Home Metamath Proof ExplorerTheorem List (p. 302 of 454) < 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 (1-28705) Hilbert Space Explorer (28706-30228) Users' Mathboxes (30229-45326)

Theorem List for Metamath Proof Explorer - 30101-30200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremssdmd2 30101 Ordering implies the dual modular pair property. Remark in [MaedaMaeda] p. 1. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐴𝐵) → (⊥‘𝐵) 𝑀 (⊥‘𝐴))

Theoremdmdsl3 30102 Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐵 𝑀* 𝐴𝐴𝐶𝐶 ⊆ (𝐴 𝐵))) → ((𝐶𝐵) ∨ 𝐴) = 𝐶)

Theoremmdsl3 30103 Sublattice mapping for a modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)
(((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀 𝐵 ∧ (𝐴𝐵) ⊆ 𝐶𝐶𝐵)) → ((𝐶 𝐴) ∩ 𝐵) = 𝐶)

Theoremmdslle1i 30104 Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝐵 𝑀* 𝐴𝐴 ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ (𝐴 𝐵)) → (𝐶𝐷 ↔ (𝐶𝐵) ⊆ (𝐷𝐵)))

Theoremmdslle2i 30105 Order preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝐴 𝑀 𝐵 ∧ (𝐴𝐵) ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ 𝐵) → (𝐶𝐷 ↔ (𝐶 𝐴) ⊆ (𝐷 𝐴)))

Theoremmdslj1i 30106 Join preservation of the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ (𝐴 ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ (𝐴 𝐵))) → ((𝐶 𝐷) ∩ 𝐵) = ((𝐶𝐵) ∨ (𝐷𝐵)))

Theoremmdslj2i 30107 Meet preservation of the reverse mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐵) ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ 𝐵)) → ((𝐶𝐷) ∨ 𝐴) = ((𝐶 𝐴) ∩ (𝐷 𝐴)))

Theoremmdsl1i 30108* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (∀𝑥C (((𝐴𝐵) ⊆ 𝑥𝑥 ⊆ (𝐴 𝐵)) → (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))) ↔ 𝐴 𝑀 𝐵)

Theoremmdsl2i 30109* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 28-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀 𝐵 ↔ ∀𝑥C (((𝐴𝐵) ⊆ 𝑥𝑥𝐵) → ((𝑥 𝐴) ∩ 𝐵) ⊆ (𝑥 (𝐴𝐵))))

Theoremmdsl2bi 30110* If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀 𝐵 ↔ ∀𝑥C (((𝐴𝐵) ⊆ 𝑥𝑥𝐵) → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))

Theoremcvmdi 30111 The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) ⋖ 𝐵𝐴 𝑀 𝐵)

Theoremmdslmd1lem1 30112 Lemma for mdslmd1i 30116. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝑅C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐶𝐴𝐷) ∧ (𝐶 ⊆ (𝐴 𝐵) ∧ 𝐷 ⊆ (𝐴 𝐵)))) → (((𝑅 𝐴) ⊆ 𝐷 → (((𝑅 𝐴) ∨ 𝐶) ∩ 𝐷) ⊆ ((𝑅 𝐴) ∨ (𝐶𝐷))) → ((((𝐶𝐵) ∩ (𝐷𝐵)) ⊆ 𝑅𝑅 ⊆ (𝐷𝐵)) → ((𝑅 (𝐶𝐵)) ∩ (𝐷𝐵)) ⊆ (𝑅 ((𝐶𝐵) ∩ (𝐷𝐵))))))

Theoremmdslmd1lem2 30113 Lemma for mdslmd1i 30116. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝑅C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐶𝐴𝐷) ∧ (𝐶 ⊆ (𝐴 𝐵) ∧ 𝐷 ⊆ (𝐴 𝐵)))) → (((𝑅𝐵) ⊆ (𝐷𝐵) → (((𝑅𝐵) ∨ (𝐶𝐵)) ∩ (𝐷𝐵)) ⊆ ((𝑅𝐵) ∨ ((𝐶𝐵) ∩ (𝐷𝐵)))) → (((𝐶𝐷) ⊆ 𝑅𝑅𝐷) → ((𝑅 𝐶) ∩ 𝐷) ⊆ (𝑅 (𝐶𝐷)))))

