Home Metamath Proof ExplorerTheorem List (p. 300 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27159) Hilbert Space Explorer (27160-28684) Users' Mathboxes (28685-42360)

Theorem List for Metamath Proof Explorer - 29901-30000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremballotlemsel1i 29901* The range (1...(𝐼𝐶)) is invariant under (𝑆𝐶). (Contributed by Thierry Arnoux, 28-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)))

Theoremballotlemsf1o 29902* The defined 𝑆 is a bijection, and an involution. (Contributed by Thierry Arnoux, 14-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))       (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ (𝑆𝐶) = (𝑆𝐶)))

Theoremballotlemsi 29903* The image by 𝑆 of the first tie pick is the first pick. (Contributed by Thierry Arnoux, 14-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))       (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶)‘(𝐼𝐶)) = 1)

Theoremballotlemsima 29904* The image by 𝑆 of an interval before the first pick. (Contributed by Thierry Arnoux, 5-May-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (1...𝐽)) = (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))

Theoremballotlemieq 29905* If two countings share the same first tie, they also have the same swap function. (Contributed by Thierry Arnoux, 18-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐷 ∈ (𝑂𝐸) ∧ (𝐼𝐶) = (𝐼𝐷)) → (𝑆𝐶) = (𝑆𝐷))

Theoremballotlemrval 29906* Value of 𝑅. (Contributed by Thierry Arnoux, 14-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) = ((𝑆𝐶) “ 𝐶))

Theoremballotlemscr 29907* The image of (𝑅𝐶) by (𝑆𝐶). (Contributed by Thierry Arnoux, 21-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ (𝑅𝐶)) = 𝐶)

Theoremballotlemrv 29908* Value of 𝑅 evaluated at 𝐽. (Contributed by Thierry Arnoux, 17-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → (𝐽 ∈ (𝑅𝐶) ↔ if(𝐽 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝐽), 𝐽) ∈ 𝐶))

Theoremballotlemrv1 29909* Value of 𝑅 before the tie. (Contributed by Thierry Arnoux, 11-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 ≤ (𝐼𝐶)) → (𝐽 ∈ (𝑅𝐶) ↔ (((𝐼𝐶) + 1) − 𝐽) ∈ 𝐶))

Theoremballotlemrv2 29910* Value of 𝑅 after the tie. (Contributed by Thierry Arnoux, 11-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ (𝐼𝐶) < 𝐽) → (𝐽 ∈ (𝑅𝐶) ↔ 𝐽𝐶))

Theoremballotlemro 29911* Range of 𝑅 is included in 𝑂. (Contributed by Thierry Arnoux, 17-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)

Theoremballotlemgval 29912* Expand the value of . (Contributed by Thierry Arnoux, 21-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))    &    = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣𝑢)) − (#‘(𝑣𝑢))))       ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 𝑉) = ((#‘(𝑉𝑈)) − (#‘(𝑉𝑈))))

Theoremballotlemgun 29913* A property of the defined operator. (Contributed by Thierry Arnoux, 26-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))    &    = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣𝑢)) − (#‘(𝑣𝑢))))    &   (𝜑𝑈 ∈ Fin)    &   (𝜑𝑉 ∈ Fin)    &   (𝜑𝑊 ∈ Fin)    &   (𝜑 → (𝑉𝑊) = ∅)       (𝜑 → (𝑈 (𝑉𝑊)) = ((𝑈 𝑉) + (𝑈 𝑊)))

Theoremballotlemfg 29914* Express the value of (𝐹𝐶) in terms of . (Contributed by Thierry Arnoux, 21-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))    &    = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣𝑢)) − (#‘(𝑣𝑢))))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘𝐽) = (𝐶 (1...𝐽)))

Theoremballotlemfrc 29915* Express the value of (𝐹‘(𝑅𝐶)) in terms of the newly defined . (Contributed by Thierry Arnoux, 21-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))    &    = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣𝑢)) − (#‘(𝑣𝑢))))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))

Theoremballotlemfrci 29916* Reverse counting preserves a tie at the first tie. (Contributed by Thierry Arnoux, 21-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))    &    = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣𝑢)) − (#‘(𝑣𝑢))))       (𝐶 ∈ (𝑂𝐸) → ((𝐹‘(𝑅𝐶))‘(𝐼𝐶)) = 0)

