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Theorem List for Metamath Proof Explorer - 15501-15600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem4sqlem15 15501* Lemma for 4sq 15506. (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 15502* Lemma for 4sq 15506. (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 15503* Lemma for 4sq 15506. (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 15504* Lemma for 4sq 15506. 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 15505* Lemma for 4sq 15506. 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 15504. 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 15496 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 15506* 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 15507 The arithmetic progression function.
class AP
 
Syntaxcvdwm 15508 The monochromatic arithmetic progression predicate.
class MonoAP
 
Syntaxcvdwp 15509 The polychromatic arithmetic progression predicate.
class PolyAP
 
Definitiondf-vdwap 15510* 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 15511* 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 15512* Define the "contains a polychromatic collection of APs" predicate. See vdwpc 15522 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 15513* 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 15514 The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
 
Theoremvdwapval 15515* 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 15516 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 15517 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 15518 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘0)𝐷) = ∅)
 
Theoremvdwap1 15519 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘1)𝐷) = {𝐴})
 
Theoremvdwmc 15520* 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 15521* 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 15522* 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 15523* Lemma for vdw 15536. (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 15524* Lemma for vdw 15536. (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 15525 Lemma for vdw 15536. (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 15526* Lemma for vdw 15536. (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 15527* Lemma for vdw 15536. (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 15528* Lemma for vdw 15536. (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 15529* Lemma for vdw 15536. (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 15530* Lemma for vdw 15536. (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 15531* Lemma for vdw 15536. (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 15532* Lemma for vdw 15536. 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 15533* Lemma for vdw 15536. (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 15534 Lemma for vdw 15536. 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 15535* Lemma for vdw 15536. 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 15536* 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 15537* Corollary of vdw 15536, and lemma for vdwnn 15540. 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 15538* Lemma for vdwnn 15540. 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 15539* Lemma for vdwnn 15540. (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 15540* 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 15541 Extend class notation with the Ramsey number function.
class Ramsey
 
Theoremramtlecl 15542* 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 15543* 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 15544* 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 15545* 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 15546* 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 15547* 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 15548* 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 15549* 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 15550* 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 15551* 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 15552* 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 15553* 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 15554* 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 15555* 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 15556* 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 15557* 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})))
 
Theoremramcl2 15558 The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞}))
 
Theoremramxrcl 15559 The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 15571.) (Contributed by Mario Carneiro, 20-Apr-2015.)
((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℝ*)
 
Theoremramubcl 15560 If the Ramsey number is upper bounded, then it is an integer. (Contributed by Mario Carneiro, 20-Apr-2015.)
(((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → (𝑀 Ramsey 𝐹) ∈ ℕ0)
 
Theoremramlb 15561* Establish a lower bound on 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)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐺:((1...𝑁)𝐶𝑀)⟶𝑅)    &   ((𝜑 ∧ (𝑐𝑅𝑥 ⊆ (1...𝑁))) → ((𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐}) → (#‘𝑥) < (𝐹𝑐)))       (𝜑𝑁 < (𝑀 Ramsey 𝐹))
 
Theorem0ram 15562* The Ramsey number when 𝑀 = 0. (Contributed by Mario Carneiro, 22-Apr-2015.)
(((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) = sup(ran 𝐹, ℝ, < ))
 
Theorem0ram2 15563 The Ramsey number when 𝑀 = 0. (Contributed by Mario Carneiro, 22-Apr-2015.)
((𝑅 ∈ Fin ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) → (0 Ramsey 𝐹) = sup(ran 𝐹, ℝ, < ))
 
Theoremram0 15564 The Ramsey number when 𝑅 = ∅. (Contributed by Mario Carneiro, 22-Apr-2015.)
(𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) = 𝑀)
 
