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Theorem List for Metamath Proof Explorer - 26901-27000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmptfcl 26901* Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( t  e.  A  |->  B ) : A --> C  ->  ( t  e.  A  ->  B  e.  C ) )
 
18.17.9  Multivariate polynomials over the integers
 
Syntaxcmzpcl 26902 Extend class notation to include pre-polynomial rings.
 class mzPolyCld
 
Syntaxcmzp 26903 Extend class notation to include polynomial rings.
 class mzPoly
 
Definitiondf-mzpcl 26904* Define the polynomially closed function rings over an arbitrary index set  v. The set  (mzPolyCld `  v
) contains all sets of functions from  ( ZZ  ^m  v
) to  ZZ 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 `  v
); see df-mzp 26905. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |- mzPolyCld  =  ( v  e.  _V  |->  { p  e.  ~P ( ZZ  ^m  ( ZZ  ^m  v ) )  |  ( ( A. i  e.  ZZ  ( ( ZZ 
 ^m  v )  X.  { i } )  e.  p  /\  A. j  e.  v  ( x  e.  ( ZZ  ^m  v
 )  |->  ( x `  j ) )  e.  p )  /\  A. f  e.  p  A. g  e.  p  (
 ( f  o F  +  g )  e.  p  /\  ( f  o F  x.  g )  e.  p ) ) } )
 
Definitiondf-mzp 26905 Polynomials over  ZZ 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  =  ( v  e.  _V  |->  |^| (mzPolyCld `
  v ) )
 
Theoremmzpclval 26906* Substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  (mzPolyCld `  V )  =  { p  e.  ~P ( ZZ  ^m  ( ZZ  ^m  V ) )  |  ( ( A. i  e.  ZZ  ( ( ZZ 
 ^m  V )  X.  { i } )  e.  p  /\  A. j  e.  V  ( x  e.  ( ZZ  ^m  V )  |->  ( x `  j ) )  e.  p )  /\  A. f  e.  p  A. g  e.  p  (
 ( f  o F  +  g )  e.  p  /\  ( f  o F  x.  g )  e.  p ) ) } )
 
Theoremelmzpcl 26907* Double substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  ( P  e.  (mzPolyCld `  V ) 
 <->  ( P  C_  ( ZZ  ^m  ( ZZ  ^m  V ) )  /\  ( ( A. i  e.  ZZ  ( ( ZZ 
 ^m  V )  X.  { i } )  e.  P  /\  A. j  e.  V  ( x  e.  ( ZZ  ^m  V )  |->  ( x `  j ) )  e.  P )  /\  A. f  e.  P  A. g  e.  P  ( ( f  o F  +  g
 )  e.  P  /\  ( f  o F  x.  g )  e.  P ) ) ) ) )
 
Theoremmzpclall 26908 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 26905 is well-defined. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  ( ZZ  ^m  ( ZZ 
 ^m  V ) )  e.  (mzPolyCld `  V ) )
 
Theoremmzpcln0 26909 Corrolary of mzpclall 26908: Polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  (mzPolyCld `  V )  =/=  (/) )
 
Theoremmzpcl1 26910 Defining property 1 of a polynomially closed function set  P: it contains all constant functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( P  e.  (mzPolyCld `  V )  /\  F  e.  ZZ )  ->  (
 ( ZZ  ^m  V )  X.  { F }
 )  e.  P )
 
Theoremmzpcl2 26911* Defining property 2 of a polynomially closed function set  P: it contains all projections. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( P  e.  (mzPolyCld `  V )  /\  F  e.  V )  ->  (
 g  e.  ( ZZ 
 ^m  V )  |->  ( g `  F ) )  e.  P )
 
Theoremmzpcl34 26912 Defining properties 3 and 4 of a polynomially closed function set  P: it is closed under pointwise addition and multiplication. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( P  e.  (mzPolyCld `  V )  /\  F  e.  P  /\  G  e.  P )  ->  ( ( F  o F  +  G )  e.  P  /\  ( F  o F  x.  G )  e.  P ) )
 