Theoremballotlemfrceq 29917* Value of 𝐹 for a reverse counting (𝑅𝐶). (Contributed by Thierry Arnoux, 27-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))    &    = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣𝑢)) − (#‘(𝑣𝑢))))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽))

Theoremballotlemfrcn0 29918* Value of 𝐹 for a reversed counting (𝑅𝐶), before the first tie, cannot be zero . (Contributed by Thierry Arnoux, 25-Apr-2017.) (Revised by AV, 6-Oct-2020.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝐹‘(𝑅𝐶))‘𝐽) ≠ 0)

Theoremballotlemrc 29919* Range of 𝑅. (Contributed by Thierry Arnoux, 19-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ (𝑂𝐸))

Theoremballotlemirc 29920* Applying 𝑅 does not change first ties. (Contributed by Thierry Arnoux, 19-Apr-2017.) (Revised by AV, 6-Oct-2020.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       (𝐶 ∈ (𝑂𝐸) → (𝐼‘(𝑅𝐶)) = (𝐼𝐶))

Theoremballotlemrinv0 29921* Lemma for ballotlemrinv 29922. (Contributed by Thierry Arnoux, 18-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐷 = ((𝑆𝐶) “ 𝐶)) → (𝐷 ∈ (𝑂𝐸) ∧ 𝐶 = ((𝑆𝐷) “ 𝐷)))

Theoremballotlemrinv 29922* 𝑅 is its own inverse : it is an involution. (Contributed by Thierry Arnoux, 10-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       𝑅 = 𝑅

Theoremballotlem1ri 29923* When the vote on the first tie is for A, the first vote is also for A on the reverse counting. (Contributed by Thierry Arnoux, 18-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       (𝐶 ∈ (𝑂𝐸) → (1 ∈ (𝑅𝐶) ↔ (𝐼𝐶) ∈ 𝐶))

Theoremballotlem7 29924* 𝑅 is a bijection between two subsets of (𝑂𝐸): one where a vote for A is picked first, and one where a vote for B is picked first. (Contributed by Thierry Arnoux, 12-Dec-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       (𝑅 ↾ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}):{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}–1-1-onto→{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}

Theoremballotlem8 29925* There are as many countings with ties starting with a ballot for A as there are starting with a ballot for B. (Contributed by Thierry Arnoux, 7-Dec-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       (#‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) = (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})

Theoremballotth 29926* Bertrand's ballot problem : the probability that A is ahead throughout the counting. This is Metamath 100 proof #30. (Contributed by Thierry Arnoux, 7-Dec-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))    &   𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))       (𝑃𝐸) = ((𝑀𝑁) / (𝑀 + 𝑁))

21.3.21  Signum (sgn or sign) function - misc. additions

Theoremsgncl 29927 Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1})

Theoremsgnclre 29928 Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(𝐴 ∈ ℝ → (sgn‘𝐴) ∈ ℝ)

Theoremsgnneg 29929 Negation of the signum. (Contributed by Thierry Arnoux, 1-Oct-2018.)
(𝐴 ∈ ℝ → (sgn‘-𝐴) = -(sgn‘𝐴))

Theoremsgn3da 29930 A conditional containing a signum is true if it is true in all three possible cases. (Contributed by Thierry Arnoux, 1-Oct-2018.)
(𝜑𝐴 ∈ ℝ*)    &   ((sgn‘𝐴) = 0 → (𝜓𝜒))    &   ((sgn‘𝐴) = 1 → (𝜓𝜃))    &   ((sgn‘𝐴) = -1 → (𝜓𝜏))    &   ((𝜑𝐴 = 0) → 𝜒)    &   ((𝜑 ∧ 0 < 𝐴) → 𝜃)    &   ((𝜑𝐴 < 0) → 𝜏)       (𝜑𝜓)

Theoremsgnmul 29931 Signum of a product. (Contributed by Thierry Arnoux, 2-Oct-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵)))

Theoremsgnmulrp2 29932 Multiplication by a positive number does not affect signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (sgn‘(𝐴 · 𝐵)) = (sgn‘𝐴))

Theoremsgnsub 29933 Subtraction of a number of opposite sign. (Contributed by Thierry Arnoux, 2-Oct-2018.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 · 𝐵) < 0) → (sgn‘(𝐴𝐵)) = (sgn‘𝐴))

