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	rmy1 42901	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 42902	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 42903	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 42904	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 42905	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 42906	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 42907	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 42908	The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 42900 and rmy1 42901. 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 42909	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 42910	"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 42911	"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 42912*	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 42913*	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 42914*	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 42915*	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 42916*	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 42917*	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 42918	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 42919	The Y-sequence is strictly monotonic on ℕ0. Strengthened by ltrmy 42923. (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 42920	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 42921	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 42922	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 42923	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 42924	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 42925	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 42926	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 42927	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 42928	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 42929	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 42930	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 42931	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 42932	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 42933	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 42934	Lemma 2.24 of [JonesMatijasevic] p. 697 extended to ℤ. Could be eliminated with a more careful proof of jm2.26lem3 42972. (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 42935	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 42936	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 42937	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 42938	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 42939	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 42940	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 42941	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 42942	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 42943*	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 42944*	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 42945	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 42946	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 42947	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 42948	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 42949	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 42950	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 42951*	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 42952	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 42953	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 42954	Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 42973. (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 42955	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 42956	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 42957	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 42958	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 42959	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 42960	Lemma for jm2.19 42964. 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 42961	Lemma for jm2.19 42964. (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 42962	Lemma for jm2.19 42964. (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 42963	Lemma for jm2.19 42964. Extend to ZZ by symmetry. TODO: use zindbi 42917. (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 42964	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 42965	Lemma for jm2.20nn 42968. 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 42966*	Lemma for jm2.20nn 42968. 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 42967	Lemma for jm2.20nn 42968. 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 42968	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 42969	Lemma for jm2.26 42973. (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 42970	Lemma for jm2.26 42973. 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 42971	Lemma for jm2.26 42973. 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 42972	Lemma for jm2.26 42973. Use acongrep 42951 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 42973	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 42974	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 42975	Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 42974 if Yrm is redefined as described in rmyluc 42908. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 − 1) ∥ ((𝐴 Yrm 𝑁) − 𝑁))
