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Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrmxm1 36401 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))))

Theoremrmym1 36402 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)))

Theoremrmxluc 36403 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))))

Theoremrmyluc 36404 The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 36396 and rmy1 36397. 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))))

Theoremrmyluc2 36405 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))))

Theoremrmxdbl 36406 "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))

Theoremrmydbl 36407 "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)))

20.24.27  Ordering and induction lemmas for the integers

Theoremmonotuz 36408* 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.)
((𝜑𝑦𝐻) → 𝐹 < 𝐺)    &   ((𝜑𝑥𝐻) → 𝐶 ∈ ℝ)    &   𝐻 = (ℤ𝐼)    &   (𝑥 = (𝑦 + 1) → 𝐶 = 𝐺)    &   (𝑥 = 𝑦𝐶 = 𝐹)    &   (𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝜑 ∧ (𝐴𝐻𝐵𝐻)) → (𝐴 < 𝐵𝐷 < 𝐸))

Theoremmonotoddzzfi 36409* A function which is odd and monotonic on 0 is monotonic on . This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.)
((𝜑𝑥 ∈ ℤ) → (𝐹𝑥) ∈ ℝ)    &   ((𝜑𝑥 ∈ ℤ) → (𝐹‘-𝑥) = -(𝐹𝑥))    &   ((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 < 𝑦 → (𝐹𝑥) < (𝐹𝑦)))       ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐹𝐴) < (𝐹𝐵)))

Theoremmonotoddzz 36410* A function (given implicitly) which is odd and monotonic on 0 is monotonic on . This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.)
((𝜑𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (𝑥 < 𝑦𝐸 < 𝐹))    &   ((𝜑𝑥 ∈ ℤ) → 𝐸 ∈ ℝ)    &   ((𝜑𝑦 ∈ ℤ) → 𝐺 = -𝐹)    &   (𝑥 = 𝐴𝐸 = 𝐶)    &   (𝑥 = 𝐵𝐸 = 𝐷)    &   (𝑥 = 𝑦𝐸 = 𝐹)    &   (𝑥 = -𝑦𝐸 = 𝐺)       ((𝜑𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵𝐶 < 𝐷))

Theoremoddcomabszz 36411* An odd function which takes nonnegative values on nonnegative arguments commutes with abs. (Contributed by Stefan O'Rear, 26-Sep-2014.)
((𝜑𝑥 ∈ ℤ) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥 ∈ ℤ ∧ 0 ≤ 𝑥) → 0 ≤ 𝐴)    &   ((𝜑𝑦 ∈ ℤ) → 𝐶 = -𝐵)    &   (𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = -𝑦𝐴 = 𝐶)    &   (𝑥 = 𝐷𝐴 = 𝐸)    &   (𝑥 = (abs‘𝐷) → 𝐴 = 𝐹)       ((𝜑𝐷 ∈ ℤ) → (abs‘𝐸) = 𝐹)

Theorem2nn0ind 36412* 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 → ((𝜃𝜏) → 𝜂))    &   (𝑥 = 0 → (𝜑𝜓))    &   (𝑥 = 1 → (𝜑𝜒))    &   (𝑥 = (𝑦 − 1) → (𝜑𝜃))    &   (𝑥 = 𝑦 → (𝜑𝜏))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜂))    &   (𝑥 = 𝐴 → (𝜑𝜌))       (𝐴 ∈ ℕ0𝜌)

Theoremzindbi 36413* 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 → (𝜓𝜒))    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜒))    &   (𝑥 = 0 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))       (𝐴 ∈ ℤ → (𝜃𝜏))

Theoremexpmordi 36414 Mantissa ordering relationship for exponentiation. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴𝐴 < 𝐵) ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) < (𝐵𝑁))

Theoremrpexpmord 36415 Mantissa ordering relationship for exponentiation of positive reals. (Contributed by Stefan O'Rear, 16-Oct-2014.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴𝑁) < (𝐵𝑁)))