Theoremsgnnbi 29934 Negative signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
(𝐴 ∈ ℝ* → ((sgn‘𝐴) = -1 ↔ 𝐴 < 0))

Theoremsgnpbi 29935 Positive signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
(𝐴 ∈ ℝ* → ((sgn‘𝐴) = 1 ↔ 0 < 𝐴))

Theoremsgn0bi 29936 Zero signum. (Contributed by Thierry Arnoux, 10-Oct-2018.)
(𝐴 ∈ ℝ* → ((sgn‘𝐴) = 0 ↔ 𝐴 = 0))

Theoremsgnsgn 29937 Signum is idempotent. (Contributed by Thierry Arnoux, 2-Oct-2018.)
(𝐴 ∈ ℝ* → (sgn‘(sgn‘𝐴)) = (sgn‘𝐴))

Theoremsgnmulsgn 29938 If two real numbers are of different signs, so are their signs. (Contributed by Thierry Arnoux, 12-Oct-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 𝐵) < 0 ↔ ((sgn‘𝐴) · (sgn‘𝐵)) < 0))

Theoremsgnmulsgp 29939 If two real numbers are of different signs, so are their signs. (Contributed by Thierry Arnoux, 12-Oct-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < (𝐴 · 𝐵) ↔ 0 < ((sgn‘𝐴) · (sgn‘𝐵))))

Theoremfzssfzo 29940 Condition for an integer interval to be a subset of an half-open integer interval. (Contributed by Thierry Arnoux, 8-Oct-2018.)
(𝐾 ∈ (𝑀..^𝑁) → (𝑀...𝐾) ⊆ (𝑀..^𝑁))

Theoremgsumncl 29941* Closure of a group sum in a non-commutative monoid. (Contributed by Thierry Arnoux, 8-Oct-2018.)
𝐾 = (Base‘𝑀)    &   (𝜑𝑀 ∈ Mnd)    &   (𝜑𝑃 ∈ (ℤ𝑁))    &   ((𝜑𝑘 ∈ (𝑁...𝑃)) → 𝐵𝐾)       (𝜑 → (𝑀 Σg (𝑘 ∈ (𝑁...𝑃) ↦ 𝐵)) ∈ 𝐾)

Theoremgsumnunsn 29942* Closure of a group sum in a non-commutative monoid. (Contributed by Thierry Arnoux, 8-Oct-2018.)
𝐾 = (Base‘𝑀)    &   (𝜑𝑀 ∈ Mnd)    &   (𝜑𝑃 ∈ (ℤ𝑁))    &   ((𝜑𝑘 ∈ (𝑁...𝑃)) → 𝐵𝐾)    &    + = (+g𝑀)    &   (𝜑𝐶𝐾)    &   ((𝜑𝑘 = (𝑃 + 1)) → 𝐵 = 𝐶)       (𝜑 → (𝑀 Σg (𝑘 ∈ (𝑁...(𝑃 + 1)) ↦ 𝐵)) = ((𝑀 Σg (𝑘 ∈ (𝑁...𝑃) ↦ 𝐵)) + 𝐶))

21.3.22  Words over a set - misc additions

Theoremwrdres 29943 Condition for the restriction of a word to be a word itself. (Contributed by Thierry Arnoux, 5-Oct-2018.)
((𝑊 ∈ Word 𝑆𝑁 ∈ (0...(#‘𝑊))) → (𝑊 ↾ (0..^𝑁)) ∈ Word 𝑆)

Theoremwrdsplex 29944* Existence of a split of a word at a given index. (Contributed by Thierry Arnoux, 11-Oct-2018.)
((𝑊 ∈ Word 𝑆𝑁 ∈ (0...(#‘𝑊))) → ∃𝑣 ∈ Word 𝑆𝑊 = ((𝑊 ↾ (0..^𝑁)) ++ 𝑣))

21.3.22.1  Operations on words

Theoremccatmulgnn0dir 29945 Concatenation of words follow the rule mulgnn0dir 17394 (although applying mulgnn0dir 17394 would require 𝑆 to be a set). In this case 𝐴 is ⟨“𝐾”⟩ to the power 𝑀 in the free monoid. (Contributed by Thierry Arnoux, 5-Oct-2018.)
𝐴 = ((0..^𝑀) × {𝐾})    &   𝐵 = ((0..^𝑁) × {𝐾})    &   𝐶 = ((0..^(𝑀 + 𝑁)) × {𝐾})    &   (𝜑𝐾𝑆)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴 ++ 𝐵) = 𝐶)

