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Theorem List for Metamath Proof Explorer - 26801-26900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempellfundex 26801 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 26791. (Contributed by Stefan O'Rear, 18-Sep-2014.)

NN PellFund Pell1QR

Theorempellfund14gap 26802 There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN Pell14QR PellFund

Theorempellfundrp 26803 The fundamental Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN PellFund

Theorempellfundne1 26804 The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN PellFund

19.16.27  Logarithm laws generalized to an arbitrary base

Theoremreglogcl 26805 General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogltb 26806 General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogleb 26807 General logarithm preserves . (Contributed by Stefan O'Rear, 19-Oct-2014.)

Theoremreglogmul 26808 Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogexp 26809 Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogbas 26810 General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglog1 26811 General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremreglogexpbas 26812 General log of a power of the base is the exponent. (Contributed by Stefan O'Rear, 19-Sep-2014.)

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

Theorempellfund14 26813* Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
NN Pell14QR PellFund

Theorempellfund14b 26814* 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

19.16.29  X and Y sequences 1: Definition and recurrence laws

Syntaxcrmx 26815 Extend class notation to include the Robertson-Matiyasevich X sequence.
Xrm

Syntaxcrmy 26816 Extend class notation to include the Robertson-Matiyasevich Y sequence.
Yrm

Definitiondf-rmx 26817* Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 26828 and rmxyval 26830 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Xrm

Definitiondf-rmy 26818* Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 26829 and rmxyval 26830 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Yrm

Theoremrmxfval 26819* Value of the X sequence. Not used after rmxyval 26830 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Xrm

Theoremrmyfval 26820* Value of the Y sequence. Not used after rmxyval 26830 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Yrm

Theoremrmspecsqrnq 26821 The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.)

Theoremrmspecnonsq 26822 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 26823 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 26824 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 26825* The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.)

Theoremrmxypairf1o 26826* The function used to extract rational and irrational parts in df-rmx 26817 and df-rmy 26818 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 26827* Lemma for frmx 26828 and frmy 26829. (Contributed by Stefan O'Rear, 22-Sep-2014.)

Theoremfrmx 26828 The X sequence is a nonnegative integer. See rmxnn 26868 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm

Theoremfrmy 26829 The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Yrm

Theoremrmxyval 26830 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 26831 The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)

Theoremrmxycomplete 26832* 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 26833 The X and Y sequences define a solution to the corresponding Pell equation. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Yrm

Theoremrmbaserp 26834 The base of exponentiation for the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)

Theoremrmxyneg 26835 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 26836 Addition formula for X and Y sequences. See rmxadd 26842 and rmyadd 26846 for most uses. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Xrm Xrm Yrm Yrm Yrm Yrm Xrm Xrm Yrm

Theoremrmxy1 26837 Value of the X and Y sequences at 1. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Yrm

Theoremrmxy0 26838 Value of the X and Y sequences at 0. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Yrm

Theoremrmxneg 26839 Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 26835, rmxyadd 26836, rmxy0 26838, and rmxy1 26837 via qirropth 26823 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 26840 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 26841 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 26842 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 26843 Negation formula for Y sequence (odd function). (Contributed by Stefan O'Rear, 22-Sep-2014.)
Yrm Yrm

Theoremrmy0 26844 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 26845 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 26846 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 26847 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 26848 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 26849 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 26850 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 26851 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 26852 The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 26844 and rmy1 26845. 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 26853 Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm Yrm Yrm

Theoremrmxdbl 26854 "Double-angle formula" for X-values. Equation 2.13 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Xrm Xrm

Theoremrmydbl 26855 "Double-angle formula" for Y-values. Equation 2.14 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Yrm Xrm Yrm

19.16.30  Ordering and induction lemmas for the integers

Theoremmonotuz 26856* 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 26857* 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 26858* 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 26859* An odd function which takes nonnegative values on nonnegative arguments commutes with . (Contributed by Stefan O'Rear, 26-Sep-2014.)

Theorem2nn0ind 26860* Induction on natural numbers with two base cases, for use with Lucas-type sequences. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremzindbi 26861* 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 26862 Mantissa ordering relationship for exponentiation. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremrpexpmord 26863 Mantissa ordering relationship for exponentiation of positive reals. (Contributed by Stefan O'Rear, 16-Oct-2014.)

