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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhfmmval 22001* Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> CC )  ->  ( A  .fn  T )  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
 
Theoremhosval 22002 Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S 
 +op  T ) `  A )  =  ( ( S `  A )  +h  ( T `  A ) ) )
 
Theoremhomval 22003 Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  =  ( A  .h  ( T `  B ) ) )
 
Theoremhodval 22004 Value of the difference of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S 
 -op  T ) `  A )  =  ( ( S `  A )  -h  ( T `  A ) ) )
 
Theoremhfsval 22005 Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( S : ~H --> CC  /\  T : ~H --> CC  /\  A  e.  ~H )  ->  ( ( S 
 +fn  T ) `  A )  =  ( ( S `  A )  +  ( T `  A ) ) )
 
Theoremhfmval 22006 Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> CC  /\  B  e.  ~H )  ->  ( ( A  .fn  T ) `  B )  =  ( A  x.  ( T `  B ) ) )
 
Theoremhoscl 22007 Closure of the sum of two Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
 |-  (
 ( ( S : ~H
 --> ~H  /\  T : ~H
 --> ~H )  /\  A  e.  ~H )  ->  (
 ( S  +op  T ) `  A )  e. 
 ~H )
 
Theoremhomcl 22008 Closure of the scalar product of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  e.  ~H )
 
Theoremhodcl 22009 Closure of the difference of two Hilbert space operators. (Contributed by NM, 15-Nov-2002.) (New usage is discouraged.)
 |-  (
 ( ( S : ~H
 --> ~H  /\  T : ~H
 --> ~H )  /\  A  e.  ~H )  ->  (
 ( S  -op  T ) `  A )  e. 
 ~H )
 
15.9.26  Commutes relation for Hilbert lattice elements
 
Definitiondf-cm 22010* Define the commutes relation (on the Hilbert lattice). Definition of commutes in [Kalmbach] p. 20, who uses the notation xCy for "x commutes with y." See cmbri 22017 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
 |-  C_H  =  { <. x ,  y >.  |  ( ( x  e.  CH  /\  y  e. 
 CH )  /\  x  =  ( ( x  i^i  y )  vH  ( x  i^i  ( _|_ `  y
 ) ) ) ) }
 
Theoremcmbr 22011 Binary relation expressing  A commutes with  B. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B ) ) ) ) )
 
Theorempjoml2i 22012 Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  ->  ( A  vH  ( ( _|_ `  A )  i^i  B ) )  =  B )
 
Theorempjoml3i 22013 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( B  C_  A  ->  ( A  i^i  ( ( _|_ `  A )  vH  B ) )  =  B )
 
Theorempjoml4i 22014 Variation of orthomodular law. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  ( B  i^i  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) ) )  =  ( A  vH  B )
 
Theorempjoml5i 22015 The orthomodular law. Remark in [Kalmbach] p. 22. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  ( ( _|_ `  A )  i^i  ( A  vH  B ) ) )  =  ( A 
 vH  B )
 
Theorempjoml6i 22016* An equivalent of the orthomodular law. Theorem 29.13(e) of [MaedaMaeda] p. 132. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  ->  E. x  e.  CH  ( A  C_  ( _|_ `  x )  /\  ( A  vH  x )  =  B )
 )
 
Theoremcmbri 22017 Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  A  =  (
 ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B )
 ) ) )
 
Theoremcmcmlem 22018 Commutation is symmetric. Theorem 3.4 of [Beran] p. 45. (Contributed by NM, 3-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  ->  B  C_H  A )
 
Theoremcmcmi 22019 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  B  C_H  A )
 
Theoremcmcm2i 22020 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  A  C_H  ( _|_ `  B ) )
 
Theoremcmcm3i 22021 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  ( _|_ `  A )  C_H  B )
 
Theoremcmcm4i 22022 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  ( _|_ `  A )  C_H  ( _|_ `  B ) )
 
Theoremcmbr2i 22023 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  A  =  (
 ( A  vH  B )  i^i  ( A  vH  ( _|_ `  B )
 ) ) )
 
Theoremcmcmii 22024 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  C_H  B   =>    |-  B  C_H  A
 
Theoremcmcm2ii 22025 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  C_H  B   =>    |-  A  C_H  ( _|_ `  B )
 
