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Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempell1234qrval 26101* Value of the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1234QR

Theoremelpell1234qr 26102* Membership in the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1234QR

Theorempell1234qrre 26103 General Pell solutions are (coded as) real numbers. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1234QR

Theorempell1234qrne0 26104 No solution to a Pell equation is zero. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1234QR

Theorempell1234qrreccl 26105 General solutions of the Pell equation are closed under reciprocals. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell1234QR Pell1234QR

Theorempell1234qrmulcl 26106 General solutions of the Pell equation are closed under multiplication. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell1234QR Pell1234QR Pell1234QR

Theorempell14qrss1234 26107 A positive Pell solution is a general Pell solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR Pell1234QR

Theorempell14qrre 26108 A positive Pell solution is a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR

Theorempell14qrne0 26109 A positive Pell solution is a nonzero number. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell14QR

Theorempell14qrgt0 26110 A positive Pell solution is a positive number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR

Theorempell14qrrp 26111 A positive Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN Pell14QR

Theorempell1234qrdich 26112 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 26113 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 26114 Positive Pell solutions are closed under multiplication. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell14QR Pell14QR Pell14QR

Theorempell14qrreccl 26115 Positive Pell solutions are closed under reciprocal. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR Pell14QR

Theorempell14qrdivcl 26116 Positive Pell solutions are closed under division. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR Pell14QR Pell14QR

Theorempell14qrexpclnn0 26117 Lemma for pell14qrexpcl 26118. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR Pell14QR

Theorempell14qrexpcl 26118 Positive Pell solutions are closed under integer powers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR Pell14QR

Theorempell1qrss14 26119 First-quadrant Pell solutions are a subset of the positive solutions. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell1QR Pell14QR

Theorempell14qrdich 26120 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 26121 A Pell solution in the first quadrant is at least 1. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1QR

Theorempell1qr1 26122 1 is a Pell solution and in the first quadrant as one. (Contributed by Stefan O'Rear, 17-Sep-2014.)
NN Pell1QR

Theoremelpell1qr2 26123 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 26124 Lemma for pell1qrgap 26125. (Contributed by Stefan O'Rear, 18-Sep-2014.)

Theorempell1qrgap 26125 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 26126 Positive Pell solutions are bounded away from 1. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR

Theorempell14qrgapw 26127 Positive Pell solutions are bounded away from 1, with a friendlier bound. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR

Theorempellqrexplicit 26128 Condition for a calculated real to be a Pell solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN Pell1QR

16.15.26  Pell equations 3: characterizing fundamental solution

Theoreminfmrgelbi 26129* 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 26130* There is a nontrivial solution of a Pell equation in the first quadrant. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell1QR

Theorempellfundval 26131* Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN PellFund Pell14QR

Theorempellfundre 26132 The fundamental solution of a Pell equation exists as a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN PellFund

Theorempellfundge 26133 Lower bound on the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN PellFund

Theorempellfundgt1 26134 Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN PellFund

Theorempellfundlb 26135 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 26136* 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 26137 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 26127. (Contributed by Stefan O'Rear, 18-Sep-2014.)

NN PellFund Pell1QR

Theorempellfund14gap 26138 There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR PellFund

Theorempellfundrp 26139 The fundamental Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN PellFund

Theorempellfundne1 26140 The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN PellFund

16.15.27  Logarithm laws generalized to an arbitrary base

Theoremreglogcl 26141 General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogltb 26142 General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogleb 26143 General logarithm preserves . (Contributed by Stefan O'Rear, 19-Oct-2014.)

Theoremreglogmul 26144 Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogexp 26145 Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogbas 26146 General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglog1 26147 General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogexpbas 26148 General log of a power of the base is the exponent. (Contributed by Stefan O'Rear, 19-Sep-2014.)

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

Theorempellfund14 26149* Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN Pell14QR PellFund

Theorempellfund14b 26150* 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

16.15.29  X and Y sequences 1: Definition and recurrence laws

Syntaxcrmx 26151 Extend class notation to include the Robertson-Matiyasevich X sequence.
Xrm

Syntaxcrmy 26152 Extend class notation to include the Robertson-Matiyasevich Y sequence.
Yrm

Definitiondf-rmx 26153* Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 26164 and rmxyval 26166 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Xrm

Definitiondf-rmy 26154* Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 26165 and rmxyval 26166 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Yrm

Theoremrmxfval 26155* Value of the X sequence. Not used after rmxyval 26166 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Xrm

Theoremrmyfval 26156* Value of the Y sequence. Not used after rmxyval 26166 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Yrm

Theoremrmspecsqrnq 26157 The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.)

Theoremrmspecnonsq 26158 The discriminant used to define the X and Y sequences is a nonsquare positive integer and thus a valid Pell equation discriminant. (Contributed by Stefan O'Rear, 21-Sep-2014.)
NN

Theoremqirropth 26159 This lemma implements the concept of "equate rational and irrational parts", used to prove many arithmetical properties of the X and Y sequences. (Contributed by Stefan O'Rear, 21-Sep-2014.)

Theoremrmspecfund 26160 The base of exponent used to define the X and Y sequences is the fundamental solution of the corresponding Pell equation. (Contributed by Stefan O'Rear, 21-Sep-2014.)
PellFund

Theoremrmxyelqirr 26161* The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.)

