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Theorem List for Metamath Proof Explorer - 40401-40500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3lexlogpow2ineq1 40401 Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
((3 / 2) < (2 logb 3) ∧ (2 logb 3) < (5 / 3))
 
Theorem3lexlogpow2ineq2 40402 Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
(2 < ((2 logb 3)↑2) ∧ ((2 logb 3)↑2) < 3)
 
Theorem3lexlogpow5ineq5 40403 Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
((2 logb 3)↑5) ≤ 15
 
21.25.5  Miscellaneous results for AKS formalisation
 
Theoremintlewftc 40404* Inequality inference by invoking fundamental theorem of calculus. (Contributed by metakunt, 22-Jul-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑𝐷 = (ℝ D 𝐹))    &   (𝜑𝐸 = (ℝ D 𝐺))    &   (𝜑𝐷 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   (𝜑𝐸 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   (𝜑𝐷 ∈ 𝐿1)    &   (𝜑𝐸 ∈ 𝐿1)    &   (𝜑𝐷 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃))    &   (𝜑𝐸 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄))    &   ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝑃𝑄)    &   (𝜑 → (𝐹𝐴) ≤ (𝐺𝐴))       (𝜑 → (𝐹𝐵) ≤ (𝐺𝐵))
 
Theoremaks4d1lem1 40405 Technical lemma to reduce proof size. (Contributed by metakunt, 14-Nov-2024.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))       (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵))
 
Theoremaks4d1p1p1 40406* Exponential law for finite products, special case. (Contributed by metakunt, 22-Jul-2024.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ∏𝑘 ∈ (1...𝑁)(𝐴𝑐𝑘) = (𝐴𝑐Σ𝑘 ∈ (1...𝑁)𝑘))
 
Theoremdvrelog2 40407* The derivative of the logarithm, ftc2 25330 version. (Contributed by metakunt, 11-Aug-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑𝐴𝐵)    &   𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥))    &   𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))       (𝜑 → (ℝ D 𝐹) = 𝐺)
 
Theoremdvrelog3 40408* The derivative of the logarithm on an open interval. (Contributed by metakunt, 11-Aug-2024.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴𝐵)    &   𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))    &   𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))       (𝜑 → (ℝ D 𝐹) = 𝐺)
 
Theoremdvrelog2b 40409* Derivative of the binary logarithm. (Contributed by metakunt, 11-Aug-2024.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴𝐵)    &   𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 logb 𝑥))    &   𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))       (𝜑 → (ℝ D 𝐹) = 𝐺)
 
Theorem0nonelalab 40410 Technical lemma for open interval. (Contributed by metakunt, 12-Aug-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))       (𝜑 → 0 ≠ 𝐶)
 
Theoremdvrelogpow2b 40411* Derivative of the power of the binary logarithm. (Contributed by metakunt, 12-Aug-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑𝐴𝐵)    &   𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 logb 𝑥)↑𝑁))    &   𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)))    &   𝐶 = (𝑁 / ((log‘2)↑𝑁))    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (ℝ D 𝐹) = 𝐺)
 
Theoremaks4d1p1p3 40412 Bound of a ceiling of the binary logarithm to the fifth power. (Contributed by metakunt, 19-Aug-2024.)
(𝜑𝑁 ∈ ℕ)    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 3 ≤ 𝑁)       (𝜑 → (𝑁𝑐(⌊‘(2 logb 𝐵))) < (𝑁𝑐(2 logb (((2 logb 𝑁)↑5) + 1))))
 
Theoremaks4d1p1p2 40413* Rewrite 𝐴 in more suitable form. (Contributed by metakunt, 19-Aug-2024.)
(𝜑𝑁 ∈ ℕ)    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 3 ≤ 𝑁)       (𝜑𝐴 < (𝑁𝑐(((2 logb (((2 logb 𝑁)↑5) + 1)) + (((2 logb 𝑁)↑2) / 2)) + (((2 logb 𝑁)↑4) / 2))))
 
