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Theorem List for Metamath Proof Explorer - 16001-16100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempcmpt2 16001* Dividing two prime count maps yields a number with all dividing primes confined to an interval. (Contributed by Mario Carneiro, 14-Mar-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛𝐴), 1))    &   (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝑛 = 𝑃𝐴 = 𝐵)    &   (𝜑𝑀 ∈ (ℤ𝑁))       (𝜑 → (𝑃 pCnt ((seq1( · , 𝐹)‘𝑀) / (seq1( · , 𝐹)‘𝑁))) = if((𝑃𝑀 ∧ ¬ 𝑃𝑁), 𝐵, 0))

Theorempcmptdvds 16002 The partial products of the prime power map form a divisibility chain. (Contributed by Mario Carneiro, 12-Mar-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛𝐴), 1))    &   (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (ℤ𝑁))       (𝜑 → (seq1( · , 𝐹)‘𝑁) ∥ (seq1( · , 𝐹)‘𝑀))

Theorempcprod 16003* The product of the primes taken to their respective powers reconstructs the original number. (Contributed by Mario Carneiro, 12-Mar-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝑁)), 1))       (𝑁 ∈ ℕ → (seq1( · , 𝐹)‘𝑁) = 𝑁)

Theoremsumhash 16004* The sum of 1 over a set is the size of the set. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 20-May-2014.)
((𝐵 ∈ Fin ∧ 𝐴𝐵) → Σ𝑘𝐵 if(𝑘𝐴, 1, 0) = (♯‘𝐴))

Theoremfldivp1 16005 The difference between the floors of adjacent fractions is either 1 or 0. (Contributed by Mario Carneiro, 8-Mar-2014.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((⌊‘((𝑀 + 1) / 𝑁)) − (⌊‘(𝑀 / 𝑁))) = if(𝑁 ∥ (𝑀 + 1), 1, 0))

Theorempcfaclem 16006 Lemma for pcfac 16007. (Contributed by Mario Carneiro, 20-May-2014.)
((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑃 ∈ ℙ) → (⌊‘(𝑁 / (𝑃𝑀))) = 0)

Theorempcfac 16007* Calculate the prime count of a factorial. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (!‘𝑁)) = Σ𝑘 ∈ (1...𝑀)(⌊‘(𝑁 / (𝑃𝑘))))

Theorempcbc 16008* Calculate the prime count of a binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...𝑁) ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt (𝑁C𝐾)) = Σ𝑘 ∈ (1...𝑁)((⌊‘(𝑁 / (𝑃𝑘))) − ((⌊‘((𝑁𝐾) / (𝑃𝑘))) + (⌊‘(𝐾 / (𝑃𝑘))))))

Theoremqexpz 16009 If a power of a rational number is an integer, then the number is an integer. In other words, all n-th roots are irrational unless they are integers (so that the original number is an n-th power). (Contributed by Mario Carneiro, 10-Aug-2015.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ ∧ (𝐴𝑁) ∈ ℤ) → 𝐴 ∈ ℤ)

Theoremexpnprm 16010 A second or higher power of a rational number is not a prime number. Or by contraposition, the n-th root of a prime number is irrational. Suggested by Norm Megill. (Contributed by Mario Carneiro, 10-Aug-2015.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ (ℤ‘2)) → ¬ (𝐴𝑁) ∈ ℙ)

Theoremoddprmdvds 16011* Every positive integer which is not a power of two is divisible by an odd prime number. (Contributed by AV, 6-Aug-2021.)
((𝐾 ∈ ℕ ∧ ¬ ∃𝑛 ∈ ℕ0 𝐾 = (2↑𝑛)) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝𝐾)

6.2.8  Pocklington's theorem

Theoremprmpwdvds 16012 A relation involving divisibility by a prime power. (Contributed by Mario Carneiro, 2-Mar-2014.)
(((𝐾 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) ∧ (𝐷 ∥ (𝐾 · (𝑃𝑁)) ∧ ¬ 𝐷 ∥ (𝐾 · (𝑃↑(𝑁 − 1))))) → (𝑃𝑁) ∥ 𝐷)

Theorempockthlem 16013 Lemma for pockthg 16014. (Contributed by Mario Carneiro, 2-Mar-2014.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐵 < 𝐴)    &   (𝜑𝑁 = ((𝐴 · 𝐵) + 1))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑃𝑁)    &   (𝜑𝑄 ∈ ℙ)    &   (𝜑 → (𝑄 pCnt 𝐴) ∈ ℕ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑 → ((𝐶↑(𝑁 − 1)) mod 𝑁) = 1)    &   (𝜑 → (((𝐶↑((𝑁 − 1) / 𝑄)) − 1) gcd 𝑁) = 1)       (𝜑 → (𝑄 pCnt 𝐴) ≤ (𝑄 pCnt (𝑃 − 1)))

