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Theorem List for Metamath Proof Explorer - 29301-29400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcmbr2i 29301 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵𝐴 = ((𝐴 𝐵) ∩ (𝐴 (⊥‘𝐵))))
 
Theoremcmcmii 29302 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴 𝐶 𝐵       𝐵 𝐶 𝐴
 
Theoremcmcm2ii 29303 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴 𝐶 𝐵       𝐴 𝐶 (⊥‘𝐵)
 
Theoremcmcm3ii 29304 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴 𝐶 𝐵       (⊥‘𝐴) 𝐶 𝐵
 
Theoremcmbr3i 29305 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵 ↔ (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = (𝐴𝐵))
 
Theoremcmbr4i 29306 Alternate definition for the commutes relation. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵 ↔ (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) ⊆ 𝐵)
 
Theoremlecmi 29307 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵𝐴 𝐶 𝐵)
 
Theoremlecmii 29308 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴𝐵       𝐴 𝐶 𝐵
 
Theoremcmj1i 29309 A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 𝐶 (𝐴 𝐵)
 
Theoremcmj2i 29310 A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐵 𝐶 (𝐴 𝐵)
 
Theoremcmm1i 29311 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴 𝐶 (𝐴𝐵)
 
Theoremcmm2i 29312 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐵 𝐶 (𝐴𝐵)
 
Theoremcmbr3 29313 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐶 𝐵 ↔ (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = (𝐴𝐵)))
 
Theoremcm0 29314 The zero Hilbert lattice element commutes with every element. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
(𝐴C → 0 𝐶 𝐴)
 
Theoremcmidi 29315 The commutes relation is reflexive. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)
𝐴C       𝐴 𝐶 𝐴
 
Theorempjoml2 29316 Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C𝐴𝐵) → (𝐴 ((⊥‘𝐴) ∩ 𝐵)) = 𝐵)
 
Theorempjoml3 29317 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐵𝐴 → (𝐴 ∩ ((⊥‘𝐴) ∨ 𝐵)) = 𝐵))
 
Theorempjoml5 29318 The orthomodular law. Remark in [Kalmbach] p. 22. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 ((⊥‘𝐴) ∩ (𝐴 𝐵))) = (𝐴 𝐵))
 
Theoremcmcm 29319 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐵 𝐶 𝐴))
 
Theoremcmcm3 29320 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐶 𝐵 ↔ (⊥‘𝐴) 𝐶 𝐵))
 
Theoremcmcm2 29321 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝐶 𝐵𝐴 𝐶 (⊥‘𝐵)))
 
Theoremlecm 29322 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵C𝐴𝐵) → 𝐴 𝐶 𝐵)
 
19.5.6  Foulis-Holland theorem
 
Theoremfh1 29323 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.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝐶 𝐵𝐴 𝐶 𝐶)) → (𝐴 ∩ (𝐵 𝐶)) = ((𝐴𝐵) ∨ (𝐴𝐶)))
 
Theoremfh2 29324 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.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝐶 𝐴𝐵 𝐶 𝐶)) → (𝐴 ∩ (𝐵 𝐶)) = ((𝐴𝐵) ∨ (𝐴𝐶)))
 
Theoremcm2j 29325 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.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝐶 𝐵𝐴 𝐶 𝐶)) → 𝐴 𝐶 (𝐵 𝐶))
 
Theoremfh1i 29326 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.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       (𝐴 ∩ (𝐵 𝐶)) = ((𝐴𝐵) ∨ (𝐴𝐶))
 
Theoremfh2i 29327 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.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       (𝐵 ∩ (𝐴 𝐶)) = ((𝐵𝐴) ∨ (𝐵𝐶))
 
Theoremfh3i 29328 Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       (𝐴 (𝐵𝐶)) = ((𝐴 𝐵) ∩ (𝐴 𝐶))
 
Theoremfh4i 29329 Variation of the Foulis-Holland Theorem. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       (𝐵 (𝐴𝐶)) = ((𝐵 𝐴) ∩ (𝐵 𝐶))
 