20.24.28  X and Y sequences 2: Order properties

Theoremrmxypos 36416 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)))

Theoremltrmynn0 36417 The Y-sequence is strictly monotonic on 0. Strengthened by ltrmy 36421. (Contributed by Stefan O'Rear, 24-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝐴 Yrm 𝑀) < (𝐴 Yrm 𝑁)))

Theoremltrmxnn0 36418 The X-sequence is strictly monotonic on 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝐴 Xrm 𝑀) < (𝐴 Xrm 𝑁)))

Theoremlermxnn0 36419 The X-sequence is monotonic on 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀𝑁 ↔ (𝐴 Xrm 𝑀) ≤ (𝐴 Xrm 𝑁)))

Theoremrmxnn 36420 The X-sequence is defined to range over 0 but never actually takes the value 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) ∈ ℕ)

Theoremltrmy 36421 The Y-sequence is strictly monotonic over . (Contributed by Stefan O'Rear, 25-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝐴 Yrm 𝑀) < (𝐴 Yrm 𝑁)))

Theoremrmyeq0 36422 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))

Theoremrmyeq 36423 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)))

Theoremlermy 36424 Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (𝐴 Yrm 𝑀) ≤ (𝐴 Yrm 𝑁)))

Theoremrmynn 36425 Yrm is positive for positive arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ) → (𝐴 Yrm 𝑁) ∈ ℕ)

Theoremrmynn0 36426 Yrm is nonnegative for nonnegative arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 Yrm 𝑁) ∈ ℕ0)

Theoremrmyabs 36427 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)))

Theoremjm2.24nn 36428 X(n) is strictly greater than Y(n) + Y(n-1). Lemma 2.24 of [JonesMatijasevic] p. 697 restricted to . (Contributed by Stefan O'Rear, 3-Oct-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ) → ((𝐴 Yrm (𝑁 − 1)) + (𝐴 Yrm 𝑁)) < (𝐴 Xrm 𝑁))

Theoremjm2.17a 36429 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)))

Theoremjm2.17b 36430 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))

Theoremjm2.17c 36431 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)))

Theoremjm2.24 36432 Lemma 2.24 of [JonesMatijasevic] p. 697 extended to . Could be eliminated with a more careful proof of jm2.26lem3 36470. (Contributed by Stefan O'Rear, 3-Oct-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Yrm (𝑁 − 1)) + (𝐴 Yrm 𝑁)) < (𝐴 Xrm 𝑁))

Theoremrmygeid 36433 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))

20.24.29  Congruential equations

Theoremcongtr 36434 A wff of the form 𝐴 ∥ (𝐵𝐶) 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 ∥ (𝐵𝐶) ∧ 𝐴 ∥ (𝐶𝐷))) → 𝐴 ∥ (𝐵𝐷))

Theoremcongadd 36435 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 ∥ (𝐵𝐶) ∧ 𝐴 ∥ (𝐷𝐸))) → 𝐴 ∥ ((𝐵 + 𝐷) − (𝐶 + 𝐸)))

Theoremcongmul 36436 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 ∥ (𝐵𝐶) ∧ 𝐴 ∥ (𝐷𝐸))) → 𝐴 ∥ ((𝐵 · 𝐷) − (𝐶 · 𝐸)))

Theoremcongsym 36437 Congruence mod 𝐴 is a symmetric/commutative relation. (Contributed by Stefan O'Rear, 1-Oct-2014.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐴 ∥ (𝐵𝐶))) → 𝐴 ∥ (𝐶𝐵))

Theoremcongneg 36438 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 ∥ (𝐵𝐶))) → 𝐴 ∥ (-𝐵 − -𝐶))

Theoremcongsub 36439 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 ∥ (𝐵𝐶) ∧ 𝐴 ∥ (𝐷𝐸))) → 𝐴 ∥ ((𝐵𝐷) − (𝐶𝐸)))

