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Theorem List for Metamath Proof Explorer - 16301-16400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvdwapf 16301 The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝐾 ∈ ℕ0 → (AP‘𝐾):(ℕ × ℕ)⟶𝒫 ℕ)

Theoremvdwapval 16302* Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐾 ∈ ℕ0𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑋 ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑋 = (𝐴 + (𝑚 · 𝐷))))

Theoremvdwapun 16303 Remove the first element of an arithmetic progression. (Contributed by Mario Carneiro, 11-Sep-2014.)
((𝐾 ∈ ℕ0𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘(𝐾 + 1))𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)))

Theoremvdwapid1 16304 The first element of an arithmetic progression. (Contributed by Mario Carneiro, 12-Sep-2014.)
((𝐾 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐴 ∈ (𝐴(AP‘𝐾)𝐷))

Theoremvdwap0 16305 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘0)𝐷) = ∅)

Theoremvdwap1 16306 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘1)𝐷) = {𝐴})

Theoremvdwmc 16307* The predicate " The 𝑅, 𝑁-coloring 𝐹 contains a monochromatic AP of length 𝐾". (Contributed by Mario Carneiro, 18-Aug-2014.)
𝑋 ∈ V    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐹:𝑋𝑅)       (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))

Theoremvdwmc2 16308* Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
𝑋 ∈ V    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐹:𝑋𝑅)    &   (𝜑𝐴𝑋)       (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})))

Theoremvdwpc 16309* The predicate " The coloring 𝐹 contains a polychromatic 𝑀-tuple of AP's of length 𝐾". A polychromatic 𝑀-tuple of AP's is a set of AP's with the same base point but different step lengths, such that each individual AP is monochromatic, but the AP's all have mutually distinct colors. (The common basepoint is not required to have the same color as any of the AP's.) (Contributed by Mario Carneiro, 18-Aug-2014.)
𝑋 ∈ V    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐹:𝑋𝑅)    &   (𝜑𝑀 ∈ ℕ)    &   𝐽 = (1...𝑀)       (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐹 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m 𝐽)(∀𝑖𝐽 ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖𝐽 ↦ (𝐹‘(𝑎 + (𝑑𝑖))))) = 𝑀)))

Theoremvdwlem1 16310* Lemma for vdw 16323. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝐹:(1...𝑊)⟶𝑅)    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐷:(1...𝑀)⟶ℕ)    &   (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐴 + (𝐷𝑖))(AP‘𝐾)(𝐷𝑖)) ⊆ (𝐹 “ {(𝐹‘(𝐴 + (𝐷𝑖)))}))    &   (𝜑𝐼 ∈ (1...𝑀))    &   (𝜑 → (𝐹𝐴) = (𝐹‘(𝐴 + (𝐷𝐼))))       (𝜑 → (𝐾 + 1) MonoAP 𝐹)

Theoremvdwlem2 16311* Lemma for vdw 16323. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)⟶𝑅)    &   (𝜑𝑀 ∈ (ℤ‘(𝑊 + 𝑁)))    &   𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))       (𝜑 → (𝐾 MonoAP 𝐺𝐾 MonoAP 𝐹))

Theoremvdwlem3 16312 Lemma for vdw 16323. (Contributed by Mario Carneiro, 13-Sep-2014.)
(𝜑𝑉 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝐴 ∈ (1...𝑉))    &   (𝜑𝐵 ∈ (1...𝑊))       (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))

Theoremvdwlem4 16313* Lemma for vdw 16323. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑉 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)    &   𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))       (𝜑𝐹:(1...𝑉)⟶(𝑅m (1...𝑊)))

Theoremvdwlem5 16314* Lemma for vdw 16323. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑉 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)    &   𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑊)⟶𝑅)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐸:(1...𝑀)⟶ℕ)    &   (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))    &   𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸𝑖))))    &   (𝜑 → (♯‘ran 𝐽) = 𝑀)    &   𝑇 = (𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)))    &   𝑃 = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)))       (𝜑𝑇 ∈ ℕ)

