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Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
16.16.4  Additional topology
 
Theoremelrfi 26101* Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (
 ( B  e.  V  /\  C  C_  ~P B )  ->  ( A  e.  ( fi `  ( { B }  u.  C ) )  <->  E. v  e.  ( ~P C  i^i  Fin ) A  =  ( B  i^i  |^| v ) ) )
 
Theoremelrfirn 26102* Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (
 ( B  e.  V  /\  F : I --> ~P B )  ->  ( A  e.  ( fi `  ( { B }  u.  ran  F ) )  <->  E. v  e.  ( ~P I  i^i  Fin ) A  =  ( B  i^i  |^|_ y  e.  v  ( F `  y ) ) ) )
 
Theoremelrfirn2 26103* Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (
 ( B  e.  V  /\  A. y  e.  I  C  C_  B )  ->  ( A  e.  ( fi `  ( { B }  u.  ran  (  y  e.  I  |->  C ) ) )  <->  E. v  e.  ( ~P I  i^i  Fin ) A  =  ( B  i^i  |^|_ y  e.  v  C ) ) )
 
Theoremcmpfiiin 26104* 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.)
 |-  X  =  U. J   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ( ph  /\  k  e.  I ) 
 ->  S  e.  ( Clsd `  J ) )   &    |-  (
 ( ph  /\  ( l 
 C_  I  /\  l  e.  Fin ) )  ->  ( X  i^i  |^|_ k  e.  l  S )  =/=  (/) )   =>    |-  ( ph  ->  ( X  i^i  |^|_ k  e.  I  S )  =/=  (/) )
 
16.16.5  Characterization of closure operators. Kuratowski closure axioms
 
Theoremismrcd1 26105* Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 13446), isotone (satisfies mrcss 13445), and idempotent (satisfies mrcidm 13448) 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 26106 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  F : ~P B --> ~P B )   &    |-  ( ( ph  /\  x  C_  B )  ->  x  C_  ( F `  x ) )   &    |-  ( ( ph  /\  x  C_  B  /\  y  C_  x )  ->  ( F `  y ) 
 C_  ( F `  x ) )   &    |-  (
 ( ph  /\  x  C_  B )  ->  ( F `
  ( F `  x ) )  =  ( F `  x ) )   =>    |-  ( ph  ->  dom  (  F  i^i  _I  )  e.  (Moore `  B )
 )
 
Theoremismrcd2 26106* Second half of ismrcd1 26105. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  F : ~P B --> ~P B )   &    |-  ( ( ph  /\  x  C_  B )  ->  x  C_  ( F `  x ) )   &    |-  ( ( ph  /\  x  C_  B  /\  y  C_  x )  ->  ( F `  y ) 
 C_  ( F `  x ) )   &    |-  (
 ( ph  /\  x  C_  B )  ->  ( F `
  ( F `  x ) )  =  ( F `  x ) )   =>    |-  ( ph  ->  F  =  (mrCls `  dom  (  F  i^i  _I  ) ) )
 
Theoremistopclsd 26107* A closure function which satisfies sscls 16720, clsidm 16731, cls0 16744, and clsun 25578 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  F : ~P B --> ~P B )   &    |-  ( ( ph  /\  x  C_  B )  ->  x  C_  ( F `  x ) )   &    |-  ( ( ph  /\  x  C_  B )  ->  ( F `  ( F `  x ) )  =  ( F `  x ) )   &    |-  ( ph  ->  ( F `  (/) )  =  (/) )   &    |-  (
 ( ph  /\  x  C_  B  /\  y  C_  B )  ->  ( F `  ( x  u.  y
 ) )  =  ( ( F `  x )  u.  ( F `  y ) ) )   &    |-  J  =  { z  e.  ~P B  |  ( F `  ( B 
 \  z ) )  =  ( B  \  z ) }   =>    |-  ( ph  ->  ( J  e.  (TopOn `  B )  /\  ( cls `  J )  =  F ) )
 
