P. 434 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43301-43400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pell1234qrmulcl 43301	General solutions of the Pell equation are closed under multiplication. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷) ∧ 𝐵 ∈ (Pell1234QR‘𝐷)) → (𝐴 · 𝐵) ∈ (Pell1234QR‘𝐷))
Theorem	pell14qrss1234 43302	A positive Pell solution is a general Pell solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ (Pell1234QR‘𝐷))
Theorem	pell14qrre 43303	A positive Pell solution is a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ)
Theorem	pell14qrne0 43304	A positive Pell solution is a nonzero number. (Contributed by Stefan O'Rear, 17-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ≠ 0)
Theorem	pell14qrgt0 43305	A positive Pell solution is a positive number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 < 𝐴)
Theorem	pell14qrrp 43306	A positive Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ+)
Theorem	pell1234qrdich 43307	A general Pell solution is either a positive solution, or its negation is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → (𝐴 ∈ (Pell14QR‘𝐷) ∨ -𝐴 ∈ (Pell14QR‘𝐷)))
Theorem	elpell14qr2 43308	A number is a positive Pell solution iff it is positive and a Pell solution, justifying our name choice. (Contributed by Stefan O'Rear, 19-Oct-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ (𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴)))
Theorem	pell14qrmulcl 43309	Positive Pell solutions are closed under multiplication. (Contributed by Stefan O'Rear, 17-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 · 𝐵) ∈ (Pell14QR‘𝐷))
Theorem	pell14qrreccl 43310	Positive Pell solutions are closed under reciprocal. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (1 / 𝐴) ∈ (Pell14QR‘𝐷))
Theorem	pell14qrdivcl 43311	Positive Pell solutions are closed under division. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 / 𝐵) ∈ (Pell14QR‘𝐷))
Theorem	pell14qrexpclnn0 43312	Lemma for pell14qrexpcl 43313. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ ℕ0) → (𝐴↑𝐵) ∈ (Pell14QR‘𝐷))
Theorem	pell14qrexpcl 43313	Positive Pell solutions are closed under integer powers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ ℤ) → (𝐴↑𝐵) ∈ (Pell14QR‘𝐷))
Theorem	pell1qrss14 43314	First-quadrant Pell solutions are a subset of the positive solutions. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
Theorem	pell14qrdich 43315	A positive Pell solution is either in the first quadrant, or its reciprocal is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 ∈ (Pell1QR‘𝐷) ∨ (1 / 𝐴) ∈ (Pell1QR‘𝐷)))
Theorem	pell1qrge1 43316	A Pell solution in the first quadrant is at least 1. (Contributed by Stefan O'Rear, 17-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1QR‘𝐷)) → 1 ≤ 𝐴)
Theorem	pell1qr1 43317	1 is a Pell solution and in the first quadrant as one. (Contributed by Stefan O'Rear, 17-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ∈ (Pell1QR‘𝐷))
Theorem	elpell1qr2 43318	The first quadrant solutions are precisely the positive Pell solutions which are at least one. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ (𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝐴)))
Theorem	pell1qrgaplem 43319	Lemma for pell1qrgap 43320. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (((𝐷 ∈ ℕ ∧ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0)) ∧ (1 < (𝐴 + ((√‘𝐷) · 𝐵)) ∧ ((𝐴↑2) − (𝐷 · (𝐵↑2))) = 1)) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (𝐴 + ((√‘𝐷) · 𝐵)))
Theorem	pell1qrgap 43320	First-quadrant Pell solutions are bounded away from 1. (This particular bound allows to prove exact values for the fundamental solution later.) (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1QR‘𝐷) ∧ 1 < 𝐴) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴)
Theorem	pell14qrgap 43321	Positive Pell solutions are bounded away from 1. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴)
Theorem	pell14qrgapw 43322	Positive Pell solutions are bounded away from 1, with a friendlier bound. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → 2 < 𝐴)
Theorem	pellqrexplicit 43323	Condition for a calculated real to be a Pell solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ((𝐴↑2) − (𝐷 · (𝐵↑2))) = 1) → (𝐴 + ((√‘𝐷) · 𝐵)) ∈ (Pell1QR‘𝐷))
21.33.23  Pell equations 3: characterizing fundamental solution
Theorem	infmrgelbi 43324*	Any lower bound of a nonempty set of real numbers is less than or equal to its infimum, one-direction version. (Contributed by Stefan O'Rear, 1-Sep-2013.) (Revised by AV, 17-Sep-2020.)