Theorem0ramcl 15565 Lemma for ramcl 15571: Existence of the Ramsey number when 𝑀 = 0. (Contributed by Mario Carneiro, 23-Apr-2015.)
((𝑅 ∈ Fin ∧ 𝐹:𝑅⟶ℕ0) → (0 Ramsey 𝐹) ∈ ℕ0)
 
Theoremramz2 15566 The Ramsey number when 𝐹 has value zero for some color 𝐶. (Contributed by Mario Carneiro, 22-Apr-2015.)
(((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → (𝑀 Ramsey 𝐹) = 0)
 
Theoremramz 15567 The Ramsey number when 𝐹 is the zero function. (Contributed by Mario Carneiro, 22-Apr-2015.)
((𝑀 ∈ ℕ0𝑅𝑉𝑅 ≠ ∅) → (𝑀 Ramsey (𝑅 × {0})) = 0)
 
Theoremramub1lem1 15568* Lemma for ramub1 15570. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:𝑅⟶ℕ)    &   𝐺 = (𝑥𝑅 ↦ (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦)))))    &   (𝜑𝐺:𝑅⟶ℕ0)    &   (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0)    &   𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})    &   (𝜑𝑆 ∈ Fin)    &   (𝜑 → (#‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))    &   (𝜑𝐾:(𝑆𝐶𝑀)⟶𝑅)    &   (𝜑𝑋𝑆)    &   𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))    &   (𝜑𝐷𝑅)    &   (𝜑𝑊 ⊆ (𝑆 ∖ {𝑋}))    &   (𝜑 → (𝐺𝐷) ≤ (#‘𝑊))    &   (𝜑 → (𝑊𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝐷}))    &   (𝜑𝐸𝑅)    &   (𝜑𝑉𝑊)    &   (𝜑 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (#‘𝑉))    &   (𝜑 → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))       (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))
 
Theoremramub1lem2 15569* Lemma for ramub1 15570. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:𝑅⟶ℕ)    &   𝐺 = (𝑥𝑅 ↦ (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦)))))    &   (𝜑𝐺:𝑅⟶ℕ0)    &   (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0)    &   𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})    &   (𝜑𝑆 ∈ Fin)    &   (𝜑 → (#‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))    &   (𝜑𝐾:(𝑆𝐶𝑀)⟶𝑅)    &   (𝜑𝑋𝑆)    &   𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))       (𝜑 → ∃𝑐𝑅𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (#‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐})))
 
Theoremramub1 15570* Inductive step for Ramsey's theorem, in the form of an explicit upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:𝑅⟶ℕ)    &   𝐺 = (𝑥𝑅 ↦ (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦)))))    &   (𝜑𝐺:𝑅⟶ℕ0)    &   (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0)       (𝜑 → (𝑀 Ramsey 𝐹) ≤ (((𝑀 − 1) Ramsey 𝐺) + 1))
 
Theoremramcl 15571 Ramsey's theorem: the Ramsey number is an integer for every finite coloring and set of upper bounds. (Contributed by Mario Carneiro, 23-Apr-2015.)
((𝑀 ∈ ℕ0𝑅 ∈ Fin ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℕ0)
 
Theoremramsey 15572* Ramsey's theorem with the definition Ramsey eliminated. If 𝑀 is an integer, 𝑅 is a specified finite set of colors, and 𝐹:𝑅⟶ℕ0 is a set of lower bounds for each color, then there is an 𝑛 such that for every set 𝑠 of size greater than 𝑛 and every coloring 𝑓 of the set (𝑠𝐶𝑀) of all 𝑀-element subsets of 𝑠, there is a color 𝑐 and a subset 𝑥𝑠 such that 𝑥 is larger than 𝐹(𝑐) and the 𝑀-element subsets of 𝑥 are monochromatic with color 𝑐. This is the hypergraph version of Ramsey's theorem; the version for simple graphs is the case 𝑀 = 2. This is Metamath 100 proof #31. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})       ((𝑀 ∈ ℕ0𝑅 ∈ Fin ∧ 𝐹:𝑅⟶ℕ0) → ∃𝑛 ∈ ℕ0𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (𝑅𝑚 (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (#‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
 
6.2.15  Primorial function

According to Wikipedia "Primorial", https://en.wikipedia.org/wiki/Primorial (28-Aug-2020): "In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying [all] positive integers [less than or equal to a given number], the function only multiplies [the] prime numbers [less than or equal to the given number]. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors."