Theoremmdslmd1lem3 30114* Lemma for mdslmd1i 30116. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝑥C ∧ ((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐶𝐴𝐷) ∧ (𝐶 ⊆ (𝐴 𝐵) ∧ 𝐷 ⊆ (𝐴 𝐵))))) → (((𝑥 𝐴) ⊆ 𝐷 → (((𝑥 𝐴) ∨ 𝐶) ∩ 𝐷) ⊆ ((𝑥 𝐴) ∨ (𝐶𝐷))) → ((((𝐶𝐵) ∩ (𝐷𝐵)) ⊆ 𝑥𝑥 ⊆ (𝐷𝐵)) → ((𝑥 (𝐶𝐵)) ∩ (𝐷𝐵)) ⊆ (𝑥 ((𝐶𝐵) ∩ (𝐷𝐵))))))

Theoremmdslmd1lem4 30115* Lemma for mdslmd1i 30116. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝑥C ∧ ((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐶𝐴𝐷) ∧ (𝐶 ⊆ (𝐴 𝐵) ∧ 𝐷 ⊆ (𝐴 𝐵))))) → (((𝑥𝐵) ⊆ (𝐷𝐵) → (((𝑥𝐵) ∨ (𝐶𝐵)) ∩ (𝐷𝐵)) ⊆ ((𝑥𝐵) ∨ ((𝐶𝐵) ∩ (𝐷𝐵)))) → (((𝐶𝐷) ⊆ 𝑥𝑥𝐷) → ((𝑥 𝐶) ∩ 𝐷) ⊆ (𝑥 (𝐶𝐷)))))

Theoremmdslmd1i 30116 Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (meet version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ (𝐴 ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ (𝐴 𝐵))) → (𝐶 𝑀 𝐷 ↔ (𝐶𝐵) 𝑀 (𝐷𝐵)))

Theoremmdslmd2i 30117 Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (join version). (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ ((𝐴𝐵) ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ 𝐵)) → (𝐶 𝑀 𝐷 ↔ (𝐶 𝐴) 𝑀 (𝐷 𝐴)))

Theoremmdsldmd1i 30118 Preservation of the dual modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵𝐵 𝑀* 𝐴) ∧ (𝐴 ⊆ (𝐶𝐷) ∧ (𝐶 𝐷) ⊆ (𝐴 𝐵))) → (𝐶 𝑀* 𝐷 ↔ (𝐶𝐵) 𝑀* (𝐷𝐵)))

Theoremmdslmd3i 30119 Modular pair conditions that imply the modular pair property in a sublattice. Lemma 1.5.1 of [MaedaMaeda] p. 2. (Contributed by NM, 23-Dec-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       (((𝐴 𝑀 𝐵 ∧ (𝐴𝐵) 𝑀 𝐶) ∧ ((𝐴𝐶) ⊆ 𝐷𝐷𝐴)) → 𝐷 𝑀 (𝐵𝐶))

Theoremmdslmd4i 30120 Modular pair condition that implies the modular pair property in a sublattice. Lemma 1.5.2 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C       ((𝐴 𝑀 𝐵 ∧ ((𝐴𝐵) ⊆ 𝐶𝐶𝐴) ∧ ((𝐴𝐵) ⊆ 𝐷𝐷𝐵)) → 𝐶 𝑀 𝐷)

Theoremcsmdsymi 30121* Cross-symmetry implies M-symmetry. Theorem 1.9.1 of [MaedaMaeda] p. 3. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((∀𝑐C (𝑐 𝑀 𝐵𝐵 𝑀* 𝑐) ∧ 𝐴 𝑀 𝐵) → 𝐵 𝑀 𝐴)

