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Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcoeq0i 26701 coeq0 26700 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremfzsplit1nn0 26702 Split a finite 1-based set of integers in the middle, allowing either end to be empty (). (Contributed by Stefan O'Rear, 8-Oct-2014.)

19.16.11  Diophantine sets 1: definitions

Syntaxcdioph 26703 Extend class notation to include the family of Diophantine sets.
Dioph

Definitiondf-dioph 26704* A Diophantine set is a set of natural numbers which is a projection of the zero set of some polynomial. This definition somewhat awkwardly mixes (via mzPoly) and (to define the zero sets); the former could be avoided by considering coincidence sets of 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 13287 that implicitly restricting variables to 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 26711. (Contributed by Stefan O'Rear, 5-Oct-2014.)
Dioph mzPoly

Theoremeldiophb 26705* Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Dioph mzPoly

Theoremeldioph 26706* Condition for a set to be Diophantine (unpacking existential quantifier) (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly Dioph

Theoremdiophrw 26707* Renaming and adding unused witness variables does not change the Diophantine set coded by a polynomial. (Contributed by Stefan O'Rear, 7-Oct-2014.)

Theoremeldioph2lem1 26708* Lemma for eldioph2 26710. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremeldioph2lem2 26709* Lemma for eldioph2 26710. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremeldioph2 26710* Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 26699. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
mzPoly Dioph

Theoremeldioph2b 26711* 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 . For instance, in diophin 26721 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Dioph mzPoly

Theoremeldiophelnn0 26712 Remove antecedent on from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph

Theoremeldioph3b 26713* 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 26705 in most cases. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph mzPoly

Theoremeldioph3 26714* Inference version of eldioph3b 26713 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly Dioph

19.16.12  Diophantine sets 2 miscellanea

Theoremellz1 26715 Membership in a set of lower integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremlzunuz 26716 A set of lower integers and upper integers which abut or overlap is all of the integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremfz1eqin 26717 Express a one-based finite range as the intersection of lower integers with . (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremlzenom 26718 Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Theoremelmapresaun 26719 fresaun 5573 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)

Theoremelmapresaunres2 26720 fresaunres2 5574 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.)

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

Theoremdiophin 26721 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.)
Dioph Dioph Dioph

Theoremdiophun 26722 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.)
Dioph Dioph Dioph

Theoremeldiophss 26723 Diophantine sets are sets of tuples of natural numbers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph

19.16.14  Diophantine sets 3: construction

Theoremdiophrex 26724* Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph

Theoremeq0rabdioph 26725* 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.)
mzPoly Dioph

Theoremeqrabdioph 26726* Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be non-negative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly mzPoly Dioph

Theorem0dioph 26727 The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph

Theoremvdioph 26728 The "universal" set (as large as possible given eldiophss 26723) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph

Theoremanrabdioph 26729* Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph

Theoremorrabdioph 26730* Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph

Theorem3anrabdioph 26731* Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph Dioph

Theorem3orrabdioph 26732* Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph Dioph

19.16.15  Diophantine sets 4 miscellanea

Theorem2sbcrex 26733* Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremsbc2rexg 26734* Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremsbc4rexg 26735* Exchange a substitution with 4 existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.)

TheoremsbcbiiiOLD 26736 Fully inferenced rewriting under an explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsbcrot3 26737* Rotate a sequence of three explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsbcrot5 26738* Rotate a sequence of five explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsbccomieg 26739* 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.)

Theoremsbcrot3gOLD 26740* Rotate a sequence of three explicit substitutions, closed theorem. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsbcrot3OLD 26741* Rotate a sequence of three explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsbcrot5OLD 26742* Rotate a sequence of five explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)

TheoremsbccomiegOLD 26743* Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)

19.16.16  Diophantine sets 4: Quantification

Theoremrexrabdioph 26744* Diophantine set builder for existential quantification. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph

Theoremrexfrabdioph 26745* Diophantine set builder for existential quantifier, explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph Dioph

Theorem2rexfrabdioph 26746* 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.)
Dioph Dioph

Theorem3rexfrabdioph 26747* 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.)
Dioph Dioph

Theorem4rexfrabdioph 26748* 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.)
Dioph Dioph

Theorem6rexfrabdioph 26749* 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.)
Dioph Dioph

Theorem7rexfrabdioph 26750* 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.)
Dioph Dioph

19.16.17  Diophantine sets 5: Arithmetic sets

Theoremrabdiophlem1 26751* Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 2741. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly

Theoremrabdiophlem2 26752* Lemma for arithmetic diophantine sets. Reuse a polynomial expression under a new quantifier. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly mzPoly

Theoremelnn0rabdioph 26753* 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.)
mzPoly Dioph

Theoremrexzrexnn0 26754* Rewrite a quantification over integers into a quantification over naturals. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremlerabdioph 26755* Diophantine set builder for the less or equals relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly mzPoly Dioph

Theoremeluzrabdioph 26756* Diophantine set builder for membership in a fixed set of upper integers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly Dioph

Theoremelnnrabdioph 26757* Diophantine set builder for positivity. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly Dioph

Theoremltrabdioph 26758* Diophantine set builder for the strict less than relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly mzPoly Dioph