Theorempockthg 16014* The generalized Pocklington's theorem. If 𝑁 − 1 = 𝐴 · 𝐵 where 𝐵 < 𝐴, then 𝑁 is prime if and only if for every prime factor 𝑝 of 𝐴, there is an 𝑥 such that 𝑥↑(𝑁 − 1) = 1( mod 𝑁) and gcd (𝑥↑((𝑁 − 1) / 𝑝) − 1, 𝑁) = 1. (Contributed by Mario Carneiro, 2-Mar-2014.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐵 < 𝐴)    &   (𝜑𝑁 = ((𝐴 · 𝐵) + 1))    &   (𝜑 → ∀𝑝 ∈ ℙ (𝑝𝐴 → ∃𝑥 ∈ ℤ (((𝑥↑(𝑁 − 1)) mod 𝑁) = 1 ∧ (((𝑥↑((𝑁 − 1) / 𝑝)) − 1) gcd 𝑁) = 1)))       (𝜑𝑁 ∈ ℙ)

Theorempockthi 16015 Pocklington's theorem, which gives a sufficient criterion for a number 𝑁 to be prime. This is the preferred method for verifying large primes, being much more efficient to compute than trial division. This form has been optimized for application to specific large primes; see pockthg 16014 for a more general closed-form version. (Contributed by Mario Carneiro, 2-Mar-2014.)
𝑃 ∈ ℙ    &   𝐺 ∈ ℕ    &   𝑀 = (𝐺 · 𝑃)    &   𝑁 = (𝑀 + 1)    &   𝐷 ∈ ℕ    &   𝐸 ∈ ℕ    &   𝐴 ∈ ℕ    &   𝑀 = (𝐷 · (𝑃𝐸))    &   𝐷 < (𝑃𝐸)    &   ((𝐴𝑀) mod 𝑁) = (1 mod 𝑁)    &   (((𝐴𝐺) − 1) gcd 𝑁) = 1       𝑁 ∈ ℙ

6.2.9  Infinite primes theorem

Theoremunbenlem 16016* Lemma for unben 16017. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)
𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) ↾ ω)       ((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → 𝐴 ≈ ω)

Theoremunben 16017* An unbounded set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)
((𝐴 ⊆ ℕ ∧ ∀𝑚 ∈ ℕ ∃𝑛𝐴 𝑚 < 𝑛) → 𝐴 ≈ ℕ)

Theoreminfpnlem1 16018* Lemma for infpn 16020. The smallest divisor (greater than 1) 𝑀 of 𝑁! + 1 is a prime greater than 𝑁. (Contributed by NM, 5-May-2005.)
𝐾 = ((!‘𝑁) + 1)       ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (((1 < 𝑀 ∧ (𝐾 / 𝑀) ∈ ℕ) ∧ ∀𝑗 ∈ ℕ ((1 < 𝑗 ∧ (𝐾 / 𝑗) ∈ ℕ) → 𝑀𝑗)) → (𝑁 < 𝑀 ∧ ∀𝑗 ∈ ℕ ((𝑀 / 𝑗) ∈ ℕ → (𝑗 = 1 ∨ 𝑗 = 𝑀)))))

Theoreminfpnlem2 16019* Lemma for infpn 16020. For any positive integer 𝑁, there exists a prime number 𝑗 greater than 𝑁. (Contributed by NM, 5-May-2005.)
𝐾 = ((!‘𝑁) + 1)       (𝑁 ∈ ℕ → ∃𝑗 ∈ ℕ (𝑁 < 𝑗 ∧ ∀𝑘 ∈ ℕ ((𝑗 / 𝑘) ∈ ℕ → (𝑘 = 1 ∨ 𝑘 = 𝑗))))

Theoreminfpn 16020* There exist infinitely many prime numbers: for any positive integer 𝑁, there exists a prime number 𝑗 greater than 𝑁. (See infpn2 16021 for the equinumerosity version.) (Contributed by NM, 1-Jun-2006.)
(𝑁 ∈ ℕ → ∃𝑗 ∈ ℕ (𝑁 < 𝑗 ∧ ∀𝑘 ∈ ℕ ((𝑗 / 𝑘) ∈ ℕ → (𝑘 = 1 ∨ 𝑘 = 𝑗))))

Theoreminfpn2 16021* There exist infinitely many prime numbers: the set of all primes 𝑆 is unbounded by infpn 16020, so by unben 16017 it is infinite. This is Metamath 100 proof #11. (Contributed by NM, 5-May-2005.)
𝑆 = {𝑛 ∈ ℕ ∣ (1 < 𝑛 ∧ ∀𝑚 ∈ ℕ ((𝑛 / 𝑚) ∈ ℕ → (𝑚 = 1 ∨ 𝑚 = 𝑛)))}       𝑆 ≈ ℕ