Theoremcm2ji 29330 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.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       𝐴 𝐶 (𝐵 𝐶)
 
Theoremcm2mi 29331 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.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 𝐶 𝐵    &   𝐴 𝐶 𝐶       𝐴 𝐶 (𝐵𝐶)
 
19.5.7  Quantum Logic Explorer axioms
 
Theoremqlax1i 29332 One of the equations showing C is an ortholattice. (This corresponds to axiom "ax-1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C       𝐴 = (⊥‘(⊥‘𝐴))
 
Theoremqlax2i 29333 One of the equations showing C is an ortholattice. (This corresponds to axiom "ax-2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵) = (𝐵 𝐴)
 
Theoremqlax3i 29334 One of the equations showing C is an ortholattice. (This corresponds to axiom "ax-3" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴 𝐵) ∨ 𝐶) = (𝐴 (𝐵 𝐶))
 
Theoremqlax4i 29335 One of the equations showing C is an ortholattice. (This corresponds to axiom "ax-4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 (𝐵 (⊥‘𝐵))) = (𝐵 (⊥‘𝐵))
 
Theoremqlax5i 29336 One of the equations showing C is an ortholattice. (This corresponds to axiom "ax-5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 (⊥‘((⊥‘𝐴) ∨ 𝐵))) = 𝐴
 
Theoremqlaxr1i 29337 One of the conditions showing C is an ortholattice. (This corresponds to axiom "ax-r1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴 = 𝐵       𝐵 = 𝐴
 
Theoremqlaxr2i 29338 One of the conditions showing C is an ortholattice. (This corresponds to axiom "ax-r2" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 = 𝐵    &   𝐵 = 𝐶       𝐴 = 𝐶
 
Theoremqlaxr4i 29339 One of the conditions showing C is an ortholattice. (This corresponds to axiom "ax-r4" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐴 = 𝐵       (⊥‘𝐴) = (⊥‘𝐵)
 
Theoremqlaxr5i 29340 One of the conditions showing C is an ortholattice. (This corresponds to axiom "ax-r5" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐴 = 𝐵       (𝐴 𝐶) = (𝐵 𝐶)
 
Theoremqlaxr3i 29341 A variation of the orthomodular law, showing C 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.)
𝐴C    &   𝐵C    &   𝐶C    &   (𝐶 (⊥‘𝐶)) = ((⊥‘((⊥‘𝐴) ∨ (⊥‘𝐵))) ∨ (⊥‘(𝐴 𝐵)))       𝐴 = 𝐵
 
19.5.8  Orthogonal subspaces
 
Theoremchscllem1 29342* Lemma for chscl 29346. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝜑𝐴C )    &   (𝜑𝐵C )    &   (𝜑𝐵 ⊆ (⊥‘𝐴))    &   (𝜑𝐻:ℕ⟶(𝐴 + 𝐵))    &   (𝜑𝐻𝑣 𝑢)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((proj𝐴)‘(𝐻𝑛)))       (𝜑𝐹:ℕ⟶𝐴)
 
Theoremchscllem2 29343* Lemma for chscl 29346. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝜑𝐴C )    &   (𝜑𝐵C )    &   (𝜑𝐵 ⊆ (⊥‘𝐴))    &   (𝜑𝐻:ℕ⟶(𝐴 + 𝐵))    &   (𝜑𝐻𝑣 𝑢)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((proj𝐴)‘(𝐻𝑛)))       (𝜑𝐹 ∈ dom ⇝𝑣 )
 
Theoremchscllem3 29344* Lemma for chscl 29346. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝜑𝐴C )    &   (𝜑𝐵C )    &   (𝜑𝐵 ⊆ (⊥‘𝐴))    &   (𝜑𝐻:ℕ⟶(𝐴 + 𝐵))    &   (𝜑𝐻𝑣 𝑢)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((proj𝐴)‘(𝐻𝑛)))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷𝐵)    &   (𝜑 → (𝐻𝑁) = (𝐶 + 𝐷))       (𝜑𝐶 = (𝐹𝑁))
 