Theoremvdwlem6 16315* Lemma for vdw 16323. (Contributed by Mario Carneiro, 13-Sep-2014.)
(𝜑𝑉 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)    &   𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑊)⟶𝑅)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐸:(1...𝑀)⟶ℕ)    &   (𝜑 → ∀𝑖 ∈ (1...𝑀)((𝐵 + (𝐸𝑖))(AP‘𝐾)(𝐸𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝐵 + (𝐸𝑖)))}))    &   𝐽 = (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝐵 + (𝐸𝑖))))    &   (𝜑 → (♯‘ran 𝐽) = 𝑀)    &   𝑇 = (𝐵 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)))    &   𝑃 = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝐸𝑗)) + (𝑊 · 𝐷)))       (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))

Theoremvdwlem7 16316* Lemma for vdw 16323. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑉 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)    &   𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑊)⟶𝑅)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))       (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐺 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))

Theoremvdwlem8 16317* Lemma for vdw 16323. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑𝐹:(1...(2 · 𝑊))⟶𝑅)    &   𝐶 ∈ V    &   (𝜑𝐴 ∈ ℕ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐺 “ {𝐶}))    &   𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑊)))       (𝜑 → ⟨1, 𝐾⟩ PolyAP 𝐹)

Theoremvdwlem9 16318* Lemma for vdw 16323. (Contributed by Mario Carneiro, 12-Sep-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑊 ∈ ℕ)    &   (𝜑 → ∀𝑔 ∈ (𝑅m (1...𝑊))(⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))    &   (𝜑𝑉 ∈ ℕ)    &   (𝜑 → ∀𝑓 ∈ ((𝑅m (1...𝑊)) ↑m (1...𝑉))𝐾 MonoAP 𝑓)    &   (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)    &   𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))       (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻))

Theoremvdwlem10 16319* Lemma for vdw 16323. Set up secondary induction on 𝑀. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓)    &   (𝜑𝑀 ∈ ℕ)       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))

Theoremvdwlem11 16320* Lemma for vdw 16323. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓)       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))(𝐾 + 1) MonoAP 𝑓)

Theoremvdwlem12 16321 Lemma for vdw 16323. 𝐾 = 2 base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)    &   (𝜑 → ¬ 2 MonoAP 𝐹)        ¬ 𝜑

Theoremvdwlem13 16322* Lemma for vdw 16323. Main induction on 𝐾; 𝐾 = 0, 𝐾 = 1 base cases. (Contributed by Mario Carneiro, 18-Aug-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐾 ∈ ℕ0)       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓)

Theoremvdw 16323* Van der Waerden's theorem. For any finite coloring 𝑅 and integer 𝐾, there is an 𝑁 such that every coloring function from 1...𝑁 to 𝑅 contains a monochromatic arithmetic progression (which written out in full means that there is a color 𝑐 and base, increment values 𝑎, 𝑑 such that all the numbers 𝑎, 𝑎 + 𝑑, ..., 𝑎 + (𝑘 − 1)𝑑 lie in the preimage of {𝑐}, i.e. they are all in 1...𝑁 and 𝑓 evaluated at each one yields 𝑐). (Contributed by Mario Carneiro, 13-Sep-2014.)
((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝑓 “ {𝑐}))

Theoremvdwnnlem1 16324* Corollary of vdw 16323, and lemma for vdwnn 16327. If 𝐹 is a coloring of the integers, then there are arbitrarily long monochromatic APs in 𝐹. (Contributed by Mario Carneiro, 13-Sep-2014.)
((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅𝐾 ∈ ℕ0) → ∃𝑐𝑅𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))

Theoremvdwnnlem2 16325* Lemma for vdwnn 16327. The set of all "bad" 𝑘 for the theorem is upwards-closed, because a long AP implies a short AP. (Contributed by Mario Carneiro, 13-Sep-2014.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:ℕ⟶𝑅)    &   𝑆 = {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})}       ((𝜑𝐵 ∈ (ℤ𝐴)) → (𝐴𝑆𝐵𝑆))

Theoremvdwnnlem3 16326* Lemma for vdwnn 16327. (Contributed by Mario Carneiro, 13-Sep-2014.) (Proof shortened by AV, 27-Sep-2020.)
(𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:ℕ⟶𝑅)    &   𝑆 = {𝑘 ∈ ℕ ∣ ¬ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐})}    &   (𝜑 → ∀𝑐𝑅 𝑆 ≠ ∅)        ¬ 𝜑

