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Theorem List for Metamath Proof Explorer - 15601-15700   *Has distinct variable group(s)
TypeLabelDescription
Statement

6.2.10  Sum of prime reciprocals

Theoremprmreclem1 15601* Lemma for prmrec 15607. 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 15602* Lemma for prmrec 15607. 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 15603* Lemma for prmrec 15607. 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 15604* Lemma for prmrec 15607. 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 15461, 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 15605* Lemma for prmrec 15607. 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 15604 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 15606* Lemma for prmrec 15607. 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 15605 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 15607* 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 15608* Lemma for 1arith 15612. (Contributed by Mario Carneiro, 30-May-2014.)
𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))       (𝑁 ∈ ℕ → (𝑀𝑁) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑁)))

Theorem1arithlem2 15609* Lemma for 1arith 15612. (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 15610* Lemma for 1arith 15612. (Contributed by Mario Carneiro, 30-May-2014.)
𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)))       (𝑁 ∈ ℕ → (𝑀𝑁):ℙ⟶ℕ0)

Theorem1arithlem4 15611* Lemma for 1arith 15612. (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 15612* 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 15613* 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 15614 Extend class notation with the set of gaussian integers.
class ℤ[i]

Definitiondf-gz 15615 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 15616 Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℤ[i] ↔ (𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ ℤ ∧ (ℑ‘𝐴) ∈ ℤ))

Theoremgzcn 15617 A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℤ[i] → 𝐴 ∈ ℂ)

Theoremzgz 15618 An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℤ → 𝐴 ∈ ℤ[i])

Theoremigz 15619 i is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)
i ∈ ℤ[i]

Theoremgznegcl 15620 The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℤ[i] → -𝐴 ∈ ℤ[i])

Theoremgzcjcl 15621 The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℤ[i] → (∗‘𝐴) ∈ ℤ[i])

Theoremgzaddcl 15622 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 15623 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 15624 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 15625 The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014.)
((𝐴 ∈ ℤ[i] ∧ 𝐵 ∈ ℤ[i]) → (𝐴𝐵) ∈ ℤ[i])

Theoremgzabssqcl 15626 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 15627 Lemma for 4sq 15649. (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 15628 Lemma for 4sq 15649. (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 15629 Lemma for 4sq 15649. (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 15630 Lemma for 4sq 15649. (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 15631 Lemma for 4sq 15649. (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 15632 Lemma for 4sq 15649. (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 15633* Lemma for 4sq 15649. 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 15634* Lemma for 4sq 15649. 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 15635* Lemma for 4sq 15649. 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 15636* Lemma for 4sqlem4 15637. (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 15637* Lemma for 4sq 15649. 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 15638* Lemma for mul4sq 15639: algebraic manipulations. The extra assumptions involving 𝑀 are for a part of 4sqlem17 15646 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 15639* 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 15638. (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 15640* Lemma for 4sq 15649. 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 15641* Lemma for 4sq 15649. 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 15642* Lemma for 4sq 15649. (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 15643* Lemma for 4sq 15649. (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 15644* Lemma for 4sq 15649. (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 15645* Lemma for 4sq 15649. (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 15646* Lemma for 4sq 15649. (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 15647* Lemma for 4sq 15649. 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 15648* Lemma for 4sq 15649. 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 15647. 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 15639 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 15649* 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 15650 The arithmetic progression function.
class AP

Syntaxcvdwm 15651 The monochromatic arithmetic progression predicate.
class MonoAP

Syntaxcvdwp 15652 The polychromatic arithmetic progression predicate.
class PolyAP

Definitiondf-vdwap 15653* 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 15654* 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 15655* Define the "contains a polychromatic collection of APs" predicate. See vdwpc 15665 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 15656* 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 15657 The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)

Theoremvdwapval 15658* 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 15659 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 15660 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 15661 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘0)𝐷) = ∅)

Theoremvdwap1 15662 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘1)𝐷) = {𝐴})

Theoremvdwmc 15663* 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 15664* 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 15665* 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 15666* Lemma for vdw 15679. (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 15667* Lemma for vdw 15679. (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 15668 Lemma for vdw 15679. (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 15669* Lemma for vdw 15679. (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 15670* Lemma for vdw 15679. (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 15671* Lemma for vdw 15679. (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 15672* Lemma for vdw 15679. (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 15673* Lemma for vdw 15679. (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 15674* Lemma for vdw 15679. (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 15675* Lemma for vdw 15679. 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 15676* Lemma for vdw 15679. (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 15677 Lemma for vdw 15679. 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

Theoremvdwlem13 15678* Lemma for vdw 15679. Main induction on 𝐾; 𝐾 = 0, 𝐾 = 1 base cases. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ ℕ0)       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓)

Theoremvdw 15679* Van der Waerden's theorem. For any finite coloring 𝑅 and integer 𝐾, there is an 𝑁 such that every coloring function from 1...𝑁 to 𝑅 contains a monochromatic arithmetic progression (which written out in full means that there is a color 𝑐 and base, increment values 𝑎, 𝑑 such that all the numbers 𝑎, 𝑎 + 𝑑, ..., 𝑎 + (𝑘 − 1)𝑑 lie in the preimage of {𝑐}, i.e. they are all in 1...𝑁 and 𝑓 evaluated at each one yields 𝑐). (Contributed by Mario Carneiro, 13-Sep-2014.)
((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝑓 “ {𝑐}))

