P. 167 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 16601-16700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	vdwapf 16601	The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)
⊢ (𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)
Theorem	vdwapval 16602*	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 16603	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 16604	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 16605	Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘0)𝐷) = ∅)
Theorem	vdwap1 16606	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 16607*	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 16608*	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 16609*	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 16610*	Lemma for vdw 16623. (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 16611*	Lemma for vdw 16623. (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 16612	Lemma for vdw 16623. (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 16613*	Lemma for vdw 16623. (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 16614*	Lemma for vdw 16623. (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 16615*	Lemma for vdw 16623. (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 16616*	Lemma for vdw 16623. (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 16617*	Lemma for vdw 16623. (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 16618*	Lemma for vdw 16623. (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 16619*	Lemma for vdw 16623. 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 16620*	Lemma for vdw 16623. (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 16621	Lemma for vdw 16623. 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 16622*	Lemma for vdw 16623. 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 16623*	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 16624*	Corollary of vdw 16623, and lemma for vdwnn 16627. 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 16625*	Lemma for vdwnn 16627. 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 16626*	Lemma for vdwnn 16627. (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 16627*	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}))
6.2.14  Ramsey's theorem
Syntax	cram 16628	Extend class notation with the Ramsey number function.
class Ramsey
Theorem	ramtlecl 16629*	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 16630*	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 16631*	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 16632*	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 16633*	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 16634*	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 16635*	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 16636*	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 16637*	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 16638*	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 16639*	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 16640*	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 16641*	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 16642*	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 16643*	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 16644*	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 16645	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 16646	The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 16658.) (Contributed by Mario Carneiro, 20-Apr-2015.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℝ*)
Theorem	ramubcl 16647	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 16648*	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 16649*	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 16650	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 16651	The Ramsey number when 𝑅 = ∅. (Contributed by Mario Carneiro, 22-Apr-2015.)
⊢ (𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) = 𝑀)
Theorem	0ramcl 16652	Lemma for ramcl 16658: 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 16653	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 16654	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 16655*	Lemma for ramub1 16657. (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 16656*	Lemma for ramub1 16657. (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 16657*	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 16658	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 16659*	Ramsey's theorem with the definition of Ramsey (df-ram 16630) 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}))))
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."
Syntax	cprmo 16660	Extend class notation to include the primorial of nonnegative integers.
class #p
Definition	df-prmo 16661*	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 28724). In the literature, the primorial function is written as a postscript hash: 6# = 30. In contrast to prmorcht 26232, 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 16662*	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 16663	Closure of the primorial function. (Contributed by AV, 28-Aug-2020.)
⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ∈ ℕ)
Theorem	prmone0 16664	The primorial function is nonzero. (Contributed by AV, 28-Aug-2020.)
⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ≠ 0)
Theorem	prmo0 16665	The primorial of 0. (Contributed by AV, 28-Aug-2020.)
⊢ (#p‘0) = 1
Theorem	prmo1 16666	The primorial of 1. (Contributed by AV, 28-Aug-2020.)
⊢ (#p‘1) = 1
Theorem	prmop1 16667	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 16668	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 16669	The primorial of 2. (Contributed by AV, 28-Aug-2020.)
⊢ (#p‘2) = 2
Theorem	prmo3 16670	The primorial of 3. (Contributed by AV, 28-Aug-2020.)
⊢ (#p‘3) = 6
Theorem	prmdvdsprmo 16671*	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 16672*	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 16673*	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 16674*	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 16675	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 16676	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 16677	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)))
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 16688. Instead of using the factorial of n (see df-fac 13916), 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 16689, or the primorial n# (the product of all prime numbers less than or equal to n), see prmgapprmo 16691, are such functions, which provide smaller values than the factorial function (see lcmflefac 16281 and prmolefac 16675 resp. prmolelcmf 16677): "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 16678	Lemma for prmgap 16688: 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 16679	Lemma for prmgap 16688: 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 16680	Lemma for prmgaplcm 16689: 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 16681	Lemma for prmgaplcm 16689: 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 16682*	Lemma for prmgap 16688. (Contributed by AV, 9-Aug-2020.)
⊢ 𝐴 = {𝑝 ∈ ℙ ∣ 𝑝 < 𝑁}    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘3) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)
Theorem	prmgaplem4 16683*	Lemma for prmgap 16688. (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 16684*	Lemma for prmgap 16688: 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 16685*	Lemma for prmgap 16688: 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 16686*	Lemma for prmgap 16688. (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 16687*	Lemma for prmgap 16688. (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 16688*	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 16689*	Alternate proof of prmgap 16688: in contrast to prmgap 16688, 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 16690	Lemma for prmgapprmo 16691: 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 16691*	Alternate proof of prmgap 16688: in contrast to prmgap 16688, 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)
6.2.17  Decimal arithmetic (cont.)
Theorem	dec2dvds 16692	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 16693	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 16694	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 16695	Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ ¬ ;𝐴5 ∈ ℙ
Theorem	dec2nprm 16696	Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ (𝐵 · 2) = 𝐶    ⇒   ⊢ ¬ ;𝐴𝐶 ∈ ℙ
Theorem	modxai 16697	Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)
⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℤ    &   ⊢ 𝐾 ∈ ℕ0    &   ⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐿 ∈ ℕ0    &   ⊢ ((𝐴↑𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   ⊢ ((𝐴↑𝐶) mod 𝑁) = (𝐿 mod 𝑁)    &   ⊢ (𝐵 + 𝐶) = 𝐸    &   ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐿)    ⇒   ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁)
Theorem	mod2xi 16698	Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℤ    &   ⊢ 𝐾 ∈ ℕ0    &   ⊢ 𝑀 ∈ ℕ0    &   ⊢ ((𝐴↑𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   ⊢ (2 · 𝐵) = 𝐸    &   ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐾)    ⇒   ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁)
Theorem	modxp1i 16699	Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
⊢ 𝑁 ∈ ℕ    &   ⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℤ    &   ⊢ 𝐾 ∈ ℕ0    &   ⊢ 𝑀 ∈ ℕ0    &   ⊢ ((𝐴↑𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   ⊢ (𝐵 + 1) = 𝐸    &   ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐴)    ⇒   ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁)
Theorem	mod2xnegi 16700	Version of mod2xi 16698 with a negative mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℤ    &   ⊢ 𝐾 ∈ ℕ    &   ⊢ 𝑀 ∈ ℕ0    &   ⊢ 𝐿 ∈ ℕ0    &   ⊢ ((𝐴↑𝐵) mod 𝑁) = (𝐿 mod 𝑁)    &   ⊢ (2 · 𝐵) = 𝐸    &   ⊢ (𝐿 + 𝐾) = 𝑁    &   ⊢ ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐾)    ⇒   ⊢ ((𝐴↑𝐸) mod 𝑁) = (𝑀 mod 𝑁)