Theoremaks4d1p1p4 40414* Technical step for inequality. The hard work is in to prove the final hypothesis. (Contributed by metakunt, 19-Aug-2024.)
(𝜑𝑁 ∈ ℕ)    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 3 ≤ 𝑁)    &   𝐶 = (2 logb (((2 logb 𝑁)↑5) + 1))    &   𝐷 = ((2 logb 𝑁)↑2)    &   𝐸 = ((2 logb 𝑁)↑4)    &   (𝜑 → ((2 · 𝐶) + 𝐷) ≤ 𝐸)       (𝜑𝐴 < (2↑𝐵))
 
Theoremdvle2 40415* Collapsed dvle 25293. (Contributed by metakunt, 19-Aug-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐸)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐹))    &   (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐺)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐻))    &   ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝐹𝐻)    &   (𝑥 = 𝐴𝐸 = 𝑃)    &   (𝑥 = 𝐴𝐺 = 𝑄)    &   (𝑥 = 𝐵𝐸 = 𝑅)    &   (𝑥 = 𝐵𝐺 = 𝑆)    &   (𝜑𝑃𝑄)    &   (𝜑𝐴𝐵)       (𝜑𝑅𝑆)
 
Theoremaks4d1p1p6 40416* Inequality lift to differentiable functions for a term in AKS inequality lemma. (Contributed by metakunt, 19-Aug-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 3 ≤ 𝐴)    &   (𝜑𝐴𝐵)       (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · (2 logb (((2 logb 𝑥)↑5) + 1))) + ((2 logb 𝑥)↑2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · ((1 / ((((2 logb 𝑥)↑5) + 1) · (log‘2))) · (((5 · ((2 logb 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥)))))
 
Theoremaks4d1p1p7 40417 Bound of intermediary of inequality step. (Contributed by metakunt, 19-Aug-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 4 ≤ 𝐴)       (𝜑 → ((2 · ((1 / ((((2 logb 𝐴)↑5) + 1) · (log‘2))) · (((5 · ((2 logb 𝐴)↑4)) · (1 / (𝐴 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝐴)↑(2 − 1)) / 𝐴))) ≤ ((4 / ((log‘2)↑4)) · (((log‘𝐴)↑3) / 𝐴)))
 
Theoremaks4d1p1p5 40418* Show inequality for existence of a non-divisor. (Contributed by metakunt, 19-Aug-2024.)
(𝜑𝑁 ∈ ℕ)    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 4 ≤ 𝑁)    &   𝐶 = (2 logb (((2 logb 𝑁)↑5) + 1))    &   𝐷 = ((2 logb 𝑁)↑2)    &   𝐸 = ((2 logb 𝑁)↑4)       (𝜑𝐴 < (2↑𝐵))
 
Theoremaks4d1p1 40419* Show inequality for existence of a non-divisor. (Contributed by metakunt, 21-Aug-2024.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))       (𝜑𝐴 < (2↑𝐵))
 
Theoremaks4d1p2 40420 Technical lemma for existence of non-divisor. (Contributed by metakunt, 27-Oct-2024.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))       (𝜑 → (2↑𝐵) ≤ (lcm‘(1...𝐵)))
 
Theoremaks4d1p3 40421* There exists a small enough number such that it does not divide 𝐴. (Contributed by metakunt, 27-Oct-2024.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))       (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟𝐴)
 
Theoremaks4d1p4 40422* There exists a small enough number such that it does not divide 𝐴. (Contributed by metakunt, 28-Oct-2024.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟𝐴}, ℝ, < )       (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅𝐴))
 
Theoremaks4d1p5 40423* Show that 𝑁 and 𝑅 are coprime for AKS existence theorem. Precondition will be eliminated in further theorem. (Contributed by metakunt, 30-Oct-2024.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟𝐴}, ℝ, < )    &   (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)       (𝜑 → (𝑁 gcd 𝑅) = 1)
 
Theoremaks4d1p6 40424* The maximal prime power exponent is smaller than the binary logarithm floor of 𝐵. (Contributed by metakunt, 30-Oct-2024.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟𝐴}, ℝ, < )    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑃𝑅)    &   𝐾 = (𝑃 pCnt 𝑅)       (𝜑𝐾 ≤ (⌊‘(2 logb 𝐵)))
 