Theoremmdexchi 30122 An exchange lemma for modular pairs. Lemma 1.6 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C       ((𝐴 𝑀 𝐵𝐶 𝑀 (𝐴 𝐵) ∧ (𝐶 ∩ (𝐴 𝐵)) ⊆ 𝐴) → ((𝐶 𝐴) 𝑀 𝐵 ∧ ((𝐶 𝐴) ∩ 𝐵) = (𝐴𝐵)))

Theoremcvmd 30123 The covering property implies the modular pair property. Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ∧ (𝐴𝐵) ⋖ 𝐵) → 𝐴 𝑀 𝐵)

Theoremcvdmd 30124 The covering property implies the dual modular pair property. Lemma 7.5.2 of [MaedaMaeda] p. 31. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C𝐵 (𝐴 𝐵)) → 𝐴 𝑀* 𝐵)

19.8.2  Atoms

Definitiondf-at 30125 Define the set of atoms in a Hilbert lattice. An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is the smallest nonzero element of the lattice. Definition of atom in [Kalmbach] p. 15. See ela 30126 and elat2 30127 for membership relations. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
HAtoms = {𝑥C ∣ 0 𝑥}

Theoremela 30126 Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))

Theoremelat2 30127* Expanded membership relation for the set of atoms, i.e. the predicate "is an atom (of the Hilbert lattice)." An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is the smallest nonzero element of the lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms ↔ (𝐴C ∧ (𝐴 ≠ 0 ∧ ∀𝑥C (𝑥𝐴 → (𝑥 = 𝐴𝑥 = 0)))))

Theoremelatcv0 30128 A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴C → (𝐴 ∈ HAtoms ↔ 0 𝐴))

Theorematcv0 30129 An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms → 0 𝐴)

Theorematssch 30130 Atoms are a subset of the Hilbert lattice. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
HAtoms ⊆ C

Theorematelch 30131 An atom is a Hilbert lattice element. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms → 𝐴C )

Theorematne0 30132 An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)
(𝐴 ∈ HAtoms → 𝐴 ≠ 0)

Theorematss 30133 A lattice element smaller than an atom is either the atom or zero. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (𝐴𝐵 → (𝐴 = 𝐵𝐴 = 0)))

Theorematsseq 30134 Two atoms in a subset relationship are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (𝐴𝐵𝐴 = 𝐵))

Theorematcveq0 30135 A Hilbert lattice element covered by an atom must be the zero subspace. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (𝐴 𝐵𝐴 = 0))

Theoremh1da 30136 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (⊥‘(⊥‘{𝐴})) ∈ HAtoms)

Theoremspansna 30137 The span of the singleton of a vector is an atom. (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (span‘{𝐴}) ∈ HAtoms)

Theoremsh1dle 30138 A 1-dimensional subspace is less than or equal to any subspace containing its generating vector. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
((𝐴S𝐵𝐴) → (⊥‘(⊥‘{𝐵})) ⊆ 𝐴)

Theoremch1dle 30139 A 1-dimensional subspace is less than or equal to any member of C containing its generating vector. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)
((𝐴C𝐵𝐴) → (⊥‘(⊥‘{𝐵})) ⊆ 𝐴)

Theorematom1d 30140* The 1-dimensional subspaces of Hilbert space are its atoms. Part of Remark 10.3.5 of [BeltramettiCassinelli] p. 107. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
(𝐴 ∈ HAtoms ↔ ∃𝑥 ∈ ℋ (𝑥 ≠ 0𝐴 = (span‘{𝑥})))