Theoremofcccat 29946 Letterwise operations on word concatenations. (Contributed by Thierry Arnoux, 5-Oct-2018.)
(𝜑𝐹 ∈ Word 𝑆)    &   (𝜑𝐺 ∈ Word 𝑆)    &   (𝜑𝐾𝑇)       (𝜑 → ((𝐹 ++ 𝐺)∘𝑓/𝑐𝑅𝐾) = ((𝐹𝑓/𝑐𝑅𝐾) ++ (𝐺𝑓/𝑐𝑅𝐾)))

Theoremofcs1 29947 Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩∘𝑓/𝑐𝑅𝐵) = ⟨“(𝐴𝑅𝐵)”⟩)

Theoremofcs2 29948 Letterwise operations on a double letter word. (Contributed by Thierry Arnoux, 9-Oct-2018.)
((𝐴𝑆𝐵𝑆𝐶𝑇) → (⟨“𝐴𝐵”⟩∘𝑓/𝑐𝑅𝐶) = ⟨“(𝐴𝑅𝐶)(𝐵𝑅𝐶)”⟩)

21.3.23  Polynomials with real coefficients - misc additions

Theoremplymul02 29949 Product of a polynomial with the zero polynomial. (Contributed by Thierry Arnoux, 26-Sep-2018.)
(𝐹 ∈ (Poly‘𝑆) → (0𝑝𝑓 · 𝐹) = 0𝑝)

Theoremplymulx0 29950* Coefficients of a polynomial multiplyed by Xp. (Contributed by Thierry Arnoux, 25-Sep-2018.)
(𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))

Theoremplymulx 29951* Coefficients of a polynomial multiplyed by Xp. (Contributed by Thierry Arnoux, 25-Sep-2018.)
(𝐹 ∈ (Poly‘ℝ) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))

Theoremplyrecld 29952 Closure of a polynomial with real coefficients. (Contributed by Thierry Arnoux, 18-Sep-2018.)
(𝜑𝐹 ∈ (Poly‘ℝ))    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → (𝐹𝑋) ∈ ℝ)

Theoremsignsplypnf 29953* The quotient of a polynomial 𝐹 by a monic monomial of same degree 𝐺 converges to the highest coefficient of 𝐹. (Contributed by Thierry Arnoux, 18-Sep-2018.)
𝐷 = (deg‘𝐹)    &   𝐶 = (coeff‘𝐹)    &   𝐵 = (𝐶𝐷)    &   𝐺 = (𝑥 ∈ ℝ+ ↦ (𝑥𝐷))       (𝐹 ∈ (Poly‘ℝ) → (𝐹𝑓 / 𝐺) ⇝𝑟 𝐵)

Theoremsignsply0 29954* Lemma for the rule of signs, based on Bolzano's intermediate value theorem for polynomials : If the lowest and highest coefficient 𝐴 and 𝐵 are of opposite signs, the polynomial admits a positive root. (Contributed by Thierry Arnoux, 19-Sep-2018.)
𝐷 = (deg‘𝐹)    &   𝐶 = (coeff‘𝐹)    &   𝐵 = (𝐶𝐷)    &   𝐴 = (𝐶‘0)    &   (𝜑𝐹 ∈ (Poly‘ℝ))    &   (𝜑𝐹 ≠ 0𝑝)    &   (𝜑 → (𝐴 · 𝐵) < 0)       (𝜑 → ∃𝑧 ∈ ℝ+ (𝐹𝑧) = 0)

21.3.24  Descartes's rule of signs

21.3.24.1  Sign changes in a word over real numbers

Theoremsignspval 29955* The value of the skipping 0 sign operation. (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))       ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → (𝑋 𝑌) = if(𝑌 = 0, 𝑋, 𝑌))

Theoremsignsw0glem 29956* Neutral element property of . (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))       𝑢 ∈ {-1, 0, 1} ((0 𝑢) = 𝑢 ∧ (𝑢 0) = 𝑢)

Theoremsignswbase 29957 The base of 𝑊 is the triplet reprensenting the possible signs. (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       {-1, 0, 1} = (Base‘𝑊)

Theoremsignswplusg 29958* The operation of 𝑊. (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}        = (+g𝑊)

