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Theorem List for Metamath Proof Explorer - 26801-26900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmzpcl2 26801* 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 26802 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 26803 Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  (mzPoly `  V )  =  |^| (mzPolyCld `
  V ) )
 
Theoremdmmzp 26804 mzPoly is defined for all index sets which are sets. This is used with elfvdm 5760 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  dom mzPoly  =  _V
 
Theoremmzpincl 26805 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 26806 Constant functions are polynomial. See also mzpconstmpt 26811. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( V  e.  _V  /\  C  e.  ZZ )  ->  ( ( ZZ  ^m  V )  X.  { C } )  e.  (mzPoly `  V ) )
 
Theoremmzpf 26807 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 26808* 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 26809 The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 26812. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  (mzPoly `  V )  /\  B  e.  (mzPoly `  V )
 )  ->  ( A  o F  +  B )  e.  (mzPoly `  V ) )
 
Theoremmzpmul 26810 The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 26813. (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 26811* A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 26812, mzpmulmpt 26813, mzpnegmpt 26815, mzpsubmpt 26814, mzpexpmpt 26816) 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 26808 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 26812* Sum of polynomial functions is polynomial. Maps-to version of mzpadd 26809. (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 26813* Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 26813. (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 26814* 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 26815* 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 26816* 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 26817* "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 26818 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 26819* 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 26820* Simplified version of mzpsubst 26819 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 26821* 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 26822* Lemma for mzpcompact2 26823. (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 26823* 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 ) ) ) ) )
 
19.16.10  Miscellanea for Diophantine sets 1
 
Theoremcoeq0 26824 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 5374 and coundir 5375 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 26825 coeq0 26824 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 26826 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 ) ) )
 
19.16.11  Diophantine sets 1: definitions
 
Syntaxcdioph 26827 Extend class notation to include the family of Diophantine sets.
 class Dioph
 
Definitiondf-dioph 26828* 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 13337 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 26835. (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 26829* 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 26830* 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 26831* 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 26832* Lemma for eldioph2 26834. 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 26833* Lemma for eldioph2 26834. 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 26834* Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 26823. (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 26835* 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 26845 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 26836 Remove antecedent on  B from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  (Dioph `  B )  ->  B  e.  NN0 )
 
Theoremeldioph3b 26837* 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 26829 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 26838* Inference version of eldioph3b 26837 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 ) )
 
19.16.12  Diophantine sets 2 miscellanea
 
Theoremellz1 26839 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 26840 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 26841 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 26842 Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( N  e.  ZZ  ->  ( ZZ  \  ( ZZ>= `  ( N  +  1
 ) ) )  ~~  om )
 
Theoremelmapresaun 26843 fresaun 5617 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 26844 fresaunres2 5618 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 )
 
19.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra
 
Theoremdiophin 26845 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 26846 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 26847 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 ) ) )
 
19.16.14  Diophantine sets 3: construction
 
Theoremdiophrex 26848* 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 26849* 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 26850* 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 26851 The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  NN0  ->  (/)  e.  (Dioph `  A ) )
 
Theoremvdioph 26852 The "universal" set (as large as possible given eldiophss 26847) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  NN0  ->  ( NN0  ^m  ( 1 ...
 A ) )  e.  (Dioph `  A )
 )
 
Theoremanrabdioph 26853* 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 26854* 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 26855* 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 26856* 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 ) )
 
19.16.15  Diophantine sets 4 miscellanea
 
Theorem2sbcrex 26857* 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 26858* 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 26859* 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 26860 Fully inferenced rewriting under an explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  ( ph  <->  ps )   =>    |-  ( [. A  /  a ]. ph  <->  [. A  /  a ]. ps )
 
Theoremsbcrot3 26861* 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 26862* 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 26863* 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 26864* Rotate a sequence of three explicit substitutions, closed theorem. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( 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 26865* Rotate a sequence of three explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. A  /  a ]. ph )
 
Theoremsbcrot5OLD 26866* Rotate a sequence of five explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  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 26867* Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 a  =  A  ->  B  =  C )   =>    |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
 
19.16.16  Diophantine sets 4: Quantification
 
Theoremrexrabdioph 26868* 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 26869* 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 26870* 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 26871* 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 26872* 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 26873* 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 26874* 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 ) )
 
19.16.17  Diophantine sets 5: Arithmetic sets
 
Theoremrabdiophlem1 26875* Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 2783. (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 26876* 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 26877* 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 26878* 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 26879* 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 26880* 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 26881* 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 26882* 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 26883* 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 26884* 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 )
 )
 
19.16.18  Diophantine sets 6 miscellanea
 
Theoremfz1ssnn 26885 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 1 ... A )  C_  NN
 
