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Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltrmy 26701 The Y-sequence is strictly monotonic over . (Contributed by Stefan O'Rear, 25-Sep-2014.)
Yrm Yrm

Theoremrmyeq0 26702 Y is zero only at zero. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm

Theoremrmyeq 26703 Y is one-to-one. (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm

Theoremlermy 26704 Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm

Theoremrmynn 26705 Yrm is positive for positive arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremrmynn0 26706 Yrm is nonnegative for nonnegative arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremrmyabs 26707 Yrm commutes with . (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm

Theoremjm2.24nn 26708 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 26709 First half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 14-Oct-2014.)
Yrm

Theoremjm2.17b 26710 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 26711 Second half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 15-Oct-2014.)
Yrm

Theoremjm2.24 26712 Lemma 2.24 of [JonesMatijasevic] p. 697 extended to . Could be eliminated with a more careful proof of jm2.26lem3 26756. (Contributed by Stefan O'Rear, 3-Oct-2014.)
Yrm Yrm Xrm

Theoremrmygeid 26713 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 26714 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 26715 If two pairs of numbers are componentwise congruent, so are their sums. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongmul 26716 If two pairs of numbers are componentwise congruent, so are their products. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongsym 26717 Congruence mod is a symmetric/commutative relation. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongneg 26718 If two integers are congruent mod , so are their negatives. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremcongsub 26719 If two pairs of numbers are componentwise congruent, so are their differences. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremcongid 26720 Every integer is congruent to itself mod every base. (Contributed by Stefan O'Rear, 1-Oct-2014.)

Theoremmzpcong 26721* Polynomials commute with congruences. (Does this characterize them?) (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly

Theoremcongrep 26722* Every integer is congruent to some number in the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremcongabseq 26723 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 26724 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 26725 Symmetry of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremacongneg2 26726 Negate right side of alternating congruence. Makes essential use of the "alternating" part. (Contributed by Stefan O'Rear, 3-Oct-2014.)

Theoremacongtr 26727 Transitivity of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremacongeq12d 26728 Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.)

Theoremacongrep 26729* 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 26730 Bound on the difference between two integers constrained to two possibly overlapping finite ranges. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Theoremfzneg 26731 Reflection of a finite range of integers about 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Theoremacongeq 26732 Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 26757 (Contributed by Stefan O'Rear, 4-Oct-2014.)

Theoremdvdsacongtr 26733 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 26734 Partial converse to bezout 12962. 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 26735 Converse of bezout 12962 for the gcd = 1 case, sufficient condition for relative primality. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremcoprmdvdsb 26736 Multiplication by a coprime number does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremzabscl 26737 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremnn0sqcl 26738 The square of a natural number is a natural number. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremdvdsleabs2 26739 Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremmodabsdifz 26740 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 26-Sep-2014.)

Theoremdvdsabsmod0 26741 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremdivalgmodcl 26742 divalgmod 12846 using a class variable. (Contributed by Stefan O'Rear, 17-Oct-2014.)

19.16.35  X and Y sequences 3: Divisibility properties

Theoremjm2.18 26743 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 26744 Lemma for jm2.19 26748. X and Y values are coprime. (Contributed by Stefan O'Rear, 23-Sep-2014.)
Xrm Yrm

Theoremjm2.19lem2 26745 Lemma for jm2.19 26748. (Contributed by Stefan O'Rear, 23-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19lem3 26746 Lemma for jm2.19 26748. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19lem4 26747 Lemma for jm2.19 26748. Extend to ZZ by symmetry. TODO: use zindbi 26693. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19 26748 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 26749 Lemma for jm2.20nn 26752. Express X and Y values as a binomial. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Xrm Yrm Xrm Yrm

Theoremjm2.22 26750* Lemma for jm2.20nn 26752. 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 26751 Lemma for jm2.20nn 26752. Truncate binomial expansion p-adicly. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Xrm Yrm

Theoremjm2.20nn 26752 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 26753 Lemma for jm2.26 26757. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremjm2.25 26754 Lemma for jm2.26 26757. Remainders mod X(2n) are negaperiodic mod 2n. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Xrm Yrm Yrm Xrm Yrm Yrm

Theoremjm2.26a 26755 Lemma for jm2.26 26757. 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 26756 Lemma for jm2.26 26757. Use acongrep 26729 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 26757 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 26758 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 26759 Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 26758 if Yrm is redefined as described in rmyluc 26684. (Contributed by Stefan O'Rear, 1-Oct-2014.)
Yrm

19.16.36  X and Y sequences 4: Diophantine representability of Y

Theoremjm2.27a 26760 Lemma for jm2.27 26763. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm        Yrm               Xrm        Yrm               Xrm        Yrm        Yrm

Theoremjm2.27b 26761 Lemma for jm2.27 26763. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm

Theoremjm2.27c 26762 Lemma for jm2.27 26763. Forward direction with substitutions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm        Xrm        Yrm        Yrm        Xrm               Yrm        Xrm

Theoremjm2.27 26763* 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 26760 and jm2.27c 26762. 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 26764* Lemma for rmydioph 26769. Subsitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem2 26765 Lemma for rmydioph 26769. This theorem is used along with the next three to efficiently infer steps like . (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem3 26766 Lemma for rmydioph 26769. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem4 26767 Lemma for rmydioph 26769. Infer -hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem5 26768 Lemma for rmydioph 26769. Used with sselii 3281 to infer membership of midpoints of range; jm2.27dlem2 26765 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremrmydioph 26769 jm2.27 26763 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 26770* X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.)
Xrm Yrm

Theoremrmxdioph 26771 X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Xrm Dioph

Theoremjm3.1lem1 26772 Lemma for jm3.1 26775. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremjm3.1lem2 26773 Lemma for jm3.1 26775. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremjm3.1lem3 26774 Lemma for jm3.1 26775. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Yrm

Theoremjm3.1 26775 Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm Xrm Yrm

Theoremexpdiophlem1 26776* Lemma for expdioph 26778. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Yrm Yrm Xrm

Theoremexpdiophlem2 26777 Lemma for expdioph 26778. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph

Theoremexpdioph 26778 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 26779* 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 7599; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremsetindtrs 26780* Epsilon induction scheme without Infinity. See comments at setindtr 26779. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3lem1 26781* Lemma for dford3 26783. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3lem2 26782* Lemma for dford3 26783. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3 26783* Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford4 26784* dford3 26783 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremwopprc 26785 Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremrpnnen3lem 26786* Lemma for rpnnen3 26787. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremrpnnen3 26787 Dedekind cut injection of into . (Contributed by Stefan O'Rear, 18-Jan-2015.)

19.16.39  More equivalents of the Axiom of Choice

Theoremaxac10 26788 Characterization of choice similar to dffin1-5 8194. (Contributed by Stefan O'Rear, 6-Jan-2015.)

Theoremharinf 26789 The Hartogs number of an infinite set is at least . MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
har

Theoremwdom2d2 26790* Deduction for weak dominance by a cross product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
*

Theoremttac 26791 Tarski's theorem about choice: infxpidm 8363 is equivalent to ax-ac 8265. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
CHOICE

Theorempw2f1ocnv 26792* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7144, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 9-Jul-2015.)

Theorempw2f1o2 26793* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7144, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theorempw2f1o2val 26794* Function value of the pw2f1o2 26793 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Theorempw2f1o2val2 26795* Membership in a mapped set under the pw2f1o2 26793 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Theoremsoeq12d 26796 Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfreq12d 26797 Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremweeq12d 26798 Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremlimsuc2 26799 Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremwepwsolem 26800* Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)

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