Theoremprmunb 16022* The primes are unbounded. (Contributed by Paul Chapman, 28-Nov-2012.)
(𝑁 ∈ ℕ → ∃𝑝 ∈ ℙ 𝑁 < 𝑝)

Theoremprminf 16023 There are an infinite number of primes. Theorem 1.7 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 28-Nov-2012.)
ℙ ≈ ℕ

6.2.10  Sum of prime reciprocals

Theoremprmreclem1 16024* Lemma for prmrec 16030. Properties of the "square part" function, which extracts the 𝑚 of the decomposition 𝑁 = 𝑟𝑚↑2, with 𝑚 maximal and 𝑟 squarefree. (Contributed by Mario Carneiro, 5-Aug-2014.)
𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))       (𝑁 ∈ ℕ → ((𝑄𝑁) ∈ ℕ ∧ ((𝑄𝑁)↑2) ∥ 𝑁 ∧ (𝐾 ∈ (ℤ‘2) → ¬ (𝐾↑2) ∥ (𝑁 / ((𝑄𝑁)↑2)))))

Theoremprmreclem2 16025* Lemma for prmrec 16030. There are at most 2↑𝐾 squarefree numbers which divide no primes larger than 𝐾. (We could strengthen this to 2↑♯(ℙ ∩ (1...𝐾)) but there's no reason to.) We establish the inequality by showing that the prime counts of the number up to 𝐾 completely determine it because all higher prime counts are zero, and they are all at most 1 because no square divides the number, so there are at most 2↑𝐾 possibilities. (Contributed by Mario Carneiro, 5-Aug-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (1 / 𝑛), 0))    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛}    &   𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))       (𝜑 → (♯‘{𝑥𝑀 ∣ (𝑄𝑥) = 1}) ≤ (2↑𝐾))

Theoremprmreclem3 16026* Lemma for prmrec 16030. The main inequality established here is 𝑀 ≤ ♯{𝑥𝑀 ∣ (𝑄𝑥) = 1} · √𝑁, where {𝑥𝑀 ∣ (𝑄𝑥) = 1} is the set of squarefree numbers in 𝑀. This is demonstrated by the map 𝑦 ↦ ⟨𝑦 / (𝑄𝑦)↑2, (𝑄𝑦)⟩ where 𝑄𝑦 is the largest number whose square divides 𝑦. (Contributed by Mario Carneiro, 5-Aug-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (1 / 𝑛), 0))    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛}    &   𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))       (𝜑 → (♯‘𝑀) ≤ ((2↑𝐾) · (√‘𝑁)))

Theoremprmreclem4 16027* Lemma for prmrec 16030. Show by induction that the indexed (nondisjoint) union 𝑊𝑘 is at most the size of the prime reciprocal series. The key counting lemma is hashdvds 15884, to show that the number of numbers in 1...𝑁 that divide 𝑘 is at most 𝑁 / 𝑘. (Contributed by Mario Carneiro, 6-Aug-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (1 / 𝑛), 0))    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛}    &   (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )    &   (𝜑 → Σ𝑘 ∈ (ℤ‘(𝐾 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) < (1 / 2))    &   𝑊 = (𝑝 ∈ ℕ ↦ {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝𝑛)})       (𝜑 → (𝑁 ∈ (ℤ𝐾) → (♯‘ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))

Theoremprmreclem5 16028* Lemma for prmrec 16030. Here we show the inequality 𝑁 / 2 < ♯𝑀 by decomposing the set (1...𝑁) into the disjoint union of the set 𝑀 of those numbers that are not divisible by any "large" primes (above 𝐾) and the indexed union over 𝐾 < 𝑘 of the numbers 𝑊𝑘 that divide the prime 𝑘. By prmreclem4 16027 the second of these has size less than 𝑁 times the prime reciprocal series, which is less than 1 / 2 by assumption, we find that the complementary part 𝑀 must be at least 𝑁 / 2 large. (Contributed by Mario Carneiro, 6-Aug-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (1 / 𝑛), 0))    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝𝑛}    &   (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )    &   (𝜑 → Σ𝑘 ∈ (ℤ‘(𝐾 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) < (1 / 2))    &   𝑊 = (𝑝 ∈ ℕ ↦ {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝𝑛)})       (𝜑 → (𝑁 / 2) < ((2↑𝐾) · (√‘𝑁)))