Theoremmzpval 26913 Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  (mzPoly `  V )  =  |^| (mzPolyCld `
  V ) )
 
Theoremdmmzp 26914 mzPoly is defined for all index sets which are sets. This is used with elfvdm 5570 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  dom mzPoly  =  _V
 
Theoremmzpincl 26915 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.)
 |-  ( V  e.  _V  ->  (mzPoly `  V )  e.  (mzPolyCld `  V ) )
 
Theoremmzpconst 26916 Constant functions are polynomial. See also mzpconstmpt 26921. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( V  e.  _V  /\  C  e.  ZZ )  ->  ( ( ZZ  ^m  V )  X.  { C } )  e.  (mzPoly `  V ) )
 
Theoremmzpf 26917 A polynomial function is a function from the coordinate space to the integers. (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |-  ( F  e.  (mzPoly `  V )  ->  F : ( ZZ  ^m  V ) --> ZZ )
 
Theoremmzpproj 26918* A projection function is polynomial. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( V  e.  _V  /\  X  e.  V ) 
 ->  ( g  e.  ( ZZ  ^m  V )  |->  ( g `  X ) )  e.  (mzPoly `  V ) )
 
Theoremmzpadd 26919 The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 26922. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  (mzPoly `  V )  /\  B  e.  (mzPoly `  V )
 )  ->  ( A  o F  +  B )  e.  (mzPoly `  V ) )
 
Theoremmzpmul 26920 The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 26923. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  (mzPoly `  V )  /\  B  e.  (mzPoly `  V )
 )  ->  ( A  o F  x.  B )  e.  (mzPoly `  V ) )
 
Theoremmzpconstmpt 26921* A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 26922, mzpmulmpt 26923, mzpnegmpt 26925, mzpsubmpt 26924, mzpexpmpt 26926) 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 26918 is already expressed using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |-  (
 ( V  e.  _V  /\  C  e.  ZZ )  ->  ( x  e.  ( ZZ  ^m  V )  |->  C )  e.  (mzPoly `  V ) )
 
Theoremmzpaddmpt 26922* Sum of polynomial functions is polynomial. Maps-to version of mzpadd 26919. (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |-  (
 ( ( x  e.  ( ZZ  ^m  V )  |->  A )  e.  (mzPoly `  V )  /\  ( x  e.  ( ZZ  ^m  V )  |->  B )  e.  (mzPoly `  V ) )  ->  ( x  e.  ( ZZ  ^m  V )  |->  ( A  +  B ) )  e.  (mzPoly `  V ) )
 
Theoremmzpmulmpt 26923* Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 26923. (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |-  (
 ( ( x  e.  ( ZZ  ^m  V )  |->  A )  e.  (mzPoly `  V )  /\  ( x  e.  ( ZZ  ^m  V )  |->  B )  e.  (mzPoly `  V ) )  ->  ( x  e.  ( ZZ  ^m  V )  |->  ( A  x.  B ) )  e.  (mzPoly `  V ) )
 
Theoremmzpsubmpt 26924* The difference of two polynomial functions is polynomial. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( ( x  e.  ( ZZ  ^m  V )  |->  A )  e.  (mzPoly `  V )  /\  ( x  e.  ( ZZ  ^m  V )  |->  B )  e.  (mzPoly `  V ) )  ->  ( x  e.  ( ZZ  ^m  V )  |->  ( A  -  B ) )  e.  (mzPoly `  V ) )
 
Theoremmzpnegmpt 26925* Negation of a polynomial function. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( x  e.  ( ZZ  ^m  V )  |->  A )  e.  (mzPoly `  V )  ->  ( x  e.  ( ZZ  ^m  V )  |->  -u A )  e.  (mzPoly `  V )
 )
 
Theoremmzpexpmpt 26926* Raise a polynomial function to a (fixed) exponent. (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |-  (
 ( ( x  e.  ( ZZ  ^m  V )  |->  A )  e.  (mzPoly `  V )  /\  D  e.  NN0 )  ->  ( x  e.  ( ZZ  ^m  V )  |->  ( A ^ D ) )  e.  (mzPoly `  V ) )
 
