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

Theoremshatomici 28601* 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 28602* 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 28603* 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 28604* 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 28605* 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 28606* 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 28607* 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 28608* A consequence of relative atomicity. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       𝐴𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥𝐴 ∧ ¬ 𝑥𝐵))

Theoremcvati 28609* 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 28610* 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 28611 Lemma for cvexchi 28612. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) ⋖ 𝐵𝐴 (𝐴 𝐵))

Theoremcvexchi 28612 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 28613* A consequence of relative atomicity. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (¬ 𝐴𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥𝐴 ∧ ¬ 𝑥𝐵)))

Theoremchrelat3 28614* A consequence of relative atomicity. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥𝐴𝑥𝐵)))

Theoremchrelat3i 28615* 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 28616* 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 28617 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 28618 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 28619 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 28620 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 28621 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 28622 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 28623 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 28624 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 28620 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 28625 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 28626 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 28627 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 28628 Lemma for atcvati 28629. (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
𝐴C       (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (𝐴 ≠ 0𝐴 ⊊ (𝐵 𝐶))) → (¬ 𝐵𝐴𝐴 ∈ HAtoms))

Theorematcvati 28629 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 28630 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 28631 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 28632 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))

20.8.5  Irreducibility

Theoremchirredlem1 28633* Lemma for chirredi 28637. (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 28634* Lemma for chirredi 28637. (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 28635* Lemma for chirredi 28637. (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 28636* Lemma for chirredi 28637. (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 28637* 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 28638* 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))

20.8.6  Atoms (cont.)

Theorematcvat3i 28639 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 28640* 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 28641 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 28642 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 28643 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 28644 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 28645 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))

20.8.7  Modular symmetry

Theoremmdsymlem1 28646* Lemma for mdsymi 28654. (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 28647* Lemma for mdsymi 28654. (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 28648* Lemma for mdsymi 28654. (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 28649* Lemma for mdsymi 28654. 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 28650* Lemma for mdsymi 28654. (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 28651* Lemma for mdsymi 28654. 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 28652* Lemma for mdsymi 28654. 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 28653* Lemma for mdsymi 28654. 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 28654 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 28655 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 28656 Dual M-symmetry of the Hilbert lattice. (Contributed by NM, 25-Jul-2007.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵𝐵 𝑀* 𝐴))

Theorematdmd2 28657 Two Hilbert lattice elements have the dual modular pair property if the second is an atom. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵 ∈ HAtoms) → 𝐴 𝑀* 𝐵)

Theoremsumdmdii 28658 If the subspace sum of two Hilbert lattice elements is closed, then the elements are a dual modular pair. Remark in [MaedaMaeda] p. 139. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴 + 𝐵) = (𝐴 𝐵) → 𝐴 𝑀* 𝐵)

Theoremcmmdi 28659 Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵𝐴 𝑀 𝐵)

Theoremcmdmdi 28660 Commuting subspaces form a dual modular pair. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵𝐴 𝑀* 𝐵)

Theoremsumdmdlem 28661 Lemma for sumdmdi 28663. The span of vector 𝐶 not in the subspace sum is "trimmed off." (Contributed by NM, 18-Dec-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐶 ∈ ℋ ∧ ¬ 𝐶 ∈ (𝐴 + 𝐵)) → ((𝐵 + (span‘{𝐶})) ∩ 𝐴) = (𝐵𝐴))

Theoremsumdmdlem2 28662* Lemma for sumdmdi 28663. (Contributed by NM, 23-Dec-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       (∀𝑥 ∈ HAtoms ((𝑥 𝐵) ∩ (𝐴 𝐵)) ⊆ (((𝑥 𝐵) ∩ 𝐴) ∨ 𝐵) → (𝐴 + 𝐵) = (𝐴 𝐵))

Theoremsumdmdi 28663 The subspace sum of two Hilbert lattice elements is closed iff the elements are a dual modular pair. Theorem 2 of [Holland] p. 1519. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴 + 𝐵) = (𝐴 𝐵) ↔ 𝐴 𝑀* 𝐵)

Theoremdmdbr4ati 28664* Dual modular pair property in terms of atoms. (Contributed by NM, 15-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀* 𝐵 ↔ ∀𝑥 ∈ HAtoms ((𝑥 𝐵) ∩ (𝐴 𝐵)) ⊆ (((𝑥 𝐵) ∩ 𝐴) ∨ 𝐵))

Theoremdmdbr5ati 28665* Dual modular pair property in terms of atoms. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀* 𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 𝐵) → 𝑥 ⊆ (((𝑥 𝐵) ∩ 𝐴) ∨ 𝐵)))

