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Theorem List for Metamath Proof Explorer - 15001-15100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrexpen 15001 The real numbers are equinumerous to their own Cartesian product, even though it is not necessarily true that is well-orderable (so we cannot use infxpidm2 8878 directly). (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.)
(ℝ × ℝ) ≈ ℝ

Theoremcpnnen 15002 The complex numbers are equinumerous to the powerset of the positive integers. (Contributed by Mario Carneiro, 16-Jun-2013.)
ℂ ≈ 𝒫 ℕ

TheoremrucALT 15003 Alternate proof of ruc 15016. This proof is a simple corollary of rpnnen 15000, which determines the exact cardinality of the reals. For an alternate proof discussed at mmcomplex.html#uncountable, see ruc 15016. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 13-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ℕ ≺ ℝ

Theoremruclem1 15004* Lemma for ruc 15016 (the reals are uncountable). Substitutions for the function 𝐷. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   𝑋 = (1st ‘(⟨𝐴, 𝐵𝐷𝑀))    &   𝑌 = (2nd ‘(⟨𝐴, 𝐵𝐷𝑀))       (𝜑 → ((⟨𝐴, 𝐵𝐷𝑀) ∈ (ℝ × ℝ) ∧ 𝑋 = if(((𝐴 + 𝐵) / 2) < 𝑀, 𝐴, ((((𝐴 + 𝐵) / 2) + 𝐵) / 2)) ∧ 𝑌 = if(((𝐴 + 𝐵) / 2) < 𝑀, ((𝐴 + 𝐵) / 2), 𝐵)))

Theoremruclem2 15005* Lemma for ruc 15016. Ordering property for the input to 𝐷. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   𝑋 = (1st ‘(⟨𝐴, 𝐵𝐷𝑀))    &   𝑌 = (2nd ‘(⟨𝐴, 𝐵𝐷𝑀))    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴𝑋𝑋 < 𝑌𝑌𝐵))

Theoremruclem3 15006* Lemma for ruc 15016. The constructed interval [𝑋, 𝑌] always excludes 𝑀. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑀 ∈ ℝ)    &   𝑋 = (1st ‘(⟨𝐴, 𝐵𝐷𝑀))    &   𝑌 = (2nd ‘(⟨𝐴, 𝐵𝐷𝑀))    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝑀 < 𝑋𝑌 < 𝑀))

Theoremruclem4 15007* Lemma for ruc 15016. Initial value of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)       (𝜑 → (𝐺‘0) = ⟨0, 1⟩)

Theoremruclem6 15008* Lemma for ruc 15016. Domain and range of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)       (𝜑𝐺:ℕ0⟶(ℝ × ℝ))

Theoremruclem7 15009* Lemma for ruc 15016. Successor value for the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)       ((𝜑𝑁 ∈ ℕ0) → (𝐺‘(𝑁 + 1)) = ((𝐺𝑁)𝐷(𝐹‘(𝑁 + 1))))

Theoremruclem8 15010* Lemma for ruc 15016. The intervals of the 𝐺 sequence are all nonempty. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)       ((𝜑𝑁 ∈ ℕ0) → (1st ‘(𝐺𝑁)) < (2nd ‘(𝐺𝑁)))

Theoremruclem9 15011* Lemma for ruc 15016. The first components of the 𝐺 sequence are increasing, and the second components are decreasing. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀))))

Theoremruclem10 15012* Lemma for ruc 15016. Every first component of the 𝐺 sequence is less than every second component. That is, the sequences form a chain a1 < a2 <... < b2 < b1, where ai are the first components and bi are the second components. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (1st ‘(𝐺𝑀)) < (2nd ‘(𝐺𝑁)))

Theoremruclem11 15013* Lemma for ruc 15016. Closure lemmas for supremum. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)       (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))

Theoremruclem12 15014* Lemma for ruc 15016. The supremum of the increasing sequence 1st𝐺 is a real number that is not in the range of 𝐹. (Contributed by Mario Carneiro, 28-May-2014.)
(𝜑𝐹:ℕ⟶ℝ)    &   (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))    &   𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)    &   𝐺 = seq0(𝐷, 𝐶)    &   𝑆 = sup(ran (1st𝐺), ℝ, < )       (𝜑𝑆 ∈ (ℝ ∖ ran 𝐹))