Theoremnerabdioph 26759* 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.)
mzPoly mzPoly Dioph

Theoremdvdsrabdioph 26760* Divisibility is a Diophantine relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly mzPoly Dioph

19.16.18  Diophantine sets 6 miscellanea

Theoremfz1ssnn 26761 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)

19.16.19  Diophantine sets 6: reusability. renumbering of variables

Theoremeldioph4b 26762* 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.)
Dioph mzPoly

Theoremeldioph4i 26763* Forward-only version of eldioph4b 26762. (Contributed by Stefan O'Rear, 16-Oct-2014.)
mzPoly Dioph

Theoremdiophren 26764* 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.)
Dioph Dioph

Theoremrabrenfdioph 26765* Change variable numbers in a Diophantine class abstraction using explicit substitution. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph Dioph

Theoremrabren3dioph 26766* Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph Dioph

19.16.20  Pigeonhole Principle and cardinality helpers

Theoremfphpd 26767* 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.)

Theoremfphpdo 26768* Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.)

Theoremctbnfien 26769 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.)

Theoremfiphp3d 26770* Infinite pigeonhole principle for partitioning an infinite set between finitely many buckets. (Contributed by Stefan O'Rear, 18-Oct-2014.)

19.16.21  A non-closed set of reals is infinite

Theoremrencldnfilem 26771* Lemma for rencldnfi 26772. (Contributed by Stefan O'Rear, 18-Oct-2014.)

Theoremrencldnfi 26772* A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 26771 using infima; this theorem removes the requirement that A be non-empty. (Contributed by Stefan O'Rear, 19-Oct-2014.)

19.16.22  Miscellanea for Lagrange's theorem

Theoremicodiamlt 26773 Two elements in a half-open interval have separation strictly less than the difference between the endpoints. (Contributed by Stefan O'Rear, 12-Sep-2014.)

Theoremmodelico 26774 Modular reduction produces a half-open interval. (Contributed by Stefan O'Rear, 12-Sep-2014.)

19.16.23  Lagrange's rational approximation theorem

Theoremirrapxlem1 26775* Lemma for irrapx1 26781. Divides the unit interval into half-open sections and using the pigeonhole principle fphpdo 26768 finds two multiples of in the same section mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.)

Theoremirrapxlem2 26776* Lemma for irrapx1 26781. Two multiples in the same bucket means they are very close mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.)

Theoremirrapxlem3 26777* Lemma for irrapx1 26781. By subtraction, there is a multiple very close to an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremirrapxlem4 26778* Lemma for irrapx1 26781. Eliminate ranges, use positivity of the input to force positivity of the output by increasing as needed. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremirrapxlem5 26779* Lemma for irrapx1 26781. Switching to real intervals and fraction syntax. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremirrapxlem6 26780* Lemma for irrapx1 26781. Explicit description of a non-closed set. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremirrapx1 26781* 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.)
denom

19.16.24  Pell equations 1: A nontrivial solution always exists

Theorempellexlem1 26782 Lemma for pellex 26788. 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.)

Theorempellexlem2 26783 Lemma for pellex 26788. Arithmetical core of pellexlem3, norm upper bound. (Contributed by Stefan O'Rear, 14-Sep-2014.)

Theorempellexlem3 26784* Lemma for pellex 26788. To each good rational approximation of , there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
denom

Theorempellexlem4 26785* Lemma for pellex 26788. Invoking irrapx1 26781, we have infinitely many near-solutions. (Contributed by Stefan O'Rear, 14-Sep-2014.)

Theorempellexlem5 26786* Lemma for pellex 26788. Invoking fiphp3d 26770, we have infinitely many near-solutions for some specific norm. (Contributed by Stefan O'Rear, 19-Oct-2014.)

Theorempellexlem6 26787* Lemma for pellex 26788. 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.)

Theorempellex 26788* Every Pell equation has a nontrivial solution. Theorem 62 in [vandenDries] p. 43. (Contributed by Stefan O'Rear, 19-Oct-2014.)

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

Syntaxcsquarenn 26789 Extend class notation to include the set of square natural numbers.
NN

Syntaxcpell1qr 26790 Extend class notation to include the class of quadrant-1 Pell solutions.
Pell1QR

Syntaxcpell1234qr 26791 Extend class notation to include the class of any-quadrant Pell solutions.
Pell1234QR

Syntaxcpell14qr 26792 Extend class notation to include the class of positive Pell solutions.
Pell14QR

Syntaxcpellfund 26793 Extend class notation to include the Pell-equation fundamental solution function.
PellFund

Definitiondf-squarenn 26794 Define the set of square natural numbers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN

Definitiondf-pell1qr 26795* 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 NN

Definitiondf-pell14qr 26796* Define the positive solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Pell14QR NN

Definitiondf-pell1234qr 26797* Define the general solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Pell1234QR NN

Definitiondf-pellfund 26798* A function mapping Pell discriminants to the corresponding fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
PellFund NN Pell14QR

Theorempell1qrval 26799* Value of the set of first-quadrant Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1QR

Theoremelpell1qr 26800* Membership in a first-quadrant Pell solution set. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1QR

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