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Type | Label | Description |
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Statement | ||
Theorem | 4sqlem19 16301* | Lemma for 4sq 16302. 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 16300. If 𝑘 is 0, 1, 2, we show 𝑘 ∈ 𝑆 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 16292 𝑘 ∈ 𝑆. (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)))} ⇒ ⊢ ℕ0 = 𝑆 | ||
Theorem | 4sq 16302* | 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)))) | ||
Syntax | cvdwa 16303 | The arithmetic progression function. |
class AP | ||
Syntax | cvdwm 16304 | The monochromatic arithmetic progression predicate. |
class MonoAP | ||
Syntax | cvdwp 16305 | The polychromatic arithmetic progression predicate. |
class PolyAP | ||
Definition | df-vdwap 16306* | 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))))) | ||
Definition | df-vdwmc 16307* | Define the "contains a monochromatic AP" predicate. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ MonoAP = {〈𝑘, 𝑓〉 ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅} | ||
Definition | df-vdwpc 16308* | Define the "contains a polychromatic collection of APs" predicate. See vdwpc 16318 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 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = 𝑚)} | ||
Theorem | vdwapfval 16309* | 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))))) | ||
Theorem | vdwapf 16310 | The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ) | ||
Theorem | vdwapval 16311* | Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑋 ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑋 = (𝐴 + (𝑚 · 𝐷)))) | ||
Theorem | vdwapun 16312 | Remove the first element of an arithmetic progression. (Contributed by Mario Carneiro, 11-Sep-2014.) |
⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘(𝐾 + 1))𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷))) | ||
Theorem | vdwapid1 16313 | 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)) | ||
Theorem | vdwap0 16314 | Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘0)𝐷) = ∅) | ||
Theorem | vdwap1 16315 | Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘1)𝐷) = {𝐴}) | ||
Theorem | vdwmc 16316* | The predicate " The 〈𝑅, 𝑁〉-coloring 𝐹 contains a monochromatic AP of length 𝐾". (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ 𝑋 ∈ V & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑅) ⇒ ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))) | ||
Theorem | vdwmc2 16317* | Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ 𝑋 ∈ V & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑅) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}))) | ||
Theorem | vdwpc 16318* | 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 & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑅) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ 𝐽 = (1...𝑀) ⇒ ⊢ (𝜑 → (〈𝑀, 𝐾〉 PolyAP 𝐹 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m 𝐽)(∀𝑖 ∈ 𝐽 ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ 𝐽 ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 𝑀))) | ||
Theorem | vdwlem1 16319* | Lemma for vdw 16332. (Contributed by Mario Carneiro, 12-Sep-2014.) |
⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑊)⟶𝑅) & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐷:(1...𝑀)⟶ℕ) & ⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐴 + (𝐷‘𝑖))(AP‘𝐾)(𝐷‘𝑖)) ⊆ (◡𝐹 “ {(𝐹‘(𝐴 + (𝐷‘𝑖)))})) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) & ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘(𝐴 + (𝐷‘𝐼)))) ⇒ ⊢ (𝜑 → (𝐾 + 1) MonoAP 𝐹) | ||
Theorem | vdwlem2 16320* | Lemma for vdw 16332. (Contributed by Mario Carneiro, 12-Sep-2014.) |
⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → 𝑊 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑀)⟶𝑅) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(𝑊 + 𝑁))) & ⊢ 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁))) ⇒ ⊢ (𝜑 → (𝐾 MonoAP 𝐺 → 𝐾 MonoAP 𝐹)) | ||
Theorem | vdwlem3 16321 | Lemma for vdw 16332. (Contributed by Mario Carneiro, 13-Sep-2014.) |
⊢ (𝜑 → 𝑉 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ (1...𝑉)) & ⊢ (𝜑 → 𝐵 ∈ (1...𝑊)) ⇒ ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉)))) | ||
Theorem | vdwlem4 16322* | Lemma for vdw 16332. (Contributed by Mario Carneiro, 12-Sep-2014.) |
⊢ (𝜑 → 𝑉 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅) & ⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))))) ⇒ ⊢ (𝜑 → 𝐹:(1...𝑉)⟶(𝑅 ↑m (1...𝑊))) | ||
Theorem | vdwlem5 16323* | Lemma for vdw 16332. (Contributed by Mario Carneiro, 12-Sep-2014.) |
⊢ (𝜑 → 𝑉 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅) & ⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℕ) & ⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (◡𝐹 “ {𝐺})) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐸:(1...𝑀)⟶ℕ) & ⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)) ⊆ (◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))})) & ⊢ 𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸‘𝑖)))) & ⊢ (𝜑 → (♯‘ran 𝐽) = 𝑀) & ⊢ 𝑇 = (𝐵 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) & ⊢ 𝑃 = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) + (𝑊 · 𝐷))) ⇒ ⊢ (𝜑 → 𝑇 ∈ ℕ) | ||
Theorem | vdwlem6 16324* | Lemma for vdw 16332. (Contributed by Mario Carneiro, 13-Sep-2014.) |
⊢ (𝜑 → 𝑉 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅) & ⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℕ) & ⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (◡𝐹 “ {𝐺})) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → 𝐸:(1...𝑀)⟶ℕ) & ⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸‘𝑖))(AP‘𝐾)(𝐸‘𝑖)) ⊆ (◡𝐺 “ {(𝐺‘(𝐵 + (𝐸‘𝑖)))})) & ⊢ 𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸‘𝑖)))) & ⊢ (𝜑 → (♯‘ran 𝐽) = 𝑀) & ⊢ 𝑇 = (𝐵 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) & ⊢ 𝑃 = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝐸‘𝑗)) + (𝑊 · 𝐷))) ⇒ ⊢ (𝜑 → (〈(𝑀 + 1), 𝐾〉 PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)) | ||
Theorem | vdwlem7 16325* | Lemma for vdw 16332. (Contributed by Mario Carneiro, 12-Sep-2014.) |
⊢ (𝜑 → 𝑉 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅) & ⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℕ) & ⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (◡𝐹 “ {𝐺})) ⇒ ⊢ (𝜑 → (〈𝑀, 𝐾〉 PolyAP 𝐺 → (〈(𝑀 + 1), 𝐾〉 PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))) | ||
Theorem | vdwlem8 16326* | Lemma for vdw 16332. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → 𝑊 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...(2 · 𝑊))⟶𝑅) & ⊢ 𝐶 ∈ V & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℕ) & ⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (◡𝐺 “ {𝐶})) & ⊢ 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑊))) ⇒ ⊢ (𝜑 → 〈1, 𝐾〉 PolyAP 𝐹) | ||
Theorem | vdwlem9 16327* | Lemma for vdw 16332. (Contributed by Mario Carneiro, 12-Sep-2014.) |
⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑m (1...𝑛))𝐾 MonoAP 𝑓) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ ℕ) & ⊢ (𝜑 → ∀𝑔 ∈ (𝑅 ↑m (1...𝑊))(〈𝑀, 𝐾〉 PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) & ⊢ (𝜑 → 𝑉 ∈ ℕ) & ⊢ (𝜑 → ∀𝑓 ∈ ((𝑅 ↑m (1...𝑊)) ↑m (1...𝑉))𝐾 MonoAP 𝑓) & ⊢ (𝜑 → 𝐻:(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)) | ||
Theorem | vdwlem10 16328* | Lemma for vdw 16332. Set up secondary induction on 𝑀. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑m (1...𝑛))𝐾 MonoAP 𝑓) & ⊢ (𝜑 → 𝑀 ∈ ℕ) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))(〈𝑀, 𝐾〉 PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) | ||
