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Statement | ||
Definition | df-mzp 39201 | Polynomials over ℤ with an arbitrary index set, that is, the smallest ring of functions containing all constant functions and all projections. This is almost the most general reasonable definition; to reach full generality, we would need to be able to replace ZZ with an arbitrary (semi)ring (and a coordinate subring), but rings have not been defined yet. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ mzPoly = (𝑣 ∈ V ↦ ∩ (mzPolyCld‘𝑣)) | ||
Theorem | mzpclval 39202* | Substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (mzPolyCld‘𝑉) = {𝑝 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝))}) | ||
Theorem | elmzpcl 39203* | Double substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑃 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))))) | ||
Theorem | mzpclall 39204 | The set of all functions with the signature of a polynomial is a polynomially closed set. This is a lemma to show that the intersection in df-mzp 39201 is well-defined. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (ℤ ↑m (ℤ ↑m 𝑉)) ∈ (mzPolyCld‘𝑉)) | ||
Theorem | mzpcln0 39205 | Corrolary of mzpclall 39204: polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (mzPolyCld‘𝑉) ≠ ∅) | ||
Theorem | mzpcl1 39206 | Defining property 1 of a polynomially closed function set 𝑃: it contains all constant functions. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ ℤ) → ((ℤ ↑m 𝑉) × {𝐹}) ∈ 𝑃) | ||
Theorem | mzpcl2 39207* | Defining property 2 of a polynomially closed function set 𝑃: it contains all projections. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝐹)) ∈ 𝑃) | ||
Theorem | mzpcl34 39208 | Defining properties 3 and 4 of a polynomially closed function set 𝑃: it is closed under pointwise addition and multiplication. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝑃 ∈ (mzPolyCld‘𝑉) ∧ 𝐹 ∈ 𝑃 ∧ 𝐺 ∈ 𝑃) → ((𝐹 ∘f + 𝐺) ∈ 𝑃 ∧ (𝐹 ∘f · 𝐺) ∈ 𝑃)) | ||
Theorem | mzpval 39209 | Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (mzPoly‘𝑉) = ∩ (mzPolyCld‘𝑉)) | ||
Theorem | dmmzp 39210 | mzPoly is defined for all index sets which are sets. This is used with elfvdm 6696 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ dom mzPoly = V | ||
Theorem | mzpincl 39211 | Polynomial closedness is a universal first-order property and passes to intersections. This is where the closure properties of the polynomial ring itself are proved. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ (𝑉 ∈ V → (mzPoly‘𝑉) ∈ (mzPolyCld‘𝑉)) | ||
Theorem | mzpconst 39212 | Constant functions are polynomial. See also mzpconstmpt 39217. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝑉 ∈ V ∧ 𝐶 ∈ ℤ) → ((ℤ ↑m 𝑉) × {𝐶}) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpf 39213 | A polynomial function is a function from the coordinate space to the integers. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ (𝐹 ∈ (mzPoly‘𝑉) → 𝐹:(ℤ ↑m 𝑉)⟶ℤ) | ||
Theorem | mzpproj 39214* | A projection function is polynomial. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝑉 ∈ V ∧ 𝑋 ∈ 𝑉) → (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑋)) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpadd 39215 | The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 39218. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → (𝐴 ∘f + 𝐵) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpmul 39216 | The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 39219. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝐴 ∈ (mzPoly‘𝑉) ∧ 𝐵 ∈ (mzPoly‘𝑉)) → (𝐴 ∘f · 𝐵) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpconstmpt 39217* | A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 39218, mzpmulmpt 39219, mzpnegmpt 39221, mzpsubmpt 39220, mzpexpmpt 39222) can be used to build proofs that functions which are "manifestly polynomial", in the sense of being a maps-to containing constants, projections, and simple arithmetic operations, are actually polynomial functions. There is no mzpprojmpt because mzpproj 39214 is already expressed using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ ((𝑉 ∈ V ∧ 𝐶 ∈ ℤ) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐶) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpaddmpt 39218* | Sum of polynomial functions is polynomial. Maps-to version of mzpadd 39215. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐵) ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴 + 𝐵)) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpmulmpt 39219* | Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 39219. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐵) ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴 · 𝐵)) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpsubmpt 39220* | The difference of two polynomial functions is polynomial. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐵) ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴 − 𝐵)) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpnegmpt 39221* | Negation of a polynomial function. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ -𝐴) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpexpmpt 39222* | Raise a polynomial function to a (fixed) exponent. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ (((𝑥 ∈ (ℤ ↑m 𝑉) ↦ 𝐴) ∈ (mzPoly‘𝑉) ∧ 𝐷 ∈ ℕ0) → (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝐴↑𝐷)) ∈ (mzPoly‘𝑉)) | ||
Theorem | mzpindd 39223* | "Structural" induction to prove properties of all polynomial functions. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → 𝜒) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑉) → 𝜃) & ⊢ ((𝜑 ∧ (𝑓:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜂)) → 𝜁) & ⊢ ((𝜑 ∧ (𝑓:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜏) ∧ (𝑔:(ℤ ↑m 𝑉)⟶ℤ ∧ 𝜂)) → 𝜎) & ⊢ (𝑥 = ((ℤ ↑m 𝑉) × {𝑓}) → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = (𝑔 ∈ (ℤ ↑m 𝑉) ↦ (𝑔‘𝑓)) → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂)) & ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁)) & ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (mzPoly‘𝑉)) → 𝜌) | ||
Theorem | mzpmfp 39224 | Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ (mzPoly‘𝐼) = ran (𝐼 eval ℤring) | ||
Theorem | mzpsubst 39225* | Substituting polynomials for the variables of a polynomial results in a polynomial. 𝐺 is expected to depend on 𝑦 and provide the polynomials which are being substituted. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) | ||
Theorem | mzprename 39226* | Simplified version of mzpsubst 39225 to simply relabel variables in a polynomial. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ 𝑅:𝑉⟶𝑊) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ∘ 𝑅))) ∈ (mzPoly‘𝑊)) | ||
Theorem | mzpresrename 39227* | A polynomial is a polynomial over all larger index sets. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.) |
⊢ ((𝑊 ∈ V ∧ 𝑉 ⊆ 𝑊 ∧ 𝐹 ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ↾ 𝑉))) ∈ (mzPoly‘𝑊)) | ||
Theorem | mzpcompact2lem 39228* | Lemma for mzpcompact2 39229. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ (mzPoly‘𝐵) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎))))) | ||
Theorem | mzpcompact2 39229* | Polynomials are finitary objects and can only reference a finite number of variables, even if the index set is infinite. Thus, every polynomial can be expressed as a (uniquely minimal, although we do not prove that) polynomial on a finite number of variables, which is then extended by adding an arbitrary set of ignored variables. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ (𝐴 ∈ (mzPoly‘𝐵) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎))))) | ||
Theorem | coeq0i 39230 | coeq0 6102 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
⊢ ((𝐴:𝐶⟶𝐷 ∧ 𝐵:𝐸⟶𝐹 ∧ (𝐶 ∩ 𝐹) = ∅) → (𝐴 ∘ 𝐵) = ∅) | ||
Theorem | fzsplit1nn0 39231 | Split a finite 1-based set of integers in the middle, allowing either end to be empty ((1...0)). (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))) | ||
Syntax | cdioph 39232 | Extend class notation to include the family of Diophantine sets. |
class Dioph | ||