Theoremmzpindd 26927* "Structural" induction to prove properties of all polynomial functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( ph  /\  f  e. 
 ZZ )  ->  ch )   &    |-  (
 ( ph  /\  f  e.  V )  ->  th )   &    |-  (
 ( ph  /\  ( f : ( ZZ  ^m  V ) --> ZZ  /\  ta )  /\  ( g : ( ZZ  ^m  V ) --> ZZ  /\  et ) )  ->  ze )   &    |-  (
 ( ph  /\  ( f : ( ZZ  ^m  V ) --> ZZ  /\  ta )  /\  ( g : ( ZZ  ^m  V ) --> ZZ  /\  et ) )  ->  si )   &    |-  ( x  =  ( ( ZZ  ^m  V )  X.  { f } )  ->  ( ps  <->  ch ) )   &    |-  ( x  =  ( g  e.  ( ZZ  ^m  V )  |->  ( g `  f ) )  ->  ( ps  <->  th ) )   &    |-  ( x  =  f  ->  ( ps  <->  ta ) )   &    |-  ( x  =  g  ->  ( ps  <->  et ) )   &    |-  ( x  =  ( f  o F  +  g
 )  ->  ( ps  <->  ze ) )   &    |-  ( x  =  ( f  o F  x.  g )  ->  ( ps 
 <-> 
 si ) )   &    |-  ( x  =  A  ->  ( ps  <->  rh ) )   =>    |-  ( ( ph  /\  A  e.  (mzPoly `  V ) )  ->  rh )
 
Theoremmzpmfp 26928 Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.)
 |-  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) )
 
Theoremmzpsubst 26929* Substituting polynomials for the variables of a polynomial results in a polynomial.  G is expected to depend on  y and provide the polynomials which are being substituted. (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |-  (
 ( W  e.  _V  /\  F  e.  (mzPoly `  V )  /\  A. y  e.  V  G  e.  (mzPoly `  W ) )  ->  ( x  e.  ( ZZ  ^m  W )  |->  ( F `  ( y  e.  V  |->  ( G `
  x ) ) ) )  e.  (mzPoly `  W ) )
 
Theoremmzprename 26930* Simplified version of mzpsubst 26929 to simply relabel variables in a polynomial. (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |-  (
 ( W  e.  _V  /\  F  e.  (mzPoly `  V )  /\  R : V --> W )  ->  ( x  e.  ( ZZ  ^m  W )  |->  ( F `
  ( x  o.  R ) ) )  e.  (mzPoly `  W ) )
 
Theoremmzpresrename 26931* 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.)
 |-  (
 ( W  e.  _V  /\  V  C_  W  /\  F  e.  (mzPoly `  V ) )  ->  ( x  e.  ( ZZ  ^m  W )  |->  ( F `
  ( x  |`  V ) ) )  e.  (mzPoly `  W ) )
 
Theoremmzpcompact2lem 26932* Lemma for mzpcompact2 26933. (Contributed by Stefan O'Rear, 9-Oct-2014.)
 |-  B  e.  _V   =>    |-  ( A  e.  (mzPoly `  B )  ->  E. a  e.  Fin  E. b  e.  (mzPoly `  a )
 ( a  C_  B  /\  A  =  ( c  e.  ( ZZ  ^m  B )  |->  ( b `
  ( c  |`  a ) ) ) ) )
 
Theoremmzpcompact2 26933* 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.)
 |-  ( A  e.  (mzPoly `  B )  ->  E. a  e.  Fin  E. b  e.  (mzPoly `  a
 ) ( a  C_  B  /\  A  =  ( c  e.  ( ZZ 
 ^m  B )  |->  ( b `  ( c  |`  a ) ) ) ) )
 
18.17.10  Miscellanea for Diophantine sets 1
 
Theoremcoeq0 26934 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 5190 and coundir 5191 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  (
 ( A  o.  B )  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
 
Theoremcoeq0i 26935 coeq0 26934 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( A  o.  B )  =  (/) )
 
