P. 431 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43001-43100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pellfundlb 43001	A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 15-Sep-2020.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴)
Theorem	pellfundglb 43002*	If a real is larger than the fundamental solution, there is a nontrivial solution less than it. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑥 ∈ (Pell1QR‘𝐷)((PellFund‘𝐷) ≤ 𝑥 ∧ 𝑥 < 𝐴))
Theorem	pellfundex 43003	The fundamental solution as an infimum is itself a solution, showing that the solution set is discrete. Since the fundamental solution is an infimum, there must be an element ge to Fund and lt 2*Fund. If this element is equal to the fundamental solution we're done, otherwise use the infimum again to find another element which must be ge Fund and lt the first element; their ratio is a group element in (1,2), contradicting pell14qrgapw 42993. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷))
Theorem	pellfund14gap 43004	There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → 𝐴 = 1)
Theorem	pellfundrp 43005	The fundamental Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ+)
Theorem	pellfundne1 43006	The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ≠ 1)
21.33.24  Logarithm laws generalized to an arbitrary base Section should be obsolete because its contents are covered by section "Logarithms to an arbitrary base" now.
Theorem	reglogcl 43007	General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbcl 26711 instead.
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1) → ((log‘𝐴) / (log‘𝐵)) ∈ ℝ)
Theorem	reglogltb 43008	General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logblt 26722 instead.
⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝐶 ∈ ℝ+ ∧ 1 < 𝐶)) → (𝐴 < 𝐵 ↔ ((log‘𝐴) / (log‘𝐶)) < ((log‘𝐵) / (log‘𝐶))))
Theorem	reglogleb 43009	General logarithm preserves ≤. (Contributed by Stefan O'Rear, 19-Oct-2014.) (New usage is discouraged.) Use logbleb 26721 instead.
⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝐶 ∈ ℝ+ ∧ 1 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ ((log‘𝐴) / (log‘𝐶)) ≤ ((log‘𝐵) / (log‘𝐶))))
Theorem	reglogmul 43010	Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbmul 26715 instead.
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ∧ (𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1)) → ((log‘(𝐴 · 𝐵)) / (log‘𝐶)) = (((log‘𝐴) / (log‘𝐶)) + ((log‘𝐵) / (log‘𝐶))))
Theorem	reglogexp 43011	Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbzexp 26714 instead.
⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ∧ (𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1)) → ((log‘(𝐴↑𝑁)) / (log‘𝐶)) = (𝑁 · ((log‘𝐴) / (log‘𝐶))))
Theorem	reglogbas 43012	General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logbid1 26706 instead.
⊢ ((𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1) → ((log‘𝐶) / (log‘𝐶)) = 1)
Theorem	reglog1 43013	General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logb1 26707 instead.
⊢ ((𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1) → ((log‘1) / (log‘𝐶)) = 0)
Theorem	reglogexpbas 43014	General log of a power of the base is the exponent. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbexp 26718 instead.
⊢ ((𝑁 ∈ ℤ ∧ (𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1)) → ((log‘(𝐶↑𝑁)) / (log‘𝐶)) = 𝑁)
21.33.25  Pell equations 4: the positive solution group is infinite cyclic
Theorem	pellfund14 43015*	Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))
Theorem	pellfund14b 43016*	The positive Pell solutions are precisely the integer powers of the fundamental solution. To get the general solution set (which we will not be using), throw in a copy of Z/2Z. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥)))
21.33.26  X and Y sequences 1: Definition and recurrence laws
Syntax	crmx 43017	Extend class notation to include the Robertson-Matiyasevich X sequence.
class Xrm
Syntax	crmy 43018	Extend class notation to include the Robertson-Matiyasevich Y sequence.
class Yrm
Definition	df-rmx 43019*	Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 43030 and rmxyval 43032 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
⊢ Xrm = (𝑎 ∈ (ℤ≥‘2), 𝑛 ∈ ℤ ↦ (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
Definition	df-rmy 43020*	Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 43031 and rmxyval 43032 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
⊢ Yrm = (𝑎 ∈ (ℤ≥‘2), 𝑛 ∈ ℤ ↦ (2nd ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd ‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))
Theorem	rmxfval 43021*	Value of the X sequence. Not used after rmxyval 43032 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) = (1st ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
Theorem	rmyfval 43022*	Value of the Y sequence. Not used after rmxyval 43032 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) = (2nd ‘(◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))
Theorem	rmspecsqrtnq 43023	The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by AV, 2-Aug-2021.)
⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))
Theorem	rmspecnonsq 43024	The discriminant used to define the X and Y sequences is a nonsquare positive integer and thus a valid Pell equation discriminant. (Contributed by Stefan O'Rear, 21-Sep-2014.)
⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN))
Theorem	qirropth 43025	This lemma implements the concept of "equate rational and irrational parts", used to prove many arithmetical properties of the X and Y sequences. (Contributed by Stefan O'Rear, 21-Sep-2014.)
⊢ ((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ((𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)) ↔ (𝐵 = 𝐷 ∧ 𝐶 = 𝐸)))
Theorem	rmspecfund 43026	The base of exponent used to define the X and Y sequences is the fundamental solution of the corresponding Pell equation. (Contributed by Stefan O'Rear, 21-Sep-2014.)
⊢ (𝐴 ∈ (ℤ≥‘2) → (PellFund‘((𝐴↑2) − 1)) = (𝐴 + (√‘((𝐴↑2) − 1))))
Theorem	rmxyelqirr 43027*	The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by SN, 23-Dec-2024.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ {𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
Theorem	rmxypairf1o 43028*	The function used to extract rational and irrational parts in df-rmx 43019 and df-rmy 43020 in fact achieves a one-to-one mapping from the quadratic irrationals to pairs of integers. (Contributed by Stefan O'Rear, 21-Sep-2014.)
⊢ (𝐴 ∈ (ℤ≥‘2) → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏)))):(ℕ0 × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
Theorem	rmxyelxp 43029*	Lemma for frmx 43030 and frmy 43031. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (◡(𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd ‘𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)) ∈ (ℕ0 × ℤ))
Theorem	frmx 43030	The X sequence is a nonnegative integer. See rmxnn 43068 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0
Theorem	frmy 43031	The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ
Theorem	rmxyval 43032	Main definition of the X and Y sequences. Compare definition 2.3 of [JonesMatijasevic] p. 694. (Contributed by Stefan O'Rear, 19-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm 𝑁) + ((√‘((𝐴↑2) − 1)) · (𝐴 Yrm 𝑁))) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))
Theorem	rmspecpos 43033	The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℝ+)
Theorem	rmxycomplete 43034*	The X and Y sequences taken together enumerate all solutions to the corresponding Pell equation in the right half-plane. This is Metamath 100 proof #39. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℕ0 ∧ 𝑌 ∈ ℤ) → (((𝑋↑2) − (((𝐴↑2) − 1) · (𝑌↑2))) = 1 ↔ ∃𝑛 ∈ ℤ (𝑋 = (𝐴 Xrm 𝑛) ∧ 𝑌 = (𝐴 Yrm 𝑛))))
Theorem	rmxynorm 43035	The X and Y sequences define a solution to the corresponding Pell equation. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (((𝐴 Xrm 𝑁)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1)
Theorem	rmbaserp 43036	The base of exponentiation for the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 + (√‘((𝐴↑2) − 1))) ∈ ℝ+)
Theorem	rmxyneg 43037	Negation law for X and Y sequences. JonesMatijasevic is inconsistent on whether the X and Y sequences have domain ℕ0 or ℤ; we use ℤ consistently to avoid the need for a separate subtraction law. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm -𝑁) = (𝐴 Xrm 𝑁) ∧ (𝐴 Yrm -𝑁) = -(𝐴 Yrm 𝑁)))
Theorem	rmxyadd 43038	Addition formula for X and Y sequences. See rmxadd 43044 and rmyadd 43048 for most uses. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm (𝑀 + 𝑁)) = (((𝐴 Xrm 𝑀) · (𝐴 Xrm 𝑁)) + (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀) · (𝐴 Yrm 𝑁)))) ∧ (𝐴 Yrm (𝑀 + 𝑁)) = (((𝐴 Yrm 𝑀) · (𝐴 Xrm 𝑁)) + ((𝐴 Xrm 𝑀) · (𝐴 Yrm 𝑁)))))