Theorem | vdwlem11 16329* | Lemma for vdw 16332. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2)) & ⊢ (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑m (1...𝑛))𝐾 MonoAP 𝑓) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))(𝐾 + 1) MonoAP 𝑓) | ||
Theorem | vdwlem12 16330 | Lemma for vdw 16332. 𝐾 = 2 base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐹:(1...((♯‘𝑅) + 1))⟶𝑅) & ⊢ (𝜑 → ¬ 2 MonoAP 𝐹) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | vdwlem13 16331* | Lemma for vdw 16332. Main induction on 𝐾; 𝐾 = 0, 𝐾 = 1 base cases. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))𝐾 MonoAP 𝑓) | ||
Theorem | vdw 16332* | 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 ∈ (𝑅 ↑m (1...𝑛))∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐})) | ||
Theorem | vdwnnlem1 16333* | Corollary of vdw 16332, and lemma for vdwnn 16336. 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⟶𝑅 ∧ 𝐾 ∈ ℕ0) → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) | ||
Theorem | vdwnnlem2 16334* | Lemma for vdwnn 16336. 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.) |
⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑅) & ⊢ 𝑆 = {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})} ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐴 ∈ 𝑆 → 𝐵 ∈ 𝑆)) | ||
Theorem | vdwnnlem3 16335* | Lemma for vdwnn 16336. (Contributed by Mario Carneiro, 13-Sep-2014.) (Proof shortened by AV, 27-Sep-2020.) |
⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑅) & ⊢ 𝑆 = {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})} & ⊢ (𝜑 → ∀𝑐 ∈ 𝑅 𝑆 ≠ ∅) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | vdwnn 16336* | 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) → ∃𝑐 ∈ 𝑅 ∀𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) | ||
Syntax | cram 16337 | Extend class notation with the Ramsey number function. |
class Ramsey | ||
Theorem | ramtlecl 16338* | 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)} ⇒ ⊢ (𝑀 ∈ 𝑇 → (ℤ≥‘𝑀) ⊆ 𝑇) | ||
Definition | df-ram 16339* | 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 𝑥 ⊆ 𝑆 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 𝑟 ↑m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)))}, ℝ*, < )) | ||
Theorem | hashbcval 16340* | 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}) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁}) | ||
Theorem | hashbccl 16341* | 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) | ||
Theorem | hashbcss 16342* | Subset relation for the binomial set. (Contributed by Mario Carneiro, 20-Apr-2015.) |
⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) ⊆ (𝐴𝐶𝑁)) | ||
Theorem | hashbc0 16343* | 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}) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴𝐶0) = {∅}) | ||
Theorem | hashbc2 16344* | 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)) | ||
Theorem | 0hashbc 16345* | 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) = ∅) | ||
Theorem | ramval 16346* | 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 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} ⇒ ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ*, < )) | ||
Theorem | ramcl2lem 16347* | 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 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} ⇒ ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = if(𝑇 = ∅, +∞, inf(𝑇, ℝ, < ))) | ||
Theorem | ramtcl 16348* | 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 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} ⇒ ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ 𝑇 ↔ 𝑇 ≠ ∅)) | ||
Theorem | ramtcl2 16349* | 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 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} ⇒ ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ ℕ0 ↔ 𝑇 ≠ ∅)) | ||