19.8.3  Superposition principle

Theoremsuperpos 30141* Superposition Principle. If 𝐴 and 𝐵 are distinct atoms, there exists a third atom, distinct from 𝐴 and 𝐵, that is the superposition of 𝐴 and 𝐵. Definition 3.4-3(a) in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ∧ 𝐴𝐵) → ∃𝑥 ∈ HAtoms (𝑥𝐴𝑥𝐵𝑥 ⊆ (𝐴 𝐵)))

19.8.4  Atoms, exchange and covering properties, atomicity

Theoremchcv1 30142 The Hilbert lattice has the covering property. Proposition 1(ii) of [Kalmbach] p. 140 (and its converse). (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (¬ 𝐵𝐴𝐴 (𝐴 𝐵)))

Theoremchcv2 30143 The Hilbert lattice has the covering property. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (𝐴 ⊊ (𝐴 𝐵) ↔ 𝐴 (𝐴 𝐵)))

Theoremchjatom 30144 The join of a closed subspace and an atom equals their subspace sum. Special case of remark in [Kalmbach] p. 65, stating that if 𝐴 or 𝐵 is finite-dimensional, then this equality holds. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (𝐴 + 𝐵) = (𝐴 𝐵))

Theoremshatomici 30145* The lattice of Hilbert subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
𝐴S       (𝐴 ≠ 0 → ∃𝑥 ∈ HAtoms 𝑥𝐴)

Theoremhatomici 30146* The Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
𝐴C       (𝐴 ≠ 0 → ∃𝑥 ∈ HAtoms 𝑥𝐴)

Theoremhatomic 30147* A Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in [Kalmbach] p. 140. Also Definition 3.4-2 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐴 ≠ 0) → ∃𝑥 ∈ HAtoms 𝑥𝐴)

Theoremshatomistici 30148* The lattice of Hilbert subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)
𝐴S       𝐴 = (span‘ {𝑥 ∈ HAtoms ∣ 𝑥𝐴})

Theoremhatomistici 30149* C is atomistic, i.e. any element is the supremum of its atoms. Remark in [Kalmbach] p. 140. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
𝐴C       𝐴 = ( ‘{𝑥 ∈ HAtoms ∣ 𝑥𝐴})

Theoremchpssati 30150* Two Hilbert lattice elements in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 → ∃𝑥 ∈ HAtoms (𝑥𝐵 ∧ ¬ 𝑥𝐴))

Theoremchrelati 30151* The Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 → ∃𝑥 ∈ HAtoms (𝐴 ⊊ (𝐴 𝑥) ∧ (𝐴 𝑥) ⊆ 𝐵))

Theoremchrelat2i 30152* A consequence of relative atomicity. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥𝐴 ∧ ¬ 𝑥𝐵))

Theoremcvati 30153* If a Hilbert lattice element covers another, it equals the other joined with some atom. This is a consequence of the relative atomicity of Hilbert space. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵 → ∃𝑥 ∈ HAtoms (𝐴 𝑥) = 𝐵)

Theoremcvbr4i 30154* An alternate way to express the covering property. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ∃𝑥 ∈ HAtoms (𝐴 𝑥) = 𝐵))

Theoremcvexchlem 30155 Lemma for cvexchi 30156. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) ⋖ 𝐵𝐴 (𝐴 𝐵))

Theoremcvexchi 30156 The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) ⋖ 𝐵𝐴 (𝐴 𝐵))

Theoremchrelat2 30157* A consequence of relative atomicity. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (¬ 𝐴𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥𝐴 ∧ ¬ 𝑥𝐵)))

Theoremchrelat3 30158* A consequence of relative atomicity. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥𝐴𝑥𝐵)))

Theoremchrelat3i 30159* A consequence of the relative atomicity of Hilbert space: the ordering of Hilbert lattice elements is completely determined by the atoms they majorize. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥𝐴𝑥𝐵))

Theoremchrelat4i 30160* A consequence of relative atomicity. Extensionality principle: two lattice elements are equal iff they majorize the same atoms. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 = 𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥𝐴𝑥𝐵))