Theoremaks4d1p7d1 40425* Technical step in AKS lemma 4.1 (Contributed by metakunt, 31-Oct-2024.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟𝐴}, ℝ, < )    &   (𝜑 → ∀𝑝 ∈ ℙ (𝑝𝑅𝑝𝑁))       (𝜑𝑅 ∥ (𝑁↑(⌊‘(2 logb 𝐵))))
 
Theoremaks4d1p7 40426* Technical step in AKS lemma 4.1 (Contributed by metakunt, 31-Oct-2024.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟𝐴}, ℝ, < )       (𝜑 → ∃𝑝 ∈ ℙ (𝑝𝑅 ∧ ¬ 𝑝𝑁))
 
Theoremaks4d1p8d1 40427 If a prime divides one number 𝑀, but not another number 𝑁, then it divides the quotient of 𝑀 and the gcd of 𝑀 and 𝑁. (Contributed by Thierry Arnoux, 10-Nov-2024.)
(𝜑𝑃 ∈ ℙ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃𝑀)    &   (𝜑 → ¬ 𝑃𝑁)       (𝜑𝑃 ∥ (𝑀 / (𝑀 gcd 𝑁)))
 
Theoremaks4d1p8d2 40428 Any prime power dividing a positive integer is less than that integer if that integer has another prime factor. (Contributed by metakunt, 13-Nov-2024.)
(𝜑𝑅 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑄 ∈ ℙ)    &   (𝜑𝑃𝑅)    &   (𝜑𝑄𝑅)    &   (𝜑 → ¬ 𝑃𝑁)    &   (𝜑𝑄𝑁)       (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)
 
Theoremaks4d1p8d3 40429 The remainder of a division with its maximal prime power is coprime with that prime power. (Contributed by metakunt, 13-Nov-2024.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑃𝑁)       (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = 1)
 
Theoremaks4d1p8 40430* Show that 𝑁 and 𝑅 are coprime for AKS existence theorem, with eliminated hypothesis. (Contributed by metakunt, 10-Nov-2024.) (Proof sketch by Thierry Arnoux.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟𝐴}, ℝ, < )       (𝜑 → (𝑁 gcd 𝑅) = 1)
 
Theoremaks4d1p9 40431* Show that the order is bound by the squared binary logarithm. (Contributed by metakunt, 14-Nov-2024.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟𝐴}, ℝ, < )       (𝜑 → ((2 logb 𝑁)↑2) < ((od𝑅)‘𝑁))
 
Theoremaks4d1 40432* Lemma 4.1 from https://www3.nd.edu/%7eandyp/notes/AKS.pdf, existence of a polynomially bounded number by the digit size of 𝑁 that asserts the polynomial subspace that we need to search to guarantee that 𝑁 is prime. Eventually we want to show that the polynomial searching space is bounded by degree 𝐵. (Contributed by metakunt, 14-Nov-2024.)
(𝜑𝑁 ∈ (ℤ‘3))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))       (𝜑 → ∃𝑟 ∈ (1...𝐵)((𝑁 gcd 𝑟) = 1 ∧ ((2 logb 𝑁)↑2) < ((od𝑟)‘𝑁)))
 
Theoremfldhmf1 40433 A field homomorphism is injective. This follows immediately from the definition of the ring homomorphism that sends the multiplicative identity to the multiplicative identity. (Contributed by metakunt, 7-Jan-2025.)
(𝜑𝐾 ∈ Field)    &   (𝜑𝐿 ∈ Field)    &   (𝜑𝐹 ∈ (𝐾 RingHom 𝐿))    &   𝐴 = (Base‘𝐾)    &   𝐵 = (Base‘𝐿)       (𝜑𝐹:𝐴1-1𝐵)
 
Theoremaks6d1c2p1 40434* In the AKS-theorem the subset defined by 𝐸 takes values in the positive integers. (Contributed by metakunt, 7-Jan-2025.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑃𝑁)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))       (𝜑𝐸:(ℕ0 × ℕ0)⟶ℕ)
 