Theoremprmreclem6 16029* Lemma for prmrec 16030. If the series 𝐹 was convergent, there would be some 𝑘 such that the sum starting from 𝑘 + 1 sums to less than 1 / 2; this is a sufficient hypothesis for prmreclem5 16028 to produce the contradictory bound 𝑁 / 2 < (2↑𝑘)√𝑁, which is false for 𝑁 = 2↑(2𝑘 + 2). (Contributed by Mario Carneiro, 6-Aug-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (1 / 𝑛), 0))        ¬ seq1( + , 𝐹) ∈ dom ⇝

Theoremprmrec 16030* The sum of the reciprocals of the primes diverges. Theorem 1.13 in [ApostolNT] p. 18. This is the "second" proof at http://en.wikipedia.org/wiki/Prime_harmonic_series, attributed to Paul Erdős. This is Metamath 100 proof #81. (Contributed by Mario Carneiro, 6-Aug-2014.)
𝐹 = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ℙ ∩ (1...𝑛))(1 / 𝑘))        ¬ 𝐹 ∈ dom ⇝

6.2.11  Fundamental theorem of arithmetic

Theorem1arithlem1 16031* Lemma for 1arith 16035. (Contributed by Mario Carneiro, 30-May-2014.)
𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))       (𝑁 ∈ ℕ → (𝑀𝑁) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑁)))

Theorem1arithlem2 16032* Lemma for 1arith 16035. (Contributed by Mario Carneiro, 30-May-2014.)
𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))       ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((𝑀𝑁)‘𝑃) = (𝑃 pCnt 𝑁))

Theorem1arithlem3 16033* Lemma for 1arith 16035. (Contributed by Mario Carneiro, 30-May-2014.)
𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))       (𝑁 ∈ ℕ → (𝑀𝑁):ℙ⟶ℕ0)

Theorem1arithlem4 16034* Lemma for 1arith 16035. (Contributed by Mario Carneiro, 30-May-2014.)
𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))    &   𝐺 = (𝑦 ∈ ℕ ↦ if(𝑦 ∈ ℙ, (𝑦↑(𝐹𝑦)), 1))    &   (𝜑𝐹:ℙ⟶ℕ0)    &   (𝜑𝑁 ∈ ℕ)    &   ((𝜑 ∧ (𝑞 ∈ ℙ ∧ 𝑁𝑞)) → (𝐹𝑞) = 0)       (𝜑 → ∃𝑥 ∈ ℕ 𝐹 = (𝑀𝑥))

Theorem1arith 16035* Fundamental theorem of arithmetic, where a prime factorization is represented as a sequence of prime exponents, for which only finitely many primes have nonzero exponent. The function 𝑀 maps the set of positive integers one-to-one onto the set of prime factorizations 𝑅. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 30-May-2014.)
𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))    &   𝑅 = {𝑒 ∈ (ℕ0𝑚 ℙ) ∣ (𝑒 “ ℕ) ∈ Fin}       𝑀:ℕ–1-1-onto𝑅

Theorem1arith2 16036* Fundamental theorem of arithmetic, where a prime factorization is represented as a finite monotonic 1-based sequence of primes. Every positive integer has a unique prime factorization. Theorem 1.10 in [ApostolNT] p. 17. This is Metamath 100 proof #80. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 30-May-2014.)
𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))    &   𝑅 = {𝑒 ∈ (ℕ0𝑚 ℙ) ∣ (𝑒 “ ℕ) ∈ Fin}       𝑧 ∈ ℕ ∃!𝑔𝑅 (𝑀𝑧) = 𝑔

6.2.12  Lagrange's four-square theorem

Syntaxcgz 16037 Extend class notation with the set of gaussian integers.
class ℤ[i]

Definitiondf-gz 16038 Define the set of gaussian integers, which are complex numbers whose real and imaginary parts are integers. (Note that the [i] is actually part of the symbol token and has no independent meaning.) (Contributed by Mario Carneiro, 14-Jul-2014.)
ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧ (ℑ‘𝑥) ∈ ℤ)}

Theoremelgz 16039 Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))

Theoremgzcn 16040 A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)

Theoremzgz 16041 An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℤ → 𝐴 ∈ ℤ[i])

Theoremigz 16042 i is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
i ∈ ℤ[i]

Theoremgznegcl 16043 The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℤ[i] → -𝐴 ∈ ℤ[i])

Theoremgzcjcl 16044 The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℤ[i] → (∗‘𝐴) ∈ ℤ[i])

Theoremgzaddcl 16045 The gaussian integers are closed under addition. (Contributed by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (𝐴 + 𝐵) ∈ ℤ[i])

Theoremgzmulcl 16046 The gaussian integers are closed under multiplication. (Contributed by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (𝐴 · 𝐵) ∈ ℤ[i])

Theoremgzreim 16047 Construct a gaussian integer from real and imaginary parts. (Contributed by Mario Carneiro, 16-Jul-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + (i · 𝐵)) ∈ ℤ[i])