Theorem | ramtub 16350* | 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 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))} ⇒ ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ 𝐴 ∈ 𝑇) → (𝑀 Ramsey 𝐹) ≤ 𝐴) | ||
Theorem | ramub 16351* | 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}) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑅⟶ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ ((𝜑 ∧ (𝑁 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) ⇒ ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁) | ||
Theorem | ramub2 16352* | 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}) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑅⟶ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ ((𝜑 ∧ ((♯‘𝑠) = 𝑁 ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐}))) ⇒ ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁) | ||
Theorem | rami 16353* | The defining property of a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.) |
⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑅⟶ℕ0) & ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ≤ (♯‘𝑆)) & ⊢ (𝜑 → 𝐺:(𝑆𝐶𝑀)⟶𝑅) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐}))) | ||
Theorem | ramcl2 16354 | 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 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞})) | ||
Theorem | ramxrcl 16355 | The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 16367.) (Contributed by Mario Carneiro, 20-Apr-2015.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℝ*) | ||
Theorem | ramubcl 16356 | If the Ramsey number is upper bounded, then it is an integer. (Contributed by Mario Carneiro, 20-Apr-2015.) |
⊢ (((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → (𝑀 Ramsey 𝐹) ∈ ℕ0) | ||
Theorem | ramlb 16357* | 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}) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑅⟶ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐺:((1...𝑁)𝐶𝑀)⟶𝑅) & ⊢ ((𝜑 ∧ (𝑐 ∈ 𝑅 ∧ 𝑥 ⊆ (1...𝑁))) → ((𝑥𝐶𝑀) ⊆ (◡𝐺 “ {𝑐}) → (♯‘𝑥) < (𝐹‘𝑐))) ⇒ ⊢ (𝜑 → 𝑁 < (𝑀 Ramsey 𝐹)) | ||
Theorem | 0ram 16358* | The Ramsey number when 𝑀 = 0. (Contributed by Mario Carneiro, 22-Apr-2015.) |
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → (0 Ramsey 𝐹) = sup(ran 𝐹, ℝ, < )) | ||
Theorem | 0ram2 16359 | 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, < )) | ||
Theorem | ram0 16360 | The Ramsey number when 𝑅 = ∅. (Contributed by Mario Carneiro, 22-Apr-2015.) |
⊢ (𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) = 𝑀) | ||
Theorem | 0ramcl 16361 | Lemma for ramcl 16367: Existence of the Ramsey number when 𝑀 = 0. (Contributed by Mario Carneiro, 23-Apr-2015.) |
⊢ ((𝑅 ∈ Fin ∧ 𝐹:𝑅⟶ℕ0) → (0 Ramsey 𝐹) ∈ ℕ0) | ||
Theorem | ramz2 16362 | The Ramsey number when 𝐹 has value zero for some color 𝐶. (Contributed by Mario Carneiro, 22-Apr-2015.) |
⊢ (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) ∧ (𝐶 ∈ 𝑅 ∧ (𝐹‘𝐶) = 0)) → (𝑀 Ramsey 𝐹) = 0) | ||
Theorem | ramz 16363 | The Ramsey number when 𝐹 is the zero function. (Contributed by Mario Carneiro, 22-Apr-2015.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝑅 ≠ ∅) → (𝑀 Ramsey (𝑅 × {0})) = 0) | ||
Theorem | ramub1lem1 16364* | Lemma for ramub1 16366. (Contributed by Mario Carneiro, 23-Apr-2015.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝑅⟶ℕ) & ⊢ 𝐺 = (𝑥 ∈ 𝑅 ↦ (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦))))) & ⊢ (𝜑 → 𝐺:𝑅⟶ℕ0) & ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0) & ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) & ⊢ (𝜑 → 𝑆 ∈ Fin) & ⊢ (𝜑 → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1)) & ⊢ (𝜑 → 𝐾:(𝑆𝐶𝑀)⟶𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋}))) & ⊢ (𝜑 → 𝐷 ∈ 𝑅) & ⊢ (𝜑 → 𝑊 ⊆ (𝑆 ∖ {𝑋})) & ⊢ (𝜑 → (𝐺‘𝐷) ≤ (♯‘𝑊)) & ⊢ (𝜑 → (𝑊𝐶(𝑀 − 1)) ⊆ (◡𝐻 “ {𝐷})) & ⊢ (𝜑 → 𝐸 ∈ 𝑅) & ⊢ (𝜑 → 𝑉 ⊆ 𝑊) & ⊢ (𝜑 → if(𝐸 = 𝐷, ((𝐹‘𝐷) − 1), (𝐹‘𝐸)) ≤ (♯‘𝑉)) & ⊢ (𝜑 → (𝑉𝐶𝑀) ⊆ (◡𝐾 “ {𝐸})) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝐸}))) | ||