Theoremcmcm3ii 22026 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  C_H  B   =>    |-  ( _|_ `  A )  C_H  B
 
Theoremcmbr3i 22027 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  ( A  i^i  ( ( _|_ `  A )  vH  B ) )  =  ( A  i^i  B ) )
 
Theoremcmbr4i 22028 Alternate definition for the commutes relation. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  <->  ( A  i^i  ( ( _|_ `  A )  vH  B ) ) 
 C_  B )
 
Theoremlecmi 22029 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  B  ->  A  C_H  B )
 
Theoremlecmii 22030 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  C_  B   =>    |-  A  C_H  B
 
Theoremcmj1i 22031 A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  A  C_H  ( A  vH  B )
 
Theoremcmj2i 22032 A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  B  C_H  ( A  vH  B )
 
Theoremcmm1i 22033 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  A  C_H  ( A  i^i  B )
 
Theoremcmm2i 22034 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  B  C_H  ( A  i^i  B )
 
Theoremcmbr3 22035 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  ( A  i^i  ( ( _|_ `  A )  vH  B ) )  =  ( A  i^i  B ) ) )
 
Theoremcm0 22036 The zero Hilbert lattice element commutes with every element. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  CH  ->  0H  C_H 
 A )
 
Theoremcmidi 22037 The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  C_H  A
 
Theorempjoml2 22038 Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  ( A  vH  ( ( _|_ `  A )  i^i  B ) )  =  B )
 
Theorempjoml3 22039 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( B  C_  A  ->  ( A  i^i  (
 ( _|_ `  A )  vH  B ) )  =  B ) )
 
Theorempjoml5 22040 The orthomodular law. Remark in [Kalmbach] p. 22. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  (
 ( _|_ `  A )  i^i  ( A  vH  B ) ) )  =  ( A  vH  B ) )
 
Theoremcmcm 22041 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  B 
 C_H  A ) )
 
Theoremcmcm3 22042 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  ( _|_ `  A )  C_H  B ) )
 
Theoremcmcm2 22043 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A 
 C_H  ( _|_ `  B ) ) )
 
Theoremlecm 22044 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  B )  ->  A  C_H  B )
 
15.9.27  Foulis-Holland theorem
 
Theoremfh1 22045 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. First of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( A  i^i  ( B  vH  C ) )  =  (
 ( A  i^i  B )  vH  ( A  i^i  C ) ) )
 
Theoremfh2 22046 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Second of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( A  i^i  ( B  vH  C ) )  =  (
 ( A  i^i  B )  vH  ( A  i^i  C ) ) )
 
Theoremcm2j 22047 A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CH 
 /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  A  C_H  ( B  vH  C ) )
 
Theoremfh1i 22048 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. First of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  ( A  i^i  ( B  vH  C ) )  =  ( ( A  i^i  B )  vH  ( A  i^i  C ) )
 
Theoremfh2i 22049 Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Second of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  ( B  i^i  ( A  vH  C ) )  =  ( ( B  i^i  A )  vH  ( B  i^i  C ) )
 
Theoremfh3i 22050 Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  ( A  vH  ( B  i^i  C ) )  =  ( ( A 
 vH  B )  i^i  ( A  vH  C ) )
 
Theoremfh4i 22051 Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  ( B  vH  ( A  i^i  C ) )  =  ( ( B 
 vH  A )  i^i  ( B  vH  C ) )
 
Theoremcm2ji 22052 A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 11-May-2009.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  A  C_H  ( B 
 vH  C )
 
Theoremcm2mi 22053 A lattice element that commutes with two others also commutes with their meet. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 11-May-2009.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  C_H  B   &    |-  A  C_H  C   =>    |-  A  C_H  ( B  i^i  C )
 
15.9.28  Quantum Logic Explorer axioms
 
Theoremqlax1i 22054 One of the equations showing  CH is an ortholattice. (This corresponds to axiom "ax-1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  =  ( _|_ `  ( _|_ `  A ) )
 
Theoremqlax2i 22055 One of the equations showing  CH is an ortholattice. (This corresponds to axiom "ax-2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  B )  =  ( B  vH  A )
 
Theoremqlax3i 22056 One of the equations showing  CH is an ortholattice. (This corresponds to axiom "ax-3" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   =>    |-  ( ( A  vH  B )  vH  C )  =  ( A  vH  ( B  vH  C ) )
 