Theoremcvexch 30161 The Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → ((𝐴𝐵) ⋖ 𝐵𝐴 (𝐴 𝐵)))

Theoremcvp 30162 The Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → ((𝐴𝐵) = 0𝐴 (𝐴 𝐵)))

Theorematnssm0 30163 The meet of a Hilbert lattice element and an incomparable atom is the zero subspace. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → (¬ 𝐵𝐴 ↔ (𝐴𝐵) = 0))

Theorematnemeq0 30164 The meet of distinct atoms is the zero subspace. (Contributed by NM, 25-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (𝐴𝐵 ↔ (𝐴𝐵) = 0))

Theorematssma 30165 The meet with an atom's superset is the atom. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵C ) → (𝐴𝐵 ↔ (𝐴𝐵) ∈ HAtoms))

Theorematcv0eq 30166 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (0 (𝐴 𝐵) ↔ 𝐴 = 𝐵))

Theorematcv1 30167 Two atoms covering the zero subspace are equal. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
(((𝐴C𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐴 (𝐵 𝐶)) → (𝐴 = 0𝐵 = 𝐶))

Theorematexch 30168 The Hilbert lattice satisfies the atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem related to vector analysis was originally proved by Hermann Grassmann in 1862. Also Definition 3.4-3(b) in [MegPav2000] p. 2345 (PDF p. 8) (use atnemeq0 30164 to obtain atom inequality). (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐵 ⊆ (𝐴 𝐶) ∧ (𝐴𝐵) = 0) → 𝐶 ⊆ (𝐴 𝐵)))

Theorematomli 30169 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition 3.2.17 of [PtakPulmannova] p. 66. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)
𝐴C       (𝐵 ∈ HAtoms → ((𝐴 𝐵) ∩ (⊥‘𝐴)) ∈ (HAtoms ∪ {0}))

Theorematoml2i 30170 An assertion holding in atomic orthomodular lattices that is equivalent to the exchange axiom. Proposition P8(ii) of [BeltramettiCassinelli1] p. 400. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
𝐴C       ((𝐵 ∈ HAtoms ∧ ¬ 𝐵𝐴) → ((𝐴 𝐵) ∩ (⊥‘𝐴)) ∈ HAtoms)

Theorematordi 30171 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
𝐴C       ((𝐵 ∈ HAtoms ∧ 𝐴 𝐶 𝐵) → (𝐵𝐴𝐵 ⊆ (⊥‘𝐴)))

Theorematcvatlem 30172 Lemma for atcvati 30173. (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
𝐴C       (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (𝐴 ≠ 0𝐴 ⊊ (𝐵 𝐶))) → (¬ 𝐵𝐴𝐴 ∈ HAtoms))

Theorematcvati 30173 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (Contributed by NM, 28-Jun-2004.) (New usage is discouraged.)
𝐴C       ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0𝐴 ⊊ (𝐵 𝐶)) → 𝐴 ∈ HAtoms))

Theorematcvat2i 30174 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
𝐴C       ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((¬ 𝐵 = 𝐶𝐴 (𝐵 𝐶)) → 𝐴 ∈ HAtoms))

Theorematord 30175 An ordering law for a Hilbert lattice atom and a commuting subspace. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms ∧ 𝐴 𝐶 𝐵) → (𝐵𝐴𝐵 ⊆ (⊥‘𝐴)))

Theorematcvat2 30176 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (Contributed by NM, 29-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((¬ 𝐵 = 𝐶𝐴 (𝐵 𝐶)) → 𝐴 ∈ HAtoms))

19.8.5  Irreducibility

Theoremchirredlem1 30177* Lemma for chirredi 30181. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
𝐴C       (((𝑝 ∈ HAtoms ∧ (𝑞C𝑞 ⊆ (⊥‘𝐴))) ∧ ((𝑟 ∈ HAtoms ∧ 𝑟𝐴) ∧ 𝑟 ⊆ (𝑝 𝑞))) → (𝑝 ∩ (⊥‘𝑟)) = 0)

