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

Theoremrmyluc2 27001 Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm Yrm Yrm

Theoremrmxdbl 27002 "Double-angle formula" for X-values. Equation 2.13 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Xrm Xrm

Theoremrmydbl 27003 "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 27004* 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 27005* 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 27006* 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 27007* An odd function which takes nonnegative values on nonnegative arguments commutes with . (Contributed by Stefan O'Rear, 26-Sep-2014.)

Theorem2nn0ind 27008* Induction on natural numbers with two base cases, for use with Lucas-type sequences. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremzindbi 27009* 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 27010 Mantissa ordering relationship for exponentiation. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremrpexpmord 27011 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 27012 For all nonnegative indices, X is positive and Y is nonnegative. (Contributed by Stefan O'Rear, 24-Sep-2014.)
Xrm Yrm

Theoremltrmynn0 27013 The Y-sequence is strictly monotonic on . Strengthened by ltrmy 27017. (Contributed by Stefan O'Rear, 24-Sep-2014.)
Yrm Yrm

Theoremltrmxnn0 27014 The X-sequence is strictly monotonic on . (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm Xrm

Theoremlermxnn0 27015 The X-sequence is monotonic on . (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm Xrm

Theoremrmxnn 27016 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 27017 The Y-sequence is strictly monotonic over . (Contributed by Stefan O'Rear, 25-Sep-2014.)
Yrm Yrm

Theoremrmyeq0 27018 Y is zero only at zero. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm

Theoremrmyeq 27019 Y is one-to-one. (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm

Theoremlermy 27020 Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm

Theoremrmynn 27021 Yrm is positive for positive arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremrmynn0 27022 Yrm is nonnegative for nonnegative arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremrmyabs 27023 Yrm commutes with . (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm

Theoremjm2.24nn 27024 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 27025 First half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 14-Oct-2014.)
Yrm

Theoremjm2.17b 27026 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 27027 Second half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 15-Oct-2014.)
Yrm

Theoremjm2.24 27028 Lemma 2.24 of [JonesMatijasevic] p. 697 extended to . Could be eliminated with a more careful proof of jm2.26lem3 27072. (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm Xrm

Theoremrmygeid 27029 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 27030 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 27031 If two pairs of numbers are componentwise congruent, so are their sums. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongmul 27032 If two pairs of numbers are componentwise congruent, so are their products. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongsym 27033 Congruence mod is a symmetric/commutative relation. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongneg 27034 If two integers are congruent mod , so are their negatives. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongsub 27035 If two pairs of numbers are componentwise congruent, so are their differences. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremcongid 27036 Every integer is congruent to itself mod every base. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremmzpcong 27037* Polynomials commute with congruences. (Does this characterize them?) (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly

Theoremcongrep 27038* Every integer is congruent to some number in the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremcongabseq 27039 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 27040 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 27041 Symmetry of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremacongneg2 27042 Negate right side of alternating congruence. Makes essential use of the "alternating" part. (Contributed by Stefan O'Rear, 3-Oct-2014.)

Theoremacongtr 27043 Transitivity of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremacongeq12d 27044 Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.)

Theoremacongrep 27045* 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 27046 Bound on the difference between two integers constrained to two possibly overlapping finite ranges. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Theoremfzneg 27047 Reflection of a finite range of integers about 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Theoremacongeq 27048 Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 27073 (Contributed by Stefan O'Rear, 4-Oct-2014.)

Theoremdvdsacongtr 27049 Alternating congruence passes from a base to a dividing base. (Contributed by Stefan O'Rear, 4-Oct-2014.)

19.16.34  Additional theorems on integer divisibility

Theorembezoutr 27050 Partial converse to bezout 13042. 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 27051 Converse of bezout 13042 for the gcd = 1 case, sufficient condition for relative primality. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremcoprmdvdsb 27052 Multiplication by a coprime number does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremzabscl 27053 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremnn0sqcl 27054 The square of a natural number is a natural number. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremdvdsleabs2 27055 Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremmodabsdifz 27056 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 26-Sep-2014.)

Theoremdvdsabsmod0 27057 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremdivalgmodcl 27058 divalgmod 12926 using a class variable. (Contributed by Stefan O'Rear, 17-Oct-2014.)