Theorem | ramub1lem2 16365* | Lemma for ramub1 16366. (Contributed by Mario Carneiro, 23-Apr-2015.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝑅⟶ℕ) & ⊢ 𝐺 = (𝑥 ∈ 𝑅 ↦ (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦))))) & ⊢ (𝜑 → 𝐺:𝑅⟶ℕ0) & ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0) & ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) & ⊢ (𝜑 → 𝑆 ∈ Fin) & ⊢ (𝜑 → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1)) & ⊢ (𝜑 → 𝐾:(𝑆𝐶𝑀)⟶𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ 𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋}))) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ 𝑅 ∃𝑧 ∈ 𝒫 𝑆((𝐹‘𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (◡𝐾 “ {𝑐}))) | ||
Theorem | ramub1 16366* | Inductive step for Ramsey's theorem, in the form of an explicit upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.) |
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝑅⟶ℕ) & ⊢ 𝐺 = (𝑥 ∈ 𝑅 ↦ (𝑀 Ramsey (𝑦 ∈ 𝑅 ↦ if(𝑦 = 𝑥, ((𝐹‘𝑥) − 1), (𝐹‘𝑦))))) & ⊢ (𝜑 → 𝐺:𝑅⟶ℕ0) & ⊢ (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑀 Ramsey 𝐹) ≤ (((𝑀 − 1) Ramsey 𝐺) + 1)) | ||
Theorem | ramcl 16367 | 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 ∈ ℕ0 ∧ 𝑅 ∈ Fin ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℕ0) | ||
Theorem | ramsey 16368* | 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 𝑥 ⊆ 𝑠 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 ∈ ℕ0 ∧ 𝑅 ∈ Fin ∧ 𝐹:𝑅⟶ℕ0) → ∃𝑛 ∈ ℕ0 ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅 ↑m (𝑠𝐶𝑀))∃𝑐 ∈ 𝑅 ∃𝑥 ∈ 𝒫 𝑠((𝐹‘𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (◡𝑓 “ {𝑐})))) | ||
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." | ||
Syntax | cprmo 16369 | Extend class notation to include the primorial of nonnegative integers. |
class #p | ||
Definition | df-prmo 16370* |
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 28240).
In the literature, the primorial function is written as a postscript hash: 6# = 30. In contrast to prmorcht 25757, 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)) | ||
Theorem | prmoval 16371* | Value of the primorial function for nonnegative integers. (Contributed by AV, 28-Aug-2020.) |
⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1)) | ||
Theorem | prmocl 16372 | Closure of the primorial function. (Contributed by AV, 28-Aug-2020.) |
⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ∈ ℕ) | ||
Theorem | prmone0 16373 | The primorial function is nonzero. (Contributed by AV, 28-Aug-2020.) |
⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ≠ 0) | ||
Theorem | prmo0 16374 | The primorial of 0. (Contributed by AV, 28-Aug-2020.) |
⊢ (#p‘0) = 1 | ||
Theorem | prmo1 16375 | The primorial of 1. (Contributed by AV, 28-Aug-2020.) |
⊢ (#p‘1) = 1 | ||
Theorem | prmop1 16376 | The primorial of a successor. (Contributed by AV, 28-Aug-2020.) |
⊢ (𝑁 ∈ ℕ0 → (#p‘(𝑁 + 1)) = if((𝑁 + 1) ∈ ℙ, ((#p‘𝑁) · (𝑁 + 1)), (#p‘𝑁))) | ||
Theorem | prmonn2 16377 | Value of the primorial function expressed recursively. (Contributed by AV, 28-Aug-2020.) |
⊢ (𝑁 ∈ ℕ → (#p‘𝑁) = if(𝑁 ∈ ℙ, ((#p‘(𝑁 − 1)) · 𝑁), (#p‘(𝑁 − 1)))) | ||
Theorem | prmo2 16378 | The primorial of 2. (Contributed by AV, 28-Aug-2020.) |
⊢ (#p‘2) = 2 | ||
Theorem | prmo3 16379 | The primorial of 3. (Contributed by AV, 28-Aug-2020.) |
⊢ (#p‘3) = 6 | ||
Theorem | prmdvdsprmo 16380* | 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 (𝑝 ≤ 𝑁 → 𝑝 ∥ (#p‘𝑁))) | ||
Theorem | prmdvdsprmop 16381* | 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 (𝑝 ≤ 𝑁 ∧ 𝑝 ∥ 𝐼 ∧ 𝑝 ∥ ((#p‘𝑁) + 𝐼))) | ||
Theorem | fvprmselelfz 16382* | 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)) → (𝐹‘𝑋) ∈ (1...𝑁)) | ||
Theorem | fvprmselgcd1 16383* | 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) ∧ 𝑋 ≠ 𝑌) → ((𝐹‘𝑋) gcd (𝐹‘𝑌)) = 1) | ||
