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Theorem List for Metamath Proof Explorer - 29801-29900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorematdmd 29801 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 29802 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 29803 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 29804 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 29805 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 29806* Lemma for mdsymi 29814. (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 29807* Lemma for mdsymi 29814. (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 29808* Lemma for mdsymi 29814. (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 29809* Lemma for mdsymi 29814. 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 29810* Lemma for mdsymi 29814. (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 29811* Lemma for mdsymi 29814. 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 29812* Lemma for mdsymi 29814. 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 29813* Lemma for mdsymi 29814. 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 29814 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 29815 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 29816 Dual M-symmetry of the Hilbert lattice. (Contributed by NM, 25-Jul-2007.) (New usage is discouraged.)
((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵𝐵 𝑀* 𝐴))

Theorematdmd2 29817 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 29818 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 29819 Commuting subspaces form a modular pair. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)
𝐴C    &   𝐵C       (𝐴 𝐶 𝐵𝐴 𝑀 𝐵)

Theoremcmdmdi 29820 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 29821 Lemma for sumdmdi 29823. 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 29822* Lemma for sumdmdi 29823. (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 29823 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 29824* 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 29825* 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 29826* 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 29827* 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 29828 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       𝐴 𝑀 (⊥‘𝐴)

Theoremmdoc2i 29829 Orthocomplements form a modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       (⊥‘𝐴) 𝑀 𝐴

Theoremdmdoc1i 29830 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       𝐴 𝑀* (⊥‘𝐴)

Theoremdmdoc2i 29831 Orthocomplements form a dual modular pair. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
𝐴C       (⊥‘𝐴) 𝑀* 𝐴

Theoremmdcompli 29832 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 29833 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 29834* 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 29835* 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 29836* 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 29837* 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 29838* Lemma for cdj3i 29844. 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 29839* Lemma for cdj3i 29844. 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 29840* Lemma for cdj3i 29844. 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 29841* Lemma for cdj3i 29844. 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 29842* Lemma for cdj3i 29844. 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 29843* Lemma for cdj3i 29844. 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 29844* 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 (USER'S MATHBOXES)

20.1  Mathboxes for user contributions

20.1.1  Mathbox guidelines

Theoremmathbox 29845 (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 27804 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 nullary class constants for new definitions, for example as in df-div 11010.

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. (Contributed by NM, 20-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

𝜑       𝜑

20.2  Mathbox for Stefan Allan

Theoremsa-abvi 29846 A theorem about the universal class. Inference associated with bj-abv 33415 (which is proved from fewer axioms). (Contributed by Stefan Allan, 9-Dec-2008.)
𝜑       V = {𝑥𝜑}

Theoremxfree 29847 A partial converse to 19.9t 2246. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)
(∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑𝜑))

Theoremxfree2 29848 A partial converse to 19.9t 2246. (Contributed by Stefan Allan, 21-Dec-2008.)
(∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

TheoremaddltmulALT 29849 A proof readability experiment for addltmul 11594. (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 29850 Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       (𝜑 → ((𝜒𝜓) ↔ (𝜒𝜃)))

Theoremor3di 29851 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
((𝜑 ∨ (𝜓𝜒𝜏)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜏)))

Theoremor3dir 29852 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
(((𝜑𝜓𝜒) ∨ 𝜏) ↔ ((𝜑𝜏) ∧ (𝜓𝜏) ∧ (𝜒𝜏)))

Theorem3o1cs 29853 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theorem3o2cs 29854 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜓𝜃)

Theorem3o3cs 29855 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜒𝜃)

20.3.2  Predicate Calculus

20.3.2.1  Predicate Calculus - misc additions

Theoremspc2ed 29856* 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 29857* Specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
𝑥𝜒    &   𝑦𝜒    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (∀𝑥𝑦𝜓𝜒))

Theoremvtocl2d 29858* Implicit substitution of two classes for two setvar variables. (Contributed by Thierry Arnoux, 25-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒))    &   (𝜑𝜓)       (𝜑𝜒)

Theoremeqri 29859 Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)
𝑥𝐴    &   𝑥𝐵    &   (𝑥𝐴𝑥𝐵)       𝐴 = 𝐵

20.3.2.2  Restricted quantification - misc additions

Theoremralcom4f 29860* 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 29861* 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 29862 A deduction version of one direction of 19.9 2247 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜓)

Theorem19.9d2r 29863* A deduction version of one direction of 19.9 2247 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜓)

Theoremr19.29ffa 29864* A commonly used pattern based on r19.29 3282, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)       ((𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → 𝜒)

20.3.2.3  Substitution (without distinct variables) - misc additions

Theoremsbceqbidf 29865 Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))

Theoremsbcies 29866* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝐴 = (𝐸𝑊)    &   (𝑎 = 𝐴 → (𝜑𝜓))       (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎]𝜓𝜑))

20.3.2.4  Existential "at most one" - misc additions

Theoremmoel 29867* "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.)
(∃*𝑥 𝑥𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥 = 𝑦)

Theoremmo5f 29868* Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)
𝑖𝜑    &   𝑗𝜑       (∃*𝑥𝜑 ↔ ∀𝑖𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))

Theoremnmo 29869* Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)
𝑦𝜑       (¬ ∃*𝑥𝜑 ↔ ∀𝑦𝑥(𝜑𝑥𝑦))

