P. 430 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42901-43000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rmspecsqrtnq 42901	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 42902	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 42903	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 42904	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 42905*	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	rmxyelqirrOLD 42906*	Obsolete version of rmxyelqirr 42905 as of 23-Dec-2024. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ {𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
Theorem	rmxypairf1o 42907*	The function used to extract rational and irrational parts in df-rmx 42897 and df-rmy 42898 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 42908*	Lemma for frmx 42909 and frmy 42910. (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 42909	The X sequence is a nonnegative integer. See rmxnn 42947 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0
Theorem	frmy 42910	The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.)
⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ
Theorem	rmxyval 42911	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 42912	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 42913*	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 42914	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 42915	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 42916	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 42917	Addition formula for X and Y sequences. See rmxadd 42923 and rmyadd 42927 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 42918	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 42919	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 42920	Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 42916, rmxyadd 42917, rmxy0 42919, and rmxy1 42918 via qirropth 42903 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 42921	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 42922	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 42923	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 42924	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 42925	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 42926	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 42927	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 42928	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 42929	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 42930	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 42931	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 42932	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 42933	The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 42925 and rmy1 42926. 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 42934	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 42935	"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 42936	"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 42937*	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 42938*	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 42939*	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 42940*	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 42941*	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 42942*	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 42943	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 42944	The Y-sequence is strictly monotonic on ℕ0. Strengthened by ltrmy 42948. (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 42945	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 42946	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 42947	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 42948	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 42949	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 42950	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 42951	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 42952	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 42953	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 42954	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 42955	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 42956	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 42957	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 42958	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 42959	Lemma 2.24 of [JonesMatijasevic] p. 697 extended to ℤ. Could be eliminated with a more careful proof of jm2.26lem3 42997. (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 42960	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 42961	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 42962	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 42963	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 42964	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 42965	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 42966	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 42967	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 42968*	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 42969*	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 42970	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 42971	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 42972	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 42973	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 42974	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 42975	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 42976*	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 42977	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 42978	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 42979	Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 42998. (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))))
