->  ( ( A 
 vH  B )  i^i  ( _|_ `  A ) )  e. HAtoms )
 
Theorematordi 23840 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  A  C_H  B ) 
 ->  ( B  C_  A  \/  B  C_  ( _|_ `  A ) ) )
 
Theorematcvatlem 23841 Lemma for atcvati 23842. (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( ( B  e. HAtoms  /\  C  e. HAtoms )  /\  ( A  =/=  0H  /\  A  C.  ( B 
 vH  C ) ) )  ->  ( -.  B  C_  A  ->  A  e. HAtoms ) )
 
Theorematcvati 23842 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (Contributed by NM, 28-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  C  e. HAtoms )  ->  (
 ( A  =/=  0H  /\  A  C.  ( B 
 vH  C ) ) 
 ->  A  e. HAtoms ) )
 
Theorematcvat2i 23843 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  C  e. HAtoms )  ->  (
 ( -.  B  =  C  /\  A  <oH  ( B 
 vH  C ) ) 
 ->  A  e. HAtoms ) )
 
Theorematord 23844 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms  /\  A  C_H  B )  ->  ( B  C_  A  \/  B  C_  ( _|_ `  A )
 ) )
 
Theorematcvat2 23845 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 29-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms ) 
 ->  ( ( -.  B  =  C  /\  A  <oH  ( B  vH  C ) )  ->  A  e. HAtoms ) )
 
18.8.5  Irreducibility
 
Theoremchirredlem1 23846* Lemma for chirredi 23850. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( ( p  e. HAtoms  /\  ( q  e. 
 CH  /\  q  C_  ( _|_ `  A ) ) )  /\  ( ( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q ) ) )  ->  ( p  i^i  ( _|_ `  r
 ) )  =  0H )
 
Theoremchirredlem2 23847* Lemma for chirredi 23850. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e.  CH  /\  q  C_  ( _|_ `  A )
 ) )  /\  (
 ( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
 ) ) )  ->  ( ( _|_ `  r
 )  i^i  ( p  vH  q ) )  =  q )
 
Theoremchirredlem3 23848* Lemma for chirredi 23850. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  ( x  e. 
 CH  ->  A  C_H  x )   =>    |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  ( r  e. HAtoms  /\  r  C_  ( p  vH  q ) ) )  ->  ( r  C_  A  ->  r  =  p ) )
 
Theoremchirredlem4 23849* Lemma for chirredi 23850. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  ( x  e. 
 CH  ->  A  C_H  x )   =>    |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  ( r  e. HAtoms  /\  r  C_  ( p  vH  q ) ) )  ->  ( r  =  p  \/  r  =  q ) )
 
Theoremchirredi 23850* The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  ( x  e. 
 CH  ->  A  C_H  x )   =>    |-  ( A  =  0H  \/  A  =  ~H )
 
Theoremchirred 23851* The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\ 
 A. x  e.  CH  A  C_H  x )  ->  ( A  =  0H  \/  A  =  ~H )
 )
 
18.8.6  Atoms (cont.)
 
Theorematcvat3i 23852 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  C  e. HAtoms )  ->  (
 ( ( -.  B  =  C  /\  -.  C  C_  A )  /\  B  C_  ( A  vH  C ) )  ->  ( A  i^i  ( B  vH  C ) )  e. HAtoms ) )
 
Theorematcvat4i 23853* A condition implying existence of an atom with the properties shown. Lemma 3.2.20 of [PtakPulmannova] p. 68. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( ( B  e. HAtoms  /\  C  e. HAtoms )  ->  (
 ( A  =/=  0H  /\  B  C_  ( A  vH  C ) )  ->  E. x  e. HAtoms  ( x 
 C_  A  /\  B  C_  ( C  vH  x ) ) ) )
 
Theorematdmd 23854 Two Hilbert lattice elements have the dual modular pair property if the first is an atom. Theorem 7.6(c) of [MaedaMaeda] p. 31. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e.  CH )  ->  A  MH* 
 B )
 