Theoremcongid 36440 Every integer is congruent to itself mod every base. (Contributed by Stefan O'Rear, 1-Oct-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∥ (𝐵𝐵))

Theoremmzpcong 36441* 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 ∧ ∀𝑘𝑉 𝑁 ∥ ((𝑋𝑘) − (𝑌𝑘)))) → 𝑁 ∥ ((𝐹𝑋) − (𝐹𝑌)))

Theoremcongrep 36442* 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 ∥ (𝑎𝑁))

Theoremcongabseq 36443 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 ∥ (𝐵𝐶)) → ((abs‘(𝐵𝐶)) < 𝐴𝐵 = 𝐶))

20.24.30  Alternating congruential equations

Theoremacongid 36444 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 ∥ (𝐵𝐵) ∨ 𝐴 ∥ (𝐵 − -𝐵)))

Theoremacongsym 36445 Symmetry of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝐴 ∥ (𝐵𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶))) → (𝐴 ∥ (𝐶𝐵) ∨ 𝐴 ∥ (𝐶 − -𝐵)))

Theoremacongneg2 36446 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 ∥ (𝐵𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶)))

Theoremacongtr 36447 Transitivity of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ ((𝐴 ∥ (𝐵𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶)) ∧ (𝐴 ∥ (𝐶𝐷) ∨ 𝐴 ∥ (𝐶 − -𝐷)))) → (𝐴 ∥ (𝐵𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)))

Theoremacongeq12d 36448 Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.)
(𝜑𝐵 = 𝐶)    &   (𝜑𝐷 = 𝐸)       (𝜑 → ((𝐴 ∥ (𝐵𝐷) ∨ 𝐴 ∥ (𝐵 − -𝐷)) ↔ (𝐴 ∥ (𝐶𝐸) ∨ 𝐴 ∥ (𝐶 − -𝐸))))

Theoremacongrep 36449* 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) ∥ (𝑎𝑁) ∨ (2 · 𝐴) ∥ (𝑎 − -𝑁)))

Theoremfzmaxdif 36450 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)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (abs‘(𝐴𝐷)) ≤ (𝐹𝐵))

Theoremfzneg 36451 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)))

Theoremacongeq 36452 Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 36471. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 = 𝐶 ↔ ((2 · 𝐴) ∥ (𝐵𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶))))

Theoremdvdsacongtr 36453 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) ∧ (𝐷𝐴 ∧ (𝐴 ∥ (𝐵𝐶) ∨ 𝐴 ∥ (𝐵 − -𝐶)))) → (𝐷 ∥ (𝐵𝐶) ∨ 𝐷 ∥ (𝐵 − -𝐶)))

20.24.31  Additional theorems on integer divisibility

Theoremcoprmdvdsb 36454 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)) → (𝐾𝑁𝐾 ∥ (𝑀 · 𝑁)))

Theoremmodabsdifz 36455 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)

Theoremdvdsabsmod0 36456 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) → (𝑀𝑁 ↔ (𝑁 mod (abs‘𝑀)) = 0))

20.24.32  X and Y sequences 3: Divisibility properties

Theoremjm2.18 36457 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 ∈ ℕ0𝑁 ∈ ℕ0) → ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1) ∥ (((𝐴 Xrm 𝑁) − ((𝐴𝐾) · (𝐴 Yrm 𝑁))) − (𝐾𝑁)))

Theoremjm2.19lem1 36458 Lemma for jm2.19 36462. 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)

Theoremjm2.19lem2 36459 Lemma for jm2.19 36462. (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))))

Theoremjm2.19lem3 36460 Lemma for jm2.19 36462. (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)))))

Theoremjm2.19lem4 36461 Lemma for jm2.19 36462. Extend to ZZ by symmetry. TODO: use zindbi 36413. (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)))))