Theoremruclem13 15015 Lemma for ruc 15016. There is no function that maps onto . (Use nex 1771 if you want this in the form ¬ ∃𝑓𝑓:ℕ–onto→ℝ.) (Contributed by NM, 14-Oct-2004.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
¬ 𝐹:ℕ–onto→ℝ

Theoremruc 15016 The set of positive integers is strictly dominated by the set of real numbers, i.e. the real numbers are uncountable. The proof consists of lemmas ruclem1 15004 through ruclem13 15015 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 15015 for the function existence version of this theorem. For an informal discussion of this proof, see mmcomplex.html#uncountable. For an alternate proof see rucALT 15003. This is Metamath 100 proof #22. (Contributed by NM, 13-Oct-2004.)
ℕ ≺ ℝ

Theoremresdomq 15017 The set of rationals is strictly less equinumerous than the set of reals ( strictly dominates ). (Contributed by NM, 18-Dec-2004.)
ℚ ≺ ℝ

Theoremaleph1re 15018 There are at least aleph-one real numbers. (Contributed by NM, 2-Feb-2005.)
(ℵ‘1𝑜) ≼ ℝ

Theoremaleph1irr 15019 There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005.)
(ℵ‘1𝑜) ≼ (ℝ ∖ ℚ)

Theoremcnso 15020 The complex numbers can be linearly ordered. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑥 𝑥 Or ℂ

PART 6  ELEMENTARY NUMBER THEORY

Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory.

6.1  Elementary properties of divisibility

6.1.1  Irrationality of square root of 2

Theoremsqrt2irrlem 15021 Lemma for sqrt2irr 15023. This is the core of the proof: if 𝐴 / 𝐵 = √(2), then 𝐴 and 𝐵 are even, so 𝐴 / 2 and 𝐵 / 2 are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). This is Metamath 100 proof #1. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) (Proof shortened by JV, 4-Jan-2022.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → (√‘2) = (𝐴 / 𝐵))       (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ))

Theoremsqrt2irrlemOLD 15022 Obsolete proof of sqrt2irrlem 15021 as of 4-Jan-2022. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → (√‘2) = (𝐴 / 𝐵))       (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ))

Theoremsqrt2irr 15023 The square root of 2 is irrational. See zsqrtelqelz 15513 for a generalization to all non-square integers. The proof's core is proven in sqrt2irrlem 15021, which shows that if 𝐴 / 𝐵 = √(2), then 𝐴 and 𝐵 are even, so 𝐴 / 2 and 𝐵 / 2 are smaller representatives, which is absurd. An older version of this proof was included in The Seventeen Provers of the World compiled by Freek Wiedijk. It is also the first of the "top 100" mathematical theorems whose formalization is tracked by Freek Wiedijk on his Formalizing 100 Theorems page at http://www.cs.ru.nl/~freek/100/. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
(√‘2) ∉ ℚ

Theoremsqrt2re 15024 The square root of 2 exists and is a real number. (Contributed by NM, 3-Dec-2004.)
(√‘2) ∈ ℝ

6.1.2  Some Number sets are chains of proper subsets

Theoremnthruc 15025 The sequence , , , , and forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to but not , one-half belongs to but not , the square root of 2 belongs to but not , and finally that the imaginary number i belongs to but not . See nthruz 15026 for a further refinement. (Contributed by NM, 12-Jan-2002.)
((ℕ ⊊ ℤ ∧ ℤ ⊊ ℚ) ∧ (ℚ ⊊ ℝ ∧ ℝ ⊊ ℂ))

Theoremnthruz 15026 The sequence , 0, and forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to 0 but not and minus one belongs to but not 0. This theorem refines the chain of proper subsets nthruc 15025. (Contributed by NM, 9-May-2004.)
(ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ)

6.1.3  The divides relation

Syntaxcdvds 15027 Extend the definition of a class to include the divides relation. See df-dvds 15028.
class

Definitiondf-dvds 15028* Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}

Theoremdivides 15029* Define the divides relation. 𝑀𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 27443). As proven in dvdsval3 15031, 𝑀𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 15029 and dvdsval2 15030 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))

Theoremdvdsval2 15030 One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.)
((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ))

Theoremdvdsval3 15031 One nonzero integer divides another integer if and only if the remainder upon division is zero, see remark in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝑁 mod 𝑀) = 0))

