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Type | Label | Description |
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Statement | ||
Theorem | hlhilphllem 35601* | Lemma for hlhil 22475. (Contributed by NM, 23-Jun-2015.) |
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Theorem | hlhilhillem 35602* | Lemma for hlhil 22475. (Contributed by NM, 23-Jun-2015.) |
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Theorem | hlathil 35603 |
Construction of a Hilbert space (df-hil 19344) ![]() ![]() ![]() ![]()
The Hilbert space
An example of involution is the complex conjugate when the division ring
is the field of complex numbers. The nature of the division ring we
constructed is indeterminate, however, until we specialize the initial
Hilbert lattice with additional conditions found by Maria Solèr
in 1995 and refined by René Mayet in 1998 that result in a
division ring isomorphic to
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Theorem | rntrclfvOAI 35604 | The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.) |
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Theorem | moxfr 35605* | Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
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Theorem | imaiinfv 35606* | Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
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Theorem | elrfi 35607* | Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
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Theorem | elrfirn 35608* | Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
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Theorem | elrfirn2 35609* | Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.) |
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Theorem | cmpfiiin 35610* | In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
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Theorem | ismrcd1 35611* | Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 15601), isotone (satisfies mrcss 15600), and idempotent (satisfies mrcidm 15603) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 35612 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
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Theorem | ismrcd2 35612* | Second half of ismrcd1 35611. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
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Theorem | istopclsd 35613* | A closure function which satisfies sscls 20148, clsidm 20160, cls0 20173, and clsun 31055 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
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Theorem | ismrc 35614* | A function is a Moore closure operator iff it satisfies mrcssid 15601, mrcss 15600, and mrcidm 15603. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
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Syntax | cnacs 35615 | Class of Noetherian closure systems. |
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Definition | df-nacs 35616* | Define a closure system of Noetherian type (not standard terminology) as an algebraic system where all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | isnacs 35617* | Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | nacsfg 35618* | In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | isnacs2 35619 | Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | mrefg2 35620* | Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | mrefg3 35621* | Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | nacsacs 35622 | A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | isnacs3 35623* | A choice-free order equivalent to the Noetherian condition on a closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | incssnn0 35624* | Transitivity induction of subsets, lemma for nacsfix 35625. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | nacsfix 35625* | An increasing sequence of closed sets in a Noetherian-type closure system eventually fixates. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
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Theorem | constmap 35626 |
A constant (represented without dummy variables) is an element of a
function set.
_Note: In the following development, we will be quite often quantifying over functions and points in N-dimensional space (which are equivalent to functions from an "index set"). Many of the following theorems exist to transfer standard facts about functions to elements of function sets._ (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
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Theorem | mapco2g 35627 | Renaming indexes in a tuple, with sethood as antecedents. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.) |
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Theorem | mapco2 35628 | Post-composition (renaming indexes) of a mapping viewed as a point. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
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Theorem | mapfzcons 35629 | Extending a one-based mapping by adding a tuple at the end results in another mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
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Theorem | mapfzcons1 35630 | Recover prefix mapping from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
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Theorem | mapfzcons1cl 35631 | A nonempty mapping has a prefix. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
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Theorem | mapfzcons2 35632 | Recover added element from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
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Theorem | mptfcl 35633* | Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Syntax | cmzpcl 35634 | Extend class notation to include pre-polynomial rings. |
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Syntax | cmzp 35635 | Extend class notation to include polynomial rings. |
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Definition | df-mzpcl 35636* |
Define the polynomially closed function rings over an arbitrary index
set ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-mzp 35637 |
Polynomials over ![]() |
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Theorem | mzpclval 35638* | Substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
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Theorem | elmzpcl 35639* | Double substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
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Theorem | mzpclall 35640 | 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 35637 is well-defined. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
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Theorem | mzpcln0 35641 | Corrolary of mzpclall 35640: polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
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Theorem | mzpcl1 35642 |
Defining property 1 of a polynomially closed function set ![]() |
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Theorem | mzpcl2 35643* |
Defining property 2 of a polynomially closed function set ![]() |
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Theorem | mzpcl34 35644 |
Defining properties 3 and 4 of a polynomially closed function set
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Theorem | mzpval 35645 | Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
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Theorem | dmmzp 35646 | mzPoly is defined for all index sets which are sets. This is used with elfvdm 5905 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
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Theorem | mzpincl 35647 | 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.) |
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Theorem | mzpconst 35648 | Constant functions are polynomial. See also mzpconstmpt 35653. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
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Theorem | mzpf 35649 | A polynomial function is a function from the coordinate space to the integers. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