Theoremrmxypairf1o 26162* The function used to extract rational and irrational parts in df-rmx 26153 and df-rmy 26154 in fact achieves a one-to-one mapping from the quadratic irrationals to pairs of integers. (Contributed by Stefan O'Rear, 21-Sep-2014.)

Theoremrmxyelxp 26163* Lemma for frmx 26164 and frmy 26165. (Contributed by Stefan O'Rear, 22-Sep-2014.)

Theoremfrmx 26164 The X sequence is a nonnegative integer. See rmxnn 26204 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm

Theoremfrmy 26165 The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Yrm

Theoremrmxyval 26166 Main definition of the X and Y sequences. Compare definition 2.3 of [JonesMatijasevic] p. 694. (Contributed by Stefan O'Rear, 19-Oct-2014.)
Xrm Yrm

Theoremrmspecpos 26167 The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)

Theoremrmxycomplete 26168* The X and Y sequences taken together enumerate all solutions to the corresponding Pell equation in the right half-plane. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Yrm

Theoremrmxynorm 26169 The X and Y sequences define a solution to the corresponding Pell equation. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Yrm

Theoremrmbaserp 26170 The base of exponentiation for the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)

Theoremrmxyneg 26171 Negation law for X and Y sequences. JonesMatijasevic is inconsistent on whether the X and Y sequences have domain or ; we use consistently to avoid the need for a separate subtraction law. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Xrm Yrm Yrm

Theoremrmxyadd 26172 Addition formula for X and Y sequences. See rmxadd 26178 and rmyadd 26182 for most uses. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Xrm Xrm Yrm Yrm Yrm Yrm Xrm Xrm Yrm

Theoremrmxy1 26173 Value of the X and Y sequences at 1. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Yrm

Theoremrmxy0 26174 Value of the X and Y sequences at 0. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Yrm

Theoremrmxneg 26175 Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 26171, rmxyadd 26172, rmxy0 26174, and rmxy1 26173 via qirropth 26159 results in two theorems at once, but typical use requires only one, so this group of theorems serves to separate the cases. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Xrm

Theoremrmx0 26176 Value of X sequence at 0. Part 1 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm

Theoremrmx1 26177 Value of X sequence at 1. Part 2 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm

Theoremrmxadd 26178 Addition formula for X sequence. Equation 2.7 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Xrm Xrm Yrm Yrm

Theoremrmyneg 26179 Negation formula for Y sequence (odd function). (Contributed by Stefan O'Rear, 22-Sep-2014.)
Yrm Yrm

Theoremrmy0 26180 Value of Y sequence at 0. Part 1 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Yrm

Theoremrmy1 26181 Value of Y sequence at 1. Part 2 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Yrm

Theoremrmyadd 26182 Addition formula for Y sequence. Equation 2.8 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Yrm Yrm Xrm Xrm Yrm

Theoremrmxp1 26183 Special addition-of-1 formula for X sequence. Part 1 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.)
Xrm Xrm Yrm

Theoremrmyp1 26184 Special addition of 1 formula for Y sequence. Part 2 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 24-Sep-2014.)
Yrm Yrm Xrm

Theoremrmxm1 26185 Subtraction of 1 formula for X sequence. Part 1 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.)
Xrm Xrm Yrm

Theoremrmym1 26186 Subtraction of 1 formula for Y sequence. Part 2 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.)
Yrm Yrm Xrm

Theoremrmxluc 26187 The X sequence is a Lucas (second-order integer recurrence) sequence. Part 3 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.)
Xrm Xrm Xrm

Theoremrmyluc 26188 The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 26180 and rmy1 26181. Part 3 of equation 2.12 of [JonesMatijasevic] p. 695. JonesMatijasevic uses this theorem to redefine the X and Y sequences to have domain , which simplifies some later theorems. It may shorten the derivation to use this as our initial definition. Incidentally, the X sequence satisfies the exact same recurrence. (Contributed by Stefan O'Rear, 1-Oct-2014.)
Yrm Yrm Yrm

Theoremrmyluc2 26189 Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm Yrm Yrm

Theoremrmxdbl 26190 "Double-angle formula" for X-values. Equation 2.13 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Xrm Xrm

Theoremrmydbl 26191 "Double-angle formula" for Y-values. Equation 2.14 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Yrm Xrm Yrm

16.15.30  Ordering and induction lemmas for the integers

Theoremmonotuz 26192* A function defined on a set of upper integers which increases at every adjacent pair is globally strictly monotonic by induction. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremmonotoddzzfi 26193* A function which is odd and monotonic on is monotonic on . This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.)

Theoremmonotoddzz 26194* A function (given implicitly) which is odd and monotonic on is monotonic on . This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.)

Theoremoddcomabszz 26195* An odd function which takes nonnegative values on nonnegative arguments commutes with . (Contributed by Stefan O'Rear, 26-Sep-2014.)

Theorem2nn0ind 26196* Induction on natural numbers with two base cases, for use with Lucas-type sequences. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremzindbi 26197* Inductively transfer a property to the integers if it holds for zero and passes between adjacent integers in either direction. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremexpmordi 26198 Mantissa ordering relationship for exponentiation. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremrpexpmord 26199 Mantissa ordering relationship for exponentiation of positive reals. (Contributed by Stefan O'Rear, 16-Oct-2014.)

16.15.31  X and Y sequences 2: Order properties

Theoremrmxypos 26200 For all nonnegative indices, X is positive and Y is nonnegative. (Contributed by Stefan O'Rear, 24-Sep-2014.)
Xrm Yrm

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