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Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltrabdioph 26301* Diophantine set builder for the strict less than relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly mzPoly Dioph

Theoremnerabdioph 26302* 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 26303* Divisibility is a Diophantine relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly mzPoly Dioph

18.17.18  Diophantine sets 6 miscellanea

Theoremfz1ssnn 26304 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremftp 26305 A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.)

18.17.19  Diophantine sets 6: reusability. renumbering of variables

Theoremeldioph4b 26306* 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 26307* Forward-only version of eldioph4b 26306. (Contributed by Stefan O'Rear, 16-Oct-2014.)
mzPoly Dioph

Theoremdiophren 26308* 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 26309* Change variable numbers in a Diophantine class abstraction using explicit substitution. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph Dioph

Theoremrabren3dioph 26310* Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph Dioph

18.17.20  Pigeonhole Principle and cardinality helpers

Theoremfphpd 26311* 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 26312* Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.)

Theoremctbnfien 26313 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 26314* Infinite pigeonhole principle for partitioning an infinite set between finitely many buckets. (Contributed by Stefan O'Rear, 18-Oct-2014.)

18.17.21  A non-closed set of reals is infinite

Theoremrencldnfilem 26315* Lemma for rencldnfi 26316. (Contributed by Stefan O'Rear, 18-Oct-2014.)

Theoremrencldnfi 26316* A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 26315 using infima; this theorem removes the requirement that A be non-empty. (Contributed by Stefan O'Rear, 19-Oct-2014.)

18.17.22  Miscellanea for Lagrange's theorem

Theoremicodiamlt 26317 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 26318 Modular reduction produces a half-open interval. (Contributed by Stefan O'Rear, 12-Sep-2014.)

18.17.23  Lagrange's rational approximation theorem

Theoremirrapxlem1 26319* Lemma for irrapx1 26325. Divides the unit interval into half-open sections and using the pigeonhole principle fphpdo 26312 finds two multiples of in the same section mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.)

Theoremirrapxlem2 26320* Lemma for irrapx1 26325. Two multiples in the same bucket means they are very close mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.)

Theoremirrapxlem3 26321* Lemma for irrapx1 26325. By subtraction, there is a multiple very close to an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremirrapxlem4 26322* Lemma for irrapx1 26325. 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 26323* Lemma for irrapx1 26325. Switching to real intervals and fraction syntax. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremirrapxlem6 26324* Lemma for irrapx1 26325. Explicit description of a non-closed set. (Contributed by Stefan O'Rear, 13-Sep-2014.)
denom

Theoremirrapx1 26325* 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

18.17.24  Pell equations 1: A nontrivial solution always exists

Theorempellexlem1 26326 Lemma for pellex 26332. 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 26327 Lemma for pellex 26332. Arithmetical core of pellexlem3, norm upper bound. (Contributed by Stefan O'Rear, 14-Sep-2014.)

Theorempellexlem3 26328* Lemma for pellex 26332. To each good rational approximation of , there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
denom

Theorempellexlem4 26329* Lemma for pellex 26332. Invoking irrapx1 26325, we have infinitely many near-solutions. (Contributed by Stefan O'Rear, 14-Sep-2014.)

Theorempellexlem5 26330* Lemma for pellex 26332. Invoking fiphp3d 26314, we have infinitely many near-solutions for some specific norm. (Contributed by Stefan O'Rear, 19-Oct-2014.)

Theorempellexlem6 26331* Lemma for pellex 26332. 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 26332* Every Pell equation has a nontrivial solution. Theorem 62 in [vandenDries] p. 43. (Contributed by Stefan O'Rear, 19-Oct-2014.)

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

Syntaxcsquarenn 26333 Extend class notation to include the set of square natural numbers.
NN

Syntaxcpell1qr 26334 Extend class notation to include the class of quadrant-1 Pell solutions.
Pell1QR

Syntaxcpell1234qr 26335 Extend class notation to include the class of any-quadrant Pell solutions.
Pell1234QR

Syntaxcpell14qr 26336 Extend class notation to include the class of positive Pell solutions.
Pell14QR

Syntaxcpellfund 26337 Extend class notation to include the Pell-equation fundamental solution function.
PellFund

Definitiondf-squarenn 26338 Define the set of square natural numbers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN

Definitiondf-pell1qr 26339* 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 26340* Define the positive solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Pell14QR NN

Definitiondf-pell1234qr 26341* Define the general solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Pell1234QR NN

Definitiondf-pellfund 26342* A function mapping Pell discriminants to the corresponding fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
PellFund NN Pell14QR

Theorempell1qrval 26343* Value of the set of first-quadrant Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1QR

Theoremelpell1qr 26344* Membership in a first-quadrant Pell solution set. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1QR

Theorempell14qrval 26345* Value of the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell14QR

Theoremelpell14qr 26346* Membership in the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell14QR

Theorempell1234qrval 26347* Value of the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1234QR

Theoremelpell1234qr 26348* Membership in the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1234QR

Theorempell1234qrre 26349 General Pell solutions are (coded as) real numbers. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1234QR

Theorempell1234qrne0 26350 No solution to a Pell equation is zero. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1234QR

Theorempell1234qrreccl 26351 General solutions of the Pell equation are closed under reciprocals. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell1234QR Pell1234QR

Theorempell1234qrmulcl 26352 General solutions of the Pell equation are closed under multiplication. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell1234QR Pell1234QR Pell1234QR