Theoremaks6d1c2p2 40435* Injective condition for countability argument assuming that 𝑁 is not a prime power. (Contributed by metakunt, 7-Jan-2025.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑃𝑁)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   (𝜑𝑄 ∈ ℙ)    &   (𝜑𝑄𝑁)    &   (𝜑𝑃𝑄)       (𝜑𝐸:(ℕ0 × ℕ0)–1-1→ℕ)
 
Theorem5bc2eq10 40436 The value of 5 choose 2. (Contributed by metakunt, 8-Jun-2024.)
(5C2) = 10
 
Theoremfacp2 40437 The factorial of a successor's successor. (Contributed by metakunt, 19-Apr-2024.)
(𝑁 ∈ ℕ0 → (!‘(𝑁 + 2)) = ((!‘𝑁) · ((𝑁 + 1) · (𝑁 + 2))))
 
Theorem2np3bcnp1 40438 Part of induction step for 2ap1caineq 40439. (Contributed by metakunt, 8-Jun-2024.)
(𝜑𝑁 ∈ ℕ0)       (𝜑 → (((2 · (𝑁 + 1)) + 1)C(𝑁 + 1)) = ((((2 · 𝑁) + 1)C𝑁) · (2 · (((2 · 𝑁) + 3) / (𝑁 + 2)))))
 
Theorem2ap1caineq 40439 Inequality for Theorem 6.6 for AKS. (Contributed by metakunt, 8-Jun-2024.)
(𝜑𝑁 ∈ ℤ)    &   (𝜑 → 2 ≤ 𝑁)       (𝜑 → (2↑(𝑁 + 1)) < (((2 · 𝑁) + 1)C𝑁))
 
21.25.6  Sticks and stones
 
Theoremsticksstones1 40440* Different strictly monotone functions have different ranges. (Contributed by metakunt, 27-Sep-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝑋𝑌)    &   𝐼 = inf({𝑧 ∈ (1...𝐾) ∣ (𝑋𝑧) ≠ (𝑌𝑧)}, ℝ, < )       (𝜑 → ran 𝑋 ≠ ran 𝑌)
 
Theoremsticksstones2 40441* The range function on strictly monotone functions with finite domain and codomain is an injective mapping onto 𝐾-elemental sets. (Contributed by metakunt, 27-Sep-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}    &   𝐹 = (𝑧𝐴 ↦ ran 𝑧)       (𝜑𝐹:𝐴1-1𝐵)
 
Theoremsticksstones3 40442* The range function on strictly monotone functions with finite domain and codomain is an surjective mapping onto 𝐾-elemental sets. (Contributed by metakunt, 28-Sep-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}    &   𝐹 = (𝑧𝐴 ↦ ran 𝑧)       (𝜑𝐹:𝐴onto𝐵)
 
Theoremsticksstones4 40443* Equinumerosity lemma for sticks and stones. (Contributed by metakunt, 28-Sep-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}       (𝜑𝐴𝐵)
 
Theoremsticksstones5 40444* Count the number of strictly monotonely increasing functions on finite domains and codomains. (Contributed by metakunt, 28-Sep-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}       (𝜑 → (♯‘𝐴) = (𝑁C𝐾))
 
Theoremsticksstones6 40445* Function induces an order isomorphism for sticks and stones theorem. (Contributed by metakunt, 1-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐺:(1...(𝐾 + 1))⟶ℕ0)    &   (𝜑𝑋 ∈ (1...𝐾))    &   (𝜑𝑌 ∈ (1...𝐾))    &   (𝜑𝑋 < 𝑌)    &   𝐹 = (𝑥 ∈ (1...𝐾) ↦ (𝑥 + Σ𝑖 ∈ (1...𝑥)(𝐺𝑖)))       (𝜑 → (𝐹𝑋) < (𝐹𝑌))
 