Theoremvdwnn 16327* Van der Waerden's theorem, infinitary version. For any finite coloring 𝐹 of the positive integers, there is a color 𝑐 that contains arbitrarily long arithmetic progressions. (Contributed by Mario Carneiro, 13-Sep-2014.)
((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅) → ∃𝑐𝑅𝑘 ∈ ℕ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝑘 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (𝐹 “ {𝑐}))

6.2.14  Ramsey's theorem

Syntaxcram 16328 Extend class notation with the Ramsey number function.
class Ramsey

Theoremramtlecl 16329* The set 𝑇 of numbers with the Ramsey number property is upward-closed. (Contributed by Mario Carneiro, 21-Apr-2015.)
𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → 𝜑)}       (𝑀𝑇 → (ℤ𝑀) ⊆ 𝑇)

Definitiondf-ram 16330* Define the Ramsey number function. The input is a number 𝑚 for the size of the edges of the hypergraph, and a tuple 𝑟 from the finite color set to lower bounds for each color. The Ramsey number (𝑀 Ramsey 𝑅) is the smallest number such that for any set 𝑆 with (𝑀 Ramsey 𝑅) ≤ ♯𝑆 and any coloring 𝐹 of the set of 𝑀-element subsets of 𝑆 (with color set dom 𝑅), there is a color 𝑐 ∈ dom 𝑅 and a subset 𝑥𝑆 such that 𝑅(𝑐) ≤ ♯𝑥 and all the hyperedges of 𝑥 (that is, subsets of 𝑥 of size 𝑀) have color 𝑐. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
Ramsey = (𝑚 ∈ ℕ0, 𝑟 ∈ V ↦ inf({𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (dom 𝑟m {𝑦 ∈ 𝒫 𝑠 ∣ (♯‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟𝑥 ∈ 𝒫 𝑠((𝑟𝑐) ≤ (♯‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((♯‘𝑦) = 𝑚 → (𝑓𝑦) = 𝑐)))}, ℝ*, < ))

Theoremhashbcval 16331* Value of the "binomial set", the set of all 𝑁-element subsets of 𝐴. (Contributed by Mario Carneiro, 20-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})       ((𝐴𝑉𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})

Theoremhashbccl 16332* The binomial set is a finite set. (Contributed by Mario Carneiro, 20-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})       ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) ∈ Fin)

Theoremhashbcss 16333* Subset relation for the binomial set. (Contributed by Mario Carneiro, 20-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})       ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) ⊆ (𝐴𝐶𝑁))

Theoremhashbc0 16334* The set of subsets of size zero is the singleton of the empty set. (Contributed by Mario Carneiro, 22-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})       (𝐴𝑉 → (𝐴𝐶0) = {∅})

Theoremhashbc2 16335* The size of the binomial set is the binomial coefficient. (Contributed by Mario Carneiro, 20-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})       ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (♯‘(𝐴𝐶𝑁)) = ((♯‘𝐴)C𝑁))

Theorem0hashbc 16336* There are no subsets of the empty set with size greater than zero. (Contributed by Mario Carneiro, 22-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})       (𝑁 ∈ ℕ → (∅𝐶𝑁) = ∅)

Theoremramval 16337* The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 14-Sep-2020.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})    &   𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}       ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = inf(𝑇, ℝ*, < ))

Theoremramcl2lem 16338* Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})    &   𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}       ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) = if(𝑇 = ∅, +∞, inf(𝑇, ℝ, < )))

Theoremramtcl 16339* The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})    &   𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}       ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ 𝑇𝑇 ≠ ∅))

Theoremramtcl2 16340* The Ramsey number is an integer iff there is a number with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})    &   𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}       ((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → ((𝑀 Ramsey 𝐹) ∈ ℕ0𝑇 ≠ ∅))

Theoremramtub 16341* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})    &   𝑇 = {𝑛 ∈ ℕ0 ∣ ∀𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))}       (((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ 𝐴𝑇) → (𝑀 Ramsey 𝐹) ≤ 𝐴)

Theoremramub 16342* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 22-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑅𝑉)    &   (𝜑𝐹:𝑅⟶ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑 ∧ (𝑁 ≤ (♯‘𝑠) ∧ 𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))       (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁)