Theoremsignsw0g 29959* The neutral element of 𝑊. (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       0 = (0g𝑊)

Theoremsignswmnd 29960* 𝑊 is a monoid structure on {-1, 0, 1} which operation retains the right side, but skips zeroes. This will be used for skipping zeroes when counting sign changes. (Contributed by Thierry Arnoux, 9-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       𝑊 ∈ Mnd

Theoremsignswrid 29961* The zero-skipping operation propagages nonzeros. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       (𝑋 ∈ {-1, 0, 1} → (𝑋 0) = 𝑋)

Theoremsignswlid 29962* The zero-skipping operation keeps nonzeros. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       (((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) ∧ 𝑌 ≠ 0) → (𝑋 𝑌) = 𝑌)

Theoremsignswn0 29963* The zero-skipping operation propagages nonzeros. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       (((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) ∧ 𝑋 ≠ 0) → (𝑋 𝑌) ≠ 0)

Theoremsignswch 29964* The zero-skipping operation changes value when the operands change signs. (Contributed by Thierry Arnoux, 9-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}       ((𝑋 ∈ {-1, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → ((𝑋 𝑌) ≠ 𝑋 ↔ (𝑋 · 𝑌) < 0))

21.3.24.2  Counting sign changes in a word over real numbers

Theoremsignslema 29965 Computational part of signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)
(𝜑𝐸 ∈ ℕ0)    &   (𝜑𝐹 ∈ ℕ0)    &   (𝜑𝐺 ∈ ℕ0)    &   (𝜑𝐻 ∈ ℕ0)    &   (𝜑 → (𝐸 < 𝐺 ∧ ¬ 2 ∥ (𝐺𝐸)))    &   (𝜑 → ((𝐻𝐺) − (𝐹𝐸)) ∈ {0, 2})       (𝜑 → (𝐹 < 𝐻 ∧ ¬ 2 ∥ (𝐻𝐹)))

Theoremsignstfv 29966* Value of the zero-skipping sign word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐹 ∈ Word ℝ → (𝑇𝐹) = (𝑛 ∈ (0..^(#‘𝐹)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝐹𝑖))))))

Theoremsignstfval 29967* Value of the zero-skipping sign word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇𝐹)‘𝑁) = (𝑊 Σg (𝑖 ∈ (0...𝑁) ↦ (sgn‘(𝐹𝑖)))))

Theoremsignstcl 29968* Closure of the zero skipping sign word. (Contributed by Thierry Arnoux, 9-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇𝐹)‘𝑁) ∈ {-1, 0, 1})

Theoremsignstf 29969* The zero skipping sign word is a word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐹 ∈ Word ℝ → (𝑇𝐹) ∈ Word ℝ)

Theoremsignstlen 29970* Length of the zero skipping sign word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐹 ∈ Word ℝ → (#‘(𝑇𝐹)) = (#‘𝐹))

Theoremsignstf0 29971* Sign of a single letter word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐾 ∈ ℝ → (𝑇‘⟨“𝐾”⟩) = ⟨“(sgn‘𝐾)”⟩)

Theoremsignstfvn 29972* Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ 𝐾 ∈ ℝ) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘(#‘𝐹)) = (((𝑇𝐹)‘((#‘𝐹) − 1)) (sgn‘𝐾)))

Theoremsignsvtn0 29973* If the last letter is non zero, then this is the zero-skipping sign. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝑁 = (#‘𝐹)       ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘(𝑁 − 1)) ≠ 0) → ((𝑇𝐹)‘(𝑁 − 1)) = (sgn‘(𝐹‘(𝑁 − 1))))

Theoremsignstfvp 29974* Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇‘(𝐹 ++ ⟨“𝐾”⟩))‘𝑁) = ((𝑇𝐹)‘𝑁))

Theoremsignstfvneq0 29975* In case the first letter is not zero, the zero skipping sign is never zero. (Contributed by Thierry Arnoux, 10-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇𝐹)‘𝑁) ≠ 0)

Theoremsignstfvcl 29976* Closure of the zero skipping sign in case the first letter is not zero. (Contributed by Thierry Arnoux, 10-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇𝐹)‘𝑁) ∈ {-1, 1})

Theoremsignstfvc 29977* Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ Word ℝ ∧ 𝐺 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(#‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇𝐹)‘𝑁))