Theorem	rmxy1 43039	Value of the X and Y sequences at 1. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 Xrm 1) = 𝐴 ∧ (𝐴 Yrm 1) = 1))
Theorem	rmxy0 43040	Value of the X and Y sequences at 0. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴 Xrm 0) = 1 ∧ (𝐴 Yrm 0) = 0))
Theorem	rmxneg 43041	Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 43037, rmxyadd 43038, rmxy0 43040, and rmxy1 43039 via qirropth 43025 results in two theorems at once, but typical use requires only one, so this group of theorems serves to separate the cases. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm -𝑁) = (𝐴 Xrm 𝑁))
Theorem	rmx0 43042	Value of X sequence at 0. Part 1 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 Xrm 0) = 1)
Theorem	rmx1 43043	Value of X sequence at 1. Part 2 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 Xrm 1) = 𝐴)
Theorem	rmxadd 43044	Addition formula for X sequence. Equation 2.7 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑀 + 𝑁)) = (((𝐴 Xrm 𝑀) · (𝐴 Xrm 𝑁)) + (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀) · (𝐴 Yrm 𝑁)))))
Theorem	rmyneg 43045	Negation formula for Y sequence (odd function). (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm -𝑁) = -(𝐴 Yrm 𝑁))
Theorem	rmy0 43046	Value of Y sequence at 0. Part 1 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 Yrm 0) = 0)
Theorem	rmy1 43047	Value of Y sequence at 1. Part 2 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 Yrm 1) = 1)
Theorem	rmyadd 43048	Addition formula for Y sequence. Equation 2.8 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑀 + 𝑁)) = (((𝐴 Yrm 𝑀) · (𝐴 Xrm 𝑁)) + ((𝐴 Xrm 𝑀) · (𝐴 Yrm 𝑁))))
Theorem	rmxp1 43049	Special addition-of-1 formula for X sequence. Part 1 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 + 1)) = (((𝐴 Xrm 𝑁) · 𝐴) + (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁))))
Theorem	rmyp1 43050	Special addition of 1 formula for Y sequence. Part 2 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 24-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 + 1)) = (((𝐴 Yrm 𝑁) · 𝐴) + (𝐴 Xrm 𝑁)))
Theorem	rmxm1 43051	Subtraction of 1 formula for X sequence. Part 1 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 − 1)) = ((𝐴 · (𝐴 Xrm 𝑁)) − (((𝐴↑2) − 1) · (𝐴 Yrm 𝑁))))
Theorem	rmym1 43052	Subtraction of 1 formula for Y sequence. Part 2 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 − 1)) = (((𝐴 Yrm 𝑁) · 𝐴) − (𝐴 Xrm 𝑁)))
Theorem	rmxluc 43053	The X sequence is a Lucas (second-order integer recurrence) sequence. Part 3 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑁 + 1)) = (((2 · 𝐴) · (𝐴 Xrm 𝑁)) − (𝐴 Xrm (𝑁 − 1))))
Theorem	rmyluc 43054	The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 43046 and rmy1 43047. Part 3 of equation 2.12 of [JonesMatijasevic] p. 695. JonesMatijasevic uses this theorem to redefine the X and Y sequences to have domain (ℤ × ℤ), which simplifies some later theorems. It may shorten the derivation to use this as our initial definition. Incidentally, the X sequence satisfies the exact same recurrence. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 + 1)) = ((2 · ((𝐴 Yrm 𝑁) · 𝐴)) − (𝐴 Yrm (𝑁 − 1))))
Theorem	rmyluc2 43055	Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (𝑁 + 1)) = (((2 · 𝐴) · (𝐴 Yrm 𝑁)) − (𝐴 Yrm (𝑁 − 1))))
Theorem	rmxdbl 43056	"Double-angle formula" for X-values. Equation 2.13 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (2 · 𝑁)) = ((2 · ((𝐴 Xrm 𝑁)↑2)) − 1))
Theorem	rmydbl 43057	"Double-angle formula" for Y-values. Equation 2.14 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm (2 · 𝑁)) = ((2 · (𝐴 Xrm 𝑁)) · (𝐴 Yrm 𝑁)))
21.33.27  Ordering and induction lemmas for the integers