Theoremsticksstones7 40446* Closure property of sticks and stones function. (Contributed by metakunt, 1-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐺:(1...(𝐾 + 1))⟶ℕ0)    &   (𝜑𝑋 ∈ (1...𝐾))    &   𝐹 = (𝑥 ∈ (1...𝐾) ↦ (𝑥 + Σ𝑖 ∈ (1...𝑥)(𝐺𝑖)))    &   (𝜑 → Σ𝑖 ∈ (1...(𝐾 + 1))(𝐺𝑖) = 𝑁)       (𝜑 → (𝐹𝑋) ∈ (1...(𝑁 + 𝐾)))
 
Theoremsticksstones8 40447* Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 1-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐹 = (𝑎𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}       (𝜑𝐹:𝐴𝐵)
 
Theoremsticksstones9 40448* Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 = 0)    &   𝐺 = (𝑏𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}       (𝜑𝐺:𝐵𝐴)
 
Theoremsticksstones10 40449* Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ)    &   𝐺 = (𝑏𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}       (𝜑𝐺:𝐵𝐴)
 
Theoremsticksstones11 40450* Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 = 0)    &   𝐹 = (𝑎𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))))    &   𝐺 = (𝑏𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}       (𝜑𝐹:𝐴1-1-onto𝐵)
 
Theoremsticksstones12a 40451* Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 11-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ)    &   𝐹 = (𝑎𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))))    &   𝐺 = (𝑏𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}       (𝜑 → ∀𝑑𝐵 (𝐹‘(𝐺𝑑)) = 𝑑)
 
Theoremsticksstones12 40452* Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ)    &   𝐹 = (𝑎𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))))    &   𝐺 = (𝑏𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}       (𝜑𝐹:𝐴1-1-onto𝐵)
 
Theoremsticksstones13 40453* Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐹 = (𝑎𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))))    &   𝐺 = (𝑏𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}       (𝜑𝐹:𝐴1-1-onto𝐵)
 
Theoremsticksstones14 40454* Sticks and stones with definitions as hypotheses. (Contributed by metakunt, 7-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐹 = (𝑎𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎𝑙))))    &   𝐺 = (𝑏𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓𝑥) < (𝑓𝑦)))}       (𝜑 → (♯‘𝐴) = ((𝑁 + 𝐾)C𝐾))
 
Theoremsticksstones15 40455* Sticks and stones with almost collapsed definitions for positive integers. (Contributed by metakunt, 7-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔𝑖) = 𝑁)}       (𝜑 → (♯‘𝐴) = ((𝑁 + 𝐾)C𝐾))
 
Theoremsticksstones16 40456* Sticks and stones with collapsed definitions for positive integers. (Contributed by metakunt, 20-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}       (𝜑 → (♯‘𝐴) = ((𝑁 + (𝐾 − 1))C(𝐾 − 1)))
 
Theoremsticksstones17 40457* Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}    &   𝐵 = { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)}    &   (𝜑𝑍:(1...𝐾)–1-1-onto𝑆)    &   𝐺 = (𝑏𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍𝑦))))       (𝜑𝐺:𝐵𝐴)
 
Theoremsticksstones18 40458* Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}    &   𝐵 = { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)}    &   (𝜑𝑍:(1...𝐾)–1-1-onto𝑆)    &   𝐹 = (𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))       (𝜑𝐹:𝐴𝐵)
 
Theoremsticksstones19 40459* Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}    &   𝐵 = { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)}    &   (𝜑𝑍:(1...𝐾)–1-1-onto𝑆)    &   𝐹 = (𝑎𝐴 ↦ (𝑥𝑆 ↦ (𝑎‘(𝑍𝑥))))    &   𝐺 = (𝑏𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍𝑦))))       (𝜑𝐹:𝐴1-1-onto𝐵)
 
Theoremsticksstones20 40460* Lift sticks and stones to arbitrary finite non-empty sets. (Contributed by metakung, 24-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ Fin)    &   (𝜑𝐾 ∈ ℕ)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔𝑖) = 𝑁)}    &   𝐵 = { ∣ (:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑖) = 𝑁)}    &   (𝜑 → (♯‘𝑆) = 𝐾)       (𝜑 → (♯‘𝐵) = ((𝑁 + (𝐾 − 1))C(𝐾 − 1)))
 