 
Syntaxcprmo 15573 Extend class notation to include the primorial of nonnegative integers.
class #p
 
Definitiondf-prmo 15574* Define the primorial function on nonnegative integers as the product of all prime numbers less than or equal to the integer. For example, (#p‘10) = 2 · 3 · 5 · 7 = 210 (see ex-prmo 26708).

In the literature, the primorial function is written as a postscript hash: 6# = 30. In contrast to prmorcht 24704, where the primorial function is defined by using the sequence builder (𝑃 = seq1( · , 𝐹)), the more specialized definition of a product of a series is used here. (Contributed by AV, 28-Aug-2020.)

#p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
 
Theoremprmoval 15575* Value of the primorial function for nonnegative integers. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
 
Theoremprmocl 15576 Closure of the primorial function. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ∈ ℕ)
 
Theoremprmone0 15577 The primorial function is nonzero. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ≠ 0)
 
Theoremprmo0 15578 The primorial of 0. (Contributed by AV, 28-Aug-2020.)
(#p‘0) = 1
 
Theoremprmo1 15579 The primorial of 1. (Contributed by AV, 28-Aug-2020.)
(#p‘1) = 1
 
Theoremprmop1 15580 The primorial of a successor. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p‘(𝑁 + 1)) = if((𝑁 + 1) ∈ ℙ, ((#p𝑁) · (𝑁 + 1)), (#p𝑁)))
 
Theoremprmonn2 15581 Value of the primorial function expressed recursively. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ → (#p𝑁) = if(𝑁 ∈ ℙ, ((#p‘(𝑁 − 1)) · 𝑁), (#p‘(𝑁 − 1))))
 
Theoremprmo2 15582 The primorial of 2. (Contributed by AV, 28-Aug-2020.)
(#p‘2) = 2
 
Theoremprmo3 15583 The primorial of 3. (Contributed by AV, 28-Aug-2020.)
(#p‘3) = 6
 
Theoremprmdvdsprmo 15584* The primorial of a number is divisible by each prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ → ∀𝑝 ∈ ℙ (𝑝𝑁𝑝 ∥ (#p𝑁)))
 
Theoremprmdvdsprmop 15585* The primorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by a prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 28-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ∃𝑝 ∈ ℙ (𝑝𝑁𝑝𝐼𝑝 ∥ ((#p𝑁) + 𝐼)))
 
Theoremfvprmselelfz 15586* The value of the prime selection function is in a finite sequence of integers if the argument is in this finite sequence of integers. (Contributed by AV, 19-Aug-2020.)
𝐹 = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))       ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁)) → (𝐹𝑋) ∈ (1...𝑁))
 
Theoremfvprmselgcd1 15587* The greatest common divisor of two values of the prime selection function for different arguments is 1. (Contributed by AV, 19-Aug-2020.)
𝐹 = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))       ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
 
Theoremprmolefac 15588 The primorial of a positive integer is less than or equal to the factorial of the integer. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ≤ (!‘𝑁))
 
Theoremprmodvdslcmf 15589 The primorial of a nonnegative integer divides the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 29-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ∥ (lcm‘(1...𝑁)))
 
Theoremprmolelcmf 15590 The primorial of a positive integer is less than or equal to the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 29-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ≤ (lcm‘(1...𝑁)))
 
6.2.16  Prime gaps

According to Wikipedia "Prime gap", https://en.wikipedia.org/wiki/Prime_gap (16-Aug-2020): "A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n+1)-th and the n-th prime numbers, i.e. gn = pn+1 - pn . We have g1 = 1, g2 = g3 = 2, and g4 = 4."