Theorem	monotuz 43058*	A function defined on an upper set of integers which increases at every adjacent pair is globally strictly monotonic by induction. (Contributed by Stefan O'Rear, 24-Sep-2014.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐻) → 𝐹 < 𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝐶 ∈ ℝ)    &   ⊢ 𝐻 = (ℤ≥‘𝐼)    &   ⊢ (𝑥 = (𝑦 + 1) → 𝐶 = 𝐺)    &   ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐹)    &   ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻)) → (𝐴 < 𝐵 ↔ 𝐷 < 𝐸))
Theorem	monotoddzzfi 43059*	A function which is odd and monotonic on ℕ0 is monotonic on ℤ. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘𝑥) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝐹‘-𝑥) = -(𝐹‘𝑥))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐹‘𝐴) < (𝐹‘𝐵)))
Theorem	monotoddzz 43060*	A function (given implicitly) which is odd and monotonic on ℕ0 is monotonic on ℤ. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → 𝐸 < 𝐹))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐸 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐺 = -𝐹)    &   ⊢ (𝑥 = 𝐴 → 𝐸 = 𝐶)    &   ⊢ (𝑥 = 𝐵 → 𝐸 = 𝐷)    &   ⊢ (𝑥 = 𝑦 → 𝐸 = 𝐹)    &   ⊢ (𝑥 = -𝑦 → 𝐸 = 𝐺)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ 𝐶 < 𝐷))
Theorem	oddcomabszz 43061*	An odd function which takes nonnegative values on nonnegative arguments commutes with abs. (Contributed by Stefan O'Rear, 26-Sep-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ ∧ 0 ≤ 𝑥) → 0 ≤ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ ℤ) → 𝐶 = -𝐵)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = -𝑦 → 𝐴 = 𝐶)    &   ⊢ (𝑥 = 𝐷 → 𝐴 = 𝐸)    &   ⊢ (𝑥 = (abs‘𝐷) → 𝐴 = 𝐹)    ⇒   ⊢ ((𝜑 ∧ 𝐷 ∈ ℤ) → (abs‘𝐸) = 𝐹)
Theorem	2nn0ind 43062*	Induction on nonnegative integers with two base cases, for use with Lucas-type sequences. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ (𝑦 ∈ ℕ → ((𝜃 ∧ 𝜏) → 𝜂))    &   ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 − 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜂))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜌))    ⇒   ⊢ (𝐴 ∈ ℕ0 → 𝜌)
Theorem	zindbi 43063*	Inductively transfer a property to the integers if it holds for zero and passes between adjacent integers in either direction. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ (𝑦 ∈ ℤ → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    ⇒   ⊢ (𝐴 ∈ ℤ → (𝜃 ↔ 𝜏))
21.33.28  X and Y sequences 2: Order properties
Theorem	rmxypos 43064	For all nonnegative indices, X is positive and Y is nonnegative. (Contributed by Stefan O'Rear, 24-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (0 < (𝐴 Xrm 𝑁) ∧ 0 ≤ (𝐴 Yrm 𝑁)))
Theorem	ltrmynn0 43065	The Y-sequence is strictly monotonic on ℕ0. Strengthened by ltrmy 43069. (Contributed by Stefan O'Rear, 24-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝐴 Yrm 𝑀) < (𝐴 Yrm 𝑁)))
Theorem	ltrmxnn0 43066	The X-sequence is strictly monotonic on ℕ0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝐴 Xrm 𝑀) < (𝐴 Xrm 𝑁)))
Theorem	lermxnn0 43067	The X-sequence is monotonic on ℕ0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝐴 Xrm 𝑀) ≤ (𝐴 Xrm 𝑁)))
Theorem	rmxnn 43068	The X-sequence is defined to range over ℕ0 but never actually takes the value 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) ∈ ℕ)
Theorem	ltrmy 43069	The Y-sequence is strictly monotonic over ℤ. (Contributed by Stefan O'Rear, 25-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝐴 Yrm 𝑀) < (𝐴 Yrm 𝑁)))
Theorem	rmyeq0 43070	Y is zero only at zero. (Contributed by Stefan O'Rear, 26-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 ↔ (𝐴 Yrm 𝑁) = 0))
Theorem	rmyeq 43071	Y is one-to-one. (Contributed by Stefan O'Rear, 3-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 = 𝑁 ↔ (𝐴 Yrm 𝑀) = (𝐴 Yrm 𝑁)))
Theorem	lermy 43072	Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝐴 Yrm 𝑀) ≤ (𝐴 Yrm 𝑁)))