Theoremsticksstones21 40461* Lift sticks and stones to arbitrary finite non-empty sets. (Contributed by metakunt, 24-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ Fin)    &   (𝜑𝑆 ≠ ∅)    &   𝐴 = {𝑓 ∣ (𝑓:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑓𝑖) = 𝑁)}       (𝜑 → (♯‘𝐴) = ((𝑁 + ((♯‘𝑆) − 1))C((♯‘𝑆) − 1)))
 
Theoremsticksstones22 40462* Non-exhaustive sticks and stones. (Contributed by metakunt, 26-Oct-2024.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ Fin)    &   (𝜑𝑆 ≠ ∅)    &   𝐴 = {𝑓 ∣ (𝑓:𝑆⟶ℕ0 ∧ Σ𝑖𝑆 (𝑓𝑖) ≤ 𝑁)}       (𝜑 → (♯‘𝐴) = ((𝑁 + (♯‘𝑆))C(♯‘𝑆)))
 
21.25.7  Permutation results
 
Theoremmetakunt1 40463* A is an endomapping. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))       (𝜑𝐴:(1...𝑀)⟶(1...𝑀))
 
Theoremmetakunt2 40464* A is an endomapping. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝐼, if(𝑥 < 𝐼, 𝑥, (𝑥 + 1))))       (𝜑𝐴:(1...𝑀)⟶(1...𝑀))
 
Theoremmetakunt3 40465* Value of A. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐴𝑋) = if(𝑋 = 𝐼, 𝑀, if(𝑋 < 𝐼, 𝑋, (𝑋 − 1))))
 
Theoremmetakunt4 40466* Value of A. (Contributed by metakunt, 23-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝐼, if(𝑥 < 𝐼, 𝑥, (𝑥 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐴𝑋) = if(𝑋 = 𝑀, 𝐼, if(𝑋 < 𝐼, 𝑋, (𝑋 + 1))))
 
Theoremmetakunt5 40467* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 = 𝐼) → (𝐶‘(𝐴𝑋)) = 𝑋)
 
Theoremmetakunt6 40468* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 < 𝐼) → (𝐶‘(𝐴𝑋)) = 𝑋)
 
Theoremmetakunt7 40469* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝐼 < 𝑋) → ((𝐴𝑋) = (𝑋 − 1) ∧ ¬ (𝐴𝑋) = 𝑀 ∧ ¬ (𝐴𝑋) < 𝐼))
 
Theoremmetakunt8 40470* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝐼 < 𝑋) → (𝐶‘(𝐴𝑋)) = 𝑋)
 
Theoremmetakunt9 40471* C is the left inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐶‘(𝐴𝑋)) = 𝑋)
 
Theoremmetakunt10 40472* C is the right inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 = 𝑀) → (𝐴‘(𝐶𝑋)) = 𝑋)
 
Theoremmetakunt11 40473* C is the right inverse for A. (Contributed by metakunt, 24-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑𝑋 < 𝐼) → (𝐴‘(𝐶𝑋)) = 𝑋)
 
Theoremmetakunt12 40474* C is the right inverse for A. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       ((𝜑 ∧ ¬ (𝑋 = 𝑀𝑋 < 𝐼)) → (𝐴‘(𝐶𝑋)) = 𝑋)
 
Theoremmetakunt13 40475* C is the right inverse for A. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐴‘(𝐶𝑋)) = 𝑋)
 
Theoremmetakunt14 40476* A is a primitive permutation that moves the I-th element to the end and C is its inverse that moves the last element back to the I-th position. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))       (𝜑 → (𝐴:(1...𝑀)–1-1-onto→(1...𝑀) ∧ 𝐴 = 𝐶))
 
Theoremmetakunt15 40477* Construction of another permutation. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐹 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))       (𝜑𝐹:(1...(𝐼 − 1))–1-1-onto→(((𝑀𝐼) + 1)...(𝑀 − 1)))
 
Theoremmetakunt16 40478* Construction of another permutation. (Contributed by metakunt, 25-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐹 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))       (𝜑𝐹:(𝐼...(𝑀 − 1))–1-1-onto→(1...(𝑀𝐼)))
 