Theoremfzsplit1nn0 26936 Split a finite 1-based set of integers in the middle, allowing either end to be empty ( ( 1 ... 0 )). (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0  /\  A  <_  B )  ->  (
 1 ... B )  =  ( ( 1 ...
 A )  u.  (
 ( A  +  1 ) ... B ) ) )
 
18.17.11  Diophantine sets 1: definitions
 
Syntaxcdioph 26937 Extend class notation to include the family of Diophantine sets.
 class Dioph
 
Definitiondf-dioph 26938* 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  ZZ (via mzPoly) and  NN0 (to define the zero sets); the former could be avoided by considering coincidence sets of  NN0 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 13027 that implicitly restricting variables to  NN0 adds no expressive power over allowing them to range over  ZZ. While this definition stipulates a specific index set for the polynomials, there is actually flexibility here, see eldioph2b 26945. (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |- Dioph  =  ( n  e.  NN0  |->  ran  (
 k  e.  ( ZZ>= `  n ) ,  p  e.  (mzPoly `  ( 1 ... k ) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... n ) )  /\  ( p `  u )  =  0 ) }
 ) )
 
Theoremeldiophb 26939* Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  ( D  e.  (Dioph `  N ) 
 <->  ( N  e.  NN0  /\ 
 E. k  e.  ( ZZ>=
 `  N ) E. p  e.  (mzPoly `  (
 1 ... k ) ) D  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `
  u )  =  0 ) } )
 )
 
Theoremeldioph 26940* Condition for a set to be Diophantine (unpacking existential quantifier) (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ...
 K ) ) ) 
 ->  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `
  u )  =  0 ) }  e.  (Dioph `  N ) )
 
Theoremdiophrw 26941* Renaming and adding unused witness variables does not change the Diophantine set coded by a polynomial. (Contributed by Stefan O'Rear, 7-Oct-2014.)
 |-  (
 ( S  e.  _V  /\  M : T -1-1-> S  /\  ( M  |`  O )  =  (  _I  |`  O ) )  ->  { a  |  E. b  e.  ( NN0  ^m  S ) ( a  =  ( b  |`  O )  /\  (
 ( d  e.  ( ZZ  ^m  S )  |->  ( P `  ( d  o.  M ) ) ) `  b )  =  0 ) }  =  { a  |  E. c  e.  ( NN0  ^m  T ) ( a  =  ( c  |`  O )  /\  ( P `
  c )  =  0 ) } )
 
Theoremeldioph2lem1 26942* Lemma for eldioph2 26944. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  A  e.  Fin  /\  ( 1 ... N )  C_  A )  ->  E. d  e.  ( ZZ>=
 `  N ) E. e  e.  _V  (
 e : ( 1
 ... d ) -1-1-onto-> A  /\  ( e  |`  ( 1
 ... N ) )  =  (  _I  |`  ( 1
 ... N ) ) ) )
 
Theoremeldioph2lem2 26943* Lemma for eldioph2 26944. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  (
 ( ( N  e.  NN0  /\  -.  S  e.  Fin )  /\  ( ( 1
 ... N )  C_  S  /\  A  e.  ( ZZ>=
 `  N ) ) )  ->  E. c
 ( c : ( 1 ... A )
 -1-1-> S  /\  ( c  |`  ( 1 ... N ) )  =  (  _I  |`  ( 1 ...
 N ) ) ) )
 
Theoremeldioph2 26944* Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 26933. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
 |-  (
 ( N  e.  NN0  /\  ( S  e.  _V  /\  ( 1 ... N )  C_  S )  /\  P  e.  (mzPoly `  S ) )  ->  { t  |  E. u  e.  ( NN0  ^m  S ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `
  u )  =  0 ) }  e.  (Dioph `  N ) )
 