Theoremdmdbr6ati 28666* Dual modular pair property in terms of atoms. The modular law takes the form of the shearing identity. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀* 𝐵 ↔ ∀𝑥 ∈ HAtoms ((𝐴 𝐵) ∩ 𝑥) = ((((𝑥 𝐵) ∩ 𝐴) ∨ 𝐵) ∩ 𝑥))

Theoremdmdbr7ati 28667* Dual modular pair property in terms of atoms. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀* 𝐵 ↔ ∀𝑥 ∈ HAtoms ((𝐴 𝐵) ∩ 𝑥) ⊆ (((𝑥 𝐵) ∩ 𝐴) ∨ 𝐵))

Theoremmdoc1i 28668 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       𝐴 𝑀 (⊥‘𝐴)

Theoremmdoc2i 28669 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       (⊥‘𝐴) 𝑀 𝐴

Theoremdmdoc1i 28670 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       𝐴 𝑀* (⊥‘𝐴)

Theoremdmdoc2i 28671 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       (⊥‘𝐴) 𝑀* 𝐴

Theoremmdcompli 28672 A condition equivalent to the modular pair property. Part of proof of Theorem 1.14 of [MaedaMaeda] p. 4. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀 𝐵 ↔ (𝐴 ∩ (⊥‘(𝐴𝐵))) 𝑀 (𝐵 ∩ (⊥‘(𝐴𝐵))))

Theoremdmdcompli 28673 A condition equivalent to the dual modular pair property. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝑀* 𝐵 ↔ (𝐴 ∩ (⊥‘(𝐴𝐵))) 𝑀* (𝐵 ∩ (⊥‘(𝐴𝐵))))

Theoremmddmdin0i 28674* If dual modular implies modular whenever meet is zero, then dual modular implies modular for arbitrary lattice elements. This theorem is needed for the remark after Lemma 7 of [Holland] p. 1524 to hold. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝑥C𝑦C ((𝑥 𝑀* 𝑦 ∧ (𝑥𝑦) = 0) → 𝑥 𝑀 𝑦)       (𝐴 𝑀* 𝐵𝐴 𝑀 𝐵)

Theoremcdjreui 28675* A member of the sum of disjoint subspaces has a unique decomposition. Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 20-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S       ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → ∃!𝑥𝐴𝑦𝐵 𝐶 = (𝑥 + 𝑦))

Theoremcdj1i 28676* Two ways to express "𝐴 and 𝐵 are completely disjoint subspaces." (1) => (2) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 21-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S       (∃𝑤 ∈ ℝ (0 < 𝑤 ∧ ∀𝑦𝐴𝑣𝐵 ((norm𝑦) + (norm𝑣)) ≤ (𝑤 · (norm‘(𝑦 + 𝑣)))) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦𝐴𝑧𝐵 ((norm𝑦) = 1 → 𝑥 ≤ (norm‘(𝑦 𝑧)))))

Theoremcdj3lem1 28677* A property of "𝐴 and 𝐵 are completely disjoint subspaces." Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S       (∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦𝐴𝑧𝐵 ((norm𝑦) + (norm𝑧)) ≤ (𝑥 · (norm‘(𝑦 + 𝑧)))) → (𝐴𝐵) = 0)

Theoremcdj3lem2 28678* Lemma for cdj3i 28684. Value of the first-component function 𝑆. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))       ((𝐶𝐴𝐷𝐵 ∧ (𝐴𝐵) = 0) → (𝑆‘(𝐶 + 𝐷)) = 𝐶)

Theoremcdj3lem2a 28679* Lemma for cdj3i 28684. Closure of the first-component function 𝑆. (Contributed by NM, 25-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))       ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → (𝑆𝐶) ∈ 𝐴)

Theoremcdj3lem2b 28680* Lemma for cdj3i 28684. The first-component function 𝑆 is bounded if the subspaces are completely disjoint. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))       (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))

Theoremcdj3lem3 28681* Lemma for cdj3i 28684. Value of the second-component function 𝑇. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))       ((𝐶𝐴𝐷𝐵 ∧ (𝐴𝐵) = 0) → (𝑇‘(𝐶 + 𝐷)) = 𝐷)

Theoremcdj3lem3a 28682* Lemma for cdj3i 28684. Closure of the second-component function 𝑇. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))       ((𝐶 ∈ (𝐴 + 𝐵) ∧ (𝐴𝐵) = 0) → (𝑇𝐶) ∈ 𝐵)

Theoremcdj3lem3b 28683* Lemma for cdj3i 28684. The second-component function 𝑇 is bounded if the subspaces are completely disjoint. (Contributed by NM, 31-May-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))       (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))

