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Theorem List for Metamath Proof Explorer - 36201-36300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmzpexpmpt 36201* 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) ↦ (𝐴𝐷)) ∈ (mzPoly‘𝑉))

Theoremmzpindd 36202* "Structural" induction to prove properties of all polynomial functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
((𝜑𝑓 ∈ ℤ) → 𝜒)    &   ((𝜑𝑓𝑉) → 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}) → (𝜓𝜒))    &   (𝑥 = (𝑔 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑔𝑓)) → (𝜓𝜃))    &   (𝑥 = 𝑓 → (𝜓𝜏))    &   (𝑥 = 𝑔 → (𝜓𝜂))    &   (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))    &   (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))    &   (𝑥 = 𝐴 → (𝜓𝜌))       ((𝜑𝐴 ∈ (mzPoly‘𝑉)) → 𝜌)

Theoremmzpmfp 36203 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)

Theoremmzpsubst 36204* 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) ∧ ∀𝑦𝑉 𝐺 ∈ (mzPoly‘𝑊)) → (𝑥 ∈ (ℤ ↑𝑚 𝑊) ↦ (𝐹‘(𝑦𝑉 ↦ (𝐺𝑥)))) ∈ (mzPoly‘𝑊))

Theoremmzprename 36205* Simplified version of mzpsubst 36204 to simply relabel variables in a polynomial. (Contributed by Stefan O'Rear, 5-Oct-2014.)
((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ 𝑅:𝑉𝑊) → (𝑥 ∈ (ℤ ↑𝑚 𝑊) ↦ (𝐹‘(𝑥𝑅))) ∈ (mzPoly‘𝑊))

Theoremmzpresrename 36206* 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 ∧ 𝑉𝑊𝐹 ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑𝑚 𝑊) ↦ (𝐹‘(𝑥𝑉))) ∈ (mzPoly‘𝑊))

Theoremmzpcompact2lem 36207* Lemma for mzpcompact2 36208. (Contributed by Stefan O'Rear, 9-Oct-2014.)
𝐵 ∈ V       (𝐴 ∈ (mzPoly‘𝐵) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎𝐵𝐴 = (𝑐 ∈ (ℤ ↑𝑚 𝐵) ↦ (𝑏‘(𝑐𝑎)))))

Theoremmzpcompact2 36208* 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)(𝑎𝐵𝐴 = (𝑐 ∈ (ℤ ↑𝑚 𝐵) ↦ (𝑏‘(𝑐𝑎)))))

20.24.9  Miscellanea for Diophantine sets 1

Theoremcoeq0i 36209 coeq0 5451 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)
((𝐴:𝐶𝐷𝐵:𝐸𝐹 ∧ (𝐶𝐹) = ∅) → (𝐴𝐵) = ∅)

Theoremfzsplit1nn0 36210 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 ∈ ℕ0𝐵 ∈ ℕ0𝐴𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))

20.24.10  Diophantine sets 1: definitions

Syntaxcdioph 36211 Extend class notation to include the family of Diophantine sets.
class Dioph

Definitiondf-dioph 36212* 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 (via mzPoly) and 0 (to define the zero sets); the former could be avoided by considering coincidence sets of 0 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 15388 that implicitly restricting variables to 0 adds no expressive power over allowing them to range over . While this definition stipulates a specific index set for the polynomials, there is actually flexibility here, see eldioph2b 36219. (Contributed by Stefan O'Rear, 5-Oct-2014.)
Dioph = (𝑛 ∈ ℕ0 ↦ ran (𝑘 ∈ (ℤ𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0)}))

Theoremeldiophb 36213* 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 ∈ (ℤ𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝐷 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))

Theoremeldioph 36214* Condition for a set to be Diophantine (unpacking existential quantifier). (Contributed by Stefan O'Rear, 5-Oct-2014.)
((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁))

Theoremdiophrw 36215* 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:𝑇1-1𝑆 ∧ (𝑀𝑂) = ( I ↾ 𝑂)) → {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 𝑆)(𝑎 = (𝑏𝑂) ∧ ((𝑑 ∈ (ℤ ↑𝑚 𝑆) ↦ (𝑃‘(𝑑𝑀)))‘𝑏) = 0)} = {𝑎 ∣ ∃𝑐 ∈ (ℕ0𝑚 𝑇)(𝑎 = (𝑐𝑂) ∧ (𝑃𝑐) = 0)})

Theoremeldioph2lem1 36216* Lemma for eldioph2 36218. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑑 ∈ (ℤ𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))