Theorematmd 23855 Two Hilbert lattice elements have the modular pair property if the first is an atom. Theorem 7.6(b) of [MaedaMaeda] p. 31. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e. HAtoms  /\  B  e.  CH )  ->  A  MH  B )
 
Theorematmd2 23856 Two Hilbert lattice elements have the dual modular pair property if the second is an atom. Part of Exercise 6 of [Kalmbach] p. 103. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  A  MH  B )
 
Theorematabsi 23857 Absorption of an incomparable atom. Similar to Exercise 7.1 of [MaedaMaeda] p. 34. (Contributed by NM, 15-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( C  e. HAtoms  ->  ( -.  C  C_  ( A  vH  B )  ->  (
 ( A  vH  C )  i^i  B )  =  ( A  i^i  B ) ) )
 
Theorematabs2i 23858 Absorption of an incomparable atom. (Contributed by NM, 18-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( C  e. HAtoms  ->  ( -.  C  C_  ( A  vH  B )  ->  (
 ( A  vH  C )  i^i  ( A  vH  B ) )  =  A ) )
 
18.8.7  Modular symmetry
 
Theoremmdsymlem1 23859* Lemma for mdsymi 23867. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( ( p  e.  CH  /\  ( B  i^i  C )  C_  A )  /\  ( B 
 MH*  A  /\  p  C_  ( A  vH  B ) ) )  ->  p  C_  A )
 
Theoremmdsymlem2 23860* Lemma for mdsymi 23867. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( ( p  e. HAtoms  /\  ( B  i^i  C )  C_  A )  /\  ( B  MH*  A  /\  p  C_  ( A 
 vH  B ) ) )  ->  ( B  =/=  0H  ->  E. r  e. HAtoms  E. q  e. HAtoms  ( p  C_  ( q  vH  r )  /\  ( q 
 C_  A  /\  r  C_  B ) ) ) )
 
Theoremmdsymlem3 23861* Lemma for mdsymi 23867. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( ( ( p  e. HAtoms  /\  -.  ( B  i^i  C )  C_  A )  /\  p  C_  ( A  vH  B ) )  /\  A  =/=  0H )  ->  E. r  e. HAtoms  E. q  e. HAtoms  ( p  C_  ( q  vH  r )  /\  ( q 
 C_  A  /\  r  C_  B ) ) )
 
Theoremmdsymlem4 23862* Lemma for mdsymi 23867. This is the forward direction of Lemma 4(i) of [Maeda] p. 168. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( p  e. HAtoms  ->  ( ( B  MH*  A  /\  ( ( A  =/=  0H 
 /\  B  =/=  0H )  /\  p  C_  ( A  vH  B ) ) )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  ( q  vH  r )  /\  ( q 
 C_  A  /\  r  C_  B ) ) ) )
 
Theoremmdsymlem5 23863* Lemma for mdsymi 23867. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( q  e. HAtoms  /\  r  e. HAtoms )  ->  ( -.  q  =  p 
 ->  ( ( p  C_  ( q  vH  r ) 
 /\  ( q  C_  A  /\  r  C_  B ) )  ->  ( ( ( c  e.  CH  /\  A  C_  c )  /\  p  e. HAtoms )  ->  ( p  C_  c  ->  p  C_  ( ( c  i^i  B )  vH  A ) ) ) ) ) )
 
Theoremmdsymlem6 23864* Lemma for mdsymi 23867. This is the converse direction of Lemma 4(i) of [Maeda] p. 168, and is based on the proof of Theorem 1(d) to (e) of [Maeda] p. 167. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( A. p  e. HAtoms  ( p  C_  ( A 
 vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  (
 q  vH  r )  /\  ( q  C_  A  /\  r  C_  B ) ) )  ->  B  MH* 
 A )
 
Theoremmdsymlem7 23865* Lemma for mdsymi 23867. Lemma 4(i) of [Maeda] p. 168. Note that Maeda's 1965 definition of dual modular pair has reversed arguments compared to the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, which is the one that we use. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( A  =/=  0H 
 /\  B  =/=  0H )  ->  ( B  MH*  A  <->  A. p  e. HAtoms  ( p 
 C_  ( A  vH  B )  ->  E. q  e. HAtoms  E. r  e. HAtoms  ( p  C_  ( q  vH  r )  /\  ( q 
 C_  A  /\  r  C_  B ) ) ) ) )
 