Theoremvdwnnlem1 15680* Corollary of vdw 15679, and lemma for vdwnn 15683. If 𝐹 is a coloring of the integers, then there are arbitrarily long monochromatic APs in 𝐹. (Contributed by Mario Carneiro, 13-Sep-2014.)
((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) → ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))

Theoremvdwnnlem2 15681* Lemma for vdwnn 15683. The set of all "bad" 𝑘 for the theorem is upwards-closed, because a long AP implies a short AP. (Contributed by Mario Carneiro, 13-Sep-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:ℕ⟶𝑅)    &   𝑆 = {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})}       ((𝜑𝐵 ∈ (ℤ𝐴)) → (𝐴𝑆𝐵𝑆))

Theoremvdwnnlem3 15682* Lemma for vdwnn 15683. (Contributed by Mario Carneiro, 13-Sep-2014.) (Proof shortened by AV, 27-Sep-2020.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:ℕ⟶𝑅)    &   𝑆 = {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})}    &   (𝜑 → ∀𝑐𝑅 𝑆 ≠ ∅)        ¬ 𝜑

Theoremvdwnn 15683* Van der Waerden's theorem, infinitary version. For any finite coloring 𝐹 of the positive integers, there is a color 𝑐 that contains arbitrarily long arithmetic progressions. (Contributed by Mario Carneiro, 13-Sep-2014.)
((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) → ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))

6.2.14  Ramsey's theorem

Syntaxcram 15684 Extend class notation with the Ramsey number function.
class Ramsey

Theoremramtlecl 15685* The set 𝑇 of numbers with the Ramsey number property is upward-closed. (Contributed by Mario Carneiro, 21-Apr-2015.)
𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → 𝜑)}       (𝑀𝑇 → (ℤ𝑀) ⊆ 𝑇)

Definitiondf-ram 15686* Define the Ramsey number function. The input is a number 𝑚 for the size of the edges of the hypergraph, and a tuple 𝑟 from the finite color set to lower bounds for each color. The Ramsey number (𝑀 Ramsey 𝑅) is the smallest number such that for any set 𝑆 with (𝑀 Ramsey 𝑅) ≤ #𝑆 and any coloring 𝐹 of the set of 𝑀-element subsets of 𝑆 (with color set dom 𝑅), there is a color 𝑐 ∈ dom 𝑅 and a subset 𝑥𝑆 such that 𝑅(𝑐) ≤ #𝑥 and all the hyperedges of 𝑥 (that is, subsets of 𝑥 of size 𝑀) have color 𝑐. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
Ramsey = (𝑚 ∈ ℕ0, 𝑟 ∈ V ↦ inf({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (dom 𝑟𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (#‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((#‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))}, ℝ*, < ))

Theoremhashbcval 15687* Value of the "binomial set", the set of all 𝑁-element subsets of 𝐴. (Contributed by Mario Carneiro, 20-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})       ((𝐴𝑉𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})

Theoremhashbccl 15688* The binomial set is a finite set. (Contributed by Mario Carneiro, 20-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})       ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) ∈ Fin)

Theoremhashbcss 15689* Subset relation for the binomial set. (Contributed by Mario Carneiro, 20-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})       ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) ⊆ (𝐴𝐶𝑁))

Theoremhashbc0 15690* The set of subsets of size zero is the singleton of the empty set. (Contributed by Mario Carneiro, 22-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})       (𝐴𝑉 → (𝐴𝐶0) = {∅})

Theoremhashbc2 15691* The size of the binomial set is the binomial coefficient. (Contributed by Mario Carneiro, 20-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})       ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (#‘(𝐴𝐶𝑁)) = ((#‘𝐴)C𝑁))

Theorem0hashbc 15692* There are no subsets of the empty set with size greater than zero. (Contributed by Mario Carneiro, 22-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})       (𝑁 ∈ ℕ → (∅𝐶𝑁) = ∅)

Theoremramval 15693* The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 14-Sep-2020.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})    &   𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}       ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ*, < ))

Theoremramcl2lem 15694* Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})    &   𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}       ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = if(𝑇 = ∅, +∞, inf(𝑇, ℝ, < )))

Theoremramtcl 15695* The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})    &   𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}       ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ 𝑇𝑇 ≠ ∅))

Theoremramtcl2 15696* The Ramsey number is an integer iff there is a number with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})    &   𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}       ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ ℕ0𝑇 ≠ ∅))

Theoremramtub 15697* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})    &   𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}       (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ 𝐴𝑇) → (𝑀 Ramsey 𝐹) ≤ 𝐴)

Theoremramub 15698* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 22-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑅𝑉)    &   (𝜑𝐹:𝑅⟶ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑 ∧ (𝑁 ≤ (#‘𝑠) ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))       (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁)

Theoremramub2 15699* It is sufficient to check the Ramsey property on finite sets of size equal to the upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑅𝑉)    &   (𝜑𝐹:𝑅⟶ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑 ∧ ((#‘𝑠) = 𝑁𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))       (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁)

Theoremrami 15700* The defining property of a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑅𝑉)    &   (𝜑𝐹:𝑅⟶ℕ0)    &   (𝜑 → (𝑀 Ramsey 𝐹) ∈ ℕ0)    &   (𝜑𝑆𝑊)    &   (𝜑 → (𝑀 Ramsey 𝐹) ≤ (#‘𝑆))    &   (𝜑𝐺:(𝑆𝐶𝑀)⟶𝑅)       (𝜑 → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐})))

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