Theoremeldioph2lem2 36217* Lemma for eldioph2 36218. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
(((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))

Theoremeldioph2 36218* Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 36208. (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 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁))

Theoremeldioph2b 36219* 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 36229 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
(((𝑁 ∈ ℕ0𝑆 ∈ V) ∧ (¬ 𝑆 ∈ Fin ∧ (1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))

Theoremeldiophelnn0 36220 Remove antecedent on 𝐵 from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ (Dioph‘𝐵) → 𝐵 ∈ ℕ0)

Theoremeldioph3b 36221* Define Diophantine sets in terms of polynomials with variables indexed by . This avoids a quantifier over the number of witness variables and will be easier to use than eldiophb 36213 in most cases. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘ℕ)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))

Theoremeldioph3 36222* Inference version of eldioph3b 36221 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)
((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁))

20.24.11  Diophantine sets 2 miscellanea

Theoremellz1 36223 Membership in a lower set of integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)
(𝐵 ∈ ℤ → (𝐴 ∈ (ℤ ∖ (ℤ‘(𝐵 + 1))) ↔ (𝐴 ∈ ℤ ∧ 𝐴𝐵)))

Theoremlzunuz 36224 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))) ∪ (ℤ𝐵)) = ℤ)

Theoremfz1eqin 36225 Express a one-based finite range as the intersection of lower integers with . (Contributed by Stefan O'Rear, 9-Oct-2014.)
(𝑁 ∈ ℕ0 → (1...𝑁) = ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ))

Theoremlzenom 36226 Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝑁 ∈ ℤ → (ℤ ∖ (ℤ‘(𝑁 + 1))) ≈ ω)

Theoremelmapresaun 36227 fresaun 5871 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) ∈ (𝐶𝑚 (𝐴𝐵)))

Theoremelmapresaunres2 36228 fresaunres2 5872 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.)
((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)

20.24.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra

Theoremdiophin 36229 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)) → (𝐴𝐵) ∈ (Dioph‘𝑁))

Theoremdiophun 36230 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)) → (𝐴𝐵) ∈ (Dioph‘𝑁))

Theoremeldiophss 36231 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 ⊆ (ℕ0𝑚 (1...𝐵)))

20.24.13  Diophantine sets 3: construction

Theoremdiophrex 36232* Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))

Theoremeq0rabdioph 36233* 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 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 0} ∈ (Dioph‘𝑁))

Theoremeqrabdioph 36234* 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 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 = 𝐵} ∈ (Dioph‘𝑁))

Theorem0dioph 36235 The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ ℕ0 → ∅ ∈ (Dioph‘𝐴))

Theoremvdioph 36236 The "universal" set (as large as possible given eldiophss 36231) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ ℕ0 → (ℕ0𝑚 (1...𝐴)) ∈ (Dioph‘𝐴))

Theoremanrabdioph 36237* Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(({𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ (𝜑𝜓)} ∈ (Dioph‘𝑁))

Theoremorrabdioph 36238* Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(({𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ (𝜑𝜓)} ∈ (Dioph‘𝑁))

Theorem3anrabdioph 36239* Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(({𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝜒} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ (𝜑𝜓𝜒)} ∈ (Dioph‘𝑁))

Theorem3orrabdioph 36240* Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(({𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝜑} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁) ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝜒} ∈ (Dioph‘𝑁)) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ (𝜑𝜓𝜒)} ∈ (Dioph‘𝑁))

20.24.14  Diophantine sets 4 miscellanea

Theorem2sbcrex 36241* 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]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)

TheoremsbcrexgOLD 36242* 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 3385 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))

Theorem2sbcrexOLD 36243* Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 6461 instead. (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       ([𝐴 / 𝑎][𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)

Theoremsbc2rex 36244* Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑)

Theoremsbc2rexgOLD 36245* Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 6461 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑))

Theoremsbc4rex 36246* Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑)

Theoremsbc4rexgOLD 36247* Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 6461 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑))

Theoremsbcrot3 36248* 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)

Theoremsbcrot5 36249* 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)

Theoremsbccomieg 36250* 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 = 𝐴𝐵 = 𝐶)       ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑)

20.24.15  Diophantine sets 4: Quantification

Theoremrexrabdioph 36251* Diophantine set builder for existential quantification. (Contributed by Stefan O'Rear, 10-Oct-2014.)
𝑀 = (𝑁 + 1)    &   (𝑣 = (𝑡𝑀) → (𝜓𝜒))    &   (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒𝜑))       ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} ∈ (Dioph‘𝑁))

