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Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrmspecpos 27001 The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)

Theoremrmxycomplete 27002* 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 27003 The X and Y sequences define a solution to the corresponding Pell equation. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Yrm

Theoremrmbaserp 27004 The base of exponentiation for the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)

Theoremrmxyneg 27005 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 27006 Addition formula for X and Y sequences. See rmxadd 27012 and rmyadd 27016 for most uses. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Xrm Xrm Yrm Yrm Yrm Yrm Xrm Xrm Yrm

Theoremrmxy1 27007 Value of the X and Y sequences at 1. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Yrm

Theoremrmxy0 27008 Value of the X and Y sequences at 0. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm Yrm

Theoremrmxneg 27009 Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 27005, rmxyadd 27006, rmxy0 27008, and rmxy1 27007 via qirropth 26993 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 27010 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 27011 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 27012 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 27013 Negation formula for Y sequence (odd function). (Contributed by Stefan O'Rear, 22-Sep-2014.)
Yrm Yrm

Theoremrmy0 27014 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 27015 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 27016 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 27017 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 27018 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 27019 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 27020 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 27021 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 27022 The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 27014 and rmy1 27015. 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 27023 Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm Yrm Yrm

Theoremrmxdbl 27024 "Double-angle formula" for X-values. Equation 2.13 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Xrm Xrm

Theoremrmydbl 27025 "Double-angle formula" for Y-values. Equation 2.14 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Yrm Xrm Yrm

18.17.30  Ordering and induction lemmas for the integers

Theoremmonotuz 27026* 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 27027* 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 27028* 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 27029* An odd function which takes nonnegative values on nonnegative arguments commutes with . (Contributed by Stefan O'Rear, 26-Sep-2014.)

Theorem2nn0ind 27030* Induction on natural numbers with two base cases, for use with Lucas-type sequences. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremzindbi 27031* 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 27032 Mantissa ordering relationship for exponentiation. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremrpexpmord 27033 Mantissa ordering relationship for exponentiation of positive reals. (Contributed by Stefan O'Rear, 16-Oct-2014.)

18.17.31  X and Y sequences 2: Order properties

Theoremrmxypos 27034 For all nonnegative indices, X is positive and Y is nonnegative. (Contributed by Stefan O'Rear, 24-Sep-2014.)
Xrm Yrm

Theoremltrmynn0 27035 The Y-sequence is strictly monotonic on . Strengthened by ltrmy 27039. (Contributed by Stefan O'Rear, 24-Sep-2014.)
Yrm Yrm

Theoremltrmxnn0 27036 The X-sequence is strictly monotonic on . (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm Xrm

Theoremlermxnn0 27037 The X-sequence is monotonic on . (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm Xrm

Theoremrmxnn 27038 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 27039 The Y-sequence is strictly monotonic over . (Contributed by Stefan O'Rear, 25-Sep-2014.)
Yrm Yrm

Theoremrmyeq0 27040 Y is zero only at zero. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm

Theoremrmyeq 27041 Y is one-to-one. (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm

Theoremlermy 27042 Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm

Theoremrmynn 27043 Yrm is positive for positive arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremrmynn0 27044 Yrm is nonnegative for nonnegative arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremrmyabs 27045 Yrm commutes with . (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm

Theoremjm2.24nn 27046 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 27047 First half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 14-Oct-2014.)
Yrm

Theoremjm2.17b 27048 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 27049 Second half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 15-Oct-2014.)
Yrm

Theoremjm2.24 27050 Lemma 2.24 of [JonesMatijasevic] p. 697 extended to . Could be eliminated with a more careful proof of jm2.26lem3 27094. (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm Xrm

Theoremrmygeid 27051 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

18.17.32  Congruential equations

Theoremcongtr 27052 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 27053 If two pairs of numbers are componentwise congruent, so are their sums. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongmul 27054 If two pairs of numbers are componentwise congruent, so are their products. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongsym 27055 Congruence mod is a symmetric/commutative relation. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongneg 27056 If two integers are congruent mod , so are their negatives. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongsub 27057 If two pairs of numbers are componentwise congruent, so are their differences. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremcongid 27058 Every integer is congruent to itself mod every base. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremmzpcong 27059* Polynomials commute with congruences. (Does this characterize them?) (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly

Theoremcongrep 27060* Every integer is congruent to some number in the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremcongabseq 27061 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.)

18.17.33  Alternating congruential equations

Theoremacongid 27062 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 27063 Symmetry of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremacongneg2 27064 Negate right side of alternating congruence. Makes essential use of the "alternating" part. (Contributed by Stefan O'Rear, 3-Oct-2014.)

Theoremacongtr 27065 Transitivity of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremacongeq12d 27066 Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.)

Theoremacongrep 27067* 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 27068 Bound on the difference between two integers constrained to two possibly overlapping finite ranges. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Theoremfzneg 27069 Reflection of a finite range of integers about 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Theoremacongeq 27070 Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 27095 (Contributed by Stefan O'Rear, 4-Oct-2014.)