Theoremeldioph2b 26945* 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  ( S  \ 
( 1 ... N
) ). For instance, in diophin 26955 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  (
 ( ( N  e.  NN0  /\  S  e.  _V )  /\  ( -.  S  e.  Fin  /\  ( 1 ... N )  C_  S ) ) 
 ->  ( A  e.  (Dioph `  N )  <->  E. p  e.  (mzPoly `  S ) A  =  { t  |  E. u  e.  ( NN0  ^m  S ) ( t  =  ( u  |`  ( 1
 ... N ) ) 
 /\  ( p `  u )  =  0
 ) } ) )
 
Theoremeldiophelnn0 26946 Remove antecedent on  B from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  (Dioph `  B )  ->  B  e.  NN0 )
 
Theoremeldioph3b 26947* Define Diophantine sets in terms of polynomials with variables indexed by  NN. This avoids a quantifier over the number of witness variables and will be easier to use than eldiophb 26939 in most cases. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  (Dioph `  N ) 
 <->  ( N  e.  NN0  /\ 
 E. p  e.  (mzPoly `  NN ) A  =  { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  ( 1
 ... N ) ) 
 /\  ( p `  u )  =  0
 ) } ) )
 
Theoremeldioph3 26948* Inference version of eldioph3b 26947 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  P  e.  (mzPoly `  NN ) )  ->  { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `
  u )  =  0 ) }  e.  (Dioph `  N ) )
 
18.17.12  Diophantine sets 2 miscellanea
 
Theoremellz1 26949 Membership in a set of lower integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)
 |-  ( B  e.  ZZ  ->  ( A  e.  ( ZZ  \  ( ZZ>= `  ( B  +  1 ) ) )  <->  ( A  e.  ZZ  /\  A  <_  B ) ) )
 
Theoremlzunuz 26950 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.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ  /\  B  <_  ( A  +  1 ) )  ->  ( ( ZZ  \  ( ZZ>= `  ( A  +  1 ) ) )  u.  ( ZZ>= `  B ) )  =  ZZ )
 
Theoremfz1eqin 26951 Express a one-based finite range as the intersection of lower integers with  NN. (Contributed by Stefan O'Rear, 9-Oct-2014.)
 |-  ( N  e.  NN0  ->  (
 1 ... N )  =  ( ( ZZ  \  ( ZZ>= `  ( N  +  1 ) ) )  i^i  NN )
 )
 
Theoremlzenom 26952 Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( N  e.  ZZ  ->  ( ZZ  \  ( ZZ>= `  ( N  +  1
 ) ) )  ~~  om )
 
Theoremelmapresaun 26953 fresaun 5428 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  (
 ( F  e.  ( C  ^m  A )  /\  G  e.  ( C  ^m  B )  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) ) 
 ->  ( F  u.  G )  e.  ( C  ^m  ( A  u.  B ) ) )
 
Theoremelmapresaunres2 26954 fresaunres2 5429 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.)
 |-  (
 ( F  e.  ( C  ^m  A )  /\  G  e.  ( C  ^m  B )  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) ) 
 ->  ( ( F  u.  G )  |`  B )  =  G )
 
18.17.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra
 
Theoremdiophin 26955 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.)
 |-  (
 ( A  e.  (Dioph `  N )  /\  B  e.  (Dioph `  N )
 )  ->  ( A  i^i  B )  e.  (Dioph `  N ) )
 
Theoremdiophun 26956 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.)
 |-  (
 ( A  e.  (Dioph `  N )  /\  B  e.  (Dioph `  N )
 )  ->  ( A  u.  B )  e.  (Dioph `  N ) )
 
Theoremeldiophss 26957 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.)
 |-  ( A  e.  (Dioph `  B )  ->  A  C_  ( NN0  ^m  ( 1 ...
 B ) ) )
 
18.17.14  Diophantine sets 3: construction
 
Theoremdiophrex 26958* Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M ) ) 
 ->  { t  |  E. u  e.  S  t  =  ( u  |`  ( 1
 ... N ) ) }  e.  (Dioph `  N ) )
 
Theoremeq0rabdioph 26959* 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.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  A  =  0 }  e.  (Dioph `  N ) )
 