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥) → 𝐵 ≤ inf(𝐴, ℝ, < ))
Theorem	pellqrex 43325*	There is a nontrivial solution of a Pell equation in the first quadrant. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ∃𝑥 ∈ (Pell1QR‘𝐷)1 < 𝑥)
Theorem	pellfundval 43326*	Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ))
Theorem	pellfundre 43327	The fundamental solution of a Pell equation exists as a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ)
Theorem	pellfundge 43328	Lower bound on the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (PellFund‘𝐷))
Theorem	pellfundgt1 43329	Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷))
Theorem	pellfundlb 43330	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 43331*	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 43332	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 43322. (Contributed by Stefan O'Rear, 18-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷))
Theorem	pellfund14gap 43333	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 43334	The fundamental Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ+)
Theorem	pellfundne1 43335	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 43336	General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbcl 26750 instead.
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1) → ((log‘𝐴) / (log‘𝐵)) ∈ ℝ)
Theorem	reglogltb 43337	General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logblt 26761 instead.
⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝐶 ∈ ℝ+ ∧ 1 < 𝐶)) → (𝐴 < 𝐵 ↔ ((log‘𝐴) / (log‘𝐶)) < ((log‘𝐵) / (log‘𝐶))))
Theorem	reglogleb 43338	General logarithm preserves ≤. (Contributed by Stefan O'Rear, 19-Oct-2014.) (New usage is discouraged.) Use logbleb 26760 instead.
⊢ (((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) ∧ (𝐶 ∈ ℝ+ ∧ 1 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ ((log‘𝐴) / (log‘𝐶)) ≤ ((log‘𝐵) / (log‘𝐶))))
Theorem	reglogmul 43339	Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbmul 26754 instead.
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ∧ (𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1)) → ((log‘(𝐴 · 𝐵)) / (log‘𝐶)) = (((log‘𝐴) / (log‘𝐶)) + ((log‘𝐵) / (log‘𝐶))))
Theorem	reglogexp 43340	Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbzexp 26753 instead.
⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ∧ (𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1)) → ((log‘(𝐴↑𝑁)) / (log‘𝐶)) = (𝑁 · ((log‘𝐴) / (log‘𝐶))))
Theorem	reglogbas 43341	General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logbid1 26745 instead.
⊢ ((𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1) → ((log‘𝐶) / (log‘𝐶)) = 1)
Theorem	reglog1 43342	General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logb1 26746 instead.
⊢ ((𝐶 ∈ ℝ+ ∧ 𝐶 ≠ 1) → ((log‘1) / (log‘𝐶)) = 0)
Theorem	reglogexpbas 43343	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 26757 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 43344*	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 43345*	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 43346	Extend class notation to include the Robertson-Matiyasevich X sequence.
class Xrm
Syntax	crmy 43347	Extend class notation to include the Robertson-Matiyasevich Y sequence.
class Yrm
Definition	df-rmx 43348*	Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 43359 and rmxyval 43361 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 43349*	Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 43360 and rmxyval 43361 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 43350*	Value of the X sequence. Not used after rmxyval 43361 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 43351*	Value of the Y sequence. Not used after rmxyval 43361 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 43352	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 43353	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 43354	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 43355	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 43356*	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 43357*	The function used to extract rational and irrational parts in df-rmx 43348 and df-rmy 43349 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 43358*	Lemma for frmx 43359 and frmy 43360. (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 43359	The X sequence is a nonnegative integer. See rmxnn 43397 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0
Theorem	frmy 43360	The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ
Theorem	rmxyval 43361	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 43362	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 43363*	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 43364	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 43365	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 43366	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 43367	Addition formula for X and Y sequences. See rmxadd 43373 and rmyadd 43377 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 43368	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 43369	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 43370	Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 43366, rmxyadd 43367, rmxy0 43369, and rmxy1 43368 via qirropth 43354 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 43371	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 43372	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 43373	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 43374	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 43375	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 43376	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 43377	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 43378	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 43379	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 43380	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 43381	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 43382	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 43383	The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 43375 and rmy1 43376. 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 43384	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 43385	"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 43386	"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 43387*	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 43388*	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 43389*	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 43390*	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 43391*	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 43392*	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 43393	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 43394	The Y-sequence is strictly monotonic on ℕ0. Strengthened by ltrmy 43398. (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 43395	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 43396	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 43397	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 43398	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 43399	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 43400	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)))