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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcvati 30301* 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 30302* 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 30303 Lemma for cvexchi 30304. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
𝐴C    &   𝐵C       ((𝐴𝐵) ⋖ 𝐵𝐴 (𝐴 𝐵))
 
Theoremcvexchi 30304 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 30305* A consequence of relative atomicity. (Contributed by NM, 1-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (¬ 𝐴𝐵 ↔ ∃𝑥 ∈ HAtoms (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
 
Theoremchrelat3 30306* A consequence of relative atomicity. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥𝐴𝑥𝐵)))
 
Theoremchrelat3i 30307* 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 30308* 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 30309 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 30310 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 30311 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 30312 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 30313 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 30314 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 30315 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 30316 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 30312 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 30317 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 30318 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 30319 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 30320 Lemma for atcvati 30321. (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
𝐴C       (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (𝐴 ≠ 0𝐴 ⊊ (𝐵 𝐶))) → (¬ 𝐵𝐴𝐴 ∈ HAtoms))
 
Theorematcvati 30321 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 30322 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 30323 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 30324 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 30325* Lemma for chirredi 30329. (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 30326* Lemma for chirredi 30329. (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 30327* Lemma for chirredi 30329. (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 30328* Lemma for chirredi 30329. (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 30329* 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 30330* 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 30331 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 30332* 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 30333 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 30334 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 30335 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 30336 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 30337 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 30338* Lemma for mdsymi 30346. (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 30339* Lemma for mdsymi 30346. (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 30340* Lemma for mdsymi 30346. (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 30341* Lemma for mdsymi 30346. 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 30342* Lemma for mdsymi 30346. (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 30343* Lemma for mdsymi 30346. 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 30344* Lemma for mdsymi 30346. 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 30345* Lemma for mdsymi 30346. 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 30346 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 30347 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 30348 Dual M-symmetry of the Hilbert lattice. (Contributed by NM, 25-Jul-2007.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵𝐵 𝑀* 𝐴))
 
Theorematdmd2 30349 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 30350 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 30351 Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵𝐴 𝑀 𝐵)
 
Theoremcmdmdi 30352 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 30353 Lemma for sumdmdi 30355. 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 30354* Lemma for sumdmdi 30355. (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 30355 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 30356* 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 30357* 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 30358* 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 30359* 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 30360 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       𝐴 𝑀 (⊥‘𝐴)
 
Theoremmdoc2i 30361 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       (⊥‘𝐴) 𝑀 𝐴
 
Theoremdmdoc1i 30362 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       𝐴 𝑀* (⊥‘𝐴)
 
Theoremdmdoc2i 30363 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       (⊥‘𝐴) 𝑀* 𝐴
 
Theoremmdcompli 30364 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 30365 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 30366* 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 30367* 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 30368* 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 30369* 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 30370* Lemma for cdj3i 30376. 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 30371* Lemma for cdj3i 30376. 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 30372* Lemma for cdj3i 30376. 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 30373* Lemma for cdj3i 30376. 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 30374* Lemma for cdj3i 30376. 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 30375* Lemma for cdj3i 30376. 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 30376* 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 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
 
20.1  Mathboxes for user contributions
 
20.1.1  Mathbox guidelines
 
Theoremmathbox 30377 (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 28337 for our general style guidelines. For contributing via GitHub, see https://github.com/metamath/set.mm/blob/develop/CONTRIBUTING.md 28337. 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 nullary class constants for new definitions, for example as in df-div 11376.

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 "MM> WRITE SOURCE set.mm / REWRAP" command will take care of 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.

6. If a contributor is no longer active, we will continue the usual maintenance edits. As time goes on, often theorems will be moved to main or removed in favor of similar replacements. But we are also willing to maintain mathboxes in place, as work by others from years ago may form the foundation of future work; you could even argue that all of mathematics is like that.