Theorempell14qrss1234 26353 A positive Pell solution is a general Pell solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR Pell1234QR

Theorempell14qrre 26354 A positive Pell solution is a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR

Theorempell14qrne0 26355 A positive Pell solution is a nonzero number. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell14QR

Theorempell14qrgt0 26356 A positive Pell solution is a positive number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR

Theorempell14qrrp 26357 A positive Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN Pell14QR

Theorempell1234qrdich 26358 A general Pell solution is either a positive solution, or its negation is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell1234QR Pell14QR Pell14QR

Theoremelpell14qr2 26359 A number is a positive Pell solution iff it is positive and a Pell solution, justifying our name choice. (Contributed by Stefan O'Rear, 19-Oct-2014.)
NN Pell14QR Pell1234QR

Theorempell14qrmulcl 26360 Positive Pell solutions are closed under multiplication. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell14QR Pell14QR Pell14QR

Theorempell14qrreccl 26361 Positive Pell solutions are closed under reciprocal. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR Pell14QR

Theorempell14qrdivcl 26362 Positive Pell solutions are closed under division. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR Pell14QR Pell14QR

Theorempell14qrexpclnn0 26363 Lemma for pell14qrexpcl 26364. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR Pell14QR

Theorempell14qrexpcl 26364 Positive Pell solutions are closed under integer powers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR Pell14QR

Theorempell1qrss14 26365 First-quadrant Pell solutions are a subset of the positive solutions. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell1QR Pell14QR

Theorempell14qrdich 26366 A positive Pell solution is either in the first quadrant, or its reciprocal is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR Pell1QR Pell1QR

Theorempell1qrge1 26367 A Pell solution in the first quadrant is at least 1. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1QR

Theorempell1qr1 26368 1 is a Pell solution and in the first quadrant as one. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1QR

Theoremelpell1qr2 26369 The first quadrant solutions are precisely the positive Pell solutions which are at least one. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell1QR Pell14QR

Theorempell1qrgaplem 26370 Lemma for pell1qrgap 26371. (Contributed by Stefan O'Rear, 18-Sep-2014.)

Theorempell1qrgap 26371 First-quadrant Pell solutions are bounded away from 1. (This particular bound allows us to prove exact values for the fundamental solution later.) (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell1QR

Theorempell14qrgap 26372 Positive Pell solutions are bounded away from 1. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR

Theorempell14qrgapw 26373 Positive Pell solutions are bounded away from 1, with a friendlier bound. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR

Theorempellqrexplicit 26374 Condition for a calculated real to be a Pell solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN Pell1QR

18.17.26  Pell equations 3: characterizing fundamental solution

Theoreminfmrgelbi 26375* Any lower bound of a nonempty set of real numbers is less than or equal to its infimum, one-direction version. (Contributed by Stefan O'Rear, 1-Sep-2013.)

Theorempellqrex 26376* There is a nontrivial solution of a Pell equation in the first quadrant. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell1QR

Theorempellfundval 26377* Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN PellFund Pell14QR

Theorempellfundre 26378 The fundamental solution of a Pell equation exists as a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN PellFund

Theorempellfundge 26379 Lower bound on the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN PellFund

Theorempellfundgt1 26380 Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN PellFund

Theorempellfundlb 26381 A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN Pell14QR PellFund

Theorempellfundglb 26382* If a real is larger than the fundamental solution, there is a nontrivial solution less than it. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN PellFund Pell1QRPellFund

Theorempellfundex 26383 The fundamental solution as an infimum is itself a solution, showing that the solution set is discrete.

Since the fundamental solution is an infimum, there must be an element ge to Fund and lt 2*Fund. If this element is equal to the fundamental solution we're done, otherwise use the infimum again to find another element which must be ge Fund and lt the first element; their ratio is a group element in (1,2), contradicting pell14qrgapw 26373. (Contributed by Stefan O'Rear, 18-Sep-2014.)

NN PellFund Pell1QR

Theorempellfund14gap 26384 There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR PellFund

Theorempellfundrp 26385 The fundamental Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN PellFund

Theorempellfundne1 26386 The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN PellFund

18.17.27  Logarithm laws generalized to an arbitrary base

Theoremreglogcl 26387 General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogltb 26388 General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogleb 26389 General logarithm preserves . (Contributed by Stefan O'Rear, 19-Oct-2014.)

Theoremreglogmul 26390 Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogexp 26391 Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogbas 26392 General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglog1 26393 General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogexpbas 26394 General log of a power of the base is the exponent. (Contributed by Stefan O'Rear, 19-Sep-2014.)

18.17.28  Pell equations 4: the positive solution group is infinite cyclic

Theorempellfund14 26395* Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN Pell14QR PellFund

Theorempellfund14b 26396* The positive Pell solutions are precisely the integer powers of the fundamental solution. To get the general solution set (which we will not be using), throw in a copy of Z/2Z. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN Pell14QR PellFund

18.17.29  X and Y sequences 1: Definition and recurrence laws

Syntaxcrmx 26397 Extend class notation to include the Robertson-Matiyasevich X sequence.
Xrm

Syntaxcrmy 26398 Extend class notation to include the Robertson-Matiyasevich Y sequence.
Yrm

Definitiondf-rmx 26399* Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 26410 and rmxyval 26412 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Xrm

Definitiondf-rmy 26400* Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 26411 and rmxyval 26412 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Yrm

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