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Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremruclem3 14801* Lemma for ruc 14811. 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 14802* Lemma for ruc 14811. 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 14803* Lemma for ruc 14811. 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 14804* Lemma for ruc 14811. 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 14805* Lemma for ruc 14811. 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 14806* Lemma for ruc 14811. 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 14807* Lemma for ruc 14811. 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 14808* Lemma for ruc 14811. 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 14809* Lemma for ruc 14811. 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 14810 Lemma for ruc 14811. There is no function that maps onto . (Use nex 1722 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 14811 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 14799 through ruclem13 14810 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 14810 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 14798. This is Metamath 100 proof #22. (Contributed by NM, 13-Oct-2004.)
ℕ ≺ ℝ

Theoremresdomq 14812 The set of rationals is strictly less equinumerous than the set of reals ( strictly dominates ). (Contributed by NM, 18-Dec-2004.)
ℚ ≺ ℝ

Theoremaleph1re 14813 There are at least aleph-one real numbers. (Contributed by NM, 2-Feb-2005.)
(ℵ‘1𝑜) ≼ ℝ

Theoremaleph1irr 14814 There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005.)
(ℵ‘1𝑜) ≼ (ℝ ∖ ℚ)

Theoremcnso 14815 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

Theoremsqr2irrlem 14816 Lemma for irrationality of square root of 2. 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.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑 → (√‘2) = (𝐴 / 𝐵))       (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ))

Theoremsqrt2irr 14817 The square root of 2 is irrational. See zsqrtelqelz 15304 for a generalization to all non-square integers. The proof's core is proven in sqr2irrlem 14816, 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 "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 14818 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 14819 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 14820 for a further refinement. (Contributed by NM, 12-Jan-2002.)
((ℕ ⊊ ℤ ∧ ℤ ⊊ ℚ) ∧ (ℚ ⊊ ℝ ∧ ℝ ⊊ ℂ))

Theoremnthruz 14820 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 14819. (Contributed by NM, 9-May-2004.)
(ℕ ⊊ ℕ0 ∧ ℕ0 ⊊ ℤ)

6.1.3  The divides relation

Syntaxcdvds 14821 Extend the definition of a class to include the divides relation. See df-dvds 14822.
class

Definitiondf-dvds 14822* 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 14823* Define the divides relation. 𝑀𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 26705). As proven in dvdsval3 14825, 𝑀𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 14823 and dvdsval2 14824 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))

Theoremdvdsval2 14824 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 14825 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 14826 Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
(𝑋𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ))

Theoremnndivdvds 14827 Strong form of dvdsval2 14824 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐵𝐴 ↔ (𝐴 / 𝐵) ∈ ℕ))

Theoremnndivides 14828* 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 14829 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 14830 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 14831* 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 14832* 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 14833 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 14834 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝑁 ∈ ℤ → 1 ∥ 𝑁)

Theoremdvds0 14835 Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝑁 ∈ ℤ → 𝑁 ∥ 0)

Theoremnegdvdsb 14836 An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ -𝑀𝑁))

Theoremdvdsnegb 14837 An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ -𝑁))

Theoremabsdvdsb 14838 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 14839 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 14840 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 14841 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 · 𝑁))

Theoremdvdsmul2 14842 An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁))

Theoremiddvdsexp 14843 An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑀 ∥ (𝑀𝑁))

Theoremmuldvds1 14844 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 14845 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 14846 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 14847 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 14848 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 14849 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 14850 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 14851 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 14852 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 14853 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 14854 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 14855 Natural deduction form of dvds2sub 14854. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝐾𝑀)    &   (𝜑𝐾𝑁)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)       (𝜑𝐾 ∥ (𝑀𝑁))

Theoremdvdstr 14856 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 14857 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 14858 Natural deduction form of dvdsmultr1 14857. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐾𝑀)       (𝜑𝐾 ∥ (𝑀 · 𝑁))

Theoremdvdsmultr2 14859 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 14860 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 14861 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 14862 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 14863 An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (𝑁 + 𝑀)))

Theoremdvdssub 14864 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (𝑀𝑁)))

Theoremdvdssubr 14865 An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁𝑀 ∥ (𝑁𝑀)))

Theoremdvdsadd2b 14866 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 14867 Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 14866 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 14868* 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 14869 Lemma for dvdsle 14870. (Contributed by Paul Chapman, 21-Mar-2011.)
𝑀 ∈ ℤ    &   𝑁 ∈ ℕ    &   𝐾 ∈ ℤ       (𝑁 < 𝑀 → (𝐾 · 𝑀) ≠ 𝑁)