Theoremgzsubcl 16048 The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (𝐴𝐵) ∈ ℤ[i])

Theoremgzabssqcl 16049 The squared norm of a gaussian integer is an integer. (Contributed by Mario Carneiro, 16-Jul-2014.)
(𝐴 ∈ ℤ[i] → ((abs‘𝐴)↑2) ∈ ℕ0)

Theorem4sqlem5 16050 Lemma for 4sq 16072. (Contributed by Mario Carneiro, 15-Jul-2014.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝑀 ∈ ℕ)    &   𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))       (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴𝐵) / 𝑀) ∈ ℤ))

Theorem4sqlem6 16051 Lemma for 4sq 16072. (Contributed by Mario Carneiro, 15-Jul-2014.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝑀 ∈ ℕ)    &   𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))       (𝜑 → (-(𝑀 / 2) ≤ 𝐵𝐵 < (𝑀 / 2)))

Theorem4sqlem7 16052 Lemma for 4sq 16072. (Contributed by Mario Carneiro, 15-Jul-2014.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝑀 ∈ ℕ)    &   𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))       (𝜑 → (𝐵↑2) ≤ (((𝑀↑2) / 2) / 2))

Theorem4sqlem8 16053 Lemma for 4sq 16072. (Contributed by Mario Carneiro, 15-Jul-2014.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝑀 ∈ ℕ)    &   𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))       (𝜑𝑀 ∥ ((𝐴↑2) − (𝐵↑2)))

Theorem4sqlem9 16054 Lemma for 4sq 16072. (Contributed by Mario Carneiro, 15-Jul-2014.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝑀 ∈ ℕ)    &   𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   ((𝜑𝜓) → (𝐵↑2) = 0)       ((𝜑𝜓) → (𝑀↑2) ∥ (𝐴↑2))

Theorem4sqlem10 16055 Lemma for 4sq 16072. (Contributed by Mario Carneiro, 16-Jul-2014.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝑀 ∈ ℕ)    &   𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   ((𝜑𝜓) → ((((𝑀↑2) / 2) / 2) − (𝐵↑2)) = 0)       ((𝜑𝜓) → (𝑀↑2) ∥ ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))

Theorem4sqlem1 16056* Lemma for 4sq 16072. The set 𝑆 is the set of all numbers that are expressible as a sum of four squares. Our goal is to show that 𝑆 = ℕ0; here we show one subset direction. (Contributed by Mario Carneiro, 14-Jul-2014.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}       𝑆 ⊆ ℕ0

Theorem4sqlem2 16057* Lemma for 4sq 16072. Change bound variables in 𝑆. (Contributed by Mario Carneiro, 14-Jul-2014.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}       (𝐴𝑆 ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))

Theorem4sqlem3 16058* Lemma for 4sq 16072. Sufficient condition to be in 𝑆. (Contributed by Mario Carneiro, 14-Jul-2014.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (((𝐴↑2) + (𝐵↑2)) + ((𝐶↑2) + (𝐷↑2))) ∈ 𝑆)

Theorem4sqlem4a 16059* Lemma for 4sqlem4 16060. (Contributed by Mario Carneiro, 14-Jul-2014.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}       ((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) ∈ 𝑆)

Theorem4sqlem4 16060* Lemma for 4sq 16072. We can express the four-square property more compactly in terms of gaussian integers, because the norms of gaussian integers are exactly sums of two squares. (Contributed by Mario Carneiro, 14-Jul-2014.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}       (𝐴𝑆 ↔ ∃𝑢 ∈ ℤ[i] ∃𝑣 ∈ ℤ[i] 𝐴 = (((abs‘𝑢)↑2) + ((abs‘𝑣)↑2)))

Theoremmul4sqlem 16061* Lemma for mul4sq 16062: algebraic manipulations. The extra assumptions involving 𝑀 are for a part of 4sqlem17 16069 which needs to know not just that the product is a sum of squares, but also that it preserves divisibility by 𝑀. (Contributed by Mario Carneiro, 14-Jul-2014.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}    &   (𝜑𝐴 ∈ ℤ[i])    &   (𝜑𝐵 ∈ ℤ[i])    &   (𝜑𝐶 ∈ ℤ[i])    &   (𝜑𝐷 ∈ ℤ[i])    &   𝑋 = (((abs‘𝐴)↑2) + ((abs‘𝐵)↑2))    &   𝑌 = (((abs‘𝐶)↑2) + ((abs‘𝐷)↑2))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → ((𝐴𝐶) / 𝑀) ∈ ℤ[i])    &   (𝜑 → ((𝐵𝐷) / 𝑀) ∈ ℤ[i])    &   (𝜑 → (𝑋 / 𝑀) ∈ ℕ0)       (𝜑 → ((𝑋 / 𝑀) · (𝑌 / 𝑀)) ∈ 𝑆)