Theoremeqrabdioph 26960* 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.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 /\  ( t  e.  ( ZZ  ^m  (
 1 ... N ) ) 
 |->  B )  e.  (mzPoly `  ( 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  A  =  B }  e.  (Dioph `  N )
 )
 
Theorem0dioph 26961 The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  NN0  ->  (/)  e.  (Dioph `  A ) )
 
Theoremvdioph 26962 The "universal" set (as large as possible given eldiophss 26957) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  NN0  ->  ( NN0  ^m  ( 1 ...
 A ) )  e.  (Dioph `  A )
 )
 
Theoremanrabdioph 26963* Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ph }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  ps }  e.  (Dioph `  N ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ( ph  /\  ps ) }  e.  (Dioph `  N ) )
 
Theoremorrabdioph 26964* Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ph }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  ps }  e.  (Dioph `  N ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ( ph  \/  ps ) }  e.  (Dioph `  N ) )
 
Theorem3anrabdioph 26965* Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ph }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  ps }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ch }  e.  (Dioph `  N ) ) 
 ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  ( ph  /\  ps  /\ 
 ch ) }  e.  (Dioph `  N ) )
 
Theorem3orrabdioph 26966* Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ph }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  ps }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ch }  e.  (Dioph `  N ) ) 
 ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  ( ph  \/  ps  \/  ch ) }  e.  (Dioph `  N ) )
 
18.17.15  Diophantine sets 4 miscellanea
 
Theorem2sbcrex 26967* Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( [. A  /  a ]. [. B  /  b ]. E. c  e.  C  ph  <->  E. c  e.  C  [. A  /  a ]. [. B  /  b ]. ph )
 
Theoremsbc2rexg 26968* Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  ( A  e.  V  ->  (
 [. A  /  a ]. E. b  e.  B  E. c  e.  C  ph  <->  E. b  e.  B  E. c  e.  C  [. A  /  a ]. ph )
 )
 
Theoremsbc4rexg 26969* Exchange a substitution with 4 existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  ( A  e.  V  ->  (
 [. A  /  a ]. E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  ph  <->  E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  [. A  /  a ]. ph ) )
 
TheoremsbcbiiiOLD 26970 Fully inferenced rewriting under an explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  _V   &    |-  ( ph  <->  ps )   =>    |-  ( [. A  /  a ]. ph  <->  [. A  /  a ]. ps )
 
Theoremsbcrot3 26971* Rotate a sequence of three explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. A  /  a ]. ph )
 
Theoremsbcrot5 26972* Rotate a sequence of five explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. [. D  /  d ]. [. E  /  e ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. D  /  d ]. [. E  /  e ]. [. A  /  a ]. ph )
 
Theoremsbccomieg 26973* 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.)
 |-  (
 a  =  A  ->  B  =  C )   =>    |-  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph )
 
Theoremsbcrot3gOLD 26974* Rotate a sequence of three explicit substitutions, closed theorem. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( A  e.  D  /\  B  e.  E  /\  A. b  C  e.  F )  ->  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. A  /  a ]. ph ) )
 
Theoremsbcrot3OLD 26975* Rotate a sequence of three explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. A  /  a ]. ph )
 
Theoremsbcrot5OLD 26976* Rotate a sequence of five explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   =>    |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. [. D  /  d ]. [. E  /  e ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. D  /  d ]. [. E  /  e ]. [. A  /  a ]. ph )
 
TheoremsbccomiegOLD 26977* Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 a  =  A  ->  B  =  C )   =>    |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
 
18.17.16  Diophantine sets 4: Quantification
 
Theoremrexrabdioph 26978* Diophantine set builder for existential quantification. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  M  =  ( N  +  1 )   &    |-  ( v  =  ( t `  M )  ->  ( ps  <->  ch ) )   &    |-  ( u  =  ( t  |`  ( 1 ... N ) )  ->  ( ch  <->  ph ) )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  ( 1 ...
 M ) )  | 
 ph }  e.  (Dioph `  M ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  ps
 }  e.  (Dioph `  N ) )
 