Theoremrexfrabdioph 36252* 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 ∈ (ℕ0𝑚 (1...𝑀)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣]𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Theorem2rexfrabdioph 36253* 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 ∈ (ℕ0𝑚 (1...𝐿)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤]𝜑} ∈ (Dioph‘𝐿)) → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Theorem3rexfrabdioph 36254* 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 ∈ (ℕ0𝑚 (1...𝐾)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥]𝜑} ∈ (Dioph‘𝐾)) → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0𝑥 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Theorem4rexfrabdioph 36255* 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 ∈ (ℕ0𝑚 (1...𝐽)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑡𝐽) / 𝑦]𝜑} ∈ (Dioph‘𝐽)) → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0𝑥 ∈ ℕ0𝑦 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Theorem6rexfrabdioph 36256* 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 ∈ (ℕ0𝑚 (1...𝐻)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑡𝐽) / 𝑦][(𝑡𝐼) / 𝑧][(𝑡𝐻) / 𝑝]𝜑} ∈ (Dioph‘𝐻)) → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0𝑥 ∈ ℕ0𝑦 ∈ ℕ0𝑧 ∈ ℕ0𝑝 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

Theorem7rexfrabdioph 36257* 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 ∈ (ℕ0𝑚 (1...𝐺)) ∣ [(𝑡 ↾ (1...𝑁)) / 𝑢][(𝑡𝑀) / 𝑣][(𝑡𝐿) / 𝑤][(𝑡𝐾) / 𝑥][(𝑡𝐽) / 𝑦][(𝑡𝐼) / 𝑧][(𝑡𝐻) / 𝑝][(𝑡𝐺) / 𝑞]𝜑} ∈ (Dioph‘𝐺)) → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0𝑤 ∈ ℕ0𝑥 ∈ ℕ0𝑦 ∈ ℕ0𝑧 ∈ ℕ0𝑝 ∈ ℕ0𝑞 ∈ ℕ0 𝜑} ∈ (Dioph‘𝑁))

20.24.16  Diophantine sets 5: Arithmetic sets

Theoremrabdiophlem1 36258* Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 2840. (Contributed by Stefan O'Rear, 10-Oct-2014.)
((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → ∀𝑡 ∈ (ℕ0𝑚 (1...𝑁))𝐴 ∈ ℤ)

Theoremrabdiophlem2 36259* 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)) / 𝑢𝐴) ∈ (mzPoly‘(1...𝑀)))

Theoremelnn0rabdioph 36260* 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 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 ∈ ℕ0} ∈ (Dioph‘𝑁))

Theoremrexzrexnn0 36261* Rewrite a quantification over integers into a quantification over naturals. (Contributed by Stefan O'Rear, 11-Oct-2014.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = -𝑦 → (𝜑𝜒))       (∃𝑥 ∈ ℤ 𝜑 ↔ ∃𝑦 ∈ ℕ0 (𝜓𝜒))

Theoremlerabdioph 36262* Diophantine set builder for the less or equals 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 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴𝐵} ∈ (Dioph‘𝑁))

Theoremeluzrabdioph 36263* Diophantine set builder for membership in a fixed upper set of integers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
((𝑁 ∈ ℕ0𝑀 ∈ ℤ ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 ∈ (ℤ𝑀)} ∈ (Dioph‘𝑁))

Theoremelnnrabdioph 36264* 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 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 ∈ ℕ} ∈ (Dioph‘𝑁))

Theoremltrabdioph 36265* 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 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴 < 𝐵} ∈ (Dioph‘𝑁))

Theoremnerabdioph 36266* 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 formulae 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 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴𝐵} ∈ (Dioph‘𝑁))

Theoremdvdsrabdioph 36267* 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 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝐴𝐵} ∈ (Dioph‘𝑁))

20.24.17  Diophantine sets 6: reusability. renumbering of variables

Theoremeldioph4b 36268* 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 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑝‘(𝑡𝑤)) = 0}))

Theoremeldioph4i 36269* Forward-only version of eldioph4b 36268. (Contributed by Stefan O'Rear, 16-Oct-2014.)
𝑊 ∈ V    &    ¬ 𝑊 ∈ Fin    &   (𝑊 ∩ ℕ) = ∅       ((𝑁 ∈ ℕ0𝑃 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))) → {𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0𝑚 𝑊)(𝑃‘(𝑡𝑤)) = 0} ∈ (Dioph‘𝑁))

Theoremdiophren 36270* 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 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0𝑚 (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))