Theoremchirredlem2 30178* Lemma for chirredi 30181. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C       ((((𝑝 ∈ HAtoms ∧ 𝑝𝐴) ∧ (𝑞C𝑞 ⊆ (⊥‘𝐴))) ∧ ((𝑟 ∈ HAtoms ∧ 𝑟𝐴) ∧ 𝑟 ⊆ (𝑝 𝑞))) → ((⊥‘𝑟) ∩ (𝑝 𝑞)) = 𝑞)

Theoremchirredlem3 30179* Lemma for chirredi 30181. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C    &   (𝑥C𝐴 𝐶 𝑥)       ((((𝑝 ∈ HAtoms ∧ 𝑝𝐴) ∧ (𝑞 ∈ HAtoms ∧ 𝑞 ⊆ (⊥‘𝐴))) ∧ (𝑟 ∈ HAtoms ∧ 𝑟 ⊆ (𝑝 𝑞))) → (𝑟𝐴𝑟 = 𝑝))

Theoremchirredlem4 30180* Lemma for chirredi 30181. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C    &   (𝑥C𝐴 𝐶 𝑥)       ((((𝑝 ∈ HAtoms ∧ 𝑝𝐴) ∧ (𝑞 ∈ HAtoms ∧ 𝑞 ⊆ (⊥‘𝐴))) ∧ (𝑟 ∈ HAtoms ∧ 𝑟 ⊆ (𝑝 𝑞))) → (𝑟 = 𝑝𝑟 = 𝑞))

Theoremchirredi 30181* The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
𝐴C    &   (𝑥C𝐴 𝐶 𝑥)       (𝐴 = 0𝐴 = ℋ)

Theoremchirred 30182* The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166. (Contributed by NM, 16-Jun-2006.) (New usage is discouraged.)
((𝐴C ∧ ∀𝑥C 𝐴 𝐶 𝑥) → (𝐴 = 0𝐴 = ℋ))

19.8.6  Atoms (cont.)

Theorematcvat3i 30183 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
𝐴C       ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶𝐴) ∧ 𝐵 ⊆ (𝐴 𝐶)) → (𝐴 ∩ (𝐵 𝐶)) ∈ HAtoms))

Theorematcvat4i 30184* A condition implying existence of an atom with the properties shown. Lemma 3.2.20 of [PtakPulmannova] p. 68. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
𝐴C       ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0𝐵 ⊆ (𝐴 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥𝐴𝐵 ⊆ (𝐶 𝑥))))

Theorematdmd 30185 Two Hilbert lattice elements have the dual modular pair property if the first is an atom. Theorem 7.6(c) of [MaedaMaeda] p. 31. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵C ) → 𝐴 𝑀* 𝐵)

Theorematmd 30186 Two Hilbert lattice elements have the modular pair property if the first is an atom. Theorem 7.6(b) of [MaedaMaeda] p. 31. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ HAtoms ∧ 𝐵C ) → 𝐴 𝑀 𝐵)

Theorematmd2 30187 Two Hilbert lattice elements have the dual modular pair property if the second is an atom. Part of Exercise 6 of [Kalmbach] p. 103. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → 𝐴 𝑀 𝐵)

Theorematabsi 30188 Absorption of an incomparable atom. Similar to Exercise 7.1 of [MaedaMaeda] p. 34. (Contributed by NM, 15-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐶 ∈ HAtoms → (¬ 𝐶 ⊆ (𝐴 𝐵) → ((𝐴 𝐶) ∩ 𝐵) = (𝐴𝐵)))

Theorematabs2i 30189 Absorption of an incomparable atom. (Contributed by NM, 18-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐶 ∈ HAtoms → (¬ 𝐶 ⊆ (𝐴 𝐵) → ((𝐴 𝐶) ∩ (𝐴 𝐵)) = 𝐴))