It can be proven that there are arbitrary large gaps, usually by showing that "in the sequence n!+2, n!+3, ..., n!+n the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of n-1 consecutive composite integers, and it must belong to a gap between primes having length at least n.", see prmgap 15601.

Instead of using the factorial of n (see df-fac 12923), any function can be chosen for which f(n) is not relatively prime to the integers 2, 3, ..., n. For example, the least common multiple of the integers 2, 3, ..., n, see prmgaplcm 15602, or the primorial n# (the product of all prime numbers less than or equal to n), see prmgapprmo 15604, are such functions, which provide smaller values than the factorial function (see lcmflefac 15199 and prmolefac 15588 resp. prmolelcmf 15590): "For instance, the first prime gap of size larger than 14 occurs between the primes 523 and 541, while 15! is the vastly larger number 1307674368000." But the least common multiple of the integers 2, 3, ..., 15 is 360360, and 15# is 30030 (p3248 = 30029 and P3249 = 30047, so g3248 = 18).

 
Theoremprmgaplem1 15591 Lemma for prmgap 15601: The factorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 13-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((!‘𝑁) + 𝐼))
 
Theoremprmgaplem2 15592 Lemma for prmgap 15601: The factorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 13-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((!‘𝑁) + 𝐼) gcd 𝐼))
 
Theoremprmgaplcmlem1 15593 Lemma for prmgaplcm 15602: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼))
 
Theoremprmgaplcmlem2 15594 Lemma for prmgaplcm 15602: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((lcm‘(1...𝑁)) + 𝐼) gcd 𝐼))
 
Theoremprmgaplem3 15595* Lemma for prmgap 15601. (Contributed by AV, 9-Aug-2020.)
𝐴 = {𝑝 ∈ ℙ ∣ 𝑝 < 𝑁}       (𝑁 ∈ (ℤ‘3) → ∃𝑥𝐴𝑦𝐴 𝑦𝑥)
 
Theoremprmgaplem4 15596* Lemma for prmgap 15601. (Contributed by AV, 10-Aug-2020.)
𝐴 = {𝑝 ∈ ℙ ∣ (𝑁 < 𝑝𝑝𝑃)}       ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑁 < 𝑃) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
 
Theoremprmgaplem5 15597* Lemma for prmgap 15601: for each integer greater than 2 there is a smaller prime closest to this integer, i.e. there is a smaller prime and no other prime is between this prime and the integer. (Contributed by AV, 9-Aug-2020.)
(𝑁 ∈ (ℤ‘3) → ∃𝑝 ∈ ℙ (𝑝 < 𝑁 ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑁)𝑧 ∉ ℙ))
 
Theoremprmgaplem6 15598* Lemma for prmgap 15601: for each positive integer there is a greater prime closest to this integer, i.e. there is a greater prime and no other prime is between this prime and the integer. (Contributed by AV, 10-Aug-2020.)
(𝑁 ∈ ℕ → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ ∀𝑧 ∈ ((𝑁 + 1)..^𝑝)𝑧 ∉ ℙ))
 
Theoremprmgaplem7 15599* Lemma for prmgap 15601. (Contributed by AV, 12-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹 ∈ (ℕ ↑𝑚 ℕ))    &   (𝜑 → ∀𝑖 ∈ (2...𝑁)1 < (((𝐹𝑁) + 𝑖) gcd 𝑖))       (𝜑 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 < ((𝐹𝑁) + 2) ∧ ((𝐹𝑁) + 𝑁) < 𝑞 ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))
 
Theoremprmgaplem8 15600* Lemma for prmgap 15601. (Contributed by AV, 13-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹 ∈ (ℕ ↑𝑚 ℕ))    &   (𝜑 → ∀𝑖 ∈ (2...𝑁)1 < (((𝐹𝑁) + 𝑖) gcd 𝑖))       (𝜑 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑁 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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