Theoremrabrenfdioph 36271* Change variable numbers in a Diophantine class abstraction using explicit substitution. (Contributed by Stefan O'Rear, 17-Oct-2014.)
((𝐵 ∈ ℕ0𝐹:(1...𝐴)⟶(1...𝐵) ∧ {𝑎 ∈ (ℕ0𝑚 (1...𝐴)) ∣ 𝜑} ∈ (Dioph‘𝐴)) → {𝑏 ∈ (ℕ0𝑚 (1...𝐵)) ∣ [(𝑏𝐹) / 𝑎]𝜑} ∈ (Dioph‘𝐵))

Theoremrabren3dioph 36272* Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.)
(((𝑎‘1) = (𝑏𝑋) ∧ (𝑎‘2) = (𝑏𝑌) ∧ (𝑎‘3) = (𝑏𝑍)) → (𝜑𝜓))    &   𝑋 ∈ (1...𝑁)    &   𝑌 ∈ (1...𝑁)    &   𝑍 ∈ (1...𝑁)       ((𝑁 ∈ ℕ0 ∧ {𝑎 ∈ (ℕ0𝑚 (1...3)) ∣ 𝜑} ∈ (Dioph‘3)) → {𝑏 ∈ (ℕ0𝑚 (1...𝑁)) ∣ 𝜓} ∈ (Dioph‘𝑁))

20.24.18  Pigeonhole Principle and cardinality helpers

Theoremfphpd 36273* 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.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   (𝑥 = 𝑦𝐶 = 𝐷)       (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝐶 = 𝐷))

Theoremfphpdo 36274* Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐵𝐴)    &   ((𝜑𝑧𝐴) → 𝐶𝐵)    &   (𝑧 = 𝑥𝐶 = 𝐷)    &   (𝑧 = 𝑦𝐶 = 𝐸)       (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸))

Theoremctbnfien 36275 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) ∧ (𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin)) → 𝐴𝑌)

Theoremfiphp3d 36276* Infinite pigeonhole principle for partitioning an infinite set between finitely many buckets. (Contributed by Stefan O'Rear, 18-Oct-2014.)
(𝜑𝐴 ≈ ℕ)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐷𝐵)       (𝜑 → ∃𝑦𝐵 {𝑥𝐴𝐷 = 𝑦} ≈ ℕ)

20.24.19  A non-closed set of reals is infinite

Theoremrencldnfilem 36277* Lemma for rencldnfi 36278. (Contributed by Stefan O'Rear, 18-Oct-2014.)
(((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ≠ ∅ ∧ ¬ 𝐵𝐴)) ∧ ∀𝑥 ∈ ℝ+𝑦𝐴 (abs‘(𝑦𝐵)) < 𝑥) → ¬ 𝐴 ∈ Fin)

Theoremrencldnfi 36278* A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 36277 using infima; this theorem removes the requirement that A be nonempty. (Contributed by Stefan O'Rear, 19-Oct-2014.)
(((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵𝐴) ∧ ∀𝑥 ∈ ℝ+𝑦𝐴 (abs‘(𝑦𝐵)) < 𝑥) → ¬ 𝐴 ∈ Fin)

20.24.20  Lagrange's rational approximation theorem

Theoremirrapxlem1 36279* Lemma for irrapx1 36285. Divides the unit interval into 𝐵 half-open sections and using the pigeonhole principle fphpdo 36274 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)))))

Theoremirrapxlem2 36280* Lemma for irrapx1 36285. 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)))

Theoremirrapxlem3 36281* Lemma for irrapx1 36285. By subtraction, there is a multiple very close to an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℕ) → ∃𝑥 ∈ (1...𝐵)∃𝑦 ∈ ℕ0 (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / 𝐵))

Theoremirrapxlem4 36282* Lemma for irrapx1 36285. Eliminate ranges, use positivity of the input to force positivity of the output by increasing 𝐵 as needed. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / if(𝑥𝐵, 𝐵, 𝑥)))

Theoremirrapxlem5 36283* Lemma for irrapx1 36285. Switching to real intervals and fraction syntax. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℚ (0 < 𝑥 ∧ (abs‘(𝑥𝐴)) < 𝐵 ∧ (abs‘(𝑥𝐴)) < ((denom‘𝑥)↑-2)))

Theoremirrapxlem6 36284* Lemma for irrapx1 36285. Explicit description of a non-closed set. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → ∃𝑥 ∈ {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦𝐴)) < ((denom‘𝑦)↑-2))} (abs‘(𝑥𝐴)) < 𝐵)