Theoremramub2 16343* It is sufficient to check the Ramsey property on finite sets of size equal to the upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑅𝑉)    &   (𝜑𝐹:𝑅⟶ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑 ∧ ((♯‘𝑠) = 𝑁𝑓:(𝑠𝐶𝑀)⟶𝑅)) → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐})))       (𝜑 → (𝑀 Ramsey 𝐹) ≤ 𝑁)

Theoremrami 16344* The defining property of a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑅𝑉)    &   (𝜑𝐹:𝑅⟶ℕ0)    &   (𝜑 → (𝑀 Ramsey 𝐹) ∈ ℕ0)    &   (𝜑𝑆𝑊)    &   (𝜑 → (𝑀 Ramsey 𝐹) ≤ (♯‘𝑆))    &   (𝜑𝐺:(𝑆𝐶𝑀)⟶𝑅)       (𝜑 → ∃𝑐𝑅𝑥 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐})))

Theoremramcl2 16345 The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ (ℕ0 ∪ {+∞}))

Theoremramxrcl 16346 The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 16358.) (Contributed by Mario Carneiro, 20-Apr-2015.)
((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℝ*)

Theoremramubcl 16347 If the Ramsey number is upper bounded, then it is an integer. (Contributed by Mario Carneiro, 20-Apr-2015.)
(((𝑀 ∈ ℕ0𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐴 ∈ ℕ0 ∧ (𝑀 Ramsey 𝐹) ≤ 𝐴)) → (𝑀 Ramsey 𝐹) ∈ ℕ0)

Theoremramlb 16348* Establish a lower bound on a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑅𝑉)    &   (𝜑𝐹:𝑅⟶ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐺:((1...𝑁)𝐶𝑀)⟶𝑅)    &   ((𝜑 ∧ (𝑐𝑅𝑥 ⊆ (1...𝑁))) → ((𝑥𝐶𝑀) ⊆ (𝐺 “ {𝑐}) → (♯‘𝑥) < (𝐹𝑐)))       (𝜑𝑁 < (𝑀 Ramsey 𝐹))

Theorem0ram 16349* The Ramsey number when 𝑀 = 0. (Contributed by Mario Carneiro, 22-Apr-2015.)
(((𝑅𝑉𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → (0 Ramsey 𝐹) = sup(ran 𝐹, ℝ, < ))

Theorem0ram2 16350 The Ramsey number when 𝑀 = 0. (Contributed by Mario Carneiro, 22-Apr-2015.)
((𝑅 ∈ Fin ∧ 𝑅 ≠ ∅ ∧ 𝐹:𝑅⟶ℕ0) → (0 Ramsey 𝐹) = sup(ran 𝐹, ℝ, < ))

Theoremram0 16351 The Ramsey number when 𝑅 = ∅. (Contributed by Mario Carneiro, 22-Apr-2015.)
(𝑀 ∈ ℕ0 → (𝑀 Ramsey ∅) = 𝑀)

Theorem0ramcl 16352 Lemma for ramcl 16358: Existence of the Ramsey number when 𝑀 = 0. (Contributed by Mario Carneiro, 23-Apr-2015.)
((𝑅 ∈ Fin ∧ 𝐹:𝑅⟶ℕ0) → (0 Ramsey 𝐹) ∈ ℕ0)

Theoremramz2 16353 The Ramsey number when 𝐹 has value zero for some color 𝐶. (Contributed by Mario Carneiro, 22-Apr-2015.)
(((𝑀 ∈ ℕ ∧ 𝑅𝑉𝐹:𝑅⟶ℕ0) ∧ (𝐶𝑅 ∧ (𝐹𝐶) = 0)) → (𝑀 Ramsey 𝐹) = 0)

Theoremramz 16354 The Ramsey number when 𝐹 is the zero function. (Contributed by Mario Carneiro, 22-Apr-2015.)
((𝑀 ∈ ℕ0𝑅𝑉𝑅 ≠ ∅) → (𝑀 Ramsey (𝑅 × {0})) = 0)

