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Theorem List for Metamath Proof Explorer - 22201-22300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcmm1i 22201 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 22202 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 22203 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 22204 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 22205 The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
 |-  A  e.  CH   =>    |-  A  C_H  A
 
Theorempjoml2 22206 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 22207 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 22208 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 22209 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 22210 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 22211 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 22212 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 )
 
17.5.6  Foulis-Holland theorem
 
Theoremfh1 22213 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 22214 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 22215 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 22216 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 22217 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 22218 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 22219 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 22220 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 22221 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 )
 
17.5.7  Quantum Logic Explorer axioms
 
Theoremqlax1i 22222 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 22223 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 22224 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 22225 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 22226 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 22227 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 22228 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 22229 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 22230 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 22231 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
 
17.5.8  Orthogonal subspaces
 
Theoremchscllem1 22232* Lemma for chscl 22236. (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 22233* Lemma for chscl 22236. (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 22234* Lemma for chscl 22236. (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 22235* Lemma for chscl 22236. (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 22236 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 22237 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 21988, although "the hard part" of this proof, chscl 22236, 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 22238 Corollary of osumi 22237. (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 22239 Corollary of osumi 22237, 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 22240 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 22241 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 22242 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 22243 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 22244 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 22245 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 22246 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 22247 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 22248 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 } ) ) ) )
 
17.5.9  Orthoarguesian laws 5OA and 3OA
 
Theorem5oalem1 22249 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 22250 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 22251 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 22252 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 22253 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 22254 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 22255 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 22256 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 22257* 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 22258* 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 22259 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 22260 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 22261 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 22262 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 22263 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 ) ) ) ) )
 
17.5.10  Projectors (cont.)
 
Theorempjorthi 22264 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 22265 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 22266 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 22267 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 22268 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 22269 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 22270 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 22271 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 22272 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 22273 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 ) )
 
Theorempjsslem 22274 Lemma for subset relationships of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( ( ( proj  h `
  ( _|_ `  H ) ) `  A )  -h  ( ( proj  h `
  ( _|_ `  G ) ) `  A ) )  =  (
 ( ( proj  h `  G ) `  A )  -h  ( ( proj  h `
  H ) `  A ) )
 
Theorempjss2i 22275 Subset relationship for projections. Theorem 4.5(i)->(ii) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( H  C_  G  ->  ( ( proj  h `  H ) `  (
 ( proj  h `  G ) `  A ) )  =  ( ( proj  h `
  H ) `  A ) )
 
Theorempjssmii 22276 Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( H  C_  G  ->  ( ( ( proj  h `
  G ) `  A )  -h  (
 ( proj  h `  H ) `  A ) )  =  ( ( proj  h `
  ( G  i^i  ( _|_ `  H )
 ) ) `  A ) )
 
Theorempjssge0ii 22277 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( ( ( (
 proj  h `  G ) `
  A )  -h  ( ( proj  h `  H ) `  A ) )  =  (
 ( proj  h `  ( G  i^i  ( _|_ `  H ) ) ) `  A )  ->  0  <_  ( ( ( (
 proj  h `  G ) `
  A )  -h  ( ( proj  h `  H ) `  A ) )  .ih  A ) )
 
Theorempjdifnormii 22278 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( 0  <_  (
 ( ( ( proj  h `
  G ) `  A )  -h  (
 ( proj  h `  H ) `  A ) ) 
 .ih  A )  <->  ( normh `  (
 ( proj  h `  H ) `  A ) ) 
 <_  ( normh `  ( ( proj  h `  G ) `
  A ) ) )
 
Theorempjcji 22279 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
 |-  H  e.  CH   &    |-  A  e.  ~H   &    |-  G  e.  CH   =>    |-  ( H  C_  ( _|_ `  G )  ->  ( ( proj  h `  ( H  vH  G ) ) `  A )  =  ( ( (
 proj  h `  H ) `
  A )  +h  ( ( proj  h `  G ) `  A ) ) )
 