Theoremqlax4i 22057 One of the equations showing  CH is an ortholattice. (This corresponds to axiom "ax-4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  ( B  vH  ( _|_ `  B )
 ) )  =  ( B  vH  ( _|_ `  B ) )
 
Theoremqlax5i 22058 One of the equations showing  CH is an ortholattice. (This corresponds to axiom "ax-5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  vH  ( _|_ `  (
 ( _|_ `  A )  vH  B ) ) )  =  A
 
Theoremqlaxr1i 22059 One of the conditions showing 
CH is an ortholattice. (This corresponds to axiom "ax-r1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  =  B   =>    |-  B  =  A
 
Theoremqlaxr2i 22060 One of the conditions showing 
CH is an ortholattice. (This corresponds to axiom "ax-r2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  =  B   &    |-  B  =  C   =>    |-  A  =  C
 
Theoremqlaxr4i 22061 One of the conditions showing 
CH is an ortholattice. (This corresponds to axiom "ax-r4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  A  =  B   =>    |-  ( _|_ `  A )  =  ( _|_ `  B )
 
Theoremqlaxr5i 22062 One of the conditions showing 
CH is an ortholattice. (This corresponds to axiom "ax-r5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  A  =  B   =>    |-  ( A  vH  C )  =  ( B  vH  C )
 
Theoremqlaxr3i 22063 A variation of the orthomodular law, showing  CH is an orthomodular lattice. (This corresponds to axiom "ax-r3" in the Quantum Logic Explorer.) (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  ( C  vH  ( _|_ `  C )
 )  =  ( ( _|_ `  ( ( _|_ `  A )  vH  ( _|_ `  B )
 ) )  vH  ( _|_ `  ( A  vH  B ) ) )   =>    |-  A  =  B
 
15.9.29  Orthogonal subspaces
 
Theoremchscllem1 22064* Lemma for chscl 22068. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  CH )   &    |-  ( ph  ->  B  e.  CH )   &    |-  ( ph  ->  B 
 C_  ( _|_ `  A ) )   &    |-  ( ph  ->  H : NN --> ( A  +H  B ) )   &    |-  ( ph  ->  H  ~~>v  u )   &    |-  F  =  ( n  e.  NN  |->  ( ( proj  h `
  A ) `  ( H `  n ) ) )   =>    |-  ( ph  ->  F : NN --> A )
 
Theoremchscllem2 22065* Lemma for chscl 22068. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  CH )   &    |-  ( ph  ->  B  e.  CH )   &    |-  ( ph  ->  B 
 C_  ( _|_ `  A ) )   &    |-  ( ph  ->  H : NN --> ( A  +H  B ) )   &    |-  ( ph  ->  H  ~~>v  u )   &    |-  F  =  ( n  e.  NN  |->  ( ( proj  h `
  A ) `  ( H `  n ) ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>v  )
 
Theoremchscllem3 22066* Lemma for chscl 22068. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  CH )   &    |-  ( ph  ->  B  e.  CH )   &    |-  ( ph  ->  B 
 C_  ( _|_ `  A ) )   &    |-  ( ph  ->  H : NN --> ( A  +H  B ) )   &    |-  ( ph  ->  H  ~~>v  u )   &    |-  F  =  ( n  e.  NN  |->  ( ( proj  h `
  A ) `  ( H `  n ) ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  D  e.  B )   &    |-  ( ph  ->  ( H `  N )  =  ( C  +h  D ) )   =>    |-  ( ph  ->  C  =  ( F `  N ) )
 
Theoremchscllem4 22067* Lemma for chscl 22068. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  CH )   &    |-  ( ph  ->  B  e.  CH )   &    |-  ( ph  ->  B 
 C_  ( _|_ `  A ) )   &    |-  ( ph  ->  H : NN --> ( A  +H  B ) )   &    |-  ( ph  ->  H  ~~>v  u )   &    |-  F  =  ( n  e.  NN  |->  ( ( proj  h `
  A ) `  ( H `  n ) ) )   &    |-  G  =  ( n  e.  NN  |->  ( ( proj  h `  B ) `  ( H `  n ) ) )   =>    |-  ( ph  ->  u  e.  ( A  +H  B ) )
 