Theoremrexfrabdioph 26979* Diophantine set builder for existential quantifier, explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  M  =  ( N  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  ( 1 ...
 M ) )  | 
 [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. ph }  e.  (Dioph `  M ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  ph
 }  e.  (Dioph `  N ) )
 
Theorem2rexfrabdioph 26980* 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.)
 |-  M  =  ( N  +  1 )   &    |-  L  =  ( M  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  ( 1 ...
 L ) )  | 
 [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ].
 ph }  e.  (Dioph `  L ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  E. w  e.  NN0  ph }  e.  (Dioph `  N ) )
 
Theorem3rexfrabdioph 26981* 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.)
 |-  M  =  ( N  +  1 )   &    |-  L  =  ( M  +  1 )   &    |-  K  =  ( L  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  (
 1 ... K ) )  |  [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ].
 [. ( t `  K )  /  x ].
 ph }  e.  (Dioph `  K ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  N ) )
 
Theorem4rexfrabdioph 26982* 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.)
 |-  M  =  ( N  +  1 )   &    |-  L  =  ( M  +  1 )   &    |-  K  =  ( L  +  1 )   &    |-  J  =  ( K  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  ( 1 ...
 J ) )  | 
 [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ].
 [. ( t `  K )  /  x ].
 [. ( t `  J )  /  y ]. ph }  e.  (Dioph `  J ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  E. y  e. 
 NN0  ph }  e.  (Dioph `  N ) )
 
Theorem6rexfrabdioph 26983* 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.)
 |-  M  =  ( N  +  1 )   &    |-  L  =  ( M  +  1 )   &    |-  K  =  ( L  +  1 )   &    |-  J  =  ( K  +  1 )   &    |-  I  =  ( J  +  1 )   &    |-  H  =  ( I  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  (
 1 ... H ) )  |  [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ].
 [. ( t `  K )  /  x ].
 [. ( t `  J )  /  y ]. [. ( t `  I )  /  z ]. [. ( t `  H )  /  p ].
 ph }  e.  (Dioph `  H ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  E. y  e. 
 NN0  E. z  e.  NN0  E. p  e.  NN0  ph }  e.  (Dioph `  N ) )
 
Theorem7rexfrabdioph 26984* 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.)
 |-  M  =  ( N  +  1 )   &    |-  L  =  ( M  +  1 )   &    |-  K  =  ( L  +  1 )   &    |-  J  =  ( K  +  1 )   &    |-  I  =  ( J  +  1 )   &    |-  H  =  ( I  +  1 )   &    |-  G  =  ( H  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  ( 1 ...
 G ) )  | 
 [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ].
 [. ( t `  K )  /  x ].
 [. ( t `  J )  /  y ]. [. ( t `  I )  /  z ]. [. ( t `  H )  /  p ].
 [. ( t `  G )  /  q ]. ph }  e.  (Dioph `  G ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  E. y  e. 
 NN0  E. z  e.  NN0  E. p  e.  NN0  E. q  e.  NN0  ph }  e.  (Dioph `  N ) )
 
18.17.17  Diophantine sets 5: Arithmetic sets
 
Theoremrabdiophlem1 26985* Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 2631. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 ->  A. t  e.  ( NN0  ^m  ( 1 ...
 N ) ) A  e.  ZZ )
 
Theoremrabdiophlem2 26986* Lemma for arithmetic diophantine sets. Reuse a polynomial expression under a new quantifier. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  M  =  ( N  +  1 )   =>    |-  ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) )  ->  ( t  e.  ( ZZ  ^m  (
 1 ... M ) ) 
 |->  [_ ( t  |`  ( 1 ... N ) )  /  u ]_ A )  e.  (mzPoly `  ( 1 ... M ) ) )
 
Theoremelnn0rabdioph 26987* 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.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  A  e.  NN0 }  e.  (Dioph `  N ) )
 
Theoremrexzrexnn0 26988* Rewrite a quantification over integers into a quantification over naturals. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  -u y  ->  ( ph  <->  ch ) )   =>    |-  ( E. x  e.  ZZ  ph  <->  E. y  e.  NN0  ( ps  \/  ch )
 )
 