Theoremjm2.19 36462 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) → (𝑀𝑁 ↔ (𝐴 Yrm 𝑀) ∥ (𝐴 Yrm 𝑁)))

Theoremjm2.21 36463 Lemma for jm2.20nn 36466. 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))

Theoremjm2.22 36464* Lemma for jm2.20nn 36466. 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)↑(𝐽𝑖)) · (((𝐴 Yrm 𝑁)↑𝑖) · (((𝐴↑2) − 1)↑((𝑖 − 1) / 2))))))

Theoremjm2.23 36465 Lemma for jm2.20nn 36466. 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)))))

Theoremjm2.20nn 36466 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))

Theoremjm2.25lem1 36467 Lemma for jm2.26 36471. (Contributed by Stefan O'Rear, 2-Oct-2014.)
(((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ (𝐴 ∥ (𝐶𝐷) ∨ 𝐴 ∥ (𝐶 − -𝐷))) → ((𝐴 ∥ (𝐷𝐵) ∨ 𝐴 ∥ (𝐷 − -𝐵)) ↔ (𝐴 ∥ (𝐶𝐵) ∨ 𝐴 ∥ (𝐶 − -𝐵))))

Theoremjm2.25 36468 Lemma for jm2.26 36471. 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))))

Theoremjm2.26a 36469 Lemma for jm2.26 36471. Reverse direction is required to prove forward direction, so do it separatly. 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) ∥ (𝐾𝑀) ∨ (2 · 𝑁) ∥ (𝐾 − -𝑀)) → ((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − -(𝐴 Yrm 𝑀)))))

Theoremjm2.26lem3 36470 Lemma for jm2.26 36471. Use acongrep 36449 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)

Theoremjm2.26 36471 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) ∥ (𝐾𝑀) ∨ (2 · 𝑁) ∥ (𝐾 − -𝑀))))

Theoremjm2.15nn0 36472 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) → (𝐴𝐵) ∥ ((𝐴 Yrm 𝑁) − (𝐵 Yrm 𝑁)))

Theoremjm2.16nn0 36473 Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 36472 if Yrm is redefined as described in rmyluc 36404. (Contributed by Stefan O'Rear, 1-Oct-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 − 1) ∥ ((𝐴 Yrm 𝑁) − 𝑁))

20.24.33  X and Y sequences 4: Diophantine representability of Y

Theoremjm2.27a 36474 Lemma for jm2.27 36477. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.)
(𝜑𝐴 ∈ (ℤ‘2))    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝐸 ∈ ℕ0)    &   (𝜑𝐹 ∈ ℕ0)    &   (𝜑𝐺 ∈ ℕ0)    &   (𝜑𝐻 ∈ ℕ0)    &   (𝜑𝐼 ∈ ℕ0)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑 → ((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1)    &   (𝜑 → ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1)    &   (𝜑𝐺 ∈ (ℤ‘2))    &   (𝜑 → ((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1)    &   (𝜑𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2))))    &   (𝜑𝐹 ∥ (𝐺𝐴))    &   (𝜑 → (2 · 𝐶) ∥ (𝐺 − 1))    &   (𝜑𝐹 ∥ (𝐻𝐶))    &   (𝜑 → (2 · 𝐶) ∥ (𝐻𝐵))    &   (𝜑𝐵𝐶)    &   (𝜑𝑃 ∈ ℤ)    &   (𝜑𝐷 = (𝐴 Xrm 𝑃))    &   (𝜑𝐶 = (𝐴 Yrm 𝑃))    &   (𝜑𝑄 ∈ ℤ)    &   (𝜑𝐹 = (𝐴 Xrm 𝑄))    &   (𝜑𝐸 = (𝐴 Yrm 𝑄))    &   (𝜑𝑅 ∈ ℤ)    &   (𝜑𝐼 = (𝐺 Xrm 𝑅))    &   (𝜑𝐻 = (𝐺 Yrm 𝑅))       (𝜑𝐶 = (𝐴 Yrm 𝐵))

