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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rmxluc 37501 | 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 37502 | The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 37494 and rmy1 37495. 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 37503 | 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 37504 | "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 37505 | "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))) | ||
Theorem | monotuz 37506* | 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 37507* | 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 37508* | 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 37509* | 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 37510* | 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 37511* | 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)) | ||
Theorem | expmordi 37512 | Mantissa ordering relationship for exponentiation. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵) ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) < (𝐵↑𝑁)) | ||
Theorem | rpexpmord 37513 | Mantissa ordering relationship for exponentiation of positive reals. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴↑𝑁) < (𝐵↑𝑁))) | ||
Theorem | rmxypos 37514 | 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 37515 | The Y-sequence is strictly monotonic on ℕ0. Strengthened by ltrmy 37519. (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 37516 | 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 37517 | 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 37518 | 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 37519 | 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 37520 | 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 37521 | 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 37522 | 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 37523 | 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 37524 | 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 37525 | 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 37526 | 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 37527 | 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 37528 | 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 37529 | 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 37530 | Lemma 2.24 of [JonesMatijasevic] p. 697 extended to ℤ. Could be eliminated with a more careful proof of jm2.26lem3 37568. (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 37531 | 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)) | ||
Theorem | congtr 37532 | 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 37533 | 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 37534 | 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 37535 | 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 37536 | 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 37537 | 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 37538 | 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 37539* | 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 37540* | 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 37541 | 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)) | ||
Theorem | acongid 37542 |
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 37543 | 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 37544 | 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 37545 | 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 37546 | 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 37547* | 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 37548 | 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 37549 | 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 37550 | Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 37569. (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 37551 | 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))) | ||
Theorem | coprmdvdsb 37552 | 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 37553 | 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 37554 | 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)) | ||
Theorem | jm2.18 37555 | 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 37556 | Lemma for jm2.19 37560. 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 37557 | Lemma for jm2.19 37560. (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 37558 | Lemma for jm2.19 37560. (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 37559 | Lemma for jm2.19 37560. Extend to ZZ by symmetry. TODO: use zindbi 37511. (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 37560 | 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 37561 | Lemma for jm2.20nn 37564. 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 37562* | Lemma for jm2.20nn 37564. 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 37563 | Lemma for jm2.20nn 37564. 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 37564 | 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 37565 | Lemma for jm2.26 37569. (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 37566 | Lemma for jm2.26 37569. 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 37567 | Lemma for jm2.26 37569. 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 37568 | Lemma for jm2.26 37569. Use acongrep 37547 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 37569 | 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 37570 | 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 37571 | Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 37570 if Yrm is redefined as described in rmyluc 37502. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 − 1) ∥ ((𝐴 Yrm 𝑁) − 𝑁)) | ||
Theorem | jm2.27a 37572 | Lemma for jm2.27 37575. 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 37573 | Lemma for jm2.27 37575. 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 37574 | Lemma for jm2.27 37575. 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 37575* | 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 37572 and jm2.27c 37574. 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 37576* | Lemma for rmydioph 37581. 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 37577 | Lemma for rmydioph 37581. 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 37578 | Lemma for rmydioph 37581. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ 𝐴 ∈ ℕ ⇒ ⊢ 𝐴 ∈ (1...𝐴) | ||
Theorem | jm2.27dlem4 37579 | Lemma for rmydioph 37581. Infer ℕ-hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 = (𝐴 + 1) ⇒ ⊢ 𝐵 ∈ ℕ | ||
Theorem | jm2.27dlem5 37580 | Lemma for rmydioph 37581. Used with sselii 3600 to infer membership of midpoints of range; jm2.27dlem2 37577 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ 𝐵 = (𝐴 + 1) & ⊢ (1...𝐵) ⊆ (1...𝐶) ⇒ ⊢ (1...𝐴) ⊆ (1...𝐶) | ||
Theorem | rmydioph 37581 | jm2.27 37575 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) | ||
Theorem | rmxdiophlem 37582* | 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 37583 | 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 37584 | Lemma for jm3.1 37587. (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 37585 | Lemma for jm3.1 37587. (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 37586 | Lemma for jm3.1 37587. (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 37587 | 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 37588* | Lemma for expdioph 37590. 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 37589 | Lemma for expdioph 37590. 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 37590 | 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) | ||
Theorem | setindtr 37591* | 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 8610; 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 37592* | Epsilon induction scheme without Infinity. See comments at setindtr 37591. (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 37593* | Lemma for dford3 37595. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦)) | ||
Theorem | dford3lem2 37594* | Lemma for dford3 37595. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
⊢ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On) | ||
Theorem | dford3 37595* | Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.) |
⊢ (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥)) | ||
Theorem | dford4 37596* | dford3 37595 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)))) | ||
Theorem | wopprc 37597 | Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1𝑜 ∈ {{{𝐴}, ∅}, {{𝐵}}}) | ||
Theorem | rpnnen3lem 37598* | Lemma for rpnnen3 37599. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏}) | ||
Theorem | rpnnen3 37599 | Dedekind cut injection of ℝ into 𝒫 ℚ. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
⊢ ℝ ≼ 𝒫 ℚ | ||
Theorem | axac10 37600 | Characterization of choice similar to dffin1-5 9210. (Contributed by Stefan O'Rear, 6-Jan-2015.) |
⊢ ( ≈ “ On) = V |
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