Theoremdvdsacongtr 27071 Alternating congruence passes from a base to a dividing base. (Contributed by Stefan O'Rear, 4-Oct-2014.)

18.17.34  Additional theorems on integer divisibility

Theorembezoutr 27072 Partial converse to bezout 12721. Existence of a linear combination does not set the GCD, but it does upper bound it. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theorembezoutr1 27073 Converse of bezout 12721 for the gcd = 1 case, sufficient condition for relative primality. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremcoprmdvdsb 27074 Multiplication by a coprime number does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremzabscl 27075 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremnn0sqcl 27076 The square of a natural number is a natural number. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremdvdsleabs2 27077 Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremmodabsdifz 27078 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 26-Sep-2014.)

Theoremdvdsabsmod0 27079 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremdivalgmodcl 27080 divalgmod 12605 using a class variable. (Contributed by Stefan O'Rear, 17-Oct-2014.)

18.17.35  X and Y sequences 3: Divisibility properties

Theoremjm2.18 27081 Theorem 2.18 of [JonesMatijasevic] p. 696. Direct relationship of the exponential function to X and Y sequences. (Contributed by Stefan O'Rear, 14-Oct-2014.)
Xrm Yrm

Theoremjm2.19lem1 27082 Lemma for jm2.19 27086. X and Y values are coprime. (Contributed by Stefan O'Rear, 23-Sep-2014.)
Xrm Yrm

Theoremjm2.19lem2 27083 Lemma for jm2.19 27086. (Contributed by Stefan O'Rear, 23-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19lem3 27084 Lemma for jm2.19 27086. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19lem4 27085 Lemma for jm2.19 27086. Extend to ZZ by symmetry. TODO: use zindbi 27031. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19 27086 Lemma 2.19 of [JonesMatijasevic] p. 696. Transfer divisibility constraints between Y-values and their indices. (Contributed by Stefan O'Rear, 24-Sep-2014.)
Yrm Yrm

Theoremjm2.21 27087 Lemma for jm2.20nn 27090. Express X and Y values as a binomial. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Xrm Yrm Xrm Yrm

Theoremjm2.22 27088* Lemma for jm2.20nn 27090. Applying binomial theorem and taking irrational part. (Contributed by Stefan O'Rear, 26-Sep-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Yrm Xrm Yrm

Theoremjm2.23 27089 Lemma for jm2.20nn 27090. Truncate binomial expansion p-adicly. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Xrm Yrm

Theoremjm2.20nn 27090 Lemma 2.20 of [JonesMatijasevic] p. 696, the "first step down lemma". (Contributed by Stefan O'Rear, 27-Sep-2014.)
Yrm Yrm Yrm

Theoremjm2.25lem1 27091 Lemma for jm2.26 27095. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremjm2.25 27092 Lemma for jm2.26 27095. Remainders mod X(2n) are negaperiodic mod 2n. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Xrm Yrm Yrm Xrm Yrm Yrm

Theoremjm2.26a 27093 Lemma for jm2.26 27095. Reverse direction is required to prove forward direction, so do it separatly. Induction on difference between K and M, together with the addition formula fact that adding 2N only inverts sign. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Xrm Yrm Yrm Xrm Yrm Yrm

Theoremjm2.26lem3 27094 Lemma for jm2.26 27095. Use acongrep 27067 to find K', M' ~ K, M in [ 0,N ]. thus Y(K') ~ Y(M') and both are small; K' = M' on pain of contradicting 2.24, so K ~ M (Contributed by Stefan O'Rear, 3-Oct-2014.)
Xrm Yrm Yrm Xrm Yrm Yrm

Theoremjm2.26 27095 Lemma 2.26 of [JonesMatijasevic] p. 697, the "second step down lemma". (Contributed by Stefan O'Rear, 2-Oct-2014.)
Xrm Yrm Yrm Xrm Yrm Yrm

Theoremjm2.15nn0 27096 Lemma 2.15 of [JonesMatijasevic] p. 695. Yrm is a polynomial for fixed N, so has the expected congruence property. (Contributed by Stefan O'Rear, 1-Oct-2014.)
Yrm Yrm

Theoremjm2.16nn0 27097 Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 27096 if Yrm is redefined as described in rmyluc 27022. (Contributed by Stefan O'Rear, 1-Oct-2014.)
Yrm

18.17.36  X and Y sequences 4: Diophantine representability of Y

Theoremjm2.27a 27098 Lemma for jm2.27 27101. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm        Yrm               Xrm        Yrm               Xrm        Yrm        Yrm

Theoremjm2.27b 27099 Lemma for jm2.27 27101. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm

Theoremjm2.27c 27100 Lemma for jm2.27 27101. Forward direction with substitutions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm        Xrm        Yrm        Yrm        Xrm               Yrm        Xrm

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