Theoremjm2.27b 36475 Lemma for jm2.27 36477. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.)
(𝜑𝐴 ∈ (ℤ‘2))    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝐸 ∈ ℕ0)    &   (𝜑𝐹 ∈ ℕ0)    &   (𝜑𝐺 ∈ ℕ0)    &   (𝜑𝐻 ∈ ℕ0)    &   (𝜑𝐼 ∈ ℕ0)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑 → ((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1)    &   (𝜑 → ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1)    &   (𝜑𝐺 ∈ (ℤ‘2))    &   (𝜑 → ((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1)    &   (𝜑𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2))))    &   (𝜑𝐹 ∥ (𝐺𝐴))    &   (𝜑 → (2 · 𝐶) ∥ (𝐺 − 1))    &   (𝜑𝐹 ∥ (𝐻𝐶))    &   (𝜑 → (2 · 𝐶) ∥ (𝐻𝐵))    &   (𝜑𝐵𝐶)       (𝜑𝐶 = (𝐴 Yrm 𝐵))

Theoremjm2.27c 36476 Lemma for jm2.27 36477. Forward direction with substitutions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
(𝜑𝐴 ∈ (ℤ‘2))    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℕ)    &   (𝜑𝐶 = (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 ∈ ℕ0𝐸 ∈ ℕ0𝐹 ∈ ℕ0) ∧ (𝐺 ∈ ℕ0𝐻 ∈ ℕ0𝐼 ∈ 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 ∥ (𝐺𝐴))) ∧ (((2 · 𝐶) ∥ (𝐺 − 1) ∧ 𝐹 ∥ (𝐻𝐶)) ∧ ((2 · 𝐶) ∥ (𝐻𝐵) ∧ 𝐵𝐶))))))

Theoremjm2.27 36477* 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 36474 and jm2.27c 36476. 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 ∈ ℕ0𝑒 ∈ ℕ0𝑓 ∈ ℕ0𝑔 ∈ ℕ0 ∈ ℕ0𝑖 ∈ ℕ0𝑗 ∈ 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) · (↑2))) = 1 ∧ 𝑒 = ((𝑗 + 1) · (2 · (𝐶↑2))) ∧ 𝑓 ∥ (𝑔𝐴))) ∧ (((2 · 𝐶) ∥ (𝑔 − 1) ∧ 𝑓 ∥ (𝐶)) ∧ ((2 · 𝐶) ∥ (𝐵) ∧ 𝐵𝐶)))))

Theoremjm2.27dlem1 36478* Lemma for rmydioph 36483. Subsitution 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)) → (𝑎𝐴) = (𝑏𝐴))

Theoremjm2.27dlem2 36479 Lemma for rmydioph 36483. 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)

Theoremjm2.27dlem3 36480 Lemma for rmydioph 36483. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.)
𝐴 ∈ ℕ       𝐴 ∈ (1...𝐴)

Theoremjm2.27dlem4 36481 Lemma for rmydioph 36483. Infer -hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
𝐴 ∈ ℕ    &   𝐵 = (𝐴 + 1)       𝐵 ∈ ℕ

Theoremjm2.27dlem5 36482 Lemma for rmydioph 36483. Used with sselii 3469 to infer membership of midpoints of range; jm2.27dlem2 36479 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
𝐵 = (𝐴 + 1)    &   (1...𝐵) ⊆ (1...𝐶)       (1...𝐴) ⊆ (1...𝐶)

Theoremrmydioph 36483 jm2.27 36477 restated in terms of Diophantine sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
{𝑎 ∈ (ℕ0𝑚 (1...3)) ∣ ((𝑎‘1) ∈ (ℤ‘2) ∧ (𝑎‘3) = ((𝑎‘1) Yrm (𝑎‘2)))} ∈ (Dioph‘3)