Theoremmul4sq 16062* Euler's four-square identity: The product of two sums of four squares is also a sum of four squares. This is usually quoted as an explicit formula involving eight real variables; we save some time by working with complex numbers (gaussian integers) instead, so that we only have to work with four variables, and also hiding the actual formula for the product in the proof of mul4sqlem 16061. (For the curious, the explicit formula that is used is ( ∣ 𝑎 ∣ ↑2 + ∣ 𝑏 ∣ ↑2)( ∣ 𝑐 ∣ ↑2 + ∣ 𝑑 ∣ ↑2) = 𝑎∗ · 𝑐 + 𝑏 · 𝑑∗ ∣ ↑2 + ∣ 𝑎∗ · 𝑑𝑏 · 𝑐∗ ∣ ↑2.) (Contributed by Mario Carneiro, 14-Jul-2014.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}       ((𝐴𝑆𝐵𝑆) → (𝐴 · 𝐵) ∈ 𝑆)

Theorem4sqlem11 16063* Lemma for 4sq 16072. Use the pigeonhole principle to show that the sets {𝑚↑2 ∣ 𝑚 ∈ (0...𝑁)} and {-1 − 𝑛↑2 ∣ 𝑛 ∈ (0...𝑁)} have a common element, mod 𝑃. (Contributed by Mario Carneiro, 15-Jul-2014.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 = ((2 · 𝑁) + 1))    &   (𝜑𝑃 ∈ ℙ)    &   𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}    &   𝐹 = (𝑣𝐴 ↦ ((𝑃 − 1) − 𝑣))       (𝜑 → (𝐴 ∩ ran 𝐹) ≠ ∅)

Theorem4sqlem12 16064* Lemma for 4sq 16072. For any odd prime 𝑃, there is a 𝑘 < 𝑃 such that 𝑘𝑃 − 1 is a sum of two squares. (Contributed by Mario Carneiro, 15-Jul-2014.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 = ((2 · 𝑁) + 1))    &   (𝜑𝑃 ∈ ℙ)    &   𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}    &   𝐹 = (𝑣𝐴 ↦ ((𝑃 − 1) − 𝑣))       (𝜑 → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃))

Theorem4sqlem13 16065* Lemma for 4sq 16072. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 = ((2 · 𝑁) + 1))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (0...(2 · 𝑁)) ⊆ 𝑆)    &   𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}    &   𝑀 = inf(𝑇, ℝ, < )       (𝜑 → (𝑇 ≠ ∅ ∧ 𝑀 < 𝑃))

Theorem4sqlem14 16066* Lemma for 4sq 16072. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 = ((2 · 𝑁) + 1))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (0...(2 · 𝑁)) ⊆ 𝑆)    &   𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}    &   𝑀 = inf(𝑇, ℝ, < )    &   (𝜑𝑀 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   𝐸 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐹 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐺 = (((𝐶 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐻 = (((𝐷 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝑅 = ((((𝐸↑2) + (𝐹↑2)) + ((𝐺↑2) + (𝐻↑2))) / 𝑀)    &   (𝜑 → (𝑀 · 𝑃) = (((𝐴↑2) + (𝐵↑2)) + ((𝐶↑2) + (𝐷↑2))))       (𝜑𝑅 ∈ ℕ0)

Theorem4sqlem15 16067* Lemma for 4sq 16072. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 = ((2 · 𝑁) + 1))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (0...(2 · 𝑁)) ⊆ 𝑆)    &   𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}    &   𝑀 = inf(𝑇, ℝ, < )    &   (𝜑𝑀 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   𝐸 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐹 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐺 = (((𝐶 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐻 = (((𝐷 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝑅 = ((((𝐸↑2) + (𝐹↑2)) + ((𝐺↑2) + (𝐻↑2))) / 𝑀)    &   (𝜑 → (𝑀 · 𝑃) = (((𝐴↑2) + (𝐵↑2)) + ((𝐶↑2) + (𝐷↑2))))       ((𝜑𝑅 = 𝑀) → ((((((𝑀↑2) / 2) / 2) − (𝐸↑2)) = 0 ∧ ((((𝑀↑2) / 2) / 2) − (𝐹↑2)) = 0) ∧ (((((𝑀↑2) / 2) / 2) − (𝐺↑2)) = 0 ∧ ((((𝑀↑2) / 2) / 2) − (𝐻↑2)) = 0)))