Theorempjadji 22280 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 6-Oct-2000.) (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 ) ) )
 
Theorempjaddi 22281 Projection of vector sum is sum of projections. (Contributed by NM, 14-Nov-2000.) (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 ) ) )
 
Theorempjinormi 22282 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, 2-Jun-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  ( ( ( proj  h `
  H ) `  A )  .ih  A )  =  ( ( normh `  ( ( proj  h `  H ) `  A ) ) ^ 2
 ) )
 
Theorempjsubi 22283 Projection of vector difference is difference of projections. (Contributed by NM, 14-Nov-2000.) (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 ) ) )
 
Theorempjmuli 22284 Projection of scalar product is scalar product of projection. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( proj  h `
  H ) `  ( A  .h  B ) )  =  ( A  .h  ( ( proj  h `
  H ) `  B ) ) )
 
Theorempjige0i 22285 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( A  e.  ~H  ->  0  <_  ( (
 ( proj  h `  H ) `  A )  .ih  A ) )
 
Theorempjige0 22286 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  0  <_  ( (
 ( proj  h `  H ) `  A )  .ih  A ) )
 
Theorempjcjt2 22287 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  G  e.  CH  /\  A  e.  ~H )  ->  ( H  C_  ( _|_ `  G )  ->  ( ( proj  h `  ( H  vH  G ) ) `  A )  =  ( ( (
 proj  h `  H ) `
  A )  +h  ( ( proj  h `  G ) `  A ) ) ) )
 
Theorempj0i 22288 The projection of the zero vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( proj  h `  H ) `  0h )  =  0h
 
Theorempjch 22289 Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  ~H )  ->  ( A  e.  H  <->  ( ( proj  h `  H ) `  A )  =  A ) )
 
Theorempjid 22290 The projection of a vector in the projection subspace is itself. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( H  e.  CH  /\  A  e.  H ) 
 ->  ( ( proj  h `  H ) `  A )  =  A )
 
Theorempjvec 22291* The set of vectors belonging to the subspace of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  H  =  { x  e.  ~H  |  ( ( proj  h `  H ) `  x )  =  x }
 )
 
Theorempjocvec 22292* The set of vectors belonging to the orthocomplemented subspace of a projection. Second part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  ( H  e.  CH  ->  ( _|_ `  H )  =  { x  e.  ~H  |  ( ( proj  h `  H ) `  x )  =  0h } )
 
Theorempjocini 22293 Membership of projection in orthocomplement of intersection. (Contributed by NM, 21-Apr-2001.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ( _|_ `  ( G  i^i  H ) )  ->  ( (
 proj  h `  G ) `
  A )  e.  ( _|_ `  ( G  i^i  H ) ) )
 
Theorempjini 22294 Membership of projection in an intersection. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  CH   =>    |-  ( A  e.  ( G  i^i  H )  ->  (
 ( proj  h `  G ) `  A )  e.  ( G  i^i  H ) )
 
Theorempjjsi 22295* A sufficient condition for subspace join to be equal to subspace sum. (Contributed by NM, 29-May-2004.) (New usage is discouraged.)
 |-  G  e.  CH   &    |-  H  e.  SH   =>    |-  ( A. x  e.  ( G  vH  H ) ( ( proj  h `  ( _|_ `  G ) ) `
  x )  e.  H  ->  ( G  vH  H )  =  ( G  +H  H ) )
 
Theorempjfni 22296 Functionality of a projection. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H )  Fn  ~H
 
Theorempjrni 22297 The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |- 
 ran  ( proj  h `  H )  =  H
 
Theorempjfoi 22298 A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H ) : ~H -onto-> H
 
Theorempjfi 22299 The mapping of a projection. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( proj  h `  H ) : ~H --> ~H
 
Theorempjvi 22300 The value of a projection in terms of components. (Contributed by NM, 28-Nov-2000.) (New usage is discouraged.)
 |-  H  e.  CH   =>    |-  ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) ) 
 ->  ( ( proj  h `  H ) `  ( A  +h  B ) )  =  A )
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