Definition | df-dioph 39233* | A Diophantine set is a set of positive integers which is a projection of the zero set of some polynomial. This definition somewhat awkwardly mixes ℤ (via mzPoly) and ℕ0 (to define the zero sets); the former could be avoided by considering coincidence sets of ℕ0 polynomials at the cost of requiring two, and the second is driven by consistency with our mu-recursive functions and the requirements of the Davis-Putnam-Robinson-Matiyasevich proof. Both are avoidable at a complexity cost. In particular, it is a consequence of 4sq 16290 that implicitly restricting variables to ℕ0 adds no expressive power over allowing them to range over ℤ. While this definition stipulates a specific index set for the polynomials, there is actually flexibility here, see eldioph2b 39240. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ Dioph = (𝑛 ∈ ℕ0 ↦ ran (𝑘 ∈ (ℤ≥‘𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝‘𝑢) = 0)})) | ||
Theorem | eldiophb 39234* | Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
⊢ (𝐷 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝐷 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) | ||
Theorem | eldioph 39235* | Condition for a set to be Diophantine (unpacking existential quantifier). (Contributed by Stefan O'Rear, 5-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁)) | ||
Theorem | diophrw 39236* | Renaming and adding unused witness variables does not change the Diophantine set coded by a polynomial. (Contributed by Stefan O'Rear, 7-Oct-2014.) |
⊢ ((𝑆 ∈ V ∧ 𝑀:𝑇–1-1→𝑆 ∧ (𝑀 ↾ 𝑂) = ( I ↾ 𝑂)) → {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m 𝑆)(𝑎 = (𝑏 ↾ 𝑂) ∧ ((𝑑 ∈ (ℤ ↑m 𝑆) ↦ (𝑃‘(𝑑 ∘ 𝑀)))‘𝑏) = 0)} = {𝑎 ∣ ∃𝑐 ∈ (ℕ0 ↑m 𝑇)(𝑎 = (𝑐 ↾ 𝑂) ∧ (𝑃‘𝑐) = 0)}) | ||
Theorem | eldioph2lem1 39237* | Lemma for eldioph2 39239. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑑 ∈ (ℤ≥‘𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto→𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) | ||
Theorem | eldioph2lem2 39238* | Lemma for eldioph2 39239. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) | ||
Theorem | eldioph2 39239* | Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 39229. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑆 ∈ V ∧ (1...𝑁) ⊆ 𝑆) ∧ 𝑃 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁)) | ||
Theorem | eldioph2b 39240* | While Diophantine sets were defined to have a finite number of witness variables consequtively following the observable variables, this is not necessary; they can equivalently be taken to use any witness set (𝑆 ∖ (1...𝑁)). For instance, in diophin 39249 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) | ||
Theorem | eldiophelnn0 39241 | Remove antecedent on 𝐵 from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝐴 ∈ (Dioph‘𝐵) → 𝐵 ∈ ℕ0) | ||
Theorem | eldioph3b 39242* | Define Diophantine sets in terms of polynomials with variables indexed by ℕ. This avoids a quantifier over the number of witness variables and will be easier to use than eldiophb 39234 in most cases. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝐴 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘ℕ)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) | ||
Theorem | eldioph3 39243* | Inference version of eldioph3b 39242 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁)) | ||
Theorem | ellz1 39244 | Membership in a lower set of integers. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ (𝐵 ∈ ℤ → (𝐴 ∈ (ℤ ∖ (ℤ≥‘(𝐵 + 1))) ↔ (𝐴 ∈ ℤ ∧ 𝐴 ≤ 𝐵))) | ||
Theorem | lzunuz 39245 | The union of a lower set of integers and an upper set of integers which abut or overlap is all of the integers. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≤ (𝐴 + 1)) → ((ℤ ∖ (ℤ≥‘(𝐴 + 1))) ∪ (ℤ≥‘𝐵)) = ℤ) | ||
Theorem | fz1eqin 39246 | Express a one-based finite range as the intersection of lower integers with ℕ. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ (𝑁 ∈ ℕ0 → (1...𝑁) = ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)) | ||
Theorem | lzenom 39247 | Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝑁 ∈ ℤ → (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ≈ ω) | ||