Theorem4sqlem16 16068* Lemma for 4sq 16072. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 = ((2 · 𝑁) + 1))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (0...(2 · 𝑁)) ⊆ 𝑆)    &   𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}    &   𝑀 = inf(𝑇, ℝ, < )    &   (𝜑𝑀 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   𝐸 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐹 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐺 = (((𝐶 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐻 = (((𝐷 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝑅 = ((((𝐸↑2) + (𝐹↑2)) + ((𝐺↑2) + (𝐻↑2))) / 𝑀)    &   (𝜑 → (𝑀 · 𝑃) = (((𝐴↑2) + (𝐵↑2)) + ((𝐶↑2) + (𝐷↑2))))       (𝜑 → (𝑅𝑀 ∧ ((𝑅 = 0 ∨ 𝑅 = 𝑀) → (𝑀↑2) ∥ (𝑀 · 𝑃))))

Theorem4sqlem17 16069* Lemma for 4sq 16072. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 = ((2 · 𝑁) + 1))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (0...(2 · 𝑁)) ⊆ 𝑆)    &   𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}    &   𝑀 = inf(𝑇, ℝ, < )    &   (𝜑𝑀 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   𝐸 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐹 = (((𝐵 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐺 = (((𝐶 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝐻 = (((𝐷 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))    &   𝑅 = ((((𝐸↑2) + (𝐹↑2)) + ((𝐺↑2) + (𝐻↑2))) / 𝑀)    &   (𝜑 → (𝑀 · 𝑃) = (((𝐴↑2) + (𝐵↑2)) + ((𝐶↑2) + (𝐷↑2))))        ¬ 𝜑

Theorem4sqlem18 16070* Lemma for 4sq 16072. Inductive step, odd prime case. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 = ((2 · 𝑁) + 1))    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (0...(2 · 𝑁)) ⊆ 𝑆)    &   𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}    &   𝑀 = inf(𝑇, ℝ, < )       (𝜑𝑃𝑆)

Theorem4sqlem19 16071* Lemma for 4sq 16072. The proof is by strong induction - we show that if all the integers less than 𝑘 are in 𝑆, then 𝑘 is as well. In this part of the proof we do the induction argument and dispense with all the cases except the odd prime case, which is sent to 4sqlem18 16070. If 𝑘 is 0, 1, 2, we show 𝑘𝑆 directly; otherwise if 𝑘 is composite, 𝑘 is the product of two numbers less than it (and hence in 𝑆 by assumption), so by mul4sq 16062 𝑘𝑆. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}       0 = 𝑆

Theorem4sq 16072* Lagrange's four-square theorem, or Bachet's conjecture: every nonnegative integer is expressible as a sum of four squares. This is Metamath 100 proof #19. (Contributed by Mario Carneiro, 16-Jul-2014.)
(𝐴 ∈ ℕ0 ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))

6.2.13  Van der Waerden's theorem

Syntaxcvdwa 16073 The arithmetic progression function.
class AP

Syntaxcvdwm 16074 The monochromatic arithmetic progression predicate.
class MonoAP

Syntaxcvdwp 16075 The polychromatic arithmetic progression predicate.
class PolyAP

Definitiondf-vdwap 16076* Define the arithmetic progression function, which takes as input a length 𝑘, a start point 𝑎, and a step 𝑑 and outputs the set of points in this progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP = (𝑘 ∈ ℕ0 ↦ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))

Definitiondf-vdwmc 16077* Define the "contains a monochromatic AP" predicate. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP = {⟨𝑘, 𝑓⟩ ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (𝑓 “ {𝑐})) ≠ ∅}

Definitiondf-vdwpc 16078* Define the "contains a polychromatic collection of APs" predicate. See vdwpc 16088 for more information. (Contributed by Mario Carneiro, 18-Aug-2014.)
PolyAP = {⟨⟨𝑚, 𝑘⟩, 𝑓⟩ ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 (1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑𝑖))(AP‘𝑘)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = 𝑚)}

Theoremvdwapfval 16079* Define the arithmetic progression function, which takes as input a length 𝑘, a start point 𝑎, and a step 𝑑 and outputs the set of points in this progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝐾 ∈ ℕ0 → (AP‘𝐾) = (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran (𝑚 ∈ (0...(𝐾 − 1)) ↦ (𝑎 + (𝑚 · 𝑑)))))

Theoremvdwapf 16080 The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)

Theoremvdwapval 16081* Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐾 ∈ ℕ0𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑋 ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑋 = (𝐴 + (𝑚 · 𝐷))))

Theoremvdwapun 16082 Remove the first element of an arithmetic progression. (Contributed by Mario Carneiro, 11-Sep-2014.)
((𝐾 ∈ ℕ0𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘(𝐾 + 1))𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)))

Theoremvdwapid1 16083 The first element of an arithmetic progression. (Contributed by Mario Carneiro, 12-Sep-2014.)
((𝐾 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐴 ∈ (𝐴(AP‘𝐾)𝐷))

Theoremvdwap0 16084 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘0)𝐷) = ∅)