Theoremramub1lem1 16355* Lemma for ramub1 16357. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:𝑅⟶ℕ)    &   𝐺 = (𝑥𝑅 ↦ (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦)))))    &   (𝜑𝐺:𝑅⟶ℕ0)    &   (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0)    &   𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})    &   (𝜑𝑆 ∈ Fin)    &   (𝜑 → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))    &   (𝜑𝐾:(𝑆𝐶𝑀)⟶𝑅)    &   (𝜑𝑋𝑆)    &   𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))    &   (𝜑𝐷𝑅)    &   (𝜑𝑊 ⊆ (𝑆 ∖ {𝑋}))    &   (𝜑 → (𝐺𝐷) ≤ (♯‘𝑊))    &   (𝜑 → (𝑊𝐶(𝑀 − 1)) ⊆ (𝐻 “ {𝐷}))    &   (𝜑𝐸𝑅)    &   (𝜑𝑉𝑊)    &   (𝜑 → if(𝐸 = 𝐷, ((𝐹𝐷) − 1), (𝐹𝐸)) ≤ (♯‘𝑉))    &   (𝜑 → (𝑉𝐶𝑀) ⊆ (𝐾 “ {𝐸}))       (𝜑 → ∃𝑧 ∈ 𝒫 𝑆((𝐹𝐸) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝐸})))

Theoremramub1lem2 16356* Lemma for ramub1 16357. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:𝑅⟶ℕ)    &   𝐺 = (𝑥𝑅 ↦ (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦)))))    &   (𝜑𝐺:𝑅⟶ℕ0)    &   (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0)    &   𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})    &   (𝜑𝑆 ∈ Fin)    &   (𝜑 → (♯‘𝑆) = (((𝑀 − 1) Ramsey 𝐺) + 1))    &   (𝜑𝐾:(𝑆𝐶𝑀)⟶𝑅)    &   (𝜑𝑋𝑆)    &   𝐻 = (𝑢 ∈ ((𝑆 ∖ {𝑋})𝐶(𝑀 − 1)) ↦ (𝐾‘(𝑢 ∪ {𝑋})))       (𝜑 → ∃𝑐𝑅𝑧 ∈ 𝒫 𝑆((𝐹𝑐) ≤ (♯‘𝑧) ∧ (𝑧𝐶𝑀) ⊆ (𝐾 “ {𝑐})))

Theoremramub1 16357* Inductive step for Ramsey's theorem, in the form of an explicit upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑅 ∈ Fin)    &   (𝜑𝐹:𝑅⟶ℕ)    &   𝐺 = (𝑥𝑅 ↦ (𝑀 Ramsey (𝑦𝑅 ↦ if(𝑦 = 𝑥, ((𝐹𝑥) − 1), (𝐹𝑦)))))    &   (𝜑𝐺:𝑅⟶ℕ0)    &   (𝜑 → ((𝑀 − 1) Ramsey 𝐺) ∈ ℕ0)       (𝜑 → (𝑀 Ramsey 𝐹) ≤ (((𝑀 − 1) Ramsey 𝐺) + 1))

Theoremramcl 16358 Ramsey's theorem: the Ramsey number is an integer for every finite coloring and set of upper bounds. (Contributed by Mario Carneiro, 23-Apr-2015.)
((𝑀 ∈ ℕ0𝑅 ∈ Fin ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℕ0)

Theoremramsey 16359* Ramsey's theorem with the definition Ramsey eliminated. If 𝑀 is an integer, 𝑅 is a specified finite set of colors, and 𝐹:𝑅⟶ℕ0 is a set of lower bounds for each color, then there is an 𝑛 such that for every set 𝑠 of size greater than 𝑛 and every coloring 𝑓 of the set (𝑠𝐶𝑀) of all 𝑀-element subsets of 𝑠, there is a color 𝑐 and a subset 𝑥𝑠 such that 𝑥 is larger than 𝐹(𝑐) and the 𝑀-element subsets of 𝑥 are monochromatic with color 𝑐. This is the hypergraph version of Ramsey's theorem; the version for simple graphs is the case 𝑀 = 2. This is Metamath 100 proof #31. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})       ((𝑀 ∈ ℕ0𝑅 ∈ Fin ∧ 𝐹:𝑅⟶ℕ0) → ∃𝑛 ∈ ℕ0𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))

6.2.15  Primorial function

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

Syntaxcprmo 16360 Extend class notation to include the primorial of nonnegative integers.
class #p

Definitiondf-prmo 16361* 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 28247).