Theorem | elmapresaunres2 39248 | fresaunres2 6544 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺) | ||
Theorem | diophin 39249 | If two sets are Diophantine, so is their intersection. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∩ 𝐵) ∈ (Dioph‘𝑁)) | ||
Theorem | diophun 39250 | If two sets are Diophantine, so is their union. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∪ 𝐵) ∈ (Dioph‘𝑁)) | ||
Theorem | eldiophss 39251 | Diophantine sets are sets of tuples of nonnegative integers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ (𝐴 ∈ (Dioph‘𝐵) → 𝐴 ⊆ (ℕ0 ↑m (1...𝐵))) | ||
Theorem | diophrex 39252* | Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁)) | ||
Theorem | eq0rabdioph 39253* | This is the first of a number of theorems which allow sets to be proven Diophantine by syntactic induction, and models the correspondence between Diophantine sets and monotone existential first-order logic. This first theorem shows that the zero set of an implicit polynomial is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 0} ∈ (Dioph‘𝑁)) | ||
Theorem | eqrabdioph 39254* | Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be nonnegative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 𝐵} ∈ (Dioph‘𝑁)) | ||
Theorem | 0dioph 39255 | The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝐴 ∈ ℕ0 → ∅ ∈ (Dioph‘𝐴)) | ||
Theorem | vdioph 39256 | The "universal" set (as large as possible given eldiophss 39251) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝐴 ∈ ℕ0 → (ℕ0 ↑m (1...𝐴)) ∈ (Dioph‘𝐴)) | ||
Theorem | anrabdioph 39257* | Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∧ 𝜓)} ∈ (Dioph‘𝑁)) | ||
Theorem | orrabdioph 39258* | Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∨ 𝜓)} ∈ (Dioph‘𝑁)) | ||
Theorem | 3anrabdioph 39259* | Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜒} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∧ 𝜓 ∧ 𝜒)} ∈ (Dioph‘𝑁)) | ||
Theorem | 3orrabdioph 39260* | Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (({𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜒} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ (𝜑 ∨ 𝜓 ∨ 𝜒)} ∈ (Dioph‘𝑁)) | ||
Theorem | 2sbcrex 39261* | Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.) |
⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑) | ||
Theorem | sbcrexgOLD 39262* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3857 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) | ||
Theorem | 2sbcrexOLD 39263* | Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7187 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑) | ||
Theorem | sbc2rex 39264* | Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.) |
⊢ ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]𝜑) | ||
Theorem | sbc2rexgOLD 39265* | Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7187 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]𝜑)) | ||
Theorem | sbc4rex 39266* | Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.) |
⊢ ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 [𝐴 / 𝑎]𝜑) | ||
Theorem | sbc4rexgOLD 39267* | Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7187 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 [𝐴 / 𝑎]𝜑)) | ||
Theorem | sbcrot3 39268* | Rotate a sequence of three explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐴 / 𝑎]𝜑) | ||
Theorem | sbcrot5 39269* | Rotate a sequence of five explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒]𝜑 ↔ [𝐵 / 𝑏][𝐶 / 𝑐][𝐷 / 𝑑][𝐸 / 𝑒][𝐴 / 𝑎]𝜑) | ||
Theorem | sbccomieg 39270* | Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ (𝑎 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑) | ||
Theorem | rexrabdioph 39271* | Diophantine set builder for existential quantification. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ (𝑣 = (𝑡‘𝑀) → (𝜓 ↔ 𝜒)) & ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒 ↔ 𝜑)) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} ∈ (Dioph‘𝑁)) | ||