Theoremchscllem4 29345* Lemma for chscl 29346. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝜑𝐴C )    &   (𝜑𝐵C )    &   (𝜑𝐵 ⊆ (⊥‘𝐴))    &   (𝜑𝐻:ℕ⟶(𝐴 + 𝐵))    &   (𝜑𝐻𝑣 𝑢)    &   𝐹 = (𝑛 ∈ ℕ ↦ ((proj𝐴)‘(𝐻𝑛)))    &   𝐺 = (𝑛 ∈ ℕ ↦ ((proj𝐵)‘(𝐻𝑛)))       (𝜑𝑢 ∈ (𝐴 + 𝐵))
 
Theoremchscl 29346 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.)
(𝜑𝐴C )    &   (𝜑𝐵C )    &   (𝜑𝐵 ⊆ (⊥‘𝐴))       (𝜑 → (𝐴 + 𝐵) ∈ C )
 
Theoremosumi 29347 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 29098, although "the hard part" of this proof, chscl 29346, requires no choice. (Contributed by NM, 28-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 ⊆ (⊥‘𝐵) → (𝐴 + 𝐵) = (𝐴 𝐵))
 
Theoremosumcori 29348 Corollary of osumi 29347. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) + (𝐴 ∩ (⊥‘𝐵))) = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))
 
Theoremosumcor2i 29349 Corollary of osumi 29347, showing it holds under the weaker hypothesis that 𝐴 and 𝐵 commute. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵 → (𝐴 + 𝐵) = (𝐴 𝐵))
 
Theoremosum 29350 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.)
((𝐴C𝐵C𝐴 ⊆ (⊥‘𝐵)) → (𝐴 + 𝐵) = (𝐴 𝐵))
 
Theoremspansnji 29351 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.)
𝐴C    &   𝐵 ∈ ℋ       (𝐴 + (span‘{𝐵})) = (𝐴 (span‘{𝐵}))
 
Theoremspansnj 29352 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.)
((𝐴C𝐵 ∈ ℋ) → (𝐴 + (span‘{𝐵})) = (𝐴 (span‘{𝐵})))
 
Theoremspansnscl 29353 The subspace sum of a closed subspace and a one-dimensional subspace is closed. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ ℋ) → (𝐴 + (span‘{𝐵})) ∈ C )
 
Theoremsumspansn 29354 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 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 + 𝐵) ∈ (span‘{𝐴}) ↔ 𝐵 ∈ (span‘{𝐴})))
 
Theoremspansnm0i 29355 The meet of different one-dimensional subspaces is the zero subspace. (Contributed by NM, 1-Nov-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       𝐴 ∈ (span‘{𝐵}) → ((span‘{𝐴}) ∩ (span‘{𝐵})) = 0)
 
Theoremnonbooli 29356 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 ((𝐻𝐹) ∨ (𝐻𝐺)) = 0 but (𝐻 ∩ (𝐹 𝐺)) ≠ 0. 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 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐹 = (span‘{𝐴})    &   𝐺 = (span‘{𝐵})    &   𝐻 = (span‘{(𝐴 + 𝐵)})       (¬ (𝐴𝐺𝐵𝐹) → (𝐻 ∩ (𝐹 𝐺)) ≠ ((𝐻𝐹) ∨ (𝐻𝐺)))
 
Theoremspansncvi 29357 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.)
𝐴C    &   𝐵C    &   𝐶 ∈ ℋ       ((𝐴𝐵𝐵 ⊆ (𝐴 (span‘{𝐶}))) → 𝐵 = (𝐴 (span‘{𝐶})))
 
Theoremspansncv 29358 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.)
((𝐴C𝐵C𝐶 ∈ ℋ) → ((𝐴𝐵𝐵 ⊆ (𝐴 (span‘{𝐶}))) → 𝐵 = (𝐴 (span‘{𝐶}))))
 
19.5.9  Orthoarguesian laws 5OA and 3OA
 
Theorem5oalem1 29359 Lemma for orthoarguesian law 5OA. (Contributed by NM, 1-Apr-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝑅S       ((((𝑥𝐴𝑦𝐵) ∧ 𝑣 = (𝑥 + 𝑦)) ∧ (𝑧𝐶 ∧ (𝑥 𝑧) ∈ 𝑅)) → 𝑣 ∈ (𝐵 + (𝐴 ∩ (𝐶 + 𝑅))))
 