Theorem | prmolefac 16384 | 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 → (#p‘𝑁) ≤ (!‘𝑁)) | ||
Theorem | prmodvdslcmf 16385 | 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 → (#p‘𝑁) ∥ (lcm‘(1...𝑁))) | ||
Theorem | prmolelcmf 16386 | 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 → (#p‘𝑁) ≤ (lcm‘(1...𝑁))) | ||
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 16397. Instead of using the factorial of n (see df-fac 13637), 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 16398, or the primorial n# (the product of all prime numbers less than or equal to n), see prmgapprmo 16400, are such functions, which provide smaller values than the factorial function (see lcmflefac 15994 and prmolefac 16384 resp. prmolelcmf 16386): "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). | ||
Theorem | prmgaplem1 16387 | Lemma for prmgap 16397: 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)) | ||
Theorem | prmgaplem2 16388 | Lemma for prmgap 16397: 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)) | ||
Theorem | prmgaplcmlem1 16389 | Lemma for prmgaplcm 16398: 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)) | ||
Theorem | prmgaplcmlem2 16390 | Lemma for prmgaplcm 16398: 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)) | ||
Theorem | prmgaplem3 16391* | Lemma for prmgap 16397. (Contributed by AV, 9-Aug-2020.) |
⊢ 𝐴 = {𝑝 ∈ ℙ ∣ 𝑝 < 𝑁} ⇒ ⊢ (𝑁 ∈ (ℤ≥‘3) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) | ||
Theorem | prmgaplem4 16392* | Lemma for prmgap 16397. (Contributed by AV, 10-Aug-2020.) |
⊢ 𝐴 = {𝑝 ∈ ℙ ∣ (𝑁 < 𝑝 ∧ 𝑝 ≤ 𝑃)} ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑁 < 𝑃) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) | ||
Theorem | prmgaplem5 16393* | Lemma for prmgap 16397: 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)) | ||
Theorem | prmgaplem6 16394* | Lemma for prmgap 16397: 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)) | ||
Theorem | prmgaplem7 16395* | Lemma for prmgap 16397. (Contributed by AV, 12-Aug-2020.) (Proof shortened by AV, 10-Jul-2022.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹 ∈ (ℕ ↑m ℕ)) & ⊢ (𝜑 → ∀𝑖 ∈ (2...𝑁)1 < (((𝐹‘𝑁) + 𝑖) gcd 𝑖)) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 < ((𝐹‘𝑁) + 2) ∧ ((𝐹‘𝑁) + 𝑁) < 𝑞 ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)) | ||
Theorem | prmgaplem8 16396* | Lemma for prmgap 16397. (Contributed by AV, 13-Aug-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹 ∈ (ℕ ↑m ℕ)) & ⊢ (𝜑 → ∀𝑖 ∈ (2...𝑁)1 < (((𝐹‘𝑁) + 𝑖) gcd 𝑖)) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑁 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)) | ||
Theorem | prmgap 16397* | The prime gap theorem: for each positive integer there are (at least) two successive primes with a difference ("gap") at least as big as the given integer. (Contributed by AV, 13-Aug-2020.) |
⊢ ∀𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ) | ||
Theorem | prmgaplcm 16398* | Alternate proof of prmgap 16397: in contrast to prmgap 16397, where the gap starts at n! , the factorial of n, the gap starts at the least common multiple of all positive integers less than or equal to n. (Contributed by AV, 13-Aug-2020.) (Revised by AV, 27-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∀𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ) | ||
Theorem | prmgapprmolem 16399 | Lemma for prmgapprmo 16400: The primorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((#p‘𝑁) + 𝐼) gcd 𝐼)) | ||
Theorem | prmgapprmo 16400* | Alternate proof of prmgap 16397: in contrast to prmgap 16397, where the gap starts at n! , the factorial of n, the gap starts at n#, the primorial of n. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∀𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ) |
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