Theorem | rexfrabdioph 39272* | Diophantine set builder for existential quantifier, explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑀)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | 2rexfrabdioph 39273* | Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ 𝐿 = (𝑀 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | 3rexfrabdioph 39274* | Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ 𝐿 = (𝑀 + 1) & ⊢ 𝐾 = (𝐿 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 ∃𝑥 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | 4rexfrabdioph 39275* | Diophantine set builder for existential quantifier, explicit substitution, four variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ 𝐿 = (𝑀 + 1) & ⊢ 𝐾 = (𝐿 + 1) & ⊢ 𝐽 = (𝐾 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐽)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑡‘𝐽) / 𝑦]𝜑} ∈ (Dioph‘𝐽)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 ∃𝑥 ∈ ℕ0 ∃𝑦 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | 6rexfrabdioph 39276* | Diophantine set builder for existential quantifier, explicit substitution, six variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ 𝐿 = (𝑀 + 1) & ⊢ 𝐾 = (𝐿 + 1) & ⊢ 𝐽 = (𝐾 + 1) & ⊢ 𝐼 = (𝐽 + 1) & ⊢ 𝐻 = (𝐼 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐻)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑡‘𝐽) / 𝑦][(𝑡‘𝐼) / 𝑧][(𝑡‘𝐻) / 𝑝]𝜑} ∈ (Dioph‘𝐻)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 ∃𝑥 ∈ ℕ0 ∃𝑦 ∈ ℕ0 ∃𝑧 ∈ ℕ0 ∃𝑝 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | 7rexfrabdioph 39277* | Diophantine set builder for existential quantifier, explicit substitution, seven variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) & ⊢ 𝐿 = (𝑀 + 1) & ⊢ 𝐾 = (𝐿 + 1) & ⊢ 𝐽 = (𝐾 + 1) & ⊢ 𝐼 = (𝐽 + 1) & ⊢ 𝐻 = (𝐼 + 1) & ⊢ 𝐺 = (𝐻 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝐺)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡‘𝑀) / 𝑣][(𝑡‘𝐿) / 𝑤][(𝑡‘𝐾) / 𝑥][(𝑡‘𝐽) / 𝑦][(𝑡‘𝐼) / 𝑧][(𝑡‘𝐻) / 𝑝][(𝑡‘𝐺) / 𝑞]𝜑} ∈ (Dioph‘𝐺)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 ∃𝑤 ∈ ℕ0 ∃𝑥 ∈ ℕ0 ∃𝑦 ∈ ℕ0 ∃𝑧 ∈ ℕ0 ∃𝑝 ∈ ℕ0 ∃𝑞 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁)) | ||
Theorem | rabdiophlem1 39278* | Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 3160. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))𝐴 ∈ ℤ) | ||
Theorem | rabdiophlem2 39279* | Lemma for arithmetic diophantine sets. Reuse a polynomial expression under a new quantifier. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑢 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑m (1...𝑀)) ↦ ⦋(𝑡 ↾ (1...𝑁)) / 𝑢⦌𝐴) ∈ (mzPoly‘(1...𝑀))) | ||
Theorem | elnn0rabdioph 39280* | Diophantine set builder for nonnegativity constraints. The first builder which uses a witness variable internally; an expression is nonnegative if there is a nonnegative integer equal to it. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ ℕ0} ∈ (Dioph‘𝑁)) | ||
Theorem | rexzrexnn0 39281* | Rewrite a quantification over integers into a quantification over naturals. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (∃𝑥 ∈ ℤ 𝜑 ↔ ∃𝑦 ∈ ℕ0 (𝜓 ∨ 𝜒)) | ||
Theorem | lerabdioph 39282* | Diophantine set builder for the "less than or equal to" relation. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ≤ 𝐵} ∈ (Dioph‘𝑁)) | ||
Theorem | eluzrabdioph 39283* | Diophantine set builder for membership in a fixed upper set of integers. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ (ℤ≥‘𝑀)} ∈ (Dioph‘𝑁)) | ||
Theorem | elnnrabdioph 39284* | Diophantine set builder for positivity. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∈ ℕ} ∈ (Dioph‘𝑁)) | ||
Theorem | ltrabdioph 39285* | Diophantine set builder for the strict less than relation. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 < 𝐵} ∈ (Dioph‘𝑁)) | ||
Theorem | nerabdioph 39286* | Diophantine set builder for inequality. This not quite trivial theorem touches on something important; Diophantine sets are not closed under negation, but they contain an important subclass that is, namely the recursive sets. With this theorem and De Morgan's laws, all quantifier-free formulas can be negated. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ≠ 𝐵} ∈ (Dioph‘𝑁)) | ||
