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Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhfmval 22101 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 22102 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 22103 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 22104 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 22105* 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 22112 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 22106 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 22107 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 22108 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 22109 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 22110 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 22111* 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 22112 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 22113 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 22114 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 22115 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 22116 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 22117 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 22118 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 22119 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 22120 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 22121 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 22122 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 22123 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 22124 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 22125 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 22126 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 22127 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 22128 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 22129 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 22130 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 22131 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 22132 The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  C_H  A
 
Theorempjoml2 22133 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 22134 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 22135 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 22136 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 22137 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 22138 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 22139 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 22140 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 22141 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 22142 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 22143 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 22144 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 22145 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 22146 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 22147 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 22148 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 22149 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 22150 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 22151 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 22152 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 22153 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 22154 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 22155 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 22156 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 22157 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 22158 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 22159* Lemma for chscl 22163. (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 22160* Lemma for chscl 22163. (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 22161* Lemma for chscl 22163. (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 22162* Lemma for chscl 22163. (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 22163 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 22164 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 21897, although "the hard part" of this proof, chscl 22163, 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 22165 Corollary of osumi 22164. (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 22166 Corollary of osumi 22164, 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 22167 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 22168 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 22169 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 22170 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 22171 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 22172 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 22173 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 22174 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 22175 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 22176 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 22177 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 22178 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 22179 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 22180 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 22181 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 22182 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 22183 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 22184* 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 22185* 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 22186 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 22187 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 22188 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 22189 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 22190 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 22191 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 22192 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 22193 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 22194 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 22195 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 )
 
Theorempjadjii 22196 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( ( ( proj  h `
  H ) `  A )  .ih  B )  =  ( A  .ih  ( ( proj  h `  H ) `  B ) )
 
Theorempjaddii 22197 Projection of vector sum is sum of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( ( proj  h `  H ) `  ( A  +h  B ) )  =  ( ( (
 proj  h `  H ) `
  A )  +h  ( ( proj  h `  H ) `  B ) )
 
Theorempjinormii 22198 The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   =>    |-  (
 ( ( proj  h `  H ) `  A )  .ih  A )  =  ( ( normh `  (
 ( proj  h `  H ) `  A ) ) ^ 2 )
 
Theorempjmulii 22199 Projection of (scalar) product is product of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  C  e.  CC   =>    |-  ( ( proj  h `  H ) `  ( C  .h  A ) )  =  ( C  .h  ( ( proj  h `  H ) `  A ) )
 
Theorempjsubii 22200 Projection of vector difference is difference of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( ( proj  h `  H ) `  ( A  -h  B ) )  =  ( ( (
 proj  h `  H ) `
  A )  -h  ( ( proj  h `  H ) `  B ) )
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