Theoremmdsymlem8 23866* Lemma for mdsymi 23867. Lemma 4(ii) of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  =  ( A  vH  p )   =>    |-  ( ( A  =/=  0H 
 /\  B  =/=  0H )  ->  ( B  MH*  A  <->  A  MH*  B ) )
 
Theoremmdsymi 23867 M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH  B  <->  B  MH  A )
 
Theoremmdsym 23868 M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  B  MH  A ) )
 
Theoremdmdsym 23869 Dual M-symmetry of the Hilbert lattice. (Contributed by NM, 25-Jul-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH*  B  <->  B 
 MH*  A ) )
 
Theorematdmd2 23870 Two Hilbert lattice elements have the dual modular pair property if the second is an atom. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e. HAtoms )  ->  A  MH* 
 B )
 
Theoremsumdmdii 23871 If the subspace sum of two Hilbert lattice elements is closed, then the elements are a dual modular pair. Remark in [MaedaMaeda] p. 139. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  +H  B )  =  ( A  vH  B )  ->  A  MH* 
 B )
 
Theoremcmmdi 23872 Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  ->  A  MH  B )
 
Theoremcmdmdi 23873 Commuting subspaces form a dual modular pair. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  ->  A  MH* 
 B )
 
Theoremsumdmdlem 23874 Lemma for sumdmdi 23876. The span of vector  C not in the subspace sum is "trimmed off." (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( C  e.  ~H  /\ 
 -.  C  e.  ( A  +H  B ) ) 
 ->  ( ( B  +H  ( span `  { C }
 ) )  i^i  A )  =  ( B  i^i  A ) )
 
Theoremsumdmdlem2 23875* Lemma for sumdmdi 23876. (Contributed by NM, 23-Dec-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A. x  e. HAtoms  ( ( x  vH  B )  i^i  ( A  vH  B ) )  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B )  ->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremsumdmdi 23876 The subspace sum of two Hilbert lattice elements is closed iff the elements are a dual modular pair. Theorem 2 of [Holland] p. 1519. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  +H  B )  =  ( A  vH  B )  <->  A  MH*  B )
 
Theoremdmdbr4ati 23877* Dual modular pair property in terms of atoms. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH*  B  <->  A. x  e. HAtoms  (
 ( x  vH  B )  i^i  ( A  vH  B ) )  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B ) )
 
Theoremdmdbr5ati 23878* Dual modular pair property in terms of atoms. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH*  B  <->  A. x  e. HAtoms  ( x  C_  ( A  vH  B )  ->  x  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B ) ) )
 
Theoremdmdbr6ati 23879* Dual modular pair property in terms of atoms. The modular law takes the form of the shearing identity. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH*  B  <->  A. x  e. HAtoms  (
 ( A  vH  B )  i^i  x )  =  ( ( ( ( x  vH  B )  i^i  A )  vH  B )  i^i  x ) )
 
Theoremdmdbr7ati 23880* Dual modular pair property in terms of atoms. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH*  B  <->  A. x  e. HAtoms  (
 ( A  vH  B )  i^i  x )  C_  ( ( ( x 
 vH  B )  i^i 
 A )  vH  B ) )
 
Theoremmdoc1i 23881 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  MH  ( _|_ `  A )
 
Theoremmdoc2i 23882 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  A )  MH  A
 
Theoremdmdoc1i 23883 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  MH*  ( _|_ `  A )
 
Theoremdmdoc2i 23884 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  ( _|_ `  A )  MH*  A
 
Theoremmdcompli 23885 A condition equivalent to the modular pair property. Part of proof of Theorem 1.14 of [MaedaMaeda] p. 4. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH  B  <->  ( A  i^i  ( _|_ `  ( A  i^i  B ) ) )  MH  ( B  i^i  ( _|_ `  ( A  i^i  B ) ) ) )
 