Theorem5oalem2 29360 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S       ((((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝑤𝐷)) ∧ (𝑥 + 𝑦) = (𝑧 + 𝑤)) → (𝑥 𝑧) ∈ ((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)))
 
Theorem5oalem3 29361 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S       (((((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝑤𝐷)) ∧ (𝑓𝐹𝑔𝐺)) ∧ ((𝑥 + 𝑦) = (𝑓 + 𝑔) ∧ (𝑧 + 𝑤) = (𝑓 + 𝑔))) → (𝑥 𝑧) ∈ (((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) + ((𝐶 + 𝐹) ∩ (𝐷 + 𝐺))))
 
Theorem5oalem4 29362 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S       (((((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝑤𝐷)) ∧ (𝑓𝐹𝑔𝐺)) ∧ ((𝑥 + 𝑦) = (𝑓 + 𝑔) ∧ (𝑧 + 𝑤) = (𝑓 + 𝑔))) → (𝑥 𝑧) ∈ (((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)) ∩ (((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) + ((𝐶 + 𝐹) ∩ (𝐷 + 𝐺)))))
 
Theorem5oalem5 29363 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S    &   𝑅S    &   𝑆S       (((((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝑤𝐷)) ∧ ((𝑓𝐹𝑔𝐺) ∧ (𝑣𝑅𝑢𝑆))) ∧ (((𝑥 + 𝑦) = (𝑣 + 𝑢) ∧ (𝑧 + 𝑤) = (𝑣 + 𝑢)) ∧ (𝑓 + 𝑔) = (𝑣 + 𝑢))) → (𝑥 𝑧) ∈ ((((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)))) ∩ ((((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))) + (((𝐶 + 𝐹) ∩ (𝐷 + 𝐺)) ∩ (((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))))))
 
Theorem5oalem6 29364 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S    &   𝑅S    &   𝑆S       (((((𝑥𝐴𝑦𝐵) ∧ = (𝑥 + 𝑦)) ∧ ((𝑧𝐶𝑤𝐷) ∧ = (𝑧 + 𝑤))) ∧ (((𝑓𝐹𝑔𝐺) ∧ = (𝑓 + 𝑔)) ∧ ((𝑣𝑅𝑢𝑆) ∧ = (𝑣 + 𝑢)))) → ∈ (𝐵 + (𝐴 ∩ (𝐶 + ((((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)))) ∩ ((((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))) + (((𝐶 + 𝐹) ∩ (𝐷 + 𝐺)) ∩ (((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆))))))))))
 
Theorem5oalem7 29365 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) TODO: replace uses of ee4anv 2364 with 4exdistrv 1948 as in 3oalem3 29369. (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S    &   𝑅S    &   𝑆S       (((𝐴 + 𝐵) ∩ (𝐶 + 𝐷)) ∩ ((𝐹 + 𝐺) ∩ (𝑅 + 𝑆))) ⊆ (𝐵 + (𝐴 ∩ (𝐶 + ((((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)))) ∩ ((((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))) + (((𝐶 + 𝐹) ∩ (𝐷 + 𝐺)) ∩ (((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))))))))
 
Theorem5oai 29366 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.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝐹C    &   𝐺C    &   𝑅C    &   𝑆C    &   𝐴 ⊆ (⊥‘𝐵)    &   𝐶 ⊆ (⊥‘𝐷)    &   𝐹 ⊆ (⊥‘𝐺)    &   𝑅 ⊆ (⊥‘𝑆)       (((𝐴 𝐵) ∩ (𝐶 𝐷)) ∩ ((𝐹 𝐺) ∩ (𝑅 𝑆))) ⊆ (𝐵 (𝐴 ∩ (𝐶 ((((𝐴 𝐶) ∩ (𝐵 𝐷)) ∩ (((𝐴 𝑅) ∩ (𝐵 𝑆)) ∨ ((𝐶 𝑅) ∩ (𝐷 𝑆)))) ∩ ((((𝐴 𝐹) ∩ (𝐵 𝐺)) ∩ (((𝐴 𝑅) ∩ (𝐵 𝑆)) ∨ ((𝐹 𝑅) ∩ (𝐺 𝑆)))) ∨ (((𝐶 𝐹) ∩ (𝐷 𝐺)) ∩ (((𝐶 𝑅) ∩ (𝐷 𝑆)) ∨ ((𝐹 𝑅) ∩ (𝐺 𝑆)))))))))
 