Theorem	rmynn 43073	Yrm is positive for positive arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) → (𝐴 Yrm 𝑁) ∈ ℕ)
Theorem	rmynn0 43074	Yrm is nonnegative for nonnegative arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 Yrm 𝑁) ∈ ℕ0)
Theorem	rmyabs 43075	Yrm commutes with abs. (Contributed by Stefan O'Rear, 26-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℤ) → (abs‘(𝐴 Yrm 𝐵)) = (𝐴 Yrm (abs‘𝐵)))
Theorem	jm2.24nn 43076	X(n) is strictly greater than Y(n) + Y(n-1). Lemma 2.24 of [JonesMatijasevic] p. 697 restricted to ℕ. (Contributed by Stefan O'Rear, 3-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) → ((𝐴 Yrm (𝑁 − 1)) + (𝐴 Yrm 𝑁)) < (𝐴 Xrm 𝑁))
Theorem	jm2.17a 43077	First half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 14-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (((2 · 𝐴) − 1)↑𝑁) ≤ (𝐴 Yrm (𝑁 + 1)))
Theorem	jm2.17b 43078	Weak form of the second half of lemma 2.17 of [JonesMatijasevic] p. 696, allowing induction to start lower. (Contributed by Stefan O'Rear, 15-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 Yrm (𝑁 + 1)) ≤ ((2 · 𝐴)↑𝑁))
Theorem	jm2.17c 43079	Second half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 15-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) → (𝐴 Yrm ((𝑁 + 1) + 1)) < ((2 · 𝐴)↑(𝑁 + 1)))
Theorem	jm2.24 43080	Lemma 2.24 of [JonesMatijasevic] p. 697 extended to ℤ. Could be eliminated with a more careful proof of jm2.26lem3 43118. (Contributed by Stefan O'Rear, 3-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm (𝑁 − 1)) + (𝐴 Yrm 𝑁)) < (𝐴 Xrm 𝑁))
Theorem	rmygeid 43081	Y(n) increases faster than n. Used implicitly without proof or comment in lemma 2.27 of [JonesMatijasevic] p. 697. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝐴 Yrm 𝑁))
21.33.29  Congruential equations
Theorem	congtr 43082	A wff of the form 𝐴 ∥ (𝐵 − 𝐶) is interpreted as a congruential equation. This is similar to (𝐵 mod 𝐴) = (𝐶 mod 𝐴), but is defined such that behavior is regular for zero and negative values of 𝐴. To use this concept effectively, we need to show that congruential equations behave similarly to normal equations; first a transitivity law. Idea for the future: If there was a congruential equation symbol, it could incorporate type constraints, so that most of these would not need them. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − 𝐶) ∧ 𝐴 ∥ (𝐶 − 𝐷))) → 𝐴 ∥ (𝐵 − 𝐷))
Theorem	congadd 43083	If two pairs of numbers are componentwise congruent, so are their sums. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝐷 ∈ ℤ ∧ 𝐸 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − 𝐶) ∧ 𝐴 ∥ (𝐷 − 𝐸))) → 𝐴 ∥ ((𝐵 + 𝐷) − (𝐶 + 𝐸)))
Theorem	congmul 43084	If two pairs of numbers are componentwise congruent, so are their products. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝐷 ∈ ℤ ∧ 𝐸 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − 𝐶) ∧ 𝐴 ∥ (𝐷 − 𝐸))) → 𝐴 ∥ ((𝐵 · 𝐷) − (𝐶 · 𝐸)))
Theorem	congsym 43085	Congruence mod 𝐴 is a symmetric/commutative relation. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐴 ∥ (𝐵 − 𝐶))) → 𝐴 ∥ (𝐶 − 𝐵))
Theorem	congneg 43086	If two integers are congruent mod 𝐴, so are their negatives. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐴 ∥ (𝐵 − 𝐶))) → 𝐴 ∥ (-𝐵 − -𝐶))
Theorem	congsub 43087	If two pairs of numbers are componentwise congruent, so are their differences. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝐷 ∈ ℤ ∧ 𝐸 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − 𝐶) ∧ 𝐴 ∥ (𝐷 − 𝐸))) → 𝐴 ∥ ((𝐵 − 𝐷) − (𝐶 − 𝐸)))
Theorem	congid 43088	Every integer is congruent to itself mod every base. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∥ (𝐵 − 𝐵))
Theorem	mzpcong 43089*	Polynomials commute with congruences. (Does this characterize them?) (Contributed by Stefan O'Rear, 5-Oct-2014.)