Theoremismrc 26108* A function is a Moore closure operator iff it satisfies mrcssid 13446, mrcss 13445, and mrcidm 13448. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( F  e.  (mrCls " (Moore `  B ) )  <->  ( B  e.  _V 
 /\  F : ~P B
 --> ~P B  /\  A. x A. y ( ( x  C_  B  /\  y  C_  x )  ->  ( x  C_  ( F `
  x )  /\  ( F `  y ) 
 C_  ( F `  x )  /\  ( F `
  ( F `  x ) )  =  ( F `  x ) ) ) ) )
 
16.16.6  Algebraic closure systems
 
Syntaxcnacs 26109 Class of Noetherian closure systems.
 class NoeACS
 
Definitiondf-nacs 26110* 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.)
 |- NoeACS  =  ( x  e.  _V  |->  { c  e.  (ACS `  x )  |  A. s  e.  c  E. g  e.  ( ~P x  i^i  Fin ) s  =  ( (mrCls `  c ) `  g ) } )
 
Theoremisnacs 26111* Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (NoeACS `  X ) 
 <->  ( C  e.  (ACS `  X )  /\  A. s  e.  C  E. g  e.  ( ~P X  i^i  Fin ) s  =  ( F `  g ) ) )
 
Theoremnacsfg 26112* In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  (
 ( C  e.  (NoeACS `  X )  /\  S  e.  C )  ->  E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g ) )
 
Theoremisnacs2 26113 Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (NoeACS `  X ) 
 <->  ( C  e.  (ACS `  X )  /\  ( F " ( ~P X  i^i  Fin ) )  =  C ) )
 
Theoremmrefg2 26114* Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  ( E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g )  <->  E. g  e.  ( ~P S  i^i  Fin ) S  =  ( F `  g ) ) )
 
Theoremmrefg3 26115* Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  (
 ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  ( E. g  e.  ( ~P X  i^i  Fin ) S  =  ( F `  g )  <->  E. g  e.  ( ~P S  i^i  Fin ) S  C_  ( F `  g ) ) )
 
Theoremnacsacs 26116 A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  ( C  e.  (NoeACS `  X )  ->  C  e.  (ACS `  X ) )
 
Theoremisnacs3 26117* A choice-free order equivalent to the Noetherian condition on a closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  ( C  e.  (NoeACS `  X ) 
 <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  s
 ) ) )
 
Theoremincssnn0 26118* Transitivity induction of subsets, lemma for nacsfix 26119. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  (
 ( A. x  e.  NN0  ( F `  x ) 
 C_  ( F `  ( x  +  1
 ) )  /\  A  e.  NN0  /\  B  e.  ( ZZ>= `  A )
 )  ->  ( F `  A )  C_  ( F `  B ) )
 
Theoremnacsfix 26119* An increasing sequence of closed sets in a Noetherian-type closure system eventually fixates. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  (
 ( C  e.  (NoeACS `  X )  /\  F : NN0 --> C  /\  A. x  e.  NN0  ( F `
  x )  C_  ( F `  ( x  +  1 ) ) )  ->  E. y  e.  NN0  A. z  e.  ( ZZ>=
 `  y ) ( F `  z )  =  ( F `  y ) )
 
16.16.7  Miscellanea 1. Map utilities
 
Theoremconstmap 26120 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.)

 |-  A  e.  _V   &    |-  C  e.  _V   =>    |-  ( B  e.  C  ->  ( A  X.  { B } )  e.  ( C  ^m  A ) )
 
Theoremelmapfun 26121 A mapping is always a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  ( A  e.  ( B  ^m  C )  ->  Fun  A )
 
Theoremmapco2g 26122 Renaming indexes in a tuple, with sethood as antecedents. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  (
 ( E  e.  _V  /\  A  e.  ( B 
 ^m  C )  /\  D : E --> C ) 
 ->  ( A  o.  D )  e.  ( B  ^m  E ) )
 
Theoremmapco2 26123 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.)
 |-  E  e.  _V   =>    |-  ( ( A  e.  ( B  ^m  C ) 
 /\  D : E --> C )  ->  ( A  o.  D )  e.  ( B  ^m  E ) )
 
