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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremballotlemsdom 30701* Domain of 𝑆 for a given counting 𝐶. (Contributed by Thierry Arnoux, 12-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...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))

Theoremballotlemsel1i 30702* 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 30703* 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 30704* 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 30705* 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 30706* 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 30707* 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 30708* 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 30709* 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 30710* 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 30711* 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 30712* 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 30713* 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 30714* 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 30715* 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 30716* 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 30717* 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 30718* 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 30719* 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 30720* 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 30721* 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 30722* Lemma for ballotlemrinv 30723. (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 30723* 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 30724* 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 30725* 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 30726* 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 30727* 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))

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

Theoremsgncl 30728 Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1})

Theoremsgnclre 30729 Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(𝐴 ∈ ℝ → (sgn‘𝐴) ∈ ℝ)

Theoremsgnneg 30730 Negation of the signum. (Contributed by Thierry Arnoux, 1-Oct-2018.)
(𝐴 ∈ ℝ → (sgn‘-𝐴) = -(sgn‘𝐴))

Theoremsgn3da 30731 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 30732 Signum of a product. (Contributed by Thierry Arnoux, 2-Oct-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵)))

Theoremsgnmulrp2 30733 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 30734 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 30735 Negative signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
(𝐴 ∈ ℝ* → ((sgn‘𝐴) = -1 ↔ 𝐴 < 0))

Theoremsgnpbi 30736 Positive signum. (Contributed by Thierry Arnoux, 2-Oct-2018.)
(𝐴 ∈ ℝ* → ((sgn‘𝐴) = 1 ↔ 0 < 𝐴))

Theoremsgn0bi 30737 Zero signum. (Contributed by Thierry Arnoux, 10-Oct-2018.)
(𝐴 ∈ ℝ* → ((sgn‘𝐴) = 0 ↔ 𝐴 = 0))

Theoremsgnsgn 30738 Signum is idempotent. (Contributed by Thierry Arnoux, 2-Oct-2018.)
(𝐴 ∈ ℝ* → (sgn‘(sgn‘𝐴)) = (sgn‘𝐴))

Theoremsgnmulsgn 30739 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 30740 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 30741 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 30742* 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 30743* 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))

20.3.23  Words over a set - misc additions

Theoremwrdfd 30744 A word is a zero-based sequence with a recoverable upper limit, deduction version. (Contributed by Thierry Arnoux, 22-Dec-2021.)
(𝜑𝑁 = (#‘𝑊))    &   (𝜑𝑊 ∈ Word 𝑆)       (𝜑𝑊:(0..^𝑁)⟶𝑆)

Theoremwrdres 30745 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 30746* 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))

20.3.23.1  Operations on words

Theoremccatmulgnn0dir 30747 Concatenation of words follow the rule mulgnn0dir 17618 (although applying mulgnn0dir 17618 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 30748 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 30749 Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩∘𝑓/𝑐𝑅𝐵) = ⟨“(𝐴𝑅𝐵)”⟩)

Theoremofcs2 30750 Letterwise operations on a double letter word. (Contributed by Thierry Arnoux, 9-Oct-2018.)
((𝐴𝑆𝐵𝑆𝐶𝑇) → (⟨“𝐴𝐵”⟩∘𝑓/𝑐𝑅𝐶) = ⟨“(𝐴𝑅𝐶)(𝐵𝑅𝐶)”⟩)

20.3.24  Polynomials with real coefficients - misc additions

Theoremplymul02 30751 Product of a polynomial with the zero polynomial. (Contributed by Thierry Arnoux, 26-Sep-2018.)
(𝐹 ∈ (Poly‘𝑆) → (0𝑝𝑓 · 𝐹) = 0𝑝)

Theoremplymulx0 30752* 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 30753* 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 30754 Closure of a polynomial with real coefficients. (Contributed by Thierry Arnoux, 18-Sep-2018.)
(𝜑𝐹 ∈ (Poly‘ℝ))    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → (𝐹𝑋) ∈ ℝ)

Theoremsignsplypnf 30755* 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 30756* 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)

20.3.25  Descartes's rule of signs

20.3.25.1  Sign changes in a word over real numbers

Theoremsignspval 30757* 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 30758* 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 30759 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 30760* 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 30761* 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 30762* 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 30763* 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 30764* 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 30765* 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 30766* 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))

20.3.25.2  Counting sign changes in a word over real numbers

Theoremsignslema 30767 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 30768* 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 30769* 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 30770* 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 30771* 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 30772* 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 30773* 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 30774* 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 30775* 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 30776* 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 30777* 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 30778* 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 30779* 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 30780* 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 30781* Lemma for signstfveq0 30782. (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 30782* 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 30783* 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 30784* 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 30785* 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 30786* 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 30787* 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 30788* 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 30789* 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 30790* 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 30791* 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 30792* 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 30793* 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 30794* 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 30795* 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 30796* 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 ≠ ∅)

20.3.26  Number Theory

Theoremefcld 30797 Closure law for the exponential function, deduction version. (Contributed by Thierry Arnoux, 1-Dec-2021.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (exp‘𝐴) ∈ ℂ)

Theoremiblidicc 30798* The identity function is integrable on any closed interval. (Contributed by Thierry Arnoux, 13-Dec-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝑥) ∈ 𝐿1)

Theoremrpsqrtcn 30799 Continuity of the real positive square root function. (Contributed by Thierry Arnoux, 20-Dec-2021.)
(√ ↾ ℝ+) ∈ (ℝ+cn→ℝ+)

Theoremdivsqrtid 30800 A real number divided by its square root. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝐴 ∈ ℝ+ → (𝐴 / (√‘𝐴)) = (√‘𝐴))

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