Theoremdvdszrcl 15032 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
(𝑋𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ))

Theoremdvdsmod0 15033 If a positive integer divides another integer, then the remainder upon division is zero. (Contributed by AV, 3-Mar-2022.)
((𝑀 ∈ ℕ ∧ 𝑀𝑁) → (𝑁 mod 𝑀) = 0)

Theoremp1modz1 15034 If a number greater than 1 divides another number, the second number increased by 1 is 1 modulo the first number. (Contributed by AV, 19-Mar-2022.)
((𝑀𝐴 ∧ 1 < 𝑀) → ((𝐴 + 1) mod 𝑀) = 1)

Theoremdvdsmodexp 15035 If a positive integer divides another integer, this other integer is equal to its positive powers modulo the positive integer. (Formerly part of the proof for fermltl 15536). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by AV, 19-Mar-2022.)
((𝑁 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁𝐴) → ((𝐴𝐵) mod 𝑁) = (𝐴 mod 𝑁))

Theoremnndivdvds 15036 Strong form of dvdsval2 15030 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵𝐴 ↔ (𝐴 / 𝐵) ∈ ℕ))

Theoremnndivides 15037* Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀𝑁 ↔ ∃𝑛 ∈ ℕ (𝑛 · 𝑀) = 𝑁))

Theoremmoddvds 15038 Two ways to say 𝐴𝐵 (mod 𝑁), see also definition in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 18-Feb-2014.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 mod 𝑁) = (𝐵 mod 𝑁) ↔ 𝑁 ∥ (𝐴𝐵)))

Theoremdvds0lem 15039 A lemma to assist theorems of with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
(((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 · 𝑀) = 𝑁) → 𝑀𝑁)

Theoremdvds1lem 15040* A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ))    &   (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))    &   ((𝜑𝑥 ∈ ℤ) → 𝑍 ∈ ℤ)    &   ((𝜑𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → (𝑍 · 𝑀) = 𝑁))       (𝜑 → (𝐽𝐾𝑀𝑁))

Theoremdvds2lem 15041* A lemma to assist theorems of with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝜑 → (𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ))    &   (𝜑 → (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ))    &   (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))    &   ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑍 ∈ ℤ)    &   ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → (𝑍 · 𝑀) = 𝑁))       (𝜑 → ((𝐼𝐽𝐾𝐿) → 𝑀𝑁))

Theoremiddvds 15042 An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
(𝑁 ∈ ℤ → 𝑁𝑁)

Theorem1dvds 15043 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝑁 ∈ ℤ → 1 ∥ 𝑁)

Theoremdvds0 15044 Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝑁 ∈ ℤ → 𝑁 ∥ 0)

Theoremnegdvdsb 15045 An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ -𝑀𝑁))

Theoremdvdsnegb 15046 An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ -𝑁))

Theoremabsdvdsb 15047 An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (abs‘𝑀) ∥ 𝑁))

Theoremdvdsabsb 15048 An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (abs‘𝑁)))

Theorem0dvds 15049 Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝑁 ∈ ℤ → (0 ∥ 𝑁𝑁 = 0))

Theoremdvdsmul1 15050 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 · 𝑁))

Theoremdvdsmul2 15051 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁))

Theoremiddvdsexp 15052 An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑀 ∥ (𝑀𝑁))

Theoremmuldvds1 15053 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) ∥ 𝑁𝐾𝑁))

Theoremmuldvds2 15054 If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) ∥ 𝑁𝑀𝑁))

Theoremdvdscmul 15055 Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀𝑁 → (𝐾 · 𝑀) ∥ (𝐾 · 𝑁)))

Theoremdvdsmulc 15056 Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀𝑁 → (𝑀 · 𝐾) ∥ (𝑁 · 𝐾)))

Theoremdvdscmulr 15057 Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝐾 ∈ ℤ ∧ 𝐾 ≠ 0)) → ((𝐾 · 𝑀) ∥ (𝐾 · 𝑁) ↔ 𝑀𝑁))

Theoremdvdsmulcr 15058 Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝐾 ∈ ℤ ∧ 𝐾 ≠ 0)) → ((𝑀 · 𝐾) ∥ (𝑁 · 𝐾) ↔ 𝑀𝑁))

Theoremsummodnegmod 15059 The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 𝐵) mod 𝑁) = 0 ↔ (𝐴 mod 𝑁) = (-𝐵 mod 𝑁)))

