 Home Metamath Proof ExplorerTheorem List (p. 287 of 429) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

Theorem List for Metamath Proof Explorer - 28601-28700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcmcm 28601 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 28602 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 28603 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 28604 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 28605 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 28606 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 28607 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 28608 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 28609 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 28610 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 28611 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 28612 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 28613 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 28614 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 28615 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 28616 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 28617 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 28618 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 28619 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 28620 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 28621 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 28622 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 28623 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 28624* Lemma for chscl 28628. (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 28625* Lemma for chscl 28628. (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 28626* Lemma for chscl 28628. (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 28627* Lemma for chscl 28628. (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 28628 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 28629 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 28380, although "the hard part" of this proof, chscl 28628, 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 28630 Corollary of osumi 28629. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) + (𝐴 ∩ (⊥‘𝐵))) = ((𝐴𝐵) ∨ (𝐴 ∩ (⊥‘𝐵)))

Theoremosumcor2i 28631 Corollary of osumi 28629, 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 28632 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 28633 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 28634 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 28635 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 28636 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 28637 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 28638 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 28639 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 28640 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 28641 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 28642 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 28643 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 28644 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 28645 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 28646 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 28647 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (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 28648 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 28649* 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 28650* 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 28651 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 28652 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))       𝑅 ⊆ (⊥‘𝐵)

Theorem3oalem5 28653 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 28654 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 28655 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 28656 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 28657 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 28658 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 28659 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 28660 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 28661 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 28662 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 28663 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 28664 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 28665 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 28666 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 28667 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 28668 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 28669 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 28670 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 28671 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 28672 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 28673 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 28674 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 28675 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 28676 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 28677 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 28678 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 28679 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 28680 The projection of the zero vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C       ((proj𝐻)‘0) = 0

Theorempjch 28681 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 28682 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)

Theorempjvec 28683* 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.)
(𝐻C𝐻 = {𝑥 ∈ ℋ ∣ ((proj𝐻)‘𝑥) = 𝑥})

Theorempjocvec 28684* 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.)
(𝐻C → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ((proj𝐻)‘𝑥) = 0})

Theorempjocini 28685 Membership of projection in orthocomplement of intersection. (Contributed by NM, 21-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ (⊥‘(𝐺𝐻)) → ((proj𝐺)‘𝐴) ∈ (⊥‘(𝐺𝐻)))

Theorempjini 28686 Membership of projection in an intersection. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ (𝐺𝐻) → ((proj𝐺)‘𝐴) ∈ (𝐺𝐻))

Theorempjjsi 28687* A sufficient condition for subspace join to be equal to subspace sum. (Contributed by NM, 29-May-2004.) (New usage is discouraged.)
𝐺C    &   𝐻S       (∀𝑥 ∈ (𝐺 𝐻)((proj‘(⊥‘𝐺))‘𝑥) ∈ 𝐻 → (𝐺 𝐻) = (𝐺 + 𝐻))

Theorempjfni 28688 Functionality of a projection. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝐻C       (proj𝐻) Fn ℋ

Theorempjrni 28689 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.)
𝐻C       ran (proj𝐻) = 𝐻

Theorempjfoi 28690 A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻): ℋ–onto𝐻

Theorempjfi 28691 The mapping of a projection. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐻C       (proj𝐻): ℋ⟶ ℋ

Theorempjvi 28692 The value of a projection in terms of components. (Contributed by NM, 28-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) → ((proj𝐻)‘(𝐴 + 𝐵)) = 𝐴)

Theorempjhfo 28693 A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻): ℋ–onto𝐻)

Theorempjrn 28694 The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → ran (proj𝐻) = 𝐻)

Theorempjhf 28695 The mapping of a projection. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻): ℋ⟶ ℋ)

Theorempjfn 28696 Functionality of a projection. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻) Fn ℋ)

Theorempjsumi 28697 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (𝐺 ⊆ (⊥‘𝐻) → ((proj‘(𝐺 + 𝐻))‘𝐴) = (((proj𝐺)‘𝐴) + ((proj𝐻)‘𝐴))))

Theorempj11i 28698 One-to-one correspondence of projection and subspace. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       ((proj𝐺) = (proj𝐻) ↔ 𝐺 = 𝐻)

Theorempjdsi 28699 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 21-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       ((𝐴 ∈ (𝐺 𝐻) ∧ 𝐺 ⊆ (⊥‘𝐻)) → 𝐴 = (((proj𝐺)‘𝐴) + ((proj𝐻)‘𝐴)))

Theorempjds3i 28700 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((𝐴 ∈ ((𝐹 𝐺) ∨ 𝐻) ∧ 𝐹 ⊆ (⊥‘𝐺)) ∧ (𝐹 ⊆ (⊥‘𝐻) ∧ 𝐺 ⊆ (⊥‘𝐻))) → 𝐴 = ((((proj𝐹)‘𝐴) + ((proj𝐺)‘𝐴)) + ((proj𝐻)‘𝐴)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 42801-42879
 Copyright terms: Public domain < Previous  Next >