19.16.19  Diophantine sets 6: reusability. renumbering of variables
 
Theoremeldioph4b 26886* 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 26887* Forward-only version of eldioph4b 26886. (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 26888* 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 26889* 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 ) )
 
Theoremrabren3dioph 26890* Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  (
 ( ( a `  1 )  =  (
 b `  X )  /\  ( a `  2
 )  =  ( b `
  Y )  /\  ( a `  3
 )  =  ( b `
  Z ) ) 
 ->  ( ph  <->  ps ) )   &    |-  X  e.  ( 1 ... N )   &    |-  Y  e.  ( 1
 ... N )   &    |-  Z  e.  ( 1 ... N )   =>    |-  ( ( N  e.  NN0  /\  { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  | 
 ph }  e.  (Dioph `  3 ) )  ->  { b  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  ps }  e.  (Dioph `  N ) )
 
19.16.20  Pigeonhole Principle and cardinality helpers
 
Theoremfphpd 26891* 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.)
 |-  ( ph  ->  B  ~<  A )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  B )   &    |-  ( x  =  y  ->  C  =  D )   =>    |-  ( ph  ->  E. x  e.  A  E. y  e.  A  ( x  =/=  y  /\  C  =  D ) )
 
Theoremfphpdo 26892* Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  B 
 ~<  A )   &    |-  ( ( ph  /\  z  e.  A ) 
 ->  C  e.  B )   &    |-  ( z  =  x  ->  C  =  D )   &    |-  ( z  =  y  ->  C  =  E )   =>    |-  ( ph  ->  E. x  e.  A  E. y  e.  A  ( x  < 
 y  /\  D  =  E ) )
 
Theoremctbnfien 26893 An infinite subset of a countable set is countable, without using choice. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  (
 ( ( X  ~~  om 
 /\  Y  ~~  om )  /\  ( A  C_  X  /\  -.  A  e.  Fin ) )  ->  A  ~~  Y )
 
Theoremfiphp3d 26894* Infinite pigeonhole principle for partitioning an infinite set between finitely many buckets. (Contributed by Stefan O'Rear, 18-Oct-2014.)
 |-  ( ph  ->  A  ~~  NN )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  D  e.  B )   =>    |-  ( ph  ->  E. y  e.  B  { x  e.  A  |  D  =  y }  ~~  NN )
 
19.16.21  A non-closed set of reals is infinite
 
Theoremrencldnfilem 26895* Lemma for rencldnfi 26896. (Contributed by Stefan O'Rear, 18-Oct-2014.)
 |-  (
 ( ( A  C_  RR  /\  B  e.  RR  /\  ( A  =/=  (/)  /\  -.  B  e.  A )
 )  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  ( y  -  B ) )  < 
 x )  ->  -.  A  e.  Fin )
 
Theoremrencldnfi 26896* A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 26895 using infima; this theorem removes the requirement that A be non-empty. (Contributed by Stefan O'Rear, 19-Oct-2014.)
 |-  (
 ( ( A  C_  RR  /\  B  e.  RR  /\ 
 -.  B  e.  A )  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  ( y  -  B ) )  < 
 x )  ->  -.  A  e.  Fin )
 
19.16.22  Miscellanea for Lagrange's theorem
 
Theoremicodiamlt 26897 Two elements in a half-open interval have separation strictly less than the difference between the endpoints. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,) B ) 
 /\  D  e.  ( A [,) B ) ) )  ->  ( abs `  ( C  -  D ) )  <  ( B  -  A ) )
 
Theoremmodelico 26898 Modular reduction produces a half-open interval. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  mod  B )  e.  ( 0 [,) B ) )
 
19.16.23  Lagrange's rational approximation theorem
 
Theoremirrapxlem1 26899* Lemma for irrapx1 26905. Divides the unit interval into  B half-open sections and using the pigeonhole principle fphpdo 26892 finds two multiples of  A in the same section mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  (
 ( A  e.  RR+  /\  B  e.  NN )  ->  E. x  e.  (
 0 ... B ) E. y  e.  ( 0 ... B ) ( x  <  y  /\  ( |_ `  ( B  x.  ( ( A  x.  x )  mod  1 ) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  y )  mod  1 ) ) ) ) )
 
Theoremirrapxlem2 26900* Lemma for irrapx1 26905. Two multiples in the same bucket means they are very close mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  (
 ( A  e.  RR+  /\  B  e.  NN )  ->  E. x  e.  (
 0 ... B ) E. y  e.  ( 0 ... B ) ( x  <  y  /\  ( abs `  ( ( ( A  x.  x ) 
 mod  1 )  -  ( ( A  x.  y )  mod  1 ) ) )  <  (
 1  /  B )
 ) )
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