Theoremmodmulconst 15060 Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) ∧ 𝑀 ∈ ℕ) → ((𝐴 mod 𝑀) = (𝐵 mod 𝑀) ↔ ((𝐶 · 𝐴) mod (𝐶 · 𝑀)) = ((𝐶 · 𝐵) mod (𝐶 · 𝑀))))

Theoremdvds2ln 15061 If an integer divides each of two other integers, it divides any linear combination of them. Theorem 1.1(c) in [ApostolNT] p. 14 (linearity property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
(((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝐾𝑀𝐾𝑁) → 𝐾 ∥ ((𝐼 · 𝑀) + (𝐽 · 𝑁))))

Theoremdvds2add 15062 If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾𝑀𝐾𝑁) → 𝐾 ∥ (𝑀 + 𝑁)))

Theoremdvds2sub 15063 If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾𝑀𝐾𝑁) → 𝐾 ∥ (𝑀𝑁)))

Theoremdvds2subd 15064 Natural deduction form of dvds2sub 15063. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝐾𝑀)    &   (𝜑𝐾𝑁)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)       (𝜑𝐾 ∥ (𝑀𝑁))

Theoremdvdstr 15065 The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14 (transitive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾𝑀𝑀𝑁) → 𝐾𝑁))

Theoremdvdsmultr1 15066 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾𝑀𝐾 ∥ (𝑀 · 𝑁)))

Theoremdvdsmultr1d 15067 Natural deduction form of dvdsmultr1 15066. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐾𝑀)       (𝜑𝐾 ∥ (𝑀 · 𝑁))

Theoremdvdsmultr2 15068 If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾𝑁𝐾 ∥ (𝑀 · 𝑁)))

Theoremordvdsmul 15069 If an integer divides either of two others, it divides their product. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾𝑀𝐾𝑁) → 𝐾 ∥ (𝑀 · 𝑁)))

Theoremdvdssub2 15070 If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.)
(((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 ∥ (𝑀𝑁)) → (𝐾𝑀𝐾𝑁))

Theoremdvdsadd 15071 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (𝑀 + 𝑁)))

Theoremdvdsaddr 15072 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (𝑁 + 𝑀)))

Theoremdvdssub 15073 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (𝑀𝑁)))

Theoremdvdssubr 15074 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (𝑁𝑀)))

Theoremdvdsadd2b 15075 Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝐶 ∈ ℤ ∧ 𝐴𝐶)) → (𝐴𝐵𝐴 ∥ (𝐶 + 𝐵)))

Theoremdvdsaddre2b 15076 Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 15075 only requiring 𝐵 to be a real number (not necessarily an integer). (Contributed by AV, 19-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℤ ∧ 𝐴𝐶)) → (𝐴𝐵𝐴 ∥ (𝐶 + 𝐵)))

Theoremfsumdvds 15077* If every term in a sum is divisible by 𝑁, then so is the sum. (Contributed by Mario Carneiro, 17-Jan-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)    &   ((𝜑𝑘𝐴) → 𝑁𝐵)       (𝜑𝑁 ∥ Σ𝑘𝐴 𝐵)

Theoremdvdslelem 15078 Lemma for dvdsle 15079. (Contributed by Paul Chapman, 21-Mar-2011.)
𝑀 ∈ ℤ    &   𝑁 ∈ ℕ    &   𝐾 ∈ ℤ       (𝑁 < 𝑀 → (𝐾 · 𝑀) ≠ 𝑁)

Theoremdvdsle 15079 The divisors of a positive integer are bounded by it. The proof does not use /. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀𝑁𝑀𝑁))

Theoremdvdsleabs 15080 The divisors of a nonzero integer are bounded by its absolute value. Theorem 1.1(i) in [ApostolNT] p. 14 (comparison property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀𝑁𝑀 ≤ (abs‘𝑁)))

Theoremdvdsleabs2 15081 Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))

Theoremdvdsabseq 15082 If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.)
((𝑀𝑁𝑁𝑀) → (abs‘𝑀) = (abs‘𝑁))

Theoremdvdseq 15083 If two nonnegative integers divide each other, they must be equal. (Contributed by Mario Carneiro, 30-May-2014.) (Proof shortened by AV, 7-Aug-2021.)
(((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀𝑁𝑁𝑀)) → 𝑀 = 𝑁)