In the literature, the primorial function is written as a postscript hash: 6# = 30. In contrast to prmorcht 25766, where the primorial function is defined by using the sequence builder (𝑃 = seq1( · , 𝐹)), the more specialized definition of a product of a series is used here. (Contributed by AV, 28-Aug-2020.)

#p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))

Theoremprmoval 16362* Value of the primorial function for nonnegative integers. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))

Theoremprmocl 16363 Closure of the primorial function. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ∈ ℕ)

Theoremprmone0 16364 The primorial function is nonzero. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ≠ 0)

Theoremprmo0 16365 The primorial of 0. (Contributed by AV, 28-Aug-2020.)
(#p‘0) = 1

Theoremprmo1 16366 The primorial of 1. (Contributed by AV, 28-Aug-2020.)
(#p‘1) = 1

Theoremprmop1 16367 The primorial of a successor. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p‘(𝑁 + 1)) = if((𝑁 + 1) ∈ ℙ, ((#p𝑁) · (𝑁 + 1)), (#p𝑁)))

Theoremprmonn2 16368 Value of the primorial function expressed recursively. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ → (#p𝑁) = if(𝑁 ∈ ℙ, ((#p‘(𝑁 − 1)) · 𝑁), (#p‘(𝑁 − 1))))

Theoremprmo2 16369 The primorial of 2. (Contributed by AV, 28-Aug-2020.)
(#p‘2) = 2

Theoremprmo3 16370 The primorial of 3. (Contributed by AV, 28-Aug-2020.)
(#p‘3) = 6

Theoremprmdvdsprmo 16371* The primorial of a number is divisible by each prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ → ∀𝑝 ∈ ℙ (𝑝𝑁𝑝 ∥ (#p𝑁)))

Theoremprmdvdsprmop 16372* The primorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by a prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 28-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ∃𝑝 ∈ ℙ (𝑝𝑁𝑝𝐼𝑝 ∥ ((#p𝑁) + 𝐼)))

Theoremfvprmselelfz 16373* The value of the prime selection function is in a finite sequence of integers if the argument is in this finite sequence of integers. (Contributed by AV, 19-Aug-2020.)
𝐹 = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))       ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁)) → (𝐹𝑋) ∈ (1...𝑁))

Theoremfvprmselgcd1 16374* The greatest common divisor of two values of the prime selection function for different arguments is 1. (Contributed by AV, 19-Aug-2020.)
𝐹 = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))       ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)

Theoremprmolefac 16375 The primorial of a positive integer is less than or equal to the factorial of the integer. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ≤ (!‘𝑁))

Theoremprmodvdslcmf 16376 The primorial of a nonnegative integer divides the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 29-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ∥ (lcm‘(1...𝑁)))

Theoremprmolelcmf 16377 The primorial of a positive integer is less than or equal to the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 29-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ≤ (lcm‘(1...𝑁)))

6.2.16  Prime gaps

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

It can be proven that there are arbitrary large gaps, usually by showing that "in the sequence n!+2, n!+3, ..., n!+n the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of n-1 consecutive composite integers, and it must belong to a gap between primes having length at least n.", see prmgap 16388.

Instead of using the factorial of n (see df-fac 13634), 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 16389, or the primorial n# (the product of all prime numbers less than or equal to n), see prmgapprmo 16391, are such functions, which provide smaller values than the factorial function (see lcmflefac 15985 and prmolefac 16375 resp. prmolelcmf 16377): "For instance, the first prime gap of size larger than 14 occurs between the primes 523 and 541, while 15! is the vastly larger number 1307674368000." But the least common multiple of the integers 2, 3, ..., 15 is 360360, and 15# is 30030 (p3248 = 30029 and P3249 = 30047, so g3248 = 18).