Theorem | dvdsrabdioph 39287* | Divisibility is a Diophantine relation. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐵) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 ∥ 𝐵} ∈ (Dioph‘𝑁)) | ||
Theorem | eldioph4b 39288* | Membership in Dioph expressed using a quantified union to add witness variables instead of a restriction to remove them. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
⊢ 𝑊 ∈ V & ⊢ ¬ 𝑊 ∈ Fin & ⊢ (𝑊 ∩ ℕ) = ∅ ⇒ ⊢ (𝑆 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0})) | ||
Theorem | eldioph4i 39289* | Forward-only version of eldioph4b 39288. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
⊢ 𝑊 ∈ V & ⊢ ¬ 𝑊 ∈ Fin & ⊢ (𝑊 ∩ ℕ) = ∅ ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} ∈ (Dioph‘𝑁)) | ||
Theorem | diophren 39290* | Change variables in a Diophantine set, using class notation. This allows already proved Diophantine sets to be reused in contexts with more variables. (Contributed by Stefan O'Rear, 16-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.) |
⊢ ((𝑆 ∈ (Dioph‘𝑁) ∧ 𝑀 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0 ↑m (1...𝑀)) ∣ (𝑎 ∘ 𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)) | ||
Theorem | rabrenfdioph 39291* | Change variable numbers in a Diophantine class abstraction using explicit substitution. (Contributed by Stefan O'Rear, 17-Oct-2014.) |
⊢ ((𝐵 ∈ ℕ0 ∧ 𝐹:(1...𝐴)⟶(1...𝐵) ∧ {𝑎 ∈ (ℕ0 ↑m (1...𝐴)) ∣ 𝜑} ∈ (Dioph‘𝐴)) → {𝑏 ∈ (ℕ0 ↑m (1...𝐵)) ∣ [(𝑏 ∘ 𝐹) / 𝑎]𝜑} ∈ (Dioph‘𝐵)) | ||
Theorem | rabren3dioph 39292* | Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.) |
⊢ (((𝑎‘1) = (𝑏‘𝑋) ∧ (𝑎‘2) = (𝑏‘𝑌) ∧ (𝑎‘3) = (𝑏‘𝑍)) → (𝜑 ↔ 𝜓)) & ⊢ 𝑋 ∈ (1...𝑁) & ⊢ 𝑌 ∈ (1...𝑁) & ⊢ 𝑍 ∈ (1...𝑁) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ 𝜑} ∈ (Dioph‘3)) → {𝑏 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) | ||
Theorem | fphpd 39293* | Pigeonhole principle expressed with implicit substitution. If the range is smaller than the domain, two inputs must be mapped to the same output. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ (𝜑 → 𝐵 ≺ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝐶 = 𝐷)) | ||
Theorem | fphpdo 39294* | Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐵 ≺ 𝐴) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ (𝑧 = 𝑥 → 𝐶 = 𝐷) & ⊢ (𝑧 = 𝑦 → 𝐶 = 𝐸) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 < 𝑦 ∧ 𝐷 = 𝐸)) | ||
Theorem | ctbnfien 39295 | An infinite subset of a countable set is countable, without using choice. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ (((𝑋 ≈ ω ∧ 𝑌 ≈ ω) ∧ (𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin)) → 𝐴 ≈ 𝑌) | ||
Theorem | fiphp3d 39296* | Infinite pigeonhole principle for partitioning an infinite set between finitely many buckets. (Contributed by Stefan O'Rear, 18-Oct-2014.) |
⊢ (𝜑 → 𝐴 ≈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ 𝐷 = 𝑦} ≈ ℕ) | ||
Theorem | rencldnfilem 39297* | Lemma for rencldnfi 39298. (Contributed by Stefan O'Rear, 18-Oct-2014.) |
⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ≠ ∅ ∧ ¬ 𝐵 ∈ 𝐴)) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐴 (abs‘(𝑦 − 𝐵)) < 𝑥) → ¬ 𝐴 ∈ Fin) | ||
Theorem | rencldnfi 39298* | A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 39297 using infima; this theorem removes the requirement that A be nonempty. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ 𝐴) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐴 (abs‘(𝑦 − 𝐵)) < 𝑥) → ¬ 𝐴 ∈ Fin) | ||
Theorem | irrapxlem1 39299* | Lemma for irrapx1 39305. Divides the unit interval into 𝐵 half-open sections and using the pigeonhole principle fphpdo 39294 finds two multiples of 𝐴 in the same section mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ (0...𝐵)∃𝑦 ∈ (0...𝐵)(𝑥 < 𝑦 ∧ (⌊‘(𝐵 · ((𝐴 · 𝑥) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑦) mod 1))))) | ||
Theorem | irrapxlem2 39300* | Lemma for irrapx1 39305. Two multiples in the same bucket means they are very close mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ (0...𝐵)∃𝑦 ∈ (0...𝐵)(𝑥 < 𝑦 ∧ (abs‘(((𝐴 · 𝑥) mod 1) − ((𝐴 · 𝑦) mod 1))) < (1 / 𝐵))) |
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