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Theorem | mzpproj 35650* | A projection function is polynomial. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
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Theorem | mzpadd 35651 | The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 35654. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
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Theorem | mzpmul 35652 | The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 35655. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
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Theorem | mzpconstmpt 35653* | A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 35654, mzpmulmpt 35655, mzpnegmpt 35657, mzpsubmpt 35656, mzpexpmpt 35658) 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 35650 is already expressed using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
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Theorem | mzpaddmpt 35654* | Sum of polynomial functions is polynomial. Maps-to version of mzpadd 35651. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
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Theorem | mzpmulmpt 35655* | Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 35655. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
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Theorem | mzpsubmpt 35656* | The difference of two polynomial functions is polynomial. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Theorem | mzpnegmpt 35657* | Negation of a polynomial function. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
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Theorem | mzpexpmpt 35658* | Raise a polynomial function to a (fixed) exponent. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
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Theorem | mzpindd 35659* | "Structural" induction to prove properties of all polynomial functions. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
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Theorem | mzpmfp 35660 | Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Revised by AV, 13-Jun-2019.) |
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Theorem | mzpsubst 35661* |
Substituting polynomials for the variables of a polynomial results in a
polynomial. ![]() ![]() |
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Theorem | mzprename 35662* | Simplified version of mzpsubst 35661 to simply relabel variables in a polynomial. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
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Theorem | mzpresrename 35663* | 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.) |
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Theorem | mzpcompact2lem 35664* | Lemma for mzpcompact2 35665. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
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Theorem | mzpcompact2 35665* | 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.) |
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Theorem | coeq0i 35666 | coeq0 5351 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
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Theorem | fzsplit1nn0 35667 |
Split a finite 1-based set of integers in the middle, allowing either end
to be empty (![]() ![]() ![]() ![]() ![]() |
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Syntax | cdioph 35668 | Extend class notation to include the family of Diophantine sets. |
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Definition | df-dioph 35669* |
A Diophantine set is a set of positive integers which is a projection of
the zero set of some polynomial. This definition somewhat awkwardly
mixes ![]() ![]() ![]() ![]() ![]() |
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Theorem | eldiophb 35670* | Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
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Theorem | eldioph 35671* | Condition for a set to be Diophantine (unpacking existential quantifier). (Contributed by Stefan O'Rear, 5-Oct-2014.) |
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Theorem | diophrw 35672* | Renaming and adding unused witness variables does not change the Diophantine set coded by a polynomial. (Contributed by Stefan O'Rear, 7-Oct-2014.) |
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Theorem | eldioph2lem1 35673* | Lemma for eldioph2 35675. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
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Theorem | eldioph2lem2 35674* | Lemma for eldioph2 35675. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
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Theorem | eldioph2 35675* | Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 35665. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.) |
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Theorem | eldioph2b 35676* |
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
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | eldiophelnn0 35677 |
Remove antecedent on ![]() |
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Theorem | eldioph3b 35678* |
Define Diophantine sets in terms of polynomials with variables indexed
by ![]() |
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Theorem | eldioph3 35679* | Inference version of eldioph3b 35678 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Theorem | ellz1 35680 | Membership in a lower set of integers. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
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Theorem | lzunuz 35681 | The union of a lower set of integers and an upper set of integers which abut or overlap is all of the integers. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
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Theorem | fz1eqin 35682 |
Express a one-based finite range as the intersection of lower integers
with ![]() |
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Theorem | lzenom 35683 | Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Theorem | elmapresaun 35684 | fresaun 5766 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
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Theorem | elmapresaunres2 35685 | fresaunres2 5767 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) |
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Theorem | diophin 35686 | 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.) |
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Theorem | diophun 35687 | 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.) |
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Theorem | eldiophss 35688 | Diophantine sets are sets of tuples of nonnegative integers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
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Theorem | diophrex 35689* | Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Theorem | eq0rabdioph 35690* | 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.) |
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Theorem | eqrabdioph 35691* | Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be nonnegative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Theorem | 0dioph 35692 | The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Theorem | vdioph 35693 | The "universal" set (as large as possible given eldiophss 35688) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Theorem | anrabdioph 35694* | Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Theorem | orrabdioph 35695* | Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Theorem | 3anrabdioph 35696* | Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Theorem | 3orrabdioph 35697* | Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
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Theorem | 2sbcrex 35698* | Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.) |
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Theorem | sbcrexgOLD 35699* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3331 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | 2sbcrexOLD 35700* | Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 6342 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
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