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Theorem List for Metamath Proof Explorer - 26901-27000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvdsacongtr 26901 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 26902 Partial converse to bezout 12983. 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 26903 Converse of bezout 12983 for the gcd = 1 case, sufficient condition for relative primality. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremcoprmdvdsb 26904 Multiplication by a coprime number does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Theoremzabscl 26905 The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremnn0sqcl 26906 The square of a natural number is a natural number. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremdvdsleabs2 26907 Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremmodabsdifz 26908 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 26-Sep-2014.)

Theoremdvdsabsmod0 26909 Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Theoremdivalgmodcl 26910 divalgmod 12867 using a class variable. (Contributed by Stefan O'Rear, 17-Oct-2014.)

19.16.35  X and Y sequences 3: Divisibility properties

Theoremjm2.18 26911 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 26912 Lemma for jm2.19 26916. X and Y values are coprime. (Contributed by Stefan O'Rear, 23-Sep-2014.)
Xrm Yrm

Theoremjm2.19lem2 26913 Lemma for jm2.19 26916. (Contributed by Stefan O'Rear, 23-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19lem3 26914 Lemma for jm2.19 26916. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19lem4 26915 Lemma for jm2.19 26916. Extend to ZZ by symmetry. TODO: use zindbi 26861. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Yrm Yrm

Theoremjm2.19 26916 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 26917 Lemma for jm2.20nn 26920. Express X and Y values as a binomial. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Xrm Yrm Xrm Yrm

Theoremjm2.22 26918* Lemma for jm2.20nn 26920. 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 26919 Lemma for jm2.20nn 26920. Truncate binomial expansion p-adicly. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Yrm Yrm Xrm Yrm

Theoremjm2.20nn 26920 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 26921 Lemma for jm2.26 26925. (Contributed by Stefan O'Rear, 2-Oct-2014.)

Theoremjm2.25 26922 Lemma for jm2.26 26925. Remainders mod X(2n) are negaperiodic mod 2n. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Xrm Yrm Yrm Xrm Yrm Yrm

Theoremjm2.26a 26923 Lemma for jm2.26 26925. 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 26924 Lemma for jm2.26 26925. Use acongrep 26897 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 26925 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 26926 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 26927 Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 26926 if Yrm is redefined as described in rmyluc 26852. (Contributed by Stefan O'Rear, 1-Oct-2014.)
Yrm

19.16.36  X and Y sequences 4: Diophantine representability of Y

Theoremjm2.27a 26928 Lemma for jm2.27 26931. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm        Yrm               Xrm        Yrm               Xrm        Yrm        Yrm

Theoremjm2.27b 26929 Lemma for jm2.27 26931. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm

Theoremjm2.27c 26930 Lemma for jm2.27 26931. Forward direction with substitutions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm        Xrm        Yrm        Yrm        Xrm               Yrm        Xrm

Theoremjm2.27 26931* 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 26928 and jm2.27c 26930. 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 26932* Lemma for rmydioph 26937. Subsitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem2 26933 Lemma for rmydioph 26937. This theorem is used along with the next three to efficiently infer steps like . (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem3 26934 Lemma for rmydioph 26937. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem4 26935 Lemma for rmydioph 26937. Infer -hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem5 26936 Lemma for rmydioph 26937. Used with sselii 3302 to infer membership of midpoints of range; jm2.27dlem2 26933 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremrmydioph 26937 jm2.27 26931 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 26938* X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.)
Xrm Yrm

Theoremrmxdioph 26939 X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Xrm Dioph

Theoremjm3.1lem1 26940 Lemma for jm3.1 26943. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremjm3.1lem2 26941 Lemma for jm3.1 26943. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremjm3.1lem3 26942 Lemma for jm3.1 26943. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Yrm

Theoremjm3.1 26943 Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm Xrm Yrm

Theoremexpdiophlem1 26944* Lemma for expdioph 26946. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Yrm Yrm Xrm

Theoremexpdiophlem2 26945 Lemma for expdioph 26946. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph

Theoremexpdioph 26946 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 26947* 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 7620; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremsetindtrs 26948* Epsilon induction scheme without Infinity. See comments at setindtr 26947. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3lem1 26949* Lemma for dford3 26951. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3lem2 26950* Lemma for dford3 26951. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3 26951* Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford4 26952* dford3 26951 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremwopprc 26953 Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremrpnnen3lem 26954* Lemma for rpnnen3 26955. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremrpnnen3 26955 Dedekind cut injection of into . (Contributed by Stefan O'Rear, 18-Jan-2015.)