Theoremlerabdioph 26989* Diophantine set builder for the less or equals relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 /\  ( t  e.  ( ZZ  ^m  (
 1 ... N ) ) 
 |->  B )  e.  (mzPoly `  ( 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  A  <_  B }  e.  (Dioph `  N )
 )
 
Theoremeluzrabdioph 26990* Diophantine set builder for membership in a fixed set of upper integers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  M  e.  ZZ  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  A  e.  ( ZZ>=
 `  M ) }  e.  (Dioph `  N )
 )
 
Theoremelnnrabdioph 26991* Diophantine set builder for positivity. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  A  e.  NN }  e.  (Dioph `  N ) )
 
Theoremltrabdioph 26992* Diophantine set builder for the strict less than relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 /\  ( t  e.  ( ZZ  ^m  (
 1 ... N ) ) 
 |->  B )  e.  (mzPoly `  ( 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  A  <  B }  e.  (Dioph `  N )
 )
 
Theoremnerabdioph 26993* 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.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 /\  ( t  e.  ( ZZ  ^m  (
 1 ... N ) ) 
 |->  B )  e.  (mzPoly `  ( 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  A  =/=  B }  e.  (Dioph `  N )
 )
 
Theoremdvdsrabdioph 26994* Divisibility is a Diophantine relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 /\  ( t  e.  ( ZZ  ^m  (
 1 ... N ) ) 
 |->  B )  e.  (mzPoly `  ( 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  A  ||  B }  e.  (Dioph `  N )
 )
 
18.17.18  Diophantine sets 6 miscellanea
 
Theoremfz1ssnn 26995 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 1 ... A )  C_  NN
 
Theoremftp 26996 A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  Z  e.  _V   &    |-  A  =/=  B   &    |-  A  =/=  C   &    |-  B  =/=  C   =>    |- 
 { <. A ,  X >. ,  <. B ,  Y >. ,  <. C ,  Z >. } : { A ,  B ,  C } --> { X ,  Y ,  Z }
 
18.17.19  Diophantine sets 6: reusability. renumbering of variables
 
Theoremeldioph4b 26997* 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.)
 |-  W  e.  _V   &    |-  -.  W  e.  Fin   &    |-  ( W  i^i  NN )  =  (/)   =>    |-  ( S  e.  (Dioph `  N )  <->  ( N  e.  NN0  /\  E. p  e.  (mzPoly `  ( W  u.  (
 1 ... N ) ) ) S  =  {
 t  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. w  e.  ( NN0  ^m  W ) ( p `  ( t  u.  w ) )  =  0 } ) )
 
Theoremeldioph4i 26998* Forward-only version of eldioph4b 26997. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  W  e.  _V   &    |-  -.  W  e.  Fin   &    |-  ( W  i^i  NN )  =  (/)   =>    |-  ( ( N  e.  NN0  /\  P  e.  (mzPoly `  ( W  u.  ( 1 ...
 N ) ) ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  E. w  e.  ( NN0  ^m  W ) ( P `  ( t  u.  w ) )  =  0 }  e.  (Dioph `  N ) )
 
Theoremdiophren 26999* 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.)
 |-  (
 ( S  e.  (Dioph `  N )  /\  M  e.  NN0  /\  F :
 ( 1 ... N )
 --> ( 1 ... M ) )  ->  { a  e.  ( NN0  ^m  (
 1 ... M ) )  |  ( a  o.  F )  e.  S }  e.  (Dioph `  M ) )
 
Theoremrabrenfdioph 27000* Change variable numbers in a Diophantine class abstraction using explicit substitution. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  (
 ( B  e.  NN0  /\  F : ( 1
 ... A ) --> ( 1
 ... B )  /\  { a  e.  ( NN0  ^m  ( 1 ... A ) )  |  ph }  e.  (Dioph `  A ) ) 
 ->  { b  e.  ( NN0  ^m  ( 1 ...
 B ) )  | 
 [. ( b  o.  F )  /  a ]. ph }  e.  (Dioph `  B ) )
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