19.16.31  X and Y sequences 2: Order properties

Theoremrmxypos 26864 For all nonnegative indices, X is positive and Y is nonnegative. (Contributed by Stefan O'Rear, 24-Sep-2014.)
Xrm Yrm

Theoremltrmynn0 26865 The Y-sequence is strictly monotonic on . Strengthened by ltrmy 26869. (Contributed by Stefan O'Rear, 24-Sep-2014.)
Yrm Yrm

Theoremltrmxnn0 26866 The X-sequence is strictly monotonic on . (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm Xrm

Theoremlermxnn0 26867 The X-sequence is monotonic on . (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm Xrm

Theoremrmxnn 26868 The X-sequence is defined to range over but never actually takes the value 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm

Theoremltrmy 26869 The Y-sequence is strictly monotonic over . (Contributed by Stefan O'Rear, 25-Sep-2014.)
Yrm Yrm

Theoremrmyeq0 26870 Y is zero only at zero. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm

Theoremrmyeq 26871 Y is one-to-one. (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm

Theoremlermy 26872 Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm

Theoremrmynn 26873 Yrm is positive for positive arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremrmynn0 26874 Yrm is nonnegative for nonnegative arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremrmyabs 26875 Yrm commutes with . (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm

Theoremjm2.24nn 26876 X(n) is strictly greater than Y(n) + Y(n-1). Lemma 2.24 of [JonesMatijasevic] p. 697 restricted to . (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm Xrm

Theoremjm2.17a 26877 First half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 14-Oct-2014.)
Yrm

Theoremjm2.17b 26878 Weak form of the second half of lemma 2.17 of [JonesMatijasevic] p. 696, allowing induction to start lower. (Contributed by Stefan O'Rear, 15-Oct-2014.)
Yrm

Theoremjm2.17c 26879 Second half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 15-Oct-2014.)
Yrm

Theoremjm2.24 26880 Lemma 2.24 of [JonesMatijasevic] p. 697 extended to . Could be eliminated with a more careful proof of jm2.26lem3 26924. (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm Xrm

Theoremrmygeid 26881 Y(n) increases faster than n. Used implicitly without proof or comment in lemma 2.27 of [JonesMatijasevic] p. 697. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm

19.16.32  Congruential equations

Theoremcongtr 26882 A wff of the form is interpreted as a congruential equation. This is similar to , but is defined such that behavior is regular for zero and negative values of . To use this concept effectively, we need to show that congruential equations behave similarly to normal equations; first a transitivity law. Idea for the future: If there was a congruential equation symbol, it could incorporate type constraints, so that most of these would not need them. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongadd 26883 If two pairs of numbers are componentwise congruent, so are their sums. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongmul 26884 If two pairs of numbers are componentwise congruent, so are their products. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongsym 26885 Congruence mod is a symmetric/commutative relation. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongneg 26886 If two integers are congruent mod , so are their negatives. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongsub 26887 If two pairs of numbers are componentwise congruent, so are their differences. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremcongid 26888 Every integer is congruent to itself mod every base. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremmzpcong 26889* Polynomials commute with congruences. (Does this characterize them?) (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly

Theoremcongrep 26890* Every integer is congruent to some number in the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremcongabseq 26891 If two integers are congruent, they are either equal or separated by at least the congruence base. (Contributed by Stefan O'Rear, 4-Oct-2014.)

19.16.33  Alternating congruential equations

Theoremacongid 26892 A wff like that in this theorem will be known as an "alternating congruence". A special symbol might be considered if more uses come up. They have many of the same properties as normal congruences, starting with reflexivity.

JonesMatijasevic uses "a ≡ ± b (mod c)" for this construction. The disjunction of divisibility constraints seems to adequately capture the concept, but it's rather verbose and somewhat inelegant. Use of an explicit equivalence relation might also work. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremacongsym 26893 Symmetry of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremacongneg2 26894 Negate right side of alternating congruence. Makes essential use of the "alternating" part. (Contributed by Stefan O'Rear, 3-Oct-2014.)

Theoremacongtr 26895 Transitivity of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremacongeq12d 26896 Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.)

Theoremacongrep 26897* Every integer is alternating-congruent to some number in the first half of the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremfzmaxdif 26898 Bound on the difference between two integers constrained to two possibly overlapping finite ranges. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Theoremfzneg 26899 Reflection of a finite range of integers about 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Theoremacongeq 26900 Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 26925 (Contributed by Stefan O'Rear, 4-Oct-2014.)

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