Theoremsignstres 29978* Restriction of a zero skipping sign to a subword. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → ((𝑇𝐹) ↾ (0..^𝑁)) = (𝑇‘(𝐹 ↾ (0..^𝑁))))

Theoremsignstfveq0a 29979* Lemma for signstfveq0 29980. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝑁 = (#‘𝐹)       (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → 𝑁 ∈ (ℤ‘2))

Theoremsignstfveq0 29980* In case the last letter is zero, the zero skipping sign is the same as the previous letter. (Contributed by Thierry Arnoux, 11-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝑁 = (#‘𝐹)       (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ (𝐹‘(𝑁 − 1)) = 0) → ((𝑇𝐹)‘(𝑁 − 1)) = ((𝑇𝐹)‘(𝑁 − 2)))

Theoremsignsvvfval 29981* The value of 𝑉, which represents the number of times the sign changes in a word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐹 ∈ Word ℝ → (𝑉𝐹) = Σ𝑗 ∈ (1..^(#‘𝐹))if(((𝑇𝐹)‘𝑗) ≠ ((𝑇𝐹)‘(𝑗 − 1)), 1, 0))

Theoremsignsvvf 29982* 𝑉 is a function. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       𝑉:Word ℝ⟶ℕ0

Theoremsignsvf0 29983* There is no change of sign in the empty word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝑉‘∅) = 0

Theoremsignsvf1 29984* In a single-letter word, which represents a constant polynomial, there is no change of sign. (Contributed by Thierry Arnoux, 8-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (𝐾 ∈ ℝ → (𝑉‘⟨“𝐾”⟩) = 0)

Theoremsignsvfn 29985* Number of changes in a word compared to a shorter word. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       (((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) ∧ 𝐾 ∈ ℝ) → (𝑉‘(𝐹 ++ ⟨“𝐾”⟩)) = ((𝑉𝐹) + if((((𝑇𝐹)‘((#‘𝐹) − 1)) · 𝐾) < 0, 1, 0)))

Theoremsignsvtp 29986* Adding a letter of the same sign as the highest coefficient does not change the sign. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   (𝜑𝐸 ∈ (Word ℝ ∖ {∅}))    &   (𝜑 → (𝐸‘0) ≠ 0)    &   (𝜑𝐹 = (𝐸 ++ ⟨“𝐴”⟩))    &   (𝜑𝐴 ∈ ℝ)    &   𝑁 = (#‘𝐸)    &   𝐵 = ((𝑇𝐸)‘(𝑁 − 1))       ((𝜑 ∧ 0 < (𝐴 · 𝐵)) → (𝑉𝐹) = (𝑉𝐸))

Theoremsignsvtn 29987* Adding a letter of a different sign as the highest coefficient changes the sign. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   (𝜑𝐸 ∈ (Word ℝ ∖ {∅}))    &   (𝜑 → (𝐸‘0) ≠ 0)    &   (𝜑𝐹 = (𝐸 ++ ⟨“𝐴”⟩))    &   (𝜑𝐴 ∈ ℝ)    &   𝑁 = (#‘𝐸)    &   𝐵 = ((𝑇𝐸)‘(𝑁 − 1))       ((𝜑 ∧ (𝐴 · 𝐵) < 0) → ((𝑉𝐹) − (𝑉𝐸)) = 1)

Theoremsignsvfpn 29988* Adding a letter of the same sign as the highest coefficient does not change the sign. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   (𝜑𝐸 ∈ (Word ℝ ∖ {∅}))    &   (𝜑 → (𝐸‘0) ≠ 0)    &   (𝜑𝐹 = (𝐸 ++ ⟨“𝐴”⟩))    &   (𝜑𝐴 ∈ ℝ)    &   𝑁 = (#‘𝐸)    &   𝐵 = (𝐸‘(𝑁 − 1))       ((𝜑 ∧ 0 < (𝐵 · 𝐴)) → (𝑉𝐹) = (𝑉𝐸))

Theoremsignsvfnn 29989* Adding a letter of a different sign as the highest coefficient changes the sign. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   (𝜑𝐸 ∈ (Word ℝ ∖ {∅}))    &   (𝜑 → (𝐸‘0) ≠ 0)    &   (𝜑𝐹 = (𝐸 ++ ⟨“𝐴”⟩))    &   (𝜑𝐴 ∈ ℝ)    &   𝑁 = (#‘𝐸)    &   𝐵 = (𝐸‘(𝑁 − 1))       ((𝜑 ∧ (𝐵 · 𝐴) < 0) → ((𝑉𝐹) − (𝑉𝐸)) = 1)

