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Type | Label | Description |
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Statement | ||
Definition | df-vdwap 17001* | 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 17002* | 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 17003* | Define the "contains a polychromatic collection of APs" predicate. See vdwpc 17013 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 17004* | 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 17005 | The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ) | ||
Theorem | vdwapval 17006* | 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 17007 | 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 17008 | 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 17009 | Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘0)𝐷) = ∅) | ||
Theorem | vdwap1 17010 | 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 17011* | 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 17012* | 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 17013* | 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 17014* | Lemma for vdw 17027. (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 17015* | Lemma for vdw 17027. (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 17016 | Lemma for vdw 17027. (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 17017* | Lemma for vdw 17027. (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 17018* | Lemma for vdw 17027. (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 17019* | Lemma for vdw 17027. (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 17020* | Lemma for vdw 17027. (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 17021* | Lemma for vdw 17027. (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 17022* | Lemma for vdw 17027. (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 17023* | Lemma for vdw 17027. 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 17024* | Lemma for vdw 17027. (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 17025 | Lemma for vdw 17027. 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 17026* | Lemma for vdw 17027. 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 17027* | 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 17028* | Corollary of vdw 17027, and lemma for vdwnn 17031. 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 17029* | Lemma for vdwnn 17031. 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 17030* | Lemma for vdwnn 17031. (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 17031* | 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 17032 | Extend class notation with the Ramsey number function. |
class Ramsey | ||
Theorem | ramtlecl 17033* | 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 17034* | 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 17035* | 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 17036* | 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 17037* | 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 17038* | 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 17039* | 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 17040* | 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 17041* | 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 17042* | 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 17043* | 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 17044* | 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 17045* | 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 17046* | 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 17047* | 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 17048* | 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 17049 | 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 17050 | The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 17062.) (Contributed by Mario Carneiro, 20-Apr-2015.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℝ*) | ||
Theorem | ramubcl 17051 | 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 17052* | 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 17053* | 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 17054 | 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 17055 | The Ramsey number when 𝑅 = ∅. (Contributed by Mario Carneiro, 22-Apr-2015.) |
⊢ (𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) = 𝑀) | ||
Theorem | 0ramcl 17056 | Lemma for ramcl 17062: 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 17057 | 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 17058 | 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 17059* | Lemma for ramub1 17061. (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 17060* | Lemma for ramub1 17061. (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 17061* | 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 17062 | 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 17063* | Ramsey's theorem with the definition of Ramsey (df-ram 17034) 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 17064 | Extend class notation to include the primorial of nonnegative integers. |
class #p | ||
Definition | df-prmo 17065* |
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 30487).
In the literature, the primorial function is written as a postscript hash: 6# = 30. In contrast to prmorcht 27235, 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 17066* | 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 17067 | Closure of the primorial function. (Contributed by AV, 28-Aug-2020.) |
⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ∈ ℕ) | ||
Theorem | prmone0 17068 | The primorial function is nonzero. (Contributed by AV, 28-Aug-2020.) |
⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ≠ 0) | ||
Theorem | prmo0 17069 | The primorial of 0. (Contributed by AV, 28-Aug-2020.) |
⊢ (#p‘0) = 1 | ||
Theorem | prmo1 17070 | The primorial of 1. (Contributed by AV, 28-Aug-2020.) |
⊢ (#p‘1) = 1 | ||
Theorem | prmop1 17071 | 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 17072 | 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 17073 | The primorial of 2. (Contributed by AV, 28-Aug-2020.) |
⊢ (#p‘2) = 2 | ||
Theorem | prmo3 17074 | The primorial of 3. (Contributed by AV, 28-Aug-2020.) |
⊢ (#p‘3) = 6 | ||
Theorem | prmdvdsprmo 17075* | 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 17076* | 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 17077* | 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 17078* | 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 17079 | 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 17080 | 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 17081 | 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 17092. Instead of using the factorial of n (see df-fac 14309), 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 17093, or the primorial n# (the product of all prime numbers less than or equal to n), see prmgapprmo 17095, are such functions, which provide smaller values than the factorial function (see lcmflefac 16681 and prmolefac 17079 resp. prmolelcmf 17081): "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 17082 | Lemma for prmgap 17092: 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 17083 | Lemma for prmgap 17092: 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 17084 | Lemma for prmgaplcm 17093: 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 17085 | Lemma for prmgaplcm 17093: 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 17086* | Lemma for prmgap 17092. (Contributed by AV, 9-Aug-2020.) |
⊢ 𝐴 = {𝑝 ∈ ℙ ∣ 𝑝 < 𝑁} ⇒ ⊢ (𝑁 ∈ (ℤ≥‘3) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) | ||
Theorem | prmgaplem4 17087* | Lemma for prmgap 17092. (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 17088* | Lemma for prmgap 17092: 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 17089* | Lemma for prmgap 17092: 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 17090* | Lemma for prmgap 17092. (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 17091* | Lemma for prmgap 17092. (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 17092* | 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 17093* | Alternate proof of prmgap 17092: in contrast to prmgap 17092, 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 17094 | Lemma for prmgapprmo 17095: 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 17095* | Alternate proof of prmgap 17092: in contrast to prmgap 17092, 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) | ||
Theorem | dec2dvds 17096 | Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ (𝐵 · 2) = 𝐶 & ⊢ 𝐷 = (𝐶 + 1) ⇒ ⊢ ¬ 2 ∥ ;𝐴𝐷 | ||
Theorem | dec5dvds 17097 | Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ & ⊢ 𝐵 < 5 ⇒ ⊢ ¬ 5 ∥ ;𝐴𝐵 | ||
Theorem | dec5dvds2 17098 | Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ & ⊢ 𝐵 < 5 & ⊢ (5 + 𝐵) = 𝐶 ⇒ ⊢ ¬ 5 ∥ ;𝐴𝐶 | ||
Theorem | dec5nprm 17099 | Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 𝐴 ∈ ℕ ⇒ ⊢ ¬ ;𝐴5 ∈ ℙ | ||
Theorem | dec2nprm 17100 | Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ0 & ⊢ (𝐵 · 2) = 𝐶 ⇒ ⊢ ¬ ;𝐴𝐶 ∈ ℙ |
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