⊢ ((𝐹 ∈ (mzPoly‘𝑉) ∧ (𝑋 ∈ (ℤ ↑m 𝑉) ∧ 𝑌 ∈ (ℤ ↑m 𝑉)) ∧ (𝑁 ∈ ℤ ∧ ∀𝑘 ∈ 𝑉 𝑁 ∥ ((𝑋‘𝑘) − (𝑌‘𝑘)))) → 𝑁 ∥ ((𝐹‘𝑋) − (𝐹‘𝑌)))
Theorem	congrep 43090*	Every integer is congruent to some number in the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℤ) → ∃𝑎 ∈ (0...(𝐴 − 1))𝐴 ∥ (𝑎 − 𝑁))
Theorem	congabseq 43091	If two integers are congruent, they are either equal or separated by at least the congruence base. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ 𝐴 ∥ (𝐵 − 𝐶)) → ((abs‘(𝐵 − 𝐶)) < 𝐴 ↔ 𝐵 = 𝐶))
21.33.30  Alternating congruential equations
Theorem	acongid 43092	A wff like that in this theorem will be known as an "alternating congruence". A special symbol might be considered if more uses come up. They have many of the same properties as normal congruences, starting with reflexivity. JonesMatijasevic uses "a ≡ ± b (mod c)" for this construction. The disjunction of divisibility constraints seems to adequately capture the concept, but it's rather verbose and somewhat inelegant. Use of an explicit equivalence relation might also work. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ (𝐵 − 𝐵) ∨ 𝐴 ∥ (𝐵 − -𝐵)))
Theorem	acongsym 43093	Symmetry of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − 𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶))) → (𝐴 ∥ (𝐶 − 𝐵) ∨ 𝐴 ∥ (𝐶 − -𝐵)))
Theorem	acongneg2 43094	Negate right side of alternating congruence. Makes essential use of the "alternating" part. (Contributed by Stefan O'Rear, 3-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝐴 ∥ (𝐵 − -𝐶) ∨ 𝐴 ∥ (𝐵 − --𝐶))) → (𝐴 ∥ (𝐵 − 𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶)))
Theorem	acongtr 43095	Transitivity of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ ((𝐴 ∥ (𝐵 − 𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶)) ∧ (𝐴 ∥ (𝐶 − 𝐷) ∨ 𝐴 ∥ (𝐶 − -𝐷)))) → (𝐴 ∥ (𝐵 − 𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)))
Theorem	acongeq12d 43096	Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.)
⊢ (𝜑 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐷 = 𝐸)    ⇒   ⊢ (𝜑 → ((𝐴 ∥ (𝐵 − 𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶 − 𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸))))
Theorem	acongrep 43097*	Every integer is alternating-congruent to some number in the first half of the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℤ) → ∃𝑎 ∈ (0...𝐴)((2 · 𝐴) ∥ (𝑎 − 𝑁) ∨ (2 · 𝐴) ∥ (𝑎 − -𝑁)))
Theorem	fzmaxdif 43098	Bound on the difference between two integers constrained to two possibly overlapping finite ranges. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶 − 𝐸) ≤ (𝐹 − 𝐵)) → (abs‘(𝐴 − 𝐷)) ≤ (𝐹 − 𝐵))
Theorem	fzneg 43099	Reflection of a finite range of integers about 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ (𝐵...𝐶) ↔ -𝐴 ∈ (-𝐶...-𝐵)))
Theorem	acongeq 43100	Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 43119. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 = 𝐶 ↔ ((2 · 𝐴) ∥ (𝐵 − 𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶))))