21.33.33  X and Y sequences 4: Diophantine representability of Y
Theorem	jm2.27a 42976	Lemma for jm2.27 42979. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐸 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐹 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐺 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐻 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐼 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1)    &   ⊢ (𝜑 → ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1)    &   ⊢ (𝜑 → 𝐺 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → ((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1)    &   ⊢ (𝜑 → 𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2))))    &   ⊢ (𝜑 → 𝐹 ∥ (𝐺 − 𝐴))    &   ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐺 − 1))    &   ⊢ (𝜑 → 𝐹 ∥ (𝐻 − 𝐶))    &   ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐻 − 𝐵))    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    &   ⊢ (𝜑 → 𝑃 ∈ ℤ)    &   ⊢ (𝜑 → 𝐷 = (𝐴 Xrm 𝑃))    &   ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝑃))    &   ⊢ (𝜑 → 𝑄 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 = (𝐴 Xrm 𝑄))    &   ⊢ (𝜑 → 𝐸 = (𝐴 Yrm 𝑄))    &   ⊢ (𝜑 → 𝑅 ∈ ℤ)    &   ⊢ (𝜑 → 𝐼 = (𝐺 Xrm 𝑅))    &   ⊢ (𝜑 → 𝐻 = (𝐺 Yrm 𝑅))    ⇒   ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝐵))
Theorem	jm2.27b 42977	Lemma for jm2.27 42979. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐸 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐹 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐺 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐻 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐼 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1)    &   ⊢ (𝜑 → ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1)    &   ⊢ (𝜑 → 𝐺 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → ((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1)    &   ⊢ (𝜑 → 𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2))))    &   ⊢ (𝜑 → 𝐹 ∥ (𝐺 − 𝐴))    &   ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐺 − 1))    &   ⊢ (𝜑 → 𝐹 ∥ (𝐻 − 𝐶))    &   ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐻 − 𝐵))    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝐵))
Theorem	jm2.27c 42978	Lemma for jm2.27 42979. Forward direction with substitutions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝐵))    &   ⊢ 𝐷 = (𝐴 Xrm 𝐵)    &   ⊢ 𝑄 = (𝐵 · (𝐴 Yrm 𝐵))    &   ⊢ 𝐸 = (𝐴 Yrm (2 · 𝑄))    &   ⊢ 𝐹 = (𝐴 Xrm (2 · 𝑄))    &   ⊢ 𝐺 = (𝐴 + ((𝐹↑2) · ((𝐹↑2) − 𝐴)))    &   ⊢ 𝐻 = (𝐺 Yrm 𝐵)    &   ⊢ 𝐼 = (𝐺 Xrm 𝐵)    &   ⊢ 𝐽 = ((𝐸 / (2 · (𝐶↑2))) − 1)    ⇒   ⊢ (𝜑 → (((𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0 ∧ 𝐹 ∈ ℕ0) ∧ (𝐺 ∈ ℕ0 ∧ 𝐻 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝐽 ∈ ℕ0 ∧ (((((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1 ∧ ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1 ∧ 𝐺 ∈ (ℤ≥‘2)) ∧ (((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1 ∧ 𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2))) ∧ 𝐹 ∥ (𝐺 − 𝐴))) ∧ (((2 · 𝐶) ∥ (𝐺 − 1) ∧ 𝐹 ∥ (𝐻 − 𝐶)) ∧ ((2 · 𝐶) ∥ (𝐻 − 𝐵) ∧ 𝐵 ≤ 𝐶))))))
Theorem	jm2.27 42979*	Lemma 2.27 of [JonesMatijasevic] p. 697; rmY is a diophantine relation. 0 was excluded from the range of B and the lower limit of G was imposed because the source proof does not seem to work otherwise; quite possible I'm just missing something. The source proof uses both i and I; i has been changed to j to avoid collision. This theorem is basically nothing but substitution instances, all the work is done in jm2.27a 42976 and jm2.27c 42978. Once Diophantine relations have been defined, the content of the theorem is "rmY is Diophantine". (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 = (𝐴 Yrm 𝐵) ↔ ∃𝑑 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∃𝑓 ∈ ℕ0 ∃𝑔 ∈ ℕ0 ∃ℎ ∈ ℕ0 ∃𝑖 ∈ ℕ0 ∃𝑗 ∈ ℕ0 (((((𝑑↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1 ∧ ((𝑓↑2) − (((𝐴↑2) − 1) · (𝑒↑2))) = 1 ∧ 𝑔 ∈ (ℤ≥‘2)) ∧ (((𝑖↑2) − (((𝑔↑2) − 1) · (ℎ↑2))) = 1 ∧ 𝑒 = ((𝑗 + 1) · (2 · (𝐶↑2))) ∧ 𝑓 ∥ (𝑔 − 𝐴))) ∧ (((2 · 𝐶) ∥ (𝑔 − 1) ∧ 𝑓 ∥ (ℎ − 𝐶)) ∧ ((2 · 𝐶) ∥ (ℎ − 𝐵) ∧ 𝐵 ≤ 𝐶)))))
Theorem	jm2.27dlem1 42980*	Lemma for rmydioph 42985. Substitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.)
⊢ 𝐴 ∈ (1...𝐵)    ⇒   ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴))
Theorem	jm2.27dlem2 42981	Lemma for rmydioph 42985. This theorem is used along with the next three to efficiently infer steps like 7 ∈ (1...;10). (Contributed by Stefan O'Rear, 11-Oct-2014.)