Theoremmetakunt17 40479 The union of three disjoint bijections is a bijection. (Contributed by metakunt, 28-May-2024.)
(𝜑𝐺:𝐴1-1-onto𝑋)    &   (𝜑𝐻:𝐵1-1-onto𝑌)    &   (𝜑𝐼:𝐶1-1-onto𝑍)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑 → (𝐴𝐶) = ∅)    &   (𝜑 → (𝐵𝐶) = ∅)    &   (𝜑 → (𝑋𝑌) = ∅)    &   (𝜑 → (𝑋𝑍) = ∅)    &   (𝜑 → (𝑌𝑍) = ∅)    &   (𝜑𝐹 = ((𝐺𝐻) ∪ 𝐼))    &   (𝜑𝐷 = ((𝐴𝐵) ∪ 𝐶))    &   (𝜑𝑊 = ((𝑋𝑌) ∪ 𝑍))       (𝜑𝐹:𝐷1-1-onto𝑊)
 
Theoremmetakunt18 40480 Disjoint domains and codomains. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)       (𝜑 → ((((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅) ∧ (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∩ (1...(𝑀𝐼))) = ∅ ∧ ((((𝑀𝐼) + 1)...(𝑀 − 1)) ∩ {𝑀}) = ∅ ∧ ((1...(𝑀𝐼)) ∩ {𝑀}) = ∅)))
 
Theoremmetakunt19 40481* Domains on restrictions of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))       (𝜑 → ((𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))) ∧ {⟨𝑀, 𝑀⟩} Fn {𝑀}))
 
Theoremmetakunt20 40482* Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))    &   (𝜑𝑋 = 𝑀)       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
 
Theoremmetakunt21 40483* Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))    &   (𝜑 → ¬ 𝑋 = 𝑀)    &   (𝜑𝑋 < 𝐼)       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
 
Theoremmetakunt22 40484* Show that B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))    &   (𝜑 → ¬ 𝑋 = 𝑀)    &   (𝜑 → ¬ 𝑋 < 𝐼)       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
 
Theoremmetakunt23 40485* B coincides on the union of bijections of functions. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))    &   𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀𝐼)))    &   𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))    &   (𝜑𝑋 ∈ (1...𝑀))       (𝜑 → (𝐵𝑋) = (((𝐶𝐷) ∪ {⟨𝑀, 𝑀⟩})‘𝑋))
 
Theoremmetakunt24 40486 Technical condition such that metakunt17 40479 holds. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)       (𝜑 → ((((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅ ∧ (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}) ∧ (1...𝑀) = (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀})))
 
Theoremmetakunt25 40487* B is a permutation. (Contributed by metakunt, 28-May-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀𝐼)), (𝑥 + (1 − 𝐼)))))       (𝜑𝐵:(1...𝑀)–1-1-onto→(1...𝑀))
 
Theoremmetakunt26 40488* Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   (𝜑𝑋 = 𝐼)       (𝜑 → (𝐶‘(𝐵‘(𝐴𝑋))) = 𝑋)
 
Theoremmetakunt27 40489* Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   (𝜑 → ¬ 𝑋 = 𝐼)    &   (𝜑𝑋 < 𝐼)       (𝜑 → (𝐵‘(𝐴𝑋)) = (𝑋 + (𝑀𝐼)))
 
Theoremmetakunt28 40490* Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   (𝜑 → ¬ 𝑋 = 𝐼)    &   (𝜑 → ¬ 𝑋 < 𝐼)       (𝜑 → (𝐵‘(𝐴𝑋)) = (𝑋𝐼))
 
Theoremmetakunt29 40491* Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   (𝜑 → ¬ 𝑋 = 𝐼)    &   (𝜑𝑋 < 𝐼)    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐻 = if(𝐼 ≤ (𝑋 + (𝑀𝐼)), 1, 0)       (𝜑 → (𝐶‘(𝐵‘(𝐴𝑋))) = ((𝑋 + (𝑀𝐼)) + 𝐻))
 