19.8.7  Modular symmetry

Theoremmdsymlem1 30190* Lemma for mdsymi 30198. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶 = (𝐴 𝑝)       (((𝑝C ∧ (𝐵𝐶) ⊆ 𝐴) ∧ (𝐵 𝑀* 𝐴𝑝 ⊆ (𝐴 𝐵))) → 𝑝𝐴)

Theoremmdsymlem2 30191* Lemma for mdsymi 30198. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶 = (𝐴 𝑝)       (((𝑝 ∈ HAtoms ∧ (𝐵𝐶) ⊆ 𝐴) ∧ (𝐵 𝑀* 𝐴𝑝 ⊆ (𝐴 𝐵))) → (𝐵 ≠ 0 → ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑞 𝑟) ∧ (𝑞𝐴𝑟𝐵))))

Theoremmdsymlem3 30192* Lemma for mdsymi 30198. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶 = (𝐴 𝑝)       ((((𝑝 ∈ HAtoms ∧ ¬ (𝐵𝐶) ⊆ 𝐴) ∧ 𝑝 ⊆ (𝐴 𝐵)) ∧ 𝐴 ≠ 0) → ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑞 𝑟) ∧ (𝑞𝐴𝑟𝐵)))

Theoremmdsymlem4 30193* Lemma for mdsymi 30198. This is the forward direction of Lemma 4(i) of [Maeda] p. 168. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶 = (𝐴 𝑝)       (𝑝 ∈ HAtoms → ((𝐵 𝑀* 𝐴 ∧ ((𝐴 ≠ 0𝐵 ≠ 0) ∧ 𝑝 ⊆ (𝐴 𝐵))) → ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 𝑟) ∧ (𝑞𝐴𝑟𝐵))))

Theoremmdsymlem5 30194* Lemma for mdsymi 30198. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶 = (𝐴 𝑝)       ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → (¬ 𝑞 = 𝑝 → ((𝑝 ⊆ (𝑞 𝑟) ∧ (𝑞𝐴𝑟𝐵)) → (((𝑐C𝐴𝑐) ∧ 𝑝 ∈ HAtoms) → (𝑝𝑐𝑝 ⊆ ((𝑐𝐵) ∨ 𝐴))))))

Theoremmdsymlem6 30195* Lemma for mdsymi 30198. This is the converse direction of Lemma 4(i) of [Maeda] p. 168, and is based on the proof of Theorem 1(d) to (e) of [Maeda] p. 167. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶 = (𝐴 𝑝)       (∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐴 𝐵) → ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 𝑟) ∧ (𝑞𝐴𝑟𝐵))) → 𝐵 𝑀* 𝐴)

Theoremmdsymlem7 30196* Lemma for mdsymi 30198. Lemma 4(i) of [Maeda] p. 168. Note that Maeda's 1965 definition of dual modular pair has reversed arguments compared to the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, which is the one that we use. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶 = (𝐴 𝑝)       ((𝐴 ≠ 0𝐵 ≠ 0) → (𝐵 𝑀* 𝐴 ↔ ∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐴 𝐵) → ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 𝑟) ∧ (𝑞𝐴𝑟𝐵)))))

Theoremmdsymlem8 30197* Lemma for mdsymi 30198. Lemma 4(ii) of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶 = (𝐴 𝑝)       ((𝐴 ≠ 0𝐵 ≠ 0) → (𝐵 𝑀* 𝐴𝐴 𝑀* 𝐵))

Theoremmdsymi 30198 M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀 𝐵𝐵 𝑀 𝐴)

Theoremmdsym 30199 M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀 𝐵𝐵 𝑀 𝐴))

Theoremdmdsym 30200 Dual M-symmetry of the Hilbert lattice. (Contributed by NM, 25-Jul-2007.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵𝐵 𝑀* 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 42801-42900 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 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45326
 Copyright terms: Public domain < Previous  Next >