7. For theorems of importance (for example, a Metamath 100 theorem or a dependency of one), we prefer to eventually move them out of mathboxes (although a mathbox is perfectly appropriate as proofs are being developed and refined). (Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

𝜑       𝜑
 
20.2  Mathbox for Stefan Allan
 
Theoremsa-abvi 30378 A theorem about the universal class. Inference associated with bj-abv 34723 (which is proved from fewer axioms). (Contributed by Stefan Allan, 9-Dec-2008.)
𝜑       V = {𝑥𝜑}
 
Theoremxfree 30379 A partial converse to 19.9t 2206. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)
(∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑𝜑))
 
Theoremxfree2 30380 A partial converse to 19.9t 2206. (Contributed by Stefan Allan, 21-Dec-2008.)
(∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
 
TheoremaddltmulALT 30381 A proof readability experiment for addltmul 11952. (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))
 
20.3  Mathbox for Thierry Arnoux
 
20.3.1  Propositional Calculus - misc additions
 
Theorembian1d 30382 Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       (𝜑 → ((𝜒𝜓) ↔ (𝜒𝜃)))
 
Theoremor3di 30383 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
((𝜑 ∨ (𝜓𝜒𝜏)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜏)))
 
Theoremor3dir 30384 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
(((𝜑𝜓𝜒) ∨ 𝜏) ↔ ((𝜑𝜏) ∧ (𝜓𝜏) ∧ (𝜒𝜏)))
 
Theorem3o1cs 30385 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theorem3o2cs 30386 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜓𝜃)
 
Theorem3o3cs 30387 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜒𝜃)
 
Theorembiadanid 30388 Deduction associated with biadani 820. Add a conjunction to an equivalence. (Contributed by Thierry Arnoux, 16-Jun-2024.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → (𝜓𝜃))       (𝜑 → (𝜓 ↔ (𝜒𝜃)))
 
20.3.2  Predicate Calculus
 
20.3.2.1  Predicate Calculus - misc additions
 
Theoremsbc2iedf 30389* Conversion of implicit substitution to explicit class substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝜒    &   𝑦𝜒    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
 
Theoremrspc2daf 30390* Double restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝜒    &   𝑦𝜒    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝑉𝑦𝑊 𝜓)       (𝜑𝜒)
 
TheoremnelbOLD 30391* Obsolete version of nelb 3178 as of 23-Jan-2024. (Contributed by Thierry Arnoux, 20-Nov-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)
 
20.3.2.2  Restricted quantification - misc additions
 
Theoremralcom4f 30392* 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)
 
Theoremrexcom4f 30393* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
𝑦𝐴       (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
 
Theorem19.9d2rf 30394 A deduction version of one direction of 19.9 2207 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜓)
 
Theorem19.9d2r 30395* A deduction version of one direction of 19.9 2207 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜓)
 
Theoremr19.29ffa 30396* A commonly used pattern based on r19.29 3167, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)       ((𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → 𝜒)
 
20.3.2.3  Equality
 
Theoremeqtrb 30397 A transposition of equality. (Contributed by Thierry Arnoux, 20-Aug-2023.)
((𝐴 = 𝐵𝐴 = 𝐶) ↔ (𝐴 = 𝐵𝐵 = 𝐶))
 
20.3.2.4  Double restricted existential uniqueness quantification
 
Theoremopsbc2ie 30398* Conversion of implicit substitution to explicit class substitution for ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))       (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜒))
 
Theoremopreu2reuALT 30399* Correspondence between uniqueness of ordered pairs and double restricted existential uniqueness quantification. Alternate proof of one direction only, use opreu2reurex 6126 instead. (Contributed by Thierry Arnoux, 4-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))       ((∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒) → ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑)
 
20.3.2.5  Double restricted existential uniqueness quantification syntax
 
Syntaxw2reu 30400 Syntax for double restricted existential uniqueness quantification.
wff ∃!𝑥𝐴 , 𝑦𝐵𝜑
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-45962
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