Theoremdmdcompli 23886 A condition equivalent to the dual modular pair property. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  MH*  B  <->  ( A  i^i  ( _|_ `  ( A  i^i  B ) ) ) 
 MH*  ( B  i^i  ( _|_ `  ( A  i^i  B ) ) ) )
 
Theoremmddmdin0i 23887* If dual modular implies modular whenever meet is zero, then dual modular implies modular for arbitrary lattice elements. This theorem is needed for the remark after Lemma 7 of [Holland] p. 1524 to hold. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A. x  e.  CH  A. y  e. 
 CH  ( ( x 
 MH*  y  /\  ( x  i^i  y )  =  0H )  ->  x  MH  y )   =>    |-  ( A  MH*  B  ->  A  MH  B )
 
Theoremcdjreui 23888* A member of the sum of disjoint subspaces has a unique decomposition. Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 20-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  (
 ( C  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  ->  E! x  e.  A  E. y  e.  B  C  =  ( x  +h  y ) )
 
Theoremcdj1i 23889* Two ways to express " A and  B are completely disjoint subspaces." (1) => (2) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 21-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( E. w  e.  RR  ( 0  <  w  /\  A. y  e.  A  A. v  e.  B  ( ( normh `  y )  +  ( normh `  v )
 )  <_  ( w  x.  ( normh `  ( y  +h  v ) ) ) )  ->  E. x  e.  RR  ( 0  < 
 x  /\  A. y  e.  A  A. z  e.  B  ( ( normh `  y )  =  1 
 ->  x  <_  ( normh `  ( y  -h  z
 ) ) ) ) )
 
Theoremcdj3lem1 23890* A property of " A and  B are completely disjoint subspaces." Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   =>    |-  ( E. x  e.  RR  ( 0  <  x  /\  A. y  e.  A  A. z  e.  B  ( ( normh `  y )  +  ( normh `  z )
 )  <_  ( x  x.  ( normh `  ( y  +h  z ) ) ) )  ->  ( A  i^i  B )  =  0H )
 
Theoremcdj3lem2 23891* Lemma for cdj3i 23897. Value of the first-component function  S. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  S  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ z  e.  A E. w  e.  B  x  =  ( z  +h  w ) ) )   =>    |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  ( S `  ( C  +h  D ) )  =  C )
 
Theoremcdj3lem2a 23892* Lemma for cdj3i 23897. Closure of the first-component function  S. (Contributed by NM, 25-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  S  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ z  e.  A E. w  e.  B  x  =  ( z  +h  w ) ) )   =>    |-  ( ( C  e.  ( A  +H  B ) 
 /\  ( A  i^i  B )  =  0H )  ->  ( S `  C )  e.  A )
 
Theoremcdj3lem2b 23893* Lemma for cdj3i 23897. The first-component function  S is bounded if the subspaces are completely disjoint. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  S  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ z  e.  A E. w  e.  B  x  =  ( z  +h  w ) ) )   =>    |-  ( E. v  e.  RR  ( 0  <  v  /\  A. x  e.  A  A. y  e.  B  ( ( normh `  x )  +  ( normh `  y )
 )  <_  ( v  x.  ( normh `  ( x  +h  y ) ) ) )  ->  E. v  e.  RR  ( 0  < 
 v  /\  A. u  e.  ( A  +H  B ) ( normh `  ( S `  u ) ) 
 <_  ( v  x.  ( normh `  u ) ) ) )
 
Theoremcdj3lem3 23894* Lemma for cdj3i 23897. Value of the second-component function  T. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  T  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) ) )   =>    |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  ( T `  ( C  +h  D ) )  =  D )
 
Theoremcdj3lem3a 23895* Lemma for cdj3i 23897. Closure of the second-component function  T. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  T  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) ) )   =>    |-  ( ( C  e.  ( A  +H  B ) 
 /\  ( A  i^i  B )  =  0H )  ->  ( T `  C )  e.  B )
 