Theoremsignlem0 29990* Adding a zero as the highest coefficient does not change the parity of the sign changes. (Contributed by Thierry Arnoux, 12-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))       ((𝐹 ∈ (Word ℝ ∖ {∅}) ∧ (𝐹‘0) ≠ 0) → (𝑉‘(𝐹 ++ ⟨“0”⟩)) = (𝑉𝐹))

Theoremsignshf 29991* 𝐻, corresponding to the word 𝐹 multiplied by (𝑥𝐶), as a function. (Contributed by Thierry Arnoux, 29-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝐻 = ((⟨“0”⟩ ++ 𝐹) ∘𝑓 − ((𝐹 ++ ⟨“0”⟩)∘𝑓/𝑐 · 𝐶))       ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → 𝐻:(0..^((#‘𝐹) + 1))⟶ℝ)

Theoremsignshwrd 29992* 𝐻, corresponding to the word 𝐹 multiplied by (𝑥𝐶), is a word. (Contributed by Thierry Arnoux, 29-Sep-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝐻 = ((⟨“0”⟩ ++ 𝐹) ∘𝑓 − ((𝐹 ++ ⟨“0”⟩)∘𝑓/𝑐 · 𝐶))       ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → 𝐻 ∈ Word ℝ)

Theoremsignshlen 29993* Length of 𝐻, corresponding to the word 𝐹 multiplied by (𝑥𝐶). (Contributed by Thierry Arnoux, 14-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝐻 = ((⟨“0”⟩ ++ 𝐹) ∘𝑓 − ((𝐹 ++ ⟨“0”⟩)∘𝑓/𝑐 · 𝐶))       ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → (#‘𝐻) = ((#‘𝐹) + 1))

Theoremsignshnz 29994* 𝐻 is not the empty word. (Contributed by Thierry Arnoux, 14-Oct-2018.)
= (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))    &   𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}    &   𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))    &   𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))    &   𝐻 = ((⟨“0”⟩ ++ 𝐹) ∘𝑓 − ((𝐹 ++ ⟨“0”⟩)∘𝑓/𝑐 · 𝐶))       ((𝐹 ∈ Word ℝ ∧ 𝐶 ∈ ℝ+) → 𝐻 ≠ ∅)

21.3.25  Elementary Geometry

21.3.25.1  Two-dimension geometry

This definition has been superseded by DimTarskiG and is no longer needed in the main part of set.mm. It is only kept here for reference.

Syntaxcstrkg2d 29995 Extends class notation with the class of geometries fulfilling the planarity axioms.
class TarskiG2D

Definitiondf-trkg2d 29996* Define the class of geometries fulfilling the lower dimension axiom, Axiom A8 of [Schwabhauser] p. 12, and the upper dimension axiom, Axiom A9 of [Schwabhauser] p. 13, for dimension 2. (Contributed by Thierry Arnoux, 14-Mar-2019.) (New usage is discouraged.)
TarskiG2D = {𝑓[(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∃𝑥𝑝𝑦𝑝𝑧𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))}

Theoremistrkg2d 29997* Property of fulfilling dimension 2 axiom. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)       (𝐺 ∈ TarskiG2D ↔ (𝐺 ∈ V ∧ (∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))

Theoremaxtglowdim2OLD 29998* Lower dimension axiom for dimension 2, Axiom A8 of [Schwabhauser] p. 13. There exist 3 non-colinear points. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG2D)       (𝜑 → ∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))

Theoremaxtgupdim2OLD 29999 Upper dimension axiom for dimension 2, Axiom A9 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑈𝑃)    &   (𝜑𝑉𝑃)    &   (𝜑𝑈𝑉)    &   (𝜑 → (𝑋 𝑈) = (𝑋 𝑉))    &   (𝜑 → (𝑌 𝑈) = (𝑌 𝑉))    &   (𝜑 → (𝑍 𝑈) = (𝑍 𝑉))    &   (𝜑𝐺 ∈ TarskiG2D)       (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))

21.3.25.2  Outer Five Segment (not used, no need to move to main)

Syntaxcafs 30000 Declare the syntax for the outer five segment configuration.
class AFS

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
 Copyright terms: Public domain < Previous  Next >