19.16.39  More equivalents of the Axiom of Choice

Theoremaxac10 26956 Characterization of choice similar to dffin1-5 8215. (Contributed by Stefan O'Rear, 6-Jan-2015.)

Theoremharinf 26957 The Hartogs number of an infinite set is at least . MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
har

Theoremwdom2d2 26958* Deduction for weak dominance by a cross product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
*

Theoremttac 26959 Tarski's theorem about choice: infxpidm 8384 is equivalent to ax-ac 8286. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
CHOICE

Theorempw2f1ocnv 26960* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7165, 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 26961* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7165, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theorempw2f1o2val 26962* Function value of the pw2f1o2 26961 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Theorempw2f1o2val2 26963* Membership in a mapped set under the pw2f1o2 26961 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Theoremsoeq12d 26964 Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfreq12d 26965 Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremweeq12d 26966 Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremlimsuc2 26967 Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremwepwsolem 26968* Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremwepwso 26969* A well-ordering induces a strict ordering on the power set. EDITORIAL: when well-orderings are set like, this can be strengthened to remove (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoreminisegn0 26970 Non-emptyness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremdnnumch1 26971* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 7858 (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch2 26972* Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch3lem 26973* Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch3 26974* Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnwech 26975* Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremfnwe2val 26976* Lemma for fnwe2 26980. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2lem1 26977* Lemma for fnwe2 26980. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2lem2 26978* Lemma for fnwe2 26980. An element which is in a minimal fiber and minimal within its fiber is minimal globally; thus is well-founded. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2lem3 26979* Lemma for fnwe2 26980. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2 26980* A well-ordering can be constructed on a partitioned set by patching together well-orderings on each partition using a well-ordering on the partitions themselves. Similar to fnwe 6412 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremaomclem1 26981* Lemma for dfac11 26990. This is the beginning of the proof that multiple choice is equivalent to choice. Our goal is to construct, by transfinite recursion, a well-ordering of . In what follows, is the index of the rank we wish to well-order, is the collection of well-orderings constructed so far, is the set of ordinal indexes of constructed ranks i.e. the next rank to construct, and is a postulated multiple-choice function.

Successor case 1, define a simple ordering from the well-ordered predecessor. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremaomclem2 26982* Lemma for dfac11 26990. Successor case 2, a choice function for subsets of . (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremaomclem3 26983* Lemma for dfac11 26990. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
recs

Theoremaomclem4 26984* Lemma for dfac11 26990. Limit case. Patch together well-orderings constructed so far using fnwe2 26980 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremaomclem5 26985* Lemma for dfac11 26990. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs

Theoremaomclem6 26986* Lemma for dfac11 26990. Transfinite induction, close over . (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs                             recs

Theoremaomclem7 26987* Lemma for dfac11 26990. is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs                             recs

Theoremsupeq123d 26988 Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremaomclem8 26989* Lemma for dfac11 26990. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremdfac11 26990* The right-hand side of this theorem (compare with ac4 8302), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 7507, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do.

This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it.

A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.)

CHOICE

Theoremkelac1 26991* Kelley's choice, basic form: if a collection of sets can be cast as closed sets in the factors of a topology, and there is a definable element in each topology (which need not be in the closed set - if it were this would be trivial), then compactness (via finite intersection) guarantees that the final product is nonempty. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremkelac2lem 26992 Lemma for kelac2 26993 and dfac21 26994: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremkelac2 26993* Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremdfac21 26994 Tychonoff's theorem is a choice equivalent. Definition AC21 of Schechter p. 461. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 27-Aug-2015.)
CHOICE

19.16.40  Finitely generated left modules

Syntaxclfig 26995 Extend class notation with the class of finitely generated left modules.
LFinGen

Definitiondf-lfig 26996 Define the class of finitely generated left modules. Finite generation of subspaces can be intepreted using ↾s. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen

Theoremislmodfg 26997* Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen

Theoremislssfg 26998* Property of a finitely generated left (sub-)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s                      LFinGen

Theoremislssfg2 26999* Property of a finitely generated left (sub-)module, with a relaxed constraint on the spanning vectors. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s                             LFinGen

Theoremislssfgi 27000 Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LFinGen

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