19.16.35  X and Y sequences 3: Divisibility properties

Theoremjm2.18 27059 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 27060 Lemma for jm2.19 27064. X and Y values are coprime. (Contributed by Stefan O'Rear, 23-Sep-2014.)
Xrm Yrm

Theoremjm2.19lem2 27061 Lemma for jm2.19 27064. (Contributed by Stefan O'Rear, 23-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19lem3 27062 Lemma for jm2.19 27064. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19lem4 27063 Lemma for jm2.19 27064. Extend to ZZ by symmetry. TODO: use zindbi 27009. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19 27064 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 27065 Lemma for jm2.20nn 27068. Express X and Y values as a binomial. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Xrm Yrm Xrm Yrm

Theoremjm2.22 27066* Lemma for jm2.20nn 27068. 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 27067 Lemma for jm2.20nn 27068. Truncate binomial expansion p-adicly. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Xrm Yrm

Theoremjm2.20nn 27068 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 27069 Lemma for jm2.26 27073. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremjm2.25 27070 Lemma for jm2.26 27073. Remainders mod X(2n) are negaperiodic mod 2n. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Xrm Yrm Yrm Xrm Yrm Yrm

Theoremjm2.26a 27071 Lemma for jm2.26 27073. 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 27072 Lemma for jm2.26 27073. Use acongrep 27045 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 27073 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 27074 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 27075 Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 27074 if Yrm is redefined as described in rmyluc 27000. (Contributed by Stefan O'Rear, 1-Oct-2014.)
Yrm

19.16.36  X and Y sequences 4: Diophantine representability of Y

Theoremjm2.27a 27076 Lemma for jm2.27 27079. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm        Yrm               Xrm        Yrm               Xrm        Yrm        Yrm

Theoremjm2.27b 27077 Lemma for jm2.27 27079. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm

Theoremjm2.27c 27078 Lemma for jm2.27 27079. Forward direction with substitutions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm        Xrm        Yrm        Yrm        Xrm               Yrm        Xrm

Theoremjm2.27 27079* Lemma 2.27 of [JonesMatijasevic] p. 697; rmY is a diophantine relation. 0 was excluded from the range of B and the lower limit of G was imposed because the source proof does not seem to work otherwise; quite possible I'm just missing something. The source proof uses both i and I; i has been changed to j to avoid collision. This theorem is basically nothing but substitution instances, all the work is done in jm2.27a 27076 and jm2.27c 27078. Once Diophantine relations have been defined, the content of the theorem is "rmY is Diophantine" (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm

Theoremjm2.27dlem1 27080* Lemma for rmydioph 27085. Subsitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem2 27081 Lemma for rmydioph 27085. This theorem is used along with the next three to efficiently infer steps like . (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem3 27082 Lemma for rmydioph 27085. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem4 27083 Lemma for rmydioph 27085. Infer -hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem5 27084 Lemma for rmydioph 27085. Used with sselii 3345 to infer membership of midpoints of range; jm2.27dlem2 27081 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremrmydioph 27085 jm2.27 27079 restated in terms of Diophantine sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Yrm Dioph

19.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C

Theoremrmxdiophlem 27086* X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.)
Xrm Yrm

Theoremrmxdioph 27087 X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Xrm Dioph

Theoremjm3.1lem1 27088 Lemma for jm3.1 27091. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremjm3.1lem2 27089 Lemma for jm3.1 27091. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremjm3.1lem3 27090 Lemma for jm3.1 27091. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Yrm

Theoremjm3.1 27091 Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm Xrm Yrm

Theoremexpdiophlem1 27092* Lemma for expdioph 27094. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Yrm Yrm Xrm

Theoremexpdiophlem2 27093 Lemma for expdioph 27094. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph

Theoremexpdioph 27094 The exponential function is Diophantine. This result completes and encapsulates our development using Pell equation solution sequences and is sometimes regarded as Matiyasevich's theorem properly. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph

19.16.38  Uncategorized stuff not associated with a major project

Theoremsetindtr 27095* Epsilon induction for sets contained in a transitive set. If we are allowed to assume Infinity, then all sets have a transitive closure and this reduces to setind 7673; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremsetindtrs 27096* Epsilon induction scheme without Infinity. See comments at setindtr 27095. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3lem1 27097* Lemma for dford3 27099. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3lem2 27098* Lemma for dford3 27099. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3 27099* Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford4 27100* dford3 27099 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)

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