Theorem3oalem1 29367* Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐵C    &   𝐶C    &   𝑅C    &   𝑆C       ((((𝑥𝐵𝑦𝑅) ∧ 𝑣 = (𝑥 + 𝑦)) ∧ ((𝑧𝐶𝑤𝑆) ∧ 𝑣 = (𝑧 + 𝑤))) → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)))
 
Theorem3oalem2 29368* Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐵C    &   𝐶C    &   𝑅C    &   𝑆C       ((((𝑥𝐵𝑦𝑅) ∧ 𝑣 = (𝑥 + 𝑦)) ∧ ((𝑧𝐶𝑤𝑆) ∧ 𝑣 = (𝑧 + 𝑤))) → 𝑣 ∈ (𝐵 + (𝑅 ∩ (𝑆 + ((𝐵 + 𝐶) ∩ (𝑅 + 𝑆))))))
 
Theorem3oalem3 29369 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐵C    &   𝐶C    &   𝑅C    &   𝑆C       ((𝐵 + 𝑅) ∩ (𝐶 + 𝑆)) ⊆ (𝐵 + (𝑅 ∩ (𝑆 + ((𝐵 + 𝐶) ∩ (𝑅 + 𝑆)))))
 
Theorem3oalem4 29370 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))       𝑅 ⊆ (⊥‘𝐵)
 
Theorem3oalem5 29371 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))    &   𝑆 = ((⊥‘𝐶) ∩ (𝐶 𝐴))       ((𝐵 + 𝑅) ∩ (𝐶 + 𝑆)) = ((𝐵 𝑅) ∩ (𝐶 𝑆))
 
Theorem3oalem6 29372 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))    &   𝑆 = ((⊥‘𝐶) ∩ (𝐶 𝐴))       (𝐵 + (𝑅 ∩ (𝑆 + ((𝐵 + 𝐶) ∩ (𝑅 + 𝑆))))) ⊆ (𝐵 (𝑅 ∩ (𝑆 ((𝐵 𝐶) ∩ (𝑅 𝑆)))))
 
Theorem3oai 29373 3OA (weak) orthoarguesian law. Equation IV of [GodowskiGreechie] p. 249. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))    &   𝑆 = ((⊥‘𝐶) ∩ (𝐶 𝐴))       ((𝐵 𝑅) ∩ (𝐶 𝑆)) ⊆ (𝐵 (𝑅 ∩ (𝑆 ((𝐵 𝐶) ∩ (𝑅 𝑆)))))
 
19.5.10  Projectors (cont.)
 
Theorempjorthi 29374 Projection components on orthocomplemented subspaces are orthogonal. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐻C → (((proj𝐻)‘𝐴) ·ih ((proj‘(⊥‘𝐻))‘𝐵)) = 0)
 
Theorempjch1 29375 Property of identity projection. Remark in [Beran] p. 111. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((proj‘ ℋ)‘𝐴) = 𝐴)
 
Theorempjo 29376 The orthogonal projection. Lemma 4.4(i) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj‘(⊥‘𝐻))‘𝐴) = (((proj‘ ℋ)‘𝐴) − ((proj𝐻)‘𝐴)))
 
Theorempjcompi 29377 Component of a projection. (Contributed by NM, 31-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝐻C       ((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) → ((proj𝐻)‘(𝐴 + 𝐵)) = 𝐴)
 
Theorempjidmi 29378 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘((proj𝐻)‘𝐴)) = ((proj𝐻)‘𝐴)
 