Theoremchscl 22068 The subspace sum of two closed orthogonal spaces is closed. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  A  e.  CH )   &    |-  ( ph  ->  B  e.  CH )   &    |-  ( ph  ->  B 
 C_  ( _|_ `  A ) )   =>    |-  ( ph  ->  ( A  +H  B )  e. 
 CH )
 
Theoremosumi 22069 If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67. Note that the (countable) Axiom of Choice is used for this proof via pjhth 21802, although "the hard part" of this proof, chscl 22068, requires no choice. (Contributed by NM, 28-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_  ( _|_ `  B )  ->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremosumcori 22070 Corollary of osumi 22069. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  (
 ( A  i^i  B )  +H  ( A  i^i  ( _|_ `  B )
 ) )  =  ( ( A  i^i  B )  vH  ( A  i^i  ( _|_ `  B )
 ) )
 
Theoremosumcor2i 22071 Corollary of osumi 22069, showing it holds under the weaker hypothesis that  A and  B commute. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   =>    |-  ( A  C_H  B  ->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremosum 22072 If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  A  C_  ( _|_ `  B ) )  ->  ( A  +H  B )  =  ( A  vH  B ) )
 
Theoremspansnji 22073 The subspace sum of a closed subspace and a one-dimensional subspace equals their join. (Proof suggested by Eric Schechter 1-Jun-2004.) (Contributed by NM, 1-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  ~H   =>    |-  ( A  +H  ( span `  { B } ) )  =  ( A  vH  ( span `  { B }
 ) )
 
Theoremspansnj 22074 The subspace sum of a closed subspace and a one-dimensional subspace equals their join. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  ~H )  ->  ( A  +H  ( span `  { B }
 ) )  =  ( A  vH  ( span ` 
 { B } )
 ) )
 
Theoremspansnscl 22075 The subspace sum of a closed subspace and a one-dimensional subspace is closed. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  ~H )  ->  ( A  +H  ( span `  { B }
 ) )  e.  CH )
 
Theoremsumspansn 22076 The sum of two vectors belong to the span of one of them iff the other vector also belongs. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  e.  ( span `  { A }
 ) 
 <->  B  e.  ( span ` 
 { A } )
 ) )
 
Theoremspansnm0i 22077 The meet of different one-dimensional subspaces is the zero subspace. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( -.  A  e.  ( span ` 
 { B } )  ->  ( ( span `  { A } )  i^i  ( span ` 
 { B } )
 )  =  0H )
 
Theoremnonbooli 22078 A Hilbert lattice with two or more dimensions fails the distributive law and therefore cannot be a Boolean algebra. This counterexample demonstrates a condition where 
( ( H  i^i  F )  vH  ( H  i^i  G ) )  =  0H but  ( H  i^i  ( F  vH  G ) )  =/=  0H. The antecedent specifies that the vectors  A and  B are nonzero and non-colinear. The last three hypotheses assign one-dimensional subspaces to  F,  G, and  H. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  F  =  ( span `  { A }
 )   &    |-  G  =  ( span ` 
 { B } )   &    |-  H  =  ( span `  { ( A  +h  B ) }
 )   =>    |-  ( -.  ( A  e.  G  \/  B  e.  F )  ->  ( H  i^i  ( F  vH  G ) )  =/=  ( ( H  i^i  F )  vH  ( H  i^i  G ) ) )
 
Theoremspansncvi 22079 Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of [Kalmbach] p. 153. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  ~H   =>    |-  ( ( A  C.  B  /\  B  C_  ( A  vH  ( span `  { C } ) ) ) 
 ->  B  =  ( A 
 vH  ( span `  { C } ) ) )
 
Theoremspansncv 22080 Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of [Kalmbach] p. 153. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
 |-  (
 ( A  e.  CH  /\  B  e.  CH  /\  C  e.  ~H )  ->  ( ( A  C.  B  /\  B  C_  ( A  vH  ( span `  { C } ) ) ) 
 ->  B  =  ( A 
 vH  ( span `  { C } ) ) ) )
 
15.9.30  Orthoarguesian laws 5OA and 3OA
 
Theorem5oalem1 22081 Lemma for orthoarguesian law 5OA. (Contributed by NM, 1-Apr-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  R  e.  SH   =>    |-  (
 ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y
 ) )  /\  (
 z  e.  C  /\  ( x  -h  z
 )  e.  R ) )  ->  v  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) ) )
 