Theoremvdwap1 16085 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘1)𝐷) = {𝐴})

Theoremvdwmc 16086* The predicate " The 𝑅, 𝑁-coloring 𝐹 contains a monochromatic AP of length 𝐾". (Contributed by Mario Carneiro, 18-Aug-2014.)
𝑋 ∈ V    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐹:𝑋𝑅)       (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))

Theoremvdwmc2 16087* Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
𝑋 ∈ V    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐹:𝑋𝑅)    &   (𝜑𝐴𝑋)       (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})))

Theoremvdwpc 16088* The predicate " The coloring 𝐹 contains a polychromatic 𝑀-tuple of AP's of length 𝐾". A polychromatic 𝑀-tuple of AP's is a set of AP's with the same base point but different step lengths, such that each individual AP is monochromatic, but the AP's all have mutually distinct colors. (The common basepoint is not required to have the same color as any of the AP's.) (Contributed by Mario Carneiro, 18-Aug-2014.)
𝑋 ∈ V    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐹:𝑋𝑅)    &   (𝜑𝑀 ∈ ℕ)    &   𝐽 = (1...𝑀)       (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 𝐽)(∀𝑖𝐽 ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀)))

Theoremvdwlem1 16089* Lemma for vdw 16102. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝐹:(1...𝑊)⟶𝑅)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐷:(1...𝑀)⟶ℕ)    &   (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐴 + (𝐷𝑖))(AP‘𝐾)(𝐷𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝐴 + (𝐷𝑖)))}))    &   (𝜑𝐼 ∈ (1...𝑀))    &   (𝜑 → (𝐹𝐴) = (𝐹‘(𝐴 + (𝐷𝐼))))       (𝜑 → (𝐾 + 1) MonoAP 𝐹)

Theoremvdwlem2 16090* Lemma for vdw 16102. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)⟶𝑅)    &   (𝜑𝑀 ∈ (ℤ‘(𝑊 + 𝑁)))    &   𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))       (𝜑 → (𝐾 MonoAP 𝐺𝐾 MonoAP 𝐹))

Theoremvdwlem3 16091 Lemma for vdw 16102. (Contributed by Mario Carneiro, 13-Sep-2014.)
(𝜑𝑉 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝐴 ∈ (1...𝑉))    &   (𝜑𝐵 ∈ (1...𝑊))       (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))

Theoremvdwlem4 16092* Lemma for vdw 16102. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑉 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)    &   𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))       (𝜑𝐹:(1...𝑉)⟶(𝑅𝑚 (1...𝑊)))

Theoremvdwlem5 16093* Lemma for vdw 16102. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑉 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)    &   𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑊)⟶𝑅)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐸:(1...𝑀)⟶ℕ)    &   (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))    &   𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸𝑖))))    &   (𝜑 → (♯‘ran 𝐽) = 𝑀)    &   𝑇 = (𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)))    &   𝑃 = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)))       (𝜑𝑇 ∈ ℕ)

Theoremvdwlem6 16094* Lemma for vdw 16102. (Contributed by Mario Carneiro, 13-Sep-2014.)
(𝜑𝑉 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)    &   𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑊)⟶𝑅)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐸:(1...𝑀)⟶ℕ)    &   (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))    &   𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸𝑖))))    &   (𝜑 → (♯‘ran 𝐽) = 𝑀)    &   𝑇 = (𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)))    &   𝑃 = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)))       (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))

Theoremvdwlem7 16095* Lemma for vdw 16102. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑉 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)    &   𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑊)⟶𝑅)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))       (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐺 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))

Theoremvdwlem8 16096* Lemma for vdw 16102. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝐹:(1...(2 · 𝑊))⟶𝑅)    &   𝐶 ∈ V    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐺 “ {𝐶}))    &   𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑊)))       (𝜑 → ⟨1, 𝐾⟩ PolyAP 𝐹)

Theoremvdwlem9 16097* Lemma for vdw 16102. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑 → ∀𝑔 ∈ (𝑅𝑚 (1...𝑊))(⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))    &   (𝜑𝑉 ∈ ℕ)    &   (𝜑 → ∀𝑓 ∈ ((𝑅𝑚 (1...𝑊)) ↑𝑚 (1...𝑉))𝐾 MonoAP 𝑓)    &   (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)    &   𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))       (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻))

Theoremvdwlem10 16098* Lemma for vdw 16102. Set up secondary induction on 𝑀. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓)    &   (𝜑𝑀 ∈ ℕ)       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))

Theoremvdwlem11 16099* Lemma for vdw 16102. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓)       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(𝐾 + 1) MonoAP 𝑓)

Theoremvdwlem12 16100 Lemma for vdw 16102. 𝐾 = 2 base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)    &   (𝜑 → ¬ 2 MonoAP 𝐹)        ¬ 𝜑

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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 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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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