Theorempjadjii 29379 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (((proj𝐻)‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((proj𝐻)‘𝐵))
 
Theorempjaddii 29380 Projection of vector sum is sum of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((proj𝐻)‘(𝐴 + 𝐵)) = (((proj𝐻)‘𝐴) + ((proj𝐻)‘𝐵))
 
Theorempjinormii 29381 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.)
𝐻C    &   𝐴 ∈ ℋ       (((proj𝐻)‘𝐴) ·ih 𝐴) = ((norm‘((proj𝐻)‘𝐴))↑2)
 
Theorempjmulii 29382 Projection of (scalar) product is product of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐶 ∈ ℂ       ((proj𝐻)‘(𝐶 · 𝐴)) = (𝐶 · ((proj𝐻)‘𝐴))
 
Theorempjsubii 29383 Projection of vector difference is difference of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((proj𝐻)‘(𝐴 𝐵)) = (((proj𝐻)‘𝐴) − ((proj𝐻)‘𝐵))
 
Theorempjsslem 29384 Lemma for subset relationships of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (((proj‘(⊥‘𝐻))‘𝐴) − ((proj‘(⊥‘𝐺))‘𝐴)) = (((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴))
 
Theorempjss2i 29385 Subset relationship for projections. Theorem 4.5(i)->(ii) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (𝐻𝐺 → ((proj𝐻)‘((proj𝐺)‘𝐴)) = ((proj𝐻)‘𝐴))
 
Theorempjssmii 29386 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.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (𝐻𝐺 → (((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴))
 
Theorempjssge0ii 29387 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) → 0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴))
 
Theorempjdifnormii 29388 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴) ↔ (norm‘((proj𝐻)‘𝐴)) ≤ (norm‘((proj𝐺)‘𝐴)))
 
Theorempjcji 29389 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (𝐻 ⊆ (⊥‘𝐺) → ((proj‘(𝐻 𝐺))‘𝐴) = (((proj𝐻)‘𝐴) + ((proj𝐺)‘𝐴)))
 
Theorempjadji 29390 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((proj𝐻)‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((proj𝐻)‘𝐵)))
 
Theorempjaddi 29391 Projection of vector sum is sum of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((proj𝐻)‘(𝐴 + 𝐵)) = (((proj𝐻)‘𝐴) + ((proj𝐻)‘𝐵)))
 
Theorempjinormi 29392 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.)
𝐻C       (𝐴 ∈ ℋ → (((proj𝐻)‘𝐴) ·ih 𝐴) = ((norm‘((proj𝐻)‘𝐴))↑2))
 
Theorempjsubi 29393 Projection of vector difference is difference of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((proj𝐻)‘(𝐴 𝐵)) = (((proj𝐻)‘𝐴) − ((proj𝐻)‘𝐵)))
 
Theorempjmuli 29394 Projection of scalar product is scalar product of projection. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((proj𝐻)‘(𝐴 · 𝐵)) = (𝐴 · ((proj𝐻)‘𝐵)))
 
Theorempjige0i 29395 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → 0 ≤ (((proj𝐻)‘𝐴) ·ih 𝐴))
 
Theorempjige0 29396 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → 0 ≤ (((proj𝐻)‘𝐴) ·ih 𝐴))
 
Theorempjcjt2 29397 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐺C𝐴 ∈ ℋ) → (𝐻 ⊆ (⊥‘𝐺) → ((proj‘(𝐻 𝐺))‘𝐴) = (((proj𝐻)‘𝐴) + ((proj𝐺)‘𝐴))))
 
Theorempj0i 29398 The projection of the zero vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C       ((proj𝐻)‘0) = 0
 
Theorempjch 29399 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.)
((𝐻C𝐴 ∈ ℋ) → (𝐴𝐻 ↔ ((proj𝐻)‘𝐴) = 𝐴))
 
Theorempjid 29400 The projection of a vector in the projection subspace is itself. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
((𝐻C𝐴𝐻) → ((proj𝐻)‘𝐴) = 𝐴)
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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