20.3.2.5  Existential uniqueness - misc additions

Theorem2reuswap2 29870* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.)
(∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))

Theoremreuxfr3d 29871* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr2d 5119. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)       (𝜑 → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐶 𝜓))

Theoremreuxfr4d 29872* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfrd 5121. (Contributed by Thierry Arnoux, 7-Apr-2017.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))

Theoremrexunirn 29873* Restricted existential quantification over the union of the range of a function. Cf. rexrn 6610 and eluni2 4662. (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑥𝐴𝐵𝑉)       (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦 ran 𝐹𝜑)

20.3.2.6  Restricted "at most one" - misc additions

Theoremrmoxfrd 29874* Transfer "at most one" restricted quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐵 𝜓 ↔ ∃*𝑦𝐶 𝜒))

Theoremssrmo 29875 "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 → (∃*𝑥𝐵 𝜑 → ∃*𝑥𝐴 𝜑))

20.3.3  General Set Theory

20.3.3.1  Class abstractions (a.k.a. class builders)

Theoremdifrab2 29876 Difference of two restricted class abstractions. Compare with difrab 4130. (Contributed by Thierry Arnoux, 3-Jan-2022.)
({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

TheoremrabexgfGS 29877 Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)
𝑥𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theoremrabsnel 29878* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.)
𝐵 ∈ V       ({𝑥𝐴𝜑} = {𝐵} → 𝐵𝐴)

Theoremrabeqsnd 29879* Conditions for a restricted class abstraction to be a singleton, in deduction form. (Contributed by Thierry Arnoux, 2-Dec-2021.)
(𝑥 = 𝐵 → (𝜓𝜒))    &   (𝜑𝐵𝐴)    &   (𝜑𝜒)    &   (((𝜑𝑥𝐴) ∧ 𝜓) → 𝑥 = 𝐵)       (𝜑 → {𝑥𝐴𝜓} = {𝐵})

Theoremforesf1o 29880* From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)

Theoremrabfodom 29881* Domination relation for restricted abstract class builders, based on a surjective function. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝜑𝑥𝐴𝑦 = (𝐹𝑥)) → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴onto𝐵)       (𝜑 → {𝑦𝐵𝜒} ≼ {𝑥𝐴𝜓})

20.3.3.2  Image Sets

Theoremabrexdomjm 29882* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑦𝐴 → ∃*𝑥𝜑)       (𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝜑} ≼ 𝐴)

Theoremabrexdom2jm 29883* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ≼ 𝐴)

Theoremabrexexd 29884* Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)
𝑥𝐴    &   (𝜑𝐴 ∈ V)       (𝜑 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)

Theoremelabreximd 29885* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑥𝜑    &   𝑥𝜒    &   (𝐴 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐶) → 𝜓)       ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)

Theoremelabreximdv 29886* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
(𝐴 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐶) → 𝜓)       ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)

Theoremabrexss 29887* A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)
𝑥𝐶       (∀𝑥𝐴 𝐵𝐶 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶)

20.3.3.3  Set relations and operations - misc additions

Theoremrabss3d 29888* Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})

Theoreminin 29889 Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Theoreminindif 29890 See inundif 4269. (Contributed by Thierry Arnoux, 13-Sep-2017.)
((𝐴𝐶) ∩ (𝐴𝐶)) = ∅

Theoremdifininv 29891 Condition for the intersections of two sets with a given set to be equal. (Contributed by Thierry Arnoux, 28-Dec-2021.)
((((𝐴𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶𝐴) ∩ 𝐵) = ∅) → (𝐴𝐵) = (𝐶𝐵))

Theoremdifeq 29892 Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
((𝐴𝐵) = 𝐶 ↔ ((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)))

Theoremindifundif 29893 A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
(((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))

Theoremelpwincl1 29894 Closure of intersection with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐴 ∈ 𝒫 𝐶)       (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)

Theoremelpwdifcl 29895 Closure of class difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐴 ∈ 𝒫 𝐶)       (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)

Theoremelpwiuncl 29896* Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ 𝒫 𝐶)       (𝜑 𝑘𝐴 𝐵 ∈ 𝒫 𝐶)

20.3.3.4  Unordered pairs

Theoremelpreq 29897 Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
(𝜑𝑋 ∈ {𝐴, 𝐵})    &   (𝜑𝑌 ∈ {𝐴, 𝐵})    &   (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))       (𝜑𝑋 = 𝑌)

20.3.3.5  Conditional operator - misc additions

Theoremifeqeqx 29898* An equality theorem tailored for ballotlemsf1o 31110. (Contributed by Thierry Arnoux, 14-Apr-2017.)
(𝑥 = 𝑋𝐴 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝑎)    &   (𝑥 = 𝑋 → (𝜒𝜃))    &   (𝑥 = 𝑌 → (𝜒𝜓))    &   (𝜑𝑎 = 𝐶)    &   ((𝜑𝜓) → 𝜃)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑊)       ((𝜑𝑥 = if(𝜓, 𝑋, 𝑌)) → 𝑎 = if(𝜒, 𝐴, 𝐵))

Theoremelimifd 29899 Elimination of a conditional operator contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒𝜃)))    &   (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒𝜏)))       (𝜑 → (𝜒 ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))

Theoremelim2if 29900 Elimination of two conditional operators contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))       (𝜒 ↔ ((𝜑𝜃) ∨ (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))))

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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 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