Theoremcdj3lem3b 23896* Lemma for cdj3i 23897. The second-component function  T is bounded if the subspaces are completely disjoint. (Contributed by NM, 31-May-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  T  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) ) )   =>    |-  ( E. v  e.  RR  ( 0  <  v  /\  A. x  e.  A  A. y  e.  B  ( ( normh `  x )  +  ( normh `  y )
 )  <_  ( v  x.  ( normh `  ( x  +h  y ) ) ) )  ->  E. v  e.  RR  ( 0  < 
 v  /\  A. u  e.  ( A  +H  B ) ( normh `  ( T `  u ) ) 
 <_  ( v  x.  ( normh `  u ) ) ) )
 
Theoremcdj3i 23897* Two ways to express " A and  B are completely disjoint subspaces." (1) <=> (3) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 1-Jun-2005.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  S  =  ( x  e.  ( A  +H  B )  |->  (
 iota_ z  e.  A E. w  e.  B  x  =  ( z  +h  w ) ) )   &    |-  T  =  ( x  e.  ( A  +H  B )  |->  ( iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) ) )   &    |-  ( ph  <->  E. v  e.  RR  ( 0  <  v  /\  A. u  e.  ( A  +H  B ) (
 normh `  ( S `  u ) )  <_  ( v  x.  ( normh `  u ) ) ) )   &    |-  ( ps  <->  E. v  e.  RR  ( 0  <  v  /\  A. u  e.  ( A  +H  B ) (
 normh `  ( T `  u ) )  <_  ( v  x.  ( normh `  u ) ) ) )   =>    |-  ( E. v  e. 
 RR  ( 0  < 
 v  /\  A. x  e.  A  A. y  e.  B  ( ( normh `  x )  +  ( normh `  y ) ) 
 <_  ( v  x.  ( normh `  ( x  +h  y ) ) ) )  <->  ( ( A  i^i  B )  =  0H  /\  ph  /\  ps ) )
 
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
 
19.1  Mathboxes for user contributions
 
19.1.1  Mathbox guidelines
 
Theoremmathbox 23898 (This theorem is a dummy placeholder for these guidelines. The name of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of set.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm.

Mathboxes are provided to help keep your work synchronized with changes in set.mm, but they shouldn't be depended on as a permanent archive. If you want to preserve your original contribution, it is your responsibility to keep your own copy of it along with the version of set.mm that works with it.

Guidelines:

1. If at all possible, please use only 0-ary class constants for new definitions, for example as in df-div 9634.

2. Try to follow the style of the rest of set.mm. Each $p and $a statement must be immediately preceded with the comment that will be shown on its web page description. The metamath program command "write source set.mm /rewrap" will take care of wrapping comment lines and indentation conventions. All mathbox content will be on public display and should hopefully reflect the overall quality of the website.

3. Before submitting a revised mathbox, please make sure it verifies against the current set.mm.

4. Mathboxes should be independent i.e. the proofs should verify with all other mathboxes removed. If you need a theorem from another mathbox, that is fine (and encouraged), but let me know, so I can move the theorem to the main section. One way avoid undesired accidental use of other mathbox theorems is to develop your mathbox using a modified set.mm that has mathboxes removed.

Notes:

1. I may decide to move some theorems to the main part of set.mm for general use. In that case, an author acknowledgment will be added to the description of the theorem.

2. I may make changes to mathboxes to maintain the overall quality of set.mm. Normally I will let you know if a change might impact what you are working on.

3. If you use theorems from another user's mathbox, I don't provide assurance that they are based on correct or consistent $a statements. (If you find such a problem, please let me know so it can be corrected.) (Contributed by NM, 20-Feb-2007.) (New usage is discouraged.)

 |-  x  =  x
 
19.2  Mathbox for Stefan Allan
 
Theoremfoo3 23899 A theorem about the universal class. (Contributed by Stefan Allan, 9-Dec-2008.)
 |-  ph   =>    |- 
 _V  =  { x  |  ph }
 
Theoremxfree 23900 A partial converse to 19.9t 1789. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( A. x ( ph  ->  A. x ph )  <->  A. x ( E. x ph  ->  ph ) )
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