⊢ 𝐴 ∈ (1...𝐵)    &   ⊢ 𝐶 = (𝐵 + 1)    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ 𝐴 ∈ (1...𝐶)
Theorem	jm2.27dlem3 42982	Lemma for rmydioph 42985. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ∈ (1...𝐴)
Theorem	jm2.27dlem4 42983	Lemma for rmydioph 42985. Infer ℕ-hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 = (𝐴 + 1)    ⇒   ⊢ 𝐵 ∈ ℕ
Theorem	jm2.27dlem5 42984	Lemma for rmydioph 42985. Used with sselii 3955 to infer membership of midpoints of range; jm2.27dlem2 42981 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
⊢ 𝐵 = (𝐴 + 1)    &   ⊢ (1...𝐵) ⊆ (1...𝐶)    ⇒   ⊢ (1...𝐴) ⊆ (1...𝐶)
Theorem	rmydioph 42985	jm2.27 42979 restated in terms of Diophantine sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ ((𝑎‘1) ∈ (ℤ≥‘2) ∧ (𝑎‘3) = ((𝑎‘1) Yrm (𝑎‘2)))} ∈ (Dioph‘3)
21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C
Theorem	rmxdiophlem 42986*	X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (𝑋 = (𝐴 Xrm 𝑁) ↔ ∃𝑦 ∈ ℕ0 (𝑦 = (𝐴 Yrm 𝑁) ∧ ((𝑋↑2) − (((𝐴↑2) − 1) · (𝑦↑2))) = 1)))
Theorem	rmxdioph 42987	X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ ((𝑎‘1) ∈ (ℤ≥‘2) ∧ (𝑎‘3) = ((𝑎‘1) Xrm (𝑎‘2)))} ∈ (Dioph‘3)
Theorem	jm3.1lem1 42988	Lemma for jm3.1 42991. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝐾↑𝑁) < 𝐴)
Theorem	jm3.1lem2 42989	Lemma for jm3.1 42991. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝐾↑𝑁) < ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1))
Theorem	jm3.1lem3 42990	Lemma for jm3.1 42991. (Contributed by Stefan O'Rear, 17-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)    ⇒   ⊢ (𝜑 → ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1) ∈ ℕ)
Theorem	jm3.1 42991	Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝐾 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) ∧ (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴) → (𝐾↑𝑁) = (((𝐴 Xrm 𝑁) − ((𝐴 − 𝐾) · (𝐴 Yrm 𝑁))) mod ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1)))
Theorem	expdiophlem1 42992*	Lemma for expdioph 42994. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
⊢ (𝐶 ∈ ℕ0 → (((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℕ) ∧ 𝐶 = (𝐴↑𝐵)) ↔ ∃𝑑 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∃𝑓 ∈ ℕ0 ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑑 = (𝐴 Yrm (𝐵 + 1))) ∧ ((𝑑 ∈ (ℤ≥‘2) ∧ 𝑒 = (𝑑 Yrm 𝐵)) ∧ ((𝑑 ∈ (ℤ≥‘2) ∧ 𝑓 = (𝑑 Xrm 𝐵)) ∧ (𝐶 < ((((2 · 𝑑) · 𝐴) − (𝐴↑2)) − 1) ∧ ((((2 · 𝑑) · 𝐴) − (𝐴↑2)) − 1) ∥ ((𝑓 − ((𝑑 − 𝐴) · 𝑒)) − 𝐶))))))))
Theorem	expdiophlem2 42993	Lemma for expdioph 42994. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ (((𝑎‘1) ∈ (ℤ≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))} ∈ (Dioph‘3)
Theorem	expdioph 42994	The exponential function is Diophantine. This result completes and encapsulates our development using Pell equation solution sequences and is sometimes regarded as Matiyasevich's theorem properly. (Contributed by Stefan O'Rear, 17-Oct-2014.)
⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))} ∈ (Dioph‘3)
21.33.35  Uncategorized stuff not associated with a major project
Theorem	setindtr 42995*	Set induction for sets contained in a transitive set. If we are allowed to assume Infinity, then all sets have a transitive closure and this reduces to setind 9746; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (∃𝑦(Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → 𝐵 ∈ 𝐴))
Theorem	setindtrs 42996*	Set induction scheme without Infinity. See comments at setindtr 42995. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝜒)
Theorem	dford3lem1 42997*	Lemma for dford3 42999. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))
Theorem	dford3lem2 42998*	Lemma for dford3 42999. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On)
Theorem	dford3 42999*	Ordinals are precisely the hereditarily transitive classes. Definition 1.2 of [Schloeder] p. 1. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥))
Theorem	dford4 43000*	dford3 42999 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (Ord 𝑁 ↔ ∀𝑎∀𝑏∀𝑐((𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑎) → (𝑏 ∈ 𝑁 ∧ (𝑐 ∈ 𝑏 → 𝑐 ∈ 𝑎))))