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

Definitiondf-mzpcl 26801* Define the polynomially closed function rings over an arbitrary index set . The set mzPolyCld contains all sets of functions from to which include all constants and projections and are closed under addition and multiplication. This is a "temporary" set used to define the polynomial function ring itself mzPoly; see df-mzp 26802. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Definitiondf-mzp 26802 Polynomials over with an arbitrary index set, that is, the smallest ring of functions containing all constant functions and all projections. This is almost the most general reasonable definition; to reach full generality, we would need to be able to replace ZZ with an arbitrary (semi-)ring (and a coordinate subring), but rings have not been defined yet. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly mzPolyCld

Theoremmzpclval 26803* Substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremelmzpcl 26804* Double substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpclall 26805 The set of all functions with the signature of a polynomial is a polynomially closed set. This is a lemma to show that the intersection in df-mzp 26802 is well defined. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpcln0 26806 Corrolary of mzpclall 26805: Polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpcl1 26807 Defining property 1 of a polynomially closed function set : it contains all constant functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpcl2 26808* Defining property 2 of a polynomially closed function set : it contains all projections. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpcl34 26809 Defining properties 3 and 4 of a polynomially closed function set : it is closed under pointwise addition and multiplication. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpval 26810 Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly mzPolyCld

Theoremdmmzp 26811 mzPoly is defined for all index sets which are sets. This is used with elfvdm 5554 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly

Theoremmzpincl 26812 Polynomial closedness is a universal first-order property and passes to intersections. This is where the closure properties of the polynomial ring itself are proved. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly mzPolyCld

Theoremmzpconst 26813 Constant functions are polynomial. See also mzpconstmpt 26818. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly

Theoremmzpf 26814 A polynomial function is a function from the coordinate space to the integers. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly

Theoremmzpproj 26815* A projection function is polynomial. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly

Theoremmzpadd 26816 The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 26819. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzpmul 26817 The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 26820. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzpconstmpt 26818* A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 26819, mzpmulmpt 26820, mzpnegmpt 26822, mzpsubmpt 26821, mzpexpmpt 26823) can be used to build proofs that functions which are "manifestly polynomial", in the sense of being a maps-to containing constants, projections, and simple arithmetic operations, are actually polynomial functions. There is no mzpprojmpt because mzpproj 26815 is already expressed using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly

Theoremmzpaddmpt 26819* Sum of polynomial functions is polynomial. Maps-to version of mzpadd 26816. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzpmulmpt 26820* Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 26820. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzpsubmpt 26821* The difference of two polynomial functions is polynomial. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzpnegmpt 26822* Negation of a polynomial function. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly mzPoly

Theoremmzpexpmpt 26823* Raise a polynomial function to a (fixed) exponent. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly mzPoly

Theoremmzpindd 26824* "Structural" induction to prove properties of all polynomial functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly

Theoremmzpmfp 26825 Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.)
mzPoly eval flds

Theoremmzpsubst 26826* Substituting polynomials for the variables of a polynomial results in a polynomial. is expected to depend on and provide the polynomials which are being substituted. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzprename 26827* Simplified version of mzpsubst 26826 to simply relabel variables in a polynomial. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly mzPoly

Theoremmzpresrename 26828* A polynomial is a polynomial over all larger index sets. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
mzPoly mzPoly

Theoremmzpcompact2lem 26829* Lemma for mzpcompact2 26830. (Contributed by Stefan O'Rear, 9-Oct-2014.)
mzPoly mzPoly

Theoremmzpcompact2 26830* Polynomials are finitary objects and can only reference a finite number of variables, even if the index set is infinite. Thus, every polynomial can be expressed as a (uniquely minimal, although we do not prove that) polynomial on a finite number of variables, which is then extended by adding an arbitrary set of ignored variables. (Contributed by Stefan O'Rear, 9-Oct-2014.)
mzPoly mzPoly

18.17.10  Miscellanea for Diophantine sets 1

Theoremcoeq0 26831 A composition of two relations is empty iff there is no overlap betwen the range of the second and the domain of the first. Useful in combination with coundi 5174 and coundir 5175 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremcoeq0i 26832 coeq0 26831 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremfzsplit1nn0 26833 Split a finite 1-based set of integers in the middle, allowing either end to be empty (). (Contributed by Stefan O'Rear, 8-Oct-2014.)