Theoremirrapx1 36285* Dirichlet's approximation theorem. Every positive irrational number has infinitely many rational approximations which are closer than the inverse squares of their reduced denominators. Lemma 61 in [vandenDries] p. 42. (Contributed by Stefan O'Rear, 14-Sep-2014.)
(𝐴 ∈ (ℝ+ ∖ ℚ) → {𝑦 ∈ ℚ ∣ (0 < 𝑦 ∧ (abs‘(𝑦𝐴)) < ((denom‘𝑦)↑-2))} ≈ ℕ)

20.24.21  Pell equations 1: A nontrivial solution always exists

Theorempellexlem1 36286 Lemma for pellex 36292. Arithmetical core of pellexlem3, norm lower bound. This begins Dirichlet's proof of the Pell equation solution existence; the proof here follows theorem 62 of [vandenDries] p. 43. (Contributed by Stefan O'Rear, 14-Sep-2014.)
(((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ¬ (√‘𝐷) ∈ ℚ) → ((𝐴↑2) − (𝐷 · (𝐵↑2))) ≠ 0)

Theorempellexlem2 36287 Lemma for pellex 36292. Arithmetical core of pellexlem3, norm upper bound. (Contributed by Stefan O'Rear, 14-Sep-2014.)
(((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) < (1 + (2 · (√‘𝐷))))

Theorempellexlem3 36288* Lemma for pellex 36292. To each good rational approximation of (√‘𝐷), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))})

Theorempellexlem4 36289* Lemma for pellex 36292. Invoking irrapx1 36285, we have infinitely many near-solutions. (Contributed by Stefan O'Rear, 14-Sep-2014.)
((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≈ ℕ)

Theorempellexlem5 36290* Lemma for pellex 36292. Invoking fiphp3d 36276, we have infinitely many near-solutions for some specific norm. (Contributed by Stefan O'Rear, 19-Oct-2014.)
((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℤ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ))

Theorempellexlem6 36291* Lemma for pellex 36292. Doing a field division between near solutions get us to norm 1, and the modularity constraint ensures we still have an integer. Returning NN guarantees that we are not returning the trivial solution (1,0). We are not explicitly defining the Pell-field, Pell-ring, and Pell-norm explicitly because after this construction is done we will never use them. This is mostly basic algebraic number theory and could be simplified if a generic framework for that were in place. (Contributed by Stefan O'Rear, 19-Oct-2014.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → ¬ (√‘𝐷) ∈ ℚ)    &   (𝜑𝐸 ∈ ℕ)    &   (𝜑𝐹 ∈ ℕ)    &   (𝜑 → ¬ (𝐴 = 𝐸𝐵 = 𝐹))    &   (𝜑𝐶 ≠ 0)    &   (𝜑 → ((𝐴↑2) − (𝐷 · (𝐵↑2))) = 𝐶)    &   (𝜑 → ((𝐸↑2) − (𝐷 · (𝐹↑2))) = 𝐶)    &   (𝜑 → (𝐴 mod (abs‘𝐶)) = (𝐸 mod (abs‘𝐶)))    &   (𝜑 → (𝐵 mod (abs‘𝐶)) = (𝐹 mod (abs‘𝐶)))       (𝜑 → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)

Theorempellex 36292* Every Pell equation has a nontrivial solution. Theorem 62 in [vandenDries] p. 43. (Contributed by Stefan O'Rear, 19-Oct-2014.)
((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)

20.24.22  Pell equations 2: Algebraic number theory of the solution set

Syntaxcsquarenn 36293 Extend class notation to include the set of square positive integers.
class NN

Syntaxcpell1qr 36294 Extend class notation to include the class of quadrant-1 Pell solutions.
class Pell1QR

Syntaxcpell1234qr 36295 Extend class notation to include the class of any-quadrant Pell solutions.
class Pell1234QR

Syntaxcpell14qr 36296 Extend class notation to include the class of positive Pell solutions.
class Pell14QR

Syntaxcpellfund 36297 Extend class notation to include the Pell-equation fundamental solution function.
class PellFund

Definitiondf-squarenn 36298 Define the set of square positive integers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN = {𝑥 ∈ ℕ ∣ (√‘𝑥) ∈ ℚ}

Definitiondf-pell1qr 36299* Define the solutions of a Pell equation in the first quadrant. To avoid pair pain, we represent this via the canonical embedding into the reals. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Pell1QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})

Definitiondf-pell14qr 36300* Define the positive solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Pell14QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})

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