Theoremdvdsle 14870 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 14871 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 14872 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 14873 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 14874 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 14875 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 14876* 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 14877* 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 14878* 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 14879 The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.)
(𝑀 ∈ ℕ0 → (𝑀 ∥ 1 ↔ 𝑀 = 1))

Theoremalzdvds 14880* Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝑁 ∈ ℤ → (∀𝑥 ∈ ℤ 𝑥𝑁𝑁 = 0))

Theoremdvdsext 14881* Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℕ0 (𝐴𝑥𝐵𝑥)))

Theoremfzm1ndvds 14882 No number between 1 and 𝑀 − 1 divides 𝑀. (Contributed by Mario Carneiro, 24-Jan-2015.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → ¬ 𝑀𝑁)

Theoremfzo0dvdseq 14883 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 14884 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 14885 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 12607 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 14886 A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.)
((𝐾 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝐾)) → 𝐾 ∥ (!‘𝑁))

Theoremdvdsexp 14887 A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.)
((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) → (𝐴𝑀) ∥ (𝐴𝑁))

Theoremdvdsmod 14888 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 14889 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.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑁𝐴 → ((𝐴 · 𝐵) mod 𝑁) = 0))

Theorem3dvds 14890* 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 14891* Obsolete version of 3dvds 14890 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)))

Theorem3dvdsdec 14892 A decimal number is divisible by three iff the sum of its two "digits" is divisible by three. The term "digits" in its narrow sense is only correct if 𝐴 and 𝐵 actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers 𝐴 and 𝐵, especially if 𝐴 is itself a decimal number, e.g. 𝐴 = 𝐶𝐷. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       (3 ∥ 𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵))

Theorem3dvdsdecOLD 14893 Obsolete proof of 3dvdsdec 14892 as of 8-Sep-2021. (Contributed by AV, 14-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       (3 ∥ 𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵))

Theorem3dvds2dec 14894 A decimal number is divisible by three iff the sum of its three "digits" is divisible by three. The term "digits" in its narrow sense is only correct if 𝐴, 𝐵 and 𝐶 actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers 𝐴, 𝐵 and 𝐶. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0       (3 ∥ 𝐴𝐵𝐶 ↔ 3 ∥ ((𝐴 + 𝐵) + 𝐶))

Theorem3dvds2decOLD 14895 Old version of 3dvds2dec 14894. Obsolete as of 1-Aug-2021. (Contributed by AV, 14-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0       (3 ∥ 𝐴𝐵𝐶 ↔ 3 ∥ ((𝐴 + 𝐵) + 𝐶))

Theoremfprodfvdvdsd 14896* A finite product of integers is divisible by any of its factors being function values. (Contributed by AV, 1-Aug-2021.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹:𝐵⟶ℤ)       (𝜑 → ∀𝑥𝐴 (𝐹𝑥) ∥ ∏𝑘𝐴 (𝐹𝑘))

Theoremfproddvdsd 14897* A finite product of integers is divisible by any of its factors. (Contributed by AV, 14-Aug-2020.) (Proof shortened by AV, 2-Aug-2021.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ ℤ)       (𝜑 → ∀𝑥𝐴 𝑥 ∥ ∏𝑘𝐴 𝑘)

6.1.4  Even and odd numbers

The set of integers can be partitioned into the set of even numbers and the set of odd numbers, see zeo4 14900. Instead of defining new class variables Even and Odd to represent these sets, we use the idiom 2 ∥ 𝑁 to say that "𝑁 is even" (which implies 𝑁 ∈ ℤ, see evenelz 14898) and ¬ 2 ∥ 𝑁 to say that "𝑁 is odd" (under the assumption that 𝑁 ∈ ℤ). The previously proven theorems about even and odd numbers, like zneo 11336, zeo 11339, zeo2 11340, etc. use different representations, which are equivalent with the representations using the divides relation, see evend2 14919 and oddp1d2 14920. The corresponding theorems are zeneo 14901, zeo3 14899 and zeo4 14900.

Theoremevenelz 14898 An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 14826. (Contributed by AV, 22-Jun-2021.)
(2 ∥ 𝑁𝑁 ∈ ℤ)

Theoremzeo3 14899 An integer is even or odd. With this representation of even and odd integers, this variant of zeo 11339 follows immediatly from the law of excluded middle, see exmidd 431. (Contributed by AV, 17-Jun-2021.)
(𝑁 ∈ ℤ → (2 ∥ 𝑁 ∨ ¬ 2 ∥ 𝑁))

Theoremzeo4 14900 An integer is even or odd but not both. With this representation of even and odd integers, this variant of zeo2 11340 follows immediatly from the principle of double negation, see notnotb 303. (Contributed by AV, 17-Jun-2021.)
(𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁))

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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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