Theoremprmgaplem1 16378 Lemma for prmgap 16388: The factorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 13-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((!‘𝑁) + 𝐼))

Theoremprmgaplem2 16379 Lemma for prmgap 16388: The factorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 13-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((!‘𝑁) + 𝐼) gcd 𝐼))

Theoremprmgaplcmlem1 16380 Lemma for prmgaplcm 16389: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼))

Theoremprmgaplcmlem2 16381 Lemma for prmgaplcm 16389: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((lcm‘(1...𝑁)) + 𝐼) gcd 𝐼))

Theoremprmgaplem3 16382* Lemma for prmgap 16388. (Contributed by AV, 9-Aug-2020.)
𝐴 = {𝑝 ∈ ℙ ∣ 𝑝 < 𝑁}       (𝑁 ∈ (ℤ‘3) → ∃𝑥𝐴𝑦𝐴 𝑦𝑥)

Theoremprmgaplem4 16383* Lemma for prmgap 16388. (Contributed by AV, 10-Aug-2020.)
𝐴 = {𝑝 ∈ ℙ ∣ (𝑁 < 𝑝𝑝𝑃)}       ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑁 < 𝑃) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)

Theoremprmgaplem5 16384* Lemma for prmgap 16388: for each integer greater than 2 there is a smaller prime closest to this integer, i.e. there is a smaller prime and no other prime is between this prime and the integer. (Contributed by AV, 9-Aug-2020.)
(𝑁 ∈ (ℤ‘3) → ∃𝑝 ∈ ℙ (𝑝 < 𝑁 ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑁)𝑧 ∉ ℙ))

Theoremprmgaplem6 16385* Lemma for prmgap 16388: for each positive integer there is a greater prime closest to this integer, i.e. there is a greater prime and no other prime is between this prime and the integer. (Contributed by AV, 10-Aug-2020.)
(𝑁 ∈ ℕ → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ ∀𝑧 ∈ ((𝑁 + 1)..^𝑝)𝑧 ∉ ℙ))

Theoremprmgaplem7 16386* Lemma for prmgap 16388. (Contributed by AV, 12-Aug-2020.) (Proof shortened by AV, 10-Jul-2022.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹 ∈ (ℕ ↑m ℕ))    &   (𝜑 → ∀𝑖 ∈ (2...𝑁)1 < (((𝐹𝑁) + 𝑖) gcd 𝑖))       (𝜑 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 < ((𝐹𝑁) + 2) ∧ ((𝐹𝑁) + 𝑁) < 𝑞 ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))

Theoremprmgaplem8 16387* Lemma for prmgap 16388. (Contributed by AV, 13-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹 ∈ (ℕ ↑m ℕ))    &   (𝜑 → ∀𝑖 ∈ (2...𝑁)1 < (((𝐹𝑁) + 𝑖) gcd 𝑖))       (𝜑 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑁 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))

Theoremprmgap 16388* 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 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)

Theoremprmgaplcm 16389* Alternate proof of prmgap 16388: in contrast to prmgap 16388, 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 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)

Theoremprmgapprmolem 16390 Lemma for prmgapprmo 16391: 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 < (((#p𝑁) + 𝐼) gcd 𝐼))

Theoremprmgapprmo 16391* Alternate proof of prmgap 16388: in contrast to prmgap 16388, 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 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)

6.2.17  Decimal arithmetic (cont.)

Theoremdec2dvds 16392 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

Theoremdec5dvds 16393 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ    &   𝐵 < 5        ¬ 5 ∥ 𝐴𝐵

Theoremdec5dvds2 16394 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

Theoremdec5nprm 16395 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ        ¬ 𝐴5 ∈ ℙ

Theoremdec2nprm 16396 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   (𝐵 · 2) = 𝐶        ¬ 𝐴𝐶 ∈ ℙ

Theoremmodxai 16397 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    &   ((𝐴𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   ((𝐴𝐶) mod 𝑁) = (𝐿 mod 𝑁)    &   (𝐵 + 𝐶) = 𝐸    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐿)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)

Theoremmod2xi 16398 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    &   ((𝐴𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   (2 · 𝐵) = 𝐸    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐾)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)

Theoremmodxp1i 16399 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    &   ((𝐴𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   (𝐵 + 1) = 𝐸    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐴)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)

Theoremmod2xnegi 16400 Version of mod2xi 16398 with a negative mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℤ    &   𝐾 ∈ ℕ    &   𝑀 ∈ ℕ0    &   𝐿 ∈ ℕ0    &   ((𝐴𝐵) mod 𝑁) = (𝐿 mod 𝑁)    &   (2 · 𝐵) = 𝐸    &   (𝐿 + 𝐾) = 𝑁    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐾)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)

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