18.17.11  Diophantine sets 1: definitions

Syntaxcdioph 26834 Extend class notation to include the family of Diophantine sets.
Dioph

Definitiondf-dioph 26835* A Diophantine set is a set of natural numbers which is a projection of the zero set of some polynomial. This definition somewhat awkwardly mixes (via mzPoly) and (to define the zero sets); the former could be avoided by considering coincidence sets of polynomials at the cost of requiring two, and the second is driven by consistency with our mu-recursive functions and the requirements of the Davis-Putnam-Robinson-Matiyasevich proof. Both are avoidable at a complexity cost. In particular, it is a consequence of 4sq 13011 that implicitly restricting variables to adds no expressive power over allowing them to range over . While this definition stipulates a specific index set for the polynomials, there is actually flexibility here, see eldioph2b 26842. (Contributed by Stefan O'Rear, 5-Oct-2014.)
Dioph mzPoly

Theoremeldiophb 26836* Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Dioph mzPoly

Theoremeldioph 26837* Condition for a set to be Diophantine (unpacking existential quantifier) (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly Dioph

Theoremdiophrw 26838* Renaming and adding unused witness variables does not change the Diophantine set coded by a polynomial. (Contributed by Stefan O'Rear, 7-Oct-2014.)

Theoremeldioph2lem1 26839* Lemma for eldioph2 26841. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremeldioph2lem2 26840* Lemma for eldioph2 26841. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremeldioph2 26841* Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 26830. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
mzPoly Dioph

Theoremeldioph2b 26842* While Diophantine sets were defined to have a finite number of witness variables consequtively following the observable variables, this is not necessary; they can equivalently be taken to use any witness set . For instance, in diophin 26852 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Dioph mzPoly

Theoremeldiophelnn0 26843 Remove antecedent on from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph

Theoremeldioph3b 26844* Define Diophantine sets in terms of polynomials with variables indexed by . This avoids a quantifier over the number of witness variables and will be easier to use than eldiophb 26836 in most cases. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph mzPoly

Theoremeldioph3 26845* Inference version of eldioph3b 26844 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly Dioph

18.17.12  Diophantine sets 2 miscellanea

Theoremellz1 26846 Membership in a set of lower integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremlzunuz 26847 A set of lower integers and upper integers which abut or overlap is all of the integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremfz1eqin 26848 Express a one-based finite range as the intersection of lower integers with . (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremlzenom 26849 Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Theoremelmapresaun 26850 fresaun 5412 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)

Theoremelmapresaunres2 26851 fresaunres2 5413 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.)

18.17.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra

Theoremdiophin 26852 If two sets are Diophantine, so is their intersection. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph Dioph Dioph

Theoremdiophun 26853 If two sets are Diophantine, so is their union. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph Dioph Dioph

Theoremeldiophss 26854 Diophantine sets are sets of tuples of natural numbers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph

18.17.14  Diophantine sets 3: construction

Theoremdiophrex 26855* Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph

Theoremeq0rabdioph 26856* This is the first of a number of theorems which allow sets to be proven Diophantine by syntactic induction, and models the correspondence between Diophantine sets and monotone existential first order logic. This first theorem shows that the zero set of an implicit polynomial is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly Dioph

Theoremeqrabdioph 26857* Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be non-negative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly mzPoly Dioph

Theorem0dioph 26858 The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph

Theoremvdioph 26859 The "universal" set (as large as possible given eldiophss 26854) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph

Theoremanrabdioph 26860* Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph

Theoremorrabdioph 26861* Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph

Theorem3anrabdioph 26862* Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph Dioph

Theorem3orrabdioph 26863* Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph Dioph

18.17.15  Diophantine sets 4 miscellanea

Theorem2sbcrex 26864* Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremsbc2rexg 26865* Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremsbc4rexg 26866* Exchange a substitution with 4 existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.)