20.24.34  X and Y sequences 5: Diophantine representability of X, ^, _C

Theoremrmxdiophlem 36484* 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 ∈ ℕ0𝑋 ∈ ℕ0) → (𝑋 = (𝐴 Xrm 𝑁) ↔ ∃𝑦 ∈ ℕ0 (𝑦 = (𝐴 Yrm 𝑁) ∧ ((𝑋↑2) − (((𝐴↑2) − 1) · (𝑦↑2))) = 1)))

Theoremrmxdioph 36485 X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
{𝑎 ∈ (ℕ0𝑚 (1...3)) ∣ ((𝑎‘1) ∈ (ℤ‘2) ∧ (𝑎‘3) = ((𝑎‘1) Xrm (𝑎‘2)))} ∈ (Dioph‘3)

Theoremjm3.1lem1 36486 Lemma for jm3.1 36489. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(𝜑𝐴 ∈ (ℤ‘2))    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)       (𝜑 → (𝐾𝑁) < 𝐴)

Theoremjm3.1lem2 36487 Lemma for jm3.1 36489. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(𝜑𝐴 ∈ (ℤ‘2))    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)       (𝜑 → (𝐾𝑁) < ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1))

Theoremjm3.1lem3 36488 Lemma for jm3.1 36489. (Contributed by Stefan O'Rear, 17-Oct-2014.)
(𝜑𝐴 ∈ (ℤ‘2))    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)       (𝜑 → ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1) ∈ ℕ)

Theoremjm3.1 36489 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) → (𝐾𝑁) = (((𝐴 Xrm 𝑁) − ((𝐴𝐾) · (𝐴 Yrm 𝑁))) mod ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1)))

Theoremexpdiophlem1 36490* Lemma for expdioph 36492. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
(𝐶 ∈ ℕ0 → (((𝐴 ∈ (ℤ‘2) ∧ 𝐵 ∈ ℕ) ∧ 𝐶 = (𝐴𝐵)) ↔ ∃𝑑 ∈ ℕ0𝑒 ∈ ℕ0𝑓 ∈ 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 − ((𝑑𝐴) · 𝑒)) − 𝐶))))))))

Theoremexpdiophlem2 36491 Lemma for expdioph 36492. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
{𝑎 ∈ (ℕ0𝑚 (1...3)) ∣ (((𝑎‘1) ∈ (ℤ‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))} ∈ (Dioph‘3)

Theoremexpdioph 36492 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 ∈ (ℕ0𝑚 (1...3)) ∣ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))} ∈ (Dioph‘3)

20.24.35  Uncategorized stuff not associated with a major project

Theoremsetindtr 36493* Epsilon 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 8373; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∀𝑥(𝑥𝐴𝑥𝐴) → (∃𝑦(Tr 𝑦𝐵𝑦) → 𝐵𝐴))

Theoremsetindtrs 36494* Epsilon induction scheme without Infinity. See comments at setindtr 36493. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∀𝑦𝑥 𝜓𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (∃𝑧(Tr 𝑧𝐵𝑧) → 𝜒)

Theoremdford3lem1 36495* Lemma for dford3 36497. (Contributed by Stefan O'Rear, 28-Oct-2014.)
((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))

Theoremdford3lem2 36496* Lemma for dford3 36497. (Contributed by Stefan O'Rear, 28-Oct-2014.)
((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → 𝑥 ∈ On)

Theoremdford3 36497* Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥))

Theoremdford4 36498* dford3 36497 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(Ord 𝑁 ↔ ∀𝑎𝑏𝑐((𝑎𝑁𝑏𝑎) → (𝑏𝑁 ∧ (𝑐𝑏𝑐𝑎))))

Theoremwopprc 36499 Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1𝑜 ∈ {{{𝐴}, ∅}, {{𝐵}}})

Theoremrpnnen3lem 36500* Lemma for rpnnen3 36501. (Contributed by Stefan O'Rear, 18-Jan-2015.)
(((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏})

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