Theoremdivconjdvds 15084 If a nonzero integer 𝑀 divides another integer 𝑁, the other integer 𝑁 divided by the nonzero integer 𝑀 (i.e. the divisor conjugate of 𝑁 to 𝑀) divides the other integer 𝑁. Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.)
((𝑀𝑁𝑀 ≠ 0) → (𝑁 / 𝑀) ∥ 𝑁)

Theoremdvdsdivcl 15085* The complement of a divisor of 𝑁 is also a divisor of 𝑁. (Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV, 9-Aug-2021.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → (𝑁 / 𝐴) ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁})

Theoremdvdsflip 15086* An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.)
𝐴 = {𝑥 ∈ ℕ ∣ 𝑥𝑁}    &   𝐹 = (𝑦𝐴 ↦ (𝑁 / 𝑦))       (𝑁 ∈ ℕ → 𝐹:𝐴1-1-onto𝐴)

Theoremdvdsssfz1 15087* The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)
(𝐴 ∈ ℕ → {𝑝 ∈ ℕ ∣ 𝑝𝐴} ⊆ (1...𝐴))

Theoremdvds1 15088 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝑀 ∈ ℕ0 → (𝑀 ∥ 1 ↔ 𝑀 = 1))

Theoremalzdvds 15089* Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝑁 ∈ ℤ → (∀𝑥 ∈ ℤ 𝑥𝑁𝑁 = 0))

Theoremdvdsext 15090* Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℕ0 (𝐴𝑥𝐵𝑥)))

Theoremfzm1ndvds 15091 No number between 1 and 𝑀 − 1 divides 𝑀. (Contributed by Mario Carneiro, 24-Jan-2015.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → ¬ 𝑀𝑁)

Theoremfzo0dvdseq 15092 Zero is the only one of the first 𝐴 nonnegative integers that is divisible by 𝐴. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(𝐵 ∈ (0..^𝐴) → (𝐴𝐵𝐵 = 0))

Theoremfzocongeq 15093 Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.)
((𝐴 ∈ (𝐶..^𝐷) ∧ 𝐵 ∈ (𝐶..^𝐷)) → ((𝐷𝐶) ∥ (𝐴𝐵) ↔ 𝐴 = 𝐵))

TheoremaddmodlteqALT 15094 Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. Shorter proof of addmodlteq 12785 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽))

Theoremdvdsfac 15095 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝐾)) → 𝐾 ∥ (!‘𝑁))

Theoremdvdsexp 15096 A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)
((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) → (𝐴𝑀) ∥ (𝐴𝑁))

Theoremdvdsmod 15097 Any number 𝐾 whose mod base 𝑁 is divisible by a divisor 𝑃 of the base is also divisible by 𝑃. This means that primes will also be relatively prime to the base when reduced mod 𝑁 for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)
(((𝑃 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) ∧ 𝑃𝑁) → (𝑃 ∥ (𝐾 mod 𝑁) ↔ 𝑃𝐾))

Theoremmulmoddvds 15098 If an integer is divisible by a positive integer, the product of this integer with another integer modulo the positive integer is 0. (Contributed by Alexander van der Vekens, 30-Aug-2018.) (Proof shortened by AV, 18-Mar-2022.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑁𝐴 → ((𝐴 · 𝐵) mod 𝑁) = 0))

Theorem3dvds 15099* A rule for divisibility by 3 of a number written in base 10. This is Metamath 100 proof #85. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.) (Revised by AV, 8-Sep-2021.)
((𝑁 ∈ ℕ0𝐹:(0...𝑁)⟶ℤ) → (3 ∥ Σ𝑘 ∈ (0...𝑁)((𝐹𝑘) · (10↑𝑘)) ↔ 3 ∥ Σ𝑘 ∈ (0...𝑁)(𝐹𝑘)))

Theorem3dvdsOLD 15100* Obsolete version of 3dvds 15099 as of 8-Sep-2021. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑁 ∈ ℕ0𝐹:(0...𝑁)⟶ℤ) → (3 ∥ Σ𝑘 ∈ (0...𝑁)((𝐹𝑘) · (10↑𝑘)) ↔ 3 ∥ Σ𝑘 ∈ (0...𝑁)(𝐹𝑘)))

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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