Theoremmetakunt30 40492* Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   (𝜑 → ¬ 𝑋 = 𝐼)    &   (𝜑 → ¬ 𝑋 < 𝐼)    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐻 = if(𝐼 ≤ (𝑋𝐼), 1, 0)       (𝜑 → (𝐶‘(𝐵‘(𝐴𝑋))) = ((𝑋𝐼) + 𝐻))
 
Theoremmetakunt31 40493* Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐺 = if(𝐼 ≤ (𝑋 + (𝑀𝐼)), 1, 0)    &   𝐻 = if(𝐼 ≤ (𝑋𝐼), 1, 0)    &   𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀𝐼)) + 𝐺), ((𝑋𝐼) + 𝐻)))       (𝜑 → (𝐶‘(𝐵‘(𝐴𝑋))) = 𝑅)
 
Theoremmetakunt32 40494* Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (1...𝑀))    &   𝐷 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑥, if(𝑥 < 𝐼, ((𝑥 + (𝑀𝐼)) + if(𝐼 ≤ (𝑥 + (𝑀𝐼)), 1, 0)), ((𝑥𝐼) + if(𝐼 ≤ (𝑥𝐼), 1, 0)))))    &   𝐺 = if(𝐼 ≤ (𝑋 + (𝑀𝐼)), 1, 0)    &   𝐻 = if(𝐼 ≤ (𝑋𝐼), 1, 0)    &   𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀𝐼)) + 𝐺), ((𝑋𝐼) + 𝐻)))       (𝜑 → (𝐷𝑋) = 𝑅)
 
Theoremmetakunt33 40495* Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀𝐼)), (𝑧 + (1 − 𝐼)))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐷 = (𝑤 ∈ (1...𝑀) ↦ if(𝑤 = 𝐼, 𝑤, if(𝑤 < 𝐼, ((𝑤 + (𝑀𝐼)) + if(𝐼 ≤ (𝑤 + (𝑀𝐼)), 1, 0)), ((𝑤𝐼) + if(𝐼 ≤ (𝑤𝐼), 1, 0)))))       (𝜑 → (𝐶 ∘ (𝐵𝐴)) = 𝐷)
 
Theoremmetakunt34 40496* 𝐷 is a permutation. (Contributed by metakunt, 18-Jul-2024.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐼𝑀)    &   𝐷 = (𝑤 ∈ (1...𝑀) ↦ if(𝑤 = 𝐼, 𝑤, if(𝑤 < 𝐼, ((𝑤 + (𝑀𝐼)) + if(𝐼 ≤ (𝑤 + (𝑀𝐼)), 1, 0)), ((𝑤𝐼) + if(𝐼 ≤ (𝑤𝐼), 1, 0)))))       (𝜑𝐷:(1...𝑀)–1-1-onto→(1...𝑀))
 
21.25.8  Unused lemmas scheduled for deletion
 
Theoremandiff 40497 Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.)
(𝜑 → (𝜒𝜃))    &   (𝜓 → (𝜃𝜒))       ((𝜑𝜓) → (𝜒𝜃))
 
Theoremfac2xp3 40498 Factorial of 2x+3, sublemma for sublemma for AKS. (Contributed by metakunt, 19-Apr-2024.)
(𝑥 ∈ ℕ0 → (!‘((2 · 𝑥) + 3)) = ((!‘((2 · 𝑥) + 1)) · (((2 · 𝑥) + 2) · ((2 · 𝑥) + 3))))
 
Theoremprodsplit 40499* Product split into two factors, original by Steven Nguyen. (Contributed by metakunt, 21-Apr-2024.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑀𝑁)    &   (𝜑𝐾 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 𝐾))) → 𝐴 ∈ ℂ)       (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 𝐾))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ∏𝑘 ∈ ((𝑁 + 1)...(𝑁 + 𝐾))𝐴))
 
Theorem2xp3dxp2ge1d 40500 2x+3 is greater than or equal to x+2 for x >= -1, a deduction version (Contributed by metakunt, 21-Apr-2024.)
(𝜑𝑋 ∈ (-1[,)+∞))       (𝜑 → 1 ≤ (((2 · 𝑋) + 3) / (𝑋 + 2)))
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