Theoremcdj3i 28684* Two ways to express "𝐴 and 𝐵 are completely disjoint subspaces." (1) <=> (3) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 1-Jun-2005.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝑆 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑧𝐴𝑤𝐵 𝑥 = (𝑧 + 𝑤)))    &   𝑇 = (𝑥 ∈ (𝐴 + 𝐵) ↦ (𝑤𝐵𝑧𝐴 𝑥 = (𝑧 + 𝑤)))    &   (𝜑 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑆𝑢)) ≤ (𝑣 · (norm𝑢))))    &   (𝜓 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 + 𝐵)(norm‘(𝑇𝑢)) ≤ (𝑣 · (norm𝑢))))       (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥𝐴𝑦𝐵 ((norm𝑥) + (norm𝑦)) ≤ (𝑣 · (norm‘(𝑥 + 𝑦)))) ↔ ((𝐴𝐵) = 0𝜑𝜓))

PART 21  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)

21.1  Mathboxes for user contributions

21.1.1  Mathbox guidelines

Theoremmathbox 28685 (This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of set.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm.

Mathboxes are provided to help keep your work synchronized with changes in set.mm while allowing you to work independently without affecting other contributors. Even though in a sense your mathbox belongs to you, it is still part of the shared body of knowledge contained in set.mm, and occasionally other people may make maintenance edits to your mathbox for things like keeping it synchronized with the rest of set.mm, reducing proof lengths, moving your theorems to the main part of set.mm when needed, and fixing typos or other errors. If you want to preserve it the way you left it, you can keep a local copy or keep track of the GitHub commit number.

Guidelines:

1. See conventions 26650 for our general style guidelines. For contributing via GitHub, see https://github.com/metamath/set.mm/blob/develop/CONTRIBUTING.md. The metamath program command "verify markup *" will check that you have followed many of of the conventions we use.

2. If at all possible, please use only 0-ary class constants for new definitions, for example as in df-div 10564.

3. Each \$p and \$a statement must be immediately preceded with the comment that will be shown on its web page description. The metamath program command "write source set.mm /rewrap" will take care of our indentation conventions and line wrapping.

4. All mathbox content will be on public display and should hopefully reflect the overall quality of the website.

5. Mathboxes must be independent from one another (checked by "verify markup *"). If you need a theorem from another mathbox, typically it is moved to the main part of set.mm. New users should consult with more experienced users before doing this. (Contributed by NM, 20-Feb-2007.) (New usage is discouraged.)

𝜑       𝜑

21.2  Mathbox for Stefan Allan

Theoremfoo3 28686 A theorem about the universal class. (Contributed by Stefan Allan, 9-Dec-2008.)
𝜑       V = {𝑥𝜑}

Theoremxfree 28687 A partial converse to 19.9t 2059. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)
(∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑𝜑))

Theoremxfree2 28688 A partial converse to 19.9t 2059. (Contributed by Stefan Allan, 21-Dec-2008.)
(∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

TheoremaddltmulALT 28689 A proof readability experiment for addltmul 11145. (Contributed by Stefan Allan, 30-Oct-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (2 < 𝐴 ∧ 2 < 𝐵)) → (𝐴 + 𝐵) < (𝐴 · 𝐵))

21.3  Mathbox for Thierry Arnoux

21.3.1  Propositional Calculus - misc additions

Theorembian1d 28690 Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       (𝜑 → ((𝜒𝜓) ↔ (𝜒𝜃)))

Theoremor3di 28691 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
((𝜑 ∨ (𝜓𝜒𝜏)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜏)))

Theoremor3dir 28692 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
(((𝜑𝜓𝜒) ∨ 𝜏) ↔ ((𝜑𝜏) ∧ (𝜓𝜏) ∧ (𝜒𝜏)))

Theorem3o1cs 28693 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theorem3o2cs 28694 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜓𝜃)

Theorem3o3cs 28695 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜒𝜃)

21.3.2  Predicate Calculus

21.3.2.1  Predicate Calculus - misc additions

Theoremspc2ed 28696* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
𝑥𝜒    &   𝑦𝜒    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (𝜒 → ∃𝑥𝑦𝜓))

Theoremspc2d 28697* Specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
𝑥𝜒    &   𝑦𝜒    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (∀𝑥𝑦𝜓𝜒))

Theoremeqvincg 28698* A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)
(𝐴𝑉 → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))

Theoremvtocl2d 28699* Implicit substitution of two classes for two setvar variables. (Contributed by Thierry Arnoux, 25-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒))    &   (𝜑𝜓)       (𝜑𝜒)

21.3.2.2  Restricted quantification - misc additions

Theoremralcom4f 28700* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
𝑦𝐴       (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)

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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 42301-42360
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