Theorem5oalem2 22082 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   =>    |-  (
 ( ( ( x  e.  A  /\  y  e.  B )  /\  (
 z  e.  C  /\  w  e.  D )
 )  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( x  -h  z )  e.  (
 ( A  +H  C )  i^i  ( B  +H  D ) ) )
 
Theorem5oalem3 22083 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   &    |-  F  e.  SH   &    |-  G  e.  SH   =>    |-  (
 ( ( ( ( x  e.  A  /\  y  e.  B )  /\  ( z  e.  C  /\  w  e.  D ) )  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  (
 f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) ) 
 ->  ( x  -h  z
 )  e.  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G ) ) ) )
 
Theorem5oalem4 22084 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   &    |-  F  e.  SH   &    |-  G  e.  SH   =>    |-  (
 ( ( ( ( x  e.  A  /\  y  e.  B )  /\  ( z  e.  C  /\  w  e.  D ) )  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  (
 f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) ) 
 ->  ( x  -h  z
 )  e.  ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  (
 ( C  +H  F )  i^i  ( D  +H  G ) ) ) ) )
 
Theorem5oalem5 22085 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-May-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   &    |-  F  e.  SH   &    |-  G  e.  SH   &    |-  R  e.  SH   &    |-  S  e.  SH   =>    |-  (
 ( ( ( ( x  e.  A  /\  y  e.  B )  /\  ( z  e.  C  /\  w  e.  D ) )  /\  ( ( f  e.  F  /\  g  e.  G )  /\  ( v  e.  R  /\  u  e.  S ) ) )  /\  ( ( ( x  +h  y )  =  ( v  +h  u )  /\  ( z  +h  w )  =  (
 v  +h  u )
 )  /\  ( f  +h  g )  =  ( v  +h  u ) ) )  ->  ( x  -h  z )  e.  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  (
 ( C  +H  R )  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
 ( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) )
 
Theorem5oalem6 22086 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   &    |-  F  e.  SH   &    |-  G  e.  SH   &    |-  R  e.  SH   &    |-  S  e.  SH   =>    |-  (
 ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) ) 
 /\  ( ( z  e.  C  /\  w  e.  D )  /\  h  =  ( z  +h  w ) ) )  /\  ( ( ( f  e.  F  /\  g  e.  G )  /\  h  =  ( f  +h  g
 ) )  /\  (
 ( v  e.  R  /\  u  e.  S )  /\  h  =  ( v  +h  u ) ) ) )  ->  h  e.  ( B  +H  ( A  i^i  ( C  +H  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  (
 ( C  +H  R )  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
 ( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) ) ) ) )
 
Theorem5oalem7 22087 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
 |-  A  e.  SH   &    |-  B  e.  SH   &    |-  C  e.  SH   &    |-  D  e.  SH   &    |-  F  e.  SH   &    |-  G  e.  SH   &    |-  R  e.  SH   &    |-  S  e.  SH   =>    |-  (
 ( ( A  +H  B )  i^i  ( C  +H  D ) )  i^i  ( ( F  +H  G )  i^i  ( R  +H  S ) ) )  C_  ( B  +H  ( A  i^i  ( C  +H  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  (
 ( C  +H  R )  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
 ( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) ) ) )
 
Theorem5oai 22088 Orthoarguesian law 5OA. This 8-variable inference is called 5OA because it can be converted to a 5-variable equation (see Quantum Logic Explorer). (Contributed by NM, 5-May-2000.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  D  e.  CH   &    |-  F  e.  CH   &    |-  G  e.  CH   &    |-  R  e.  CH   &    |-  S  e.  CH   &    |-  A  C_  ( _|_ `  B )   &    |-  C  C_  ( _|_ `  D )   &    |-  F  C_  ( _|_ `  G )   &    |-  R  C_  ( _|_ `  S )   =>    |-  ( ( ( A 
 vH  B )  i^i  ( C  vH  D ) )  i^i  ( ( F  vH  G )  i^i  ( R  vH  S ) ) ) 
 C_  ( B  vH  ( A  i^i  ( C 
 vH  ( ( ( ( A  vH  C )  i^i  ( B  vH  D ) )  i^i  ( ( ( A 
 vH  R )  i^i  ( B  vH  S ) )  vH  ( ( C  vH  R )  i^i  ( D  vH  S ) ) ) )  i^i  ( ( ( ( A  vH  F )  i^i  ( B 
 vH  G ) )  i^i  ( ( ( A  vH  R )  i^i  ( B  vH  S ) )  vH  ( ( F  vH  R )  i^i  ( G 
 vH  S ) ) ) )  vH  (
 ( ( C  vH  F )  i^i  ( D 
 vH  G ) )  i^i  ( ( ( C  vH  R )  i^i  ( D  vH  S ) )  vH  ( ( F  vH  R )  i^i  ( G 
 vH  S ) ) ) ) ) ) ) ) )
 