Theorem	dvdsacongtr 42980	Alternating congruence passes from a base to a dividing base. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ (𝐷 ∥ 𝐴 ∧ (𝐴 ∥ (𝐵 − 𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶)))) → (𝐷 ∥ (𝐵 − 𝐶) ∨ 𝐷 ∥ (𝐵 − -𝐶)))
21.33.31  Additional theorems on integer divisibility
Theorem	coprmdvdsb 42981	Multiplication by a coprime number does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ (𝐾 gcd 𝑀) = 1)) → (𝐾 ∥ 𝑁 ↔ 𝐾 ∥ (𝑀 · 𝑁)))
Theorem	modabsdifz 42982	Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 26-Sep-2014.)
⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑀 ≠ 0) → ((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℤ)
Theorem	dvdsabsmod0 42983	Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.) (Proof shortened by OpenAI, 3-Jul-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 ∥ 𝑁 ↔ (𝑁 mod (abs‘𝑀)) = 0))
21.33.32  X and Y sequences 3: Divisibility properties
Theorem	jm2.18 42984	Theorem 2.18 of [JonesMatijasevic] p. 696. Direct relationship of the exponential function to X and Y sequences. (Contributed by Stefan O'Rear, 14-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1) ∥ (((𝐴 Xrm 𝑁) − ((𝐴 − 𝐾) · (𝐴 Yrm 𝑁))) − (𝐾↑𝑁)))
Theorem	jm2.19lem1 42985	Lemma for jm2.19 42989. X and Y values are coprime. (Contributed by Stefan O'Rear, 23-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ) → ((𝐴 Xrm 𝑀) gcd (𝐴 Yrm 𝑀)) = 1)
Theorem	jm2.19lem2 42986	Lemma for jm2.19 42989. (Contributed by Stefan O'Rear, 23-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁) ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm (𝑁 + 𝑀))))
Theorem	jm2.19lem3 42987	Lemma for jm2.19 42989. (Contributed by Stefan O'Rear, 26-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐼 ∈ ℕ0) → ((𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁) ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm (𝑁 + (𝐼 · 𝑀)))))
Theorem	jm2.19lem4 42988	Lemma for jm2.19 42989. Extend to ZZ by symmetry. TODO: use zindbi 42942. (Contributed by Stefan O'Rear, 26-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐼 ∈ ℤ) → ((𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁) ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm (𝑁 + (𝐼 · 𝑀)))))
Theorem	jm2.19 42989	Lemma 2.19 of [JonesMatijasevic] p. 696. Transfer divisibility constraints between Y-values and their indices. (Contributed by Stefan O'Rear, 24-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁)))
Theorem	jm2.21 42990	Lemma for jm2.20nn 42993. Express X and Y values as a binomial. (Contributed by Stefan O'Rear, 26-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐽 ∈ ℤ) → ((𝐴 Xrm (𝑁 · 𝐽)) + ((√‘((𝐴↑2) − 1)) · (𝐴 Yrm (𝑁 · 𝐽)))) = (((𝐴 Xrm 𝑁) + ((√‘((𝐴↑2) − 1)) · (𝐴 Yrm 𝑁)))↑𝐽))
Theorem	jm2.22 42991*	Lemma for jm2.20nn 42993. Applying binomial theorem and taking irrational part. (Contributed by Stefan O'Rear, 26-Sep-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐽 ∈ ℕ0) → (𝐴 Yrm (𝑁 · 𝐽)) = Σ𝑖 ∈ {𝑥 ∈ (0...𝐽) ∣ ¬ 2 ∥ 𝑥} ((𝐽C𝑖) · (((𝐴 Xrm 𝑁)↑(𝐽 − 𝑖)) · (((𝐴 Yrm 𝑁)↑𝑖) · (((𝐴↑2) − 1)↑((𝑖 − 1) / 2))))))
Theorem	jm2.23 42992	Lemma for jm2.20nn 42993. Truncate binomial expansion p-adicly. (Contributed by Stefan O'Rear, 26-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ ∧ 𝐽 ∈ ℕ) → ((𝐴 Yrm 𝑁)↑3) ∥ ((𝐴 Yrm (𝑁 · 𝐽)) − (𝐽 · (((𝐴 Xrm 𝑁)↑(𝐽 − 1)) · (𝐴 Yrm 𝑁)))))
Theorem	jm2.20nn 42993	Lemma 2.20 of [JonesMatijasevic] p. 696, the "first step down lemma". (Contributed by Stefan O'Rear, 27-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝐴 Yrm 𝑁)↑2) ∥ (𝐴 Yrm 𝑀) ↔ (𝑁 · (𝐴 Yrm 𝑁)) ∥ 𝑀))
Theorem	jm2.25lem1 42994	Lemma for jm2.26 42998. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ (𝐴 ∥ (𝐶 − 𝐷) ∨ 𝐴 ∥ (𝐶 − -𝐷))) → ((𝐴 ∥ (𝐷 − 𝐵) ∨ 𝐴 ∥ (𝐷 − -𝐵)) ↔ (𝐴 ∥ (𝐶 − 𝐵) ∨ 𝐴 ∥ (𝐶 − -𝐵))))
Theorem	jm2.25 42995	Lemma for jm2.26 42998. Remainders mod X(2n) are negaperiodic mod 2n. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐼 ∈ ℤ) → ((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm (𝑀 + (𝐼 · (2 · 𝑁)))) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm (𝑀 + (𝐼 · (2 · 𝑁)))) − -(𝐴 Yrm 𝑀))))
Theorem	jm2.26a 42996	Lemma for jm2.26 42998. Reverse direction is required to prove forward direction, so do it separately. Induction on difference between K and M, together with the addition formula fact that adding 2N only inverts sign. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((2 · 𝑁) ∥ (𝐾 − 𝑀) ∨ (2 · 𝑁) ∥ (𝐾 − -𝑀)) → ((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − -(𝐴 Yrm 𝑀)))))
Theorem	jm2.26lem3 42997	Lemma for jm2.26 42998. Use acongrep 42976 to find K', M' ~ K, M in [ 0,N ]. Thus Y(K') ~ Y(M') and both are small; K' = M' on pain of contradicting 2.24, so K ~ M. (Contributed by Stefan O'Rear, 3-Oct-2014.)
⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ 𝑀 ∈ (0...𝑁)) ∧ ((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − -(𝐴 Yrm 𝑀)))) → 𝐾 = 𝑀)
Theorem	jm2.26 42998	Lemma 2.26 of [JonesMatijasevic] p. 697, the "second step down lemma". (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − -(𝐴 Yrm 𝑀))) ↔ ((2 · 𝑁) ∥ (𝐾 − 𝑀) ∨ (2 · 𝑁) ∥ (𝐾 − -𝑀))))
Theorem	jm2.15nn0 42999	Lemma 2.15 of [JonesMatijasevic] p. 695. Yrm is a polynomial for fixed N, so has the expected congruence property. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 − 𝐵) ∥ ((𝐴 Yrm 𝑁) − (𝐵 Yrm 𝑁)))
Theorem	jm2.16nn0 43000	Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 42999 if Yrm is redefined as described in rmyluc 42933. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 − 1) ∥ ((𝐴 Yrm 𝑁) − 𝑁))