TheoremsbcbiiiOLD 26867 Fully inferenced rewriting under an explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremsbcrot3 26868* Rotate a sequence of three explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsbcrot5 26869* Rotate a sequence of five explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsbccomieg 26870* Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsbcrot3gOLD 26871* Rotate a sequence of three explicit substitutions, closed theorem. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremsbcrot3OLD 26872* Rotate a sequence of three explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremsbcrot5OLD 26873* Rotate a sequence of five explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.)

TheoremsbccomiegOLD 26874* Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.)

18.17.16  Diophantine sets 4: Quantification

Theoremrexrabdioph 26875* Diophantine set builder for existential quantification. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph

Theoremrexfrabdioph 26876* Diophantine set builder for existential quantifier, explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph Dioph

Theorem2rexfrabdioph 26877* Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph Dioph

Theorem3rexfrabdioph 26878* Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph Dioph

Theorem4rexfrabdioph 26879* Diophantine set builder for existential quantifier, explicit substitution, four variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph Dioph

Theorem6rexfrabdioph 26880* Diophantine set builder for existential quantifier, explicit substitution, six variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph Dioph

Theorem7rexfrabdioph 26881* Diophantine set builder for existential quantifier, explicit substitution, seven variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph Dioph

18.17.17  Diophantine sets 5: Arithmetic sets

Theoremrabdiophlem1 26882* Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 2618. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly

Theoremrabdiophlem2 26883* Lemma for arithmetic diophantine sets. Reuse a polynomial expression under a new quantifier. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly mzPoly

Theoremelnn0rabdioph 26884* Diophantine set builder for nonnegativity constraints. The first builder which uses a witness variable internally; an expression is nonnegative if there is a nonnegative integer equal to it. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly Dioph

Theoremrexzrexnn0 26885* Rewrite a quantification over integers into a quantification over naturals. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremlerabdioph 26886* Diophantine set builder for the less or equals relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly mzPoly Dioph

Theoremeluzrabdioph 26887* Diophantine set builder for membership in a fixed set of upper integers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly Dioph

Theoremelnnrabdioph 26888* Diophantine set builder for positivity. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly Dioph

Theoremltrabdioph 26889* Diophantine set builder for the strict less than relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly mzPoly Dioph

Theoremnerabdioph 26890* Diophantine set builder for inequality. This not quite trivial theorem touches on something important; Diophantine sets are not closed under negation, but they contain an important subclass that is, namely the recursive sets. With this theorem and De Morgan's laws, all quantifier-free formulae can be negated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly mzPoly Dioph

Theoremdvdsrabdioph 26891* Divisibility is a Diophantine relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly mzPoly Dioph

18.17.18  Diophantine sets 6 miscellanea

Theoremfz1ssnn 26892 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremftp 26893 A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.)

18.17.19  Diophantine sets 6: reusability. renumbering of variables

Theoremeldioph4b 26894* Membership in Dioph expressed using a quantified union to add witness variables instead of a restriction to remove them. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Dioph mzPoly

Theoremeldioph4i 26895* Forward-only version of eldioph4b 26894. (Contributed by Stefan O'Rear, 16-Oct-2014.)
mzPoly Dioph

Theoremdiophren 26896* Change variables in a Diophantine set, using class notation. This allows already proved Diophantine sets to be reused in contexts with more variables. (Contributed by Stefan O'Rear, 16-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
Dioph Dioph

Theoremrabrenfdioph 26897* Change variable numbers in a Diophantine class abstraction using explicit substitution. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph Dioph

Theoremrabren3dioph 26898* Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph Dioph

18.17.20  Pigeonhole Principle and cardinality helpers

Theoremfphpd 26899* Pigeonhole principle expressed with implicit substitution. If the range is smaller than the domain, two inputs must be mapped to the same output. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)

Theoremfphpdo 26900* Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.)

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