Theorem3oalem1 22089* Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  e.  CH   &    |-  S  e.  CH   =>    |-  (
 ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y
 ) )  /\  (
 ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  (
 ( ( x  e. 
 ~H  /\  y  e.  ~H )  /\  v  e. 
 ~H )  /\  (
 z  e.  ~H  /\  w  e.  ~H )
 ) )
 
Theorem3oalem2 22090* Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  e.  CH   &    |-  S  e.  CH   =>    |-  (
 ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y
 ) )  /\  (
 ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
 
Theorem3oalem3 22091 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  e.  CH   &    |-  S  e.  CH   =>    |-  (
 ( B  +H  R )  i^i  ( C  +H  S ) )  C_  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) )
 
Theorem3oalem4 22092 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  R  =  ( ( _|_ `  B )  i^i  ( B  vH  A ) )   =>    |-  R  C_  ( _|_ `  B )
 
Theorem3oalem5 22093 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  =  ( ( _|_ `  B )  i^i  ( B  vH  A ) )   &    |-  S  =  ( ( _|_ `  C )  i^i  ( C  vH  A ) )   =>    |-  ( ( B  +H  R )  i^i  ( C  +H  S ) )  =  (
 ( B  vH  R )  i^i  ( C  vH  S ) )
 
Theorem3oalem6 22094 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  =  ( ( _|_ `  B )  i^i  ( B  vH  A ) )   &    |-  S  =  ( ( _|_ `  C )  i^i  ( C  vH  A ) )   =>    |-  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) )  C_  ( B  vH  ( R  i^i  ( S  vH  ( ( B 
 vH  C )  i^i  ( R  vH  S ) ) ) ) )
 
Theorem3oai 22095 3OA (weak) orthoarguesian law. Equation IV of [GodowskiGreechie] p. 249. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  CH   &    |-  B  e.  CH   &    |-  C  e.  CH   &    |-  R  =  ( ( _|_ `  B )  i^i  ( B  vH  A ) )   &    |-  S  =  ( ( _|_ `  C )  i^i  ( C  vH  A ) )   =>    |-  ( ( B 
 vH  R )  i^i  ( C  vH  S ) )  C_  ( B 
 vH  ( R  i^i  ( S  vH  ( ( B  vH  C )  i^i  ( R  vH  S ) ) ) ) )
 
15.9.31  Projectors (cont.)
 
Theorempjorthi 22096 Projection components on orthocomplemented subspaces are orthogonal. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( H  e.  CH  ->  (
 ( ( proj  h `  H ) `  A )  .ih  ( ( proj  h `
  ( _|_ `  H ) ) `  B ) )  =  0
 )
 
Theorempjch1 22097 Property of identity projection. Remark in [Beran] p. 111. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( ( proj  h `  ~H ) `  A )  =  A )
 
Theorempjo 22098 The orthogonal projection. Lemma 4.4(i) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( ( proj  h `  ( _|_ `  H )
 ) `  A )  =  ( ( ( proj  h `
  ~H ) `  A )  -h  (
 ( proj  h `  H ) `  A ) ) )
 
Theorempjcompi 22099 Component of a projection. (Contributed by NM, 31-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) ) 
 ->  ( ( proj  h `  H ) `  ( A  +h  B ) )  =  A )
 
Theorempjidmi 22100 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  (
 ( proj  h `  H ) `  ( ( proj  h `
  H ) `  A ) )  =  ( ( proj  h `  H ) `  A )
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