Theoremelmapssres 26124 A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  (
 ( A  e.  ( B  ^m  C )  /\  D  C_  C )  ->  ( A  |`  D )  e.  ( B  ^m  D ) )
 
Theoremmapfzcons 26125 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.)
 |-  M  =  ( N  +  1 )   =>    |-  ( ( N  e.  NN0  /\  A  e.  ( B 
 ^m  ( 1 ...
 N ) )  /\  C  e.  B )  ->  ( A  u.  { <. M ,  C >. } )  e.  ( B 
 ^m  ( 1 ...
 M ) ) )
 
Theoremmapfzcons1 26126 Recover prefix mapping from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  M  =  ( N  +  1 )   =>    |-  ( A  e.  ( B  ^m  ( 1 ...
 N ) )  ->  ( ( A  u.  {
 <. M ,  C >. } )  |`  ( 1 ... N ) )  =  A )
 
Theoremmapfzcons1cl 26127 A nonempty mapping has a prefix. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  M  =  ( N  +  1 )   =>    |-  ( A  e.  ( B  ^m  ( 1 ...
 M ) )  ->  ( A  |`  ( 1
 ... N ) )  e.  ( B  ^m  ( 1 ... N ) ) )
 
Theoremmapfzcons2 26128 Recover added element from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  M  =  ( N  +  1 )   =>    |-  ( ( A  e.  ( B  ^m  ( 1
 ... N ) ) 
 /\  C  e.  B )  ->  ( ( A  u.  { <. M ,  C >. } ) `  M )  =  C )
 
16.16.8  Miscellanea for polynomials
 
Theoremofmpteq 26129* Value of a pointwise operation on two functions defined using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |-  (
 ( A  e.  V  /\  ( x  e.  A  |->  B )  Fn  A  /\  ( x  e.  A  |->  C )  Fn  A )  ->  ( ( x  e.  A  |->  B )  o F R ( x  e.  A  |->  C ) )  =  ( x  e.  A  |->  ( B R C ) ) )
 
Theoremmptfcl 26130* 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 ) )
 
16.16.9  Multivariate polynomials over the integers
 
Syntaxcmzpcl 26131 Extend class notation to include pre-polynomial rings.
 class mzPolyCld
 
Syntaxcmzp 26132 Extend class notation to include polynomial rings.
 class mzPoly
 
Definitiondf-mzpcl 26133* 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 26134. (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 26134 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 26135* 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 26136* 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 26137 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 26134 is well defined. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  ( ZZ  ^m  ( ZZ 
 ^m  V ) )  e.  (mzPolyCld `  V ) )
 
Theoremmzpcln0 26138 Corrolary of mzpclall 26137: Polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  (mzPolyCld `  V )  =/=  (/) )
 
Theoremmzpcl1 26139 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 26140* 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 26141 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 26142 Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  (mzPoly `  V )  =  |^| (mzPolyCld `
  V ) )
 
Theoremdmmzp 26143 mzPoly is defined for all index sets which are sets. This is used with elfvdm 5453 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  dom mzPoly  =  _V
 
Theoremmzpincl 26144 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 26145 Constant functions are polynomial. See also mzpconstmpt 26150. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( V  e.  _V  /\  C  e.  ZZ )  ->  ( ( ZZ  ^m  V )  X.  { C } )  e.  (mzPoly `  V ) )
 
Theoremmzpf 26146 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 26147* 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 26148 The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 26151. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  (mzPoly `  V )  /\  B  e.  (mzPoly `  V )
 )  ->  ( A  o F  +  B )  e.  (mzPoly `  V ) )
 
Theoremmzpmul 26149 The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 26152. (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 26150* A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 26151, mzpmulmpt 26152, mzpnegmpt 26154, mzpsubmpt 26153, mzpexpmpt 26155) 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 26147 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 26151* Sum of polynomial functions is polynomial. Maps-to version of mzpadd 26148. (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 26152* Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 26152. (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 26153* 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 26154* 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 26155* 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 26156* "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 26157 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 26158* 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 26159* Simplified version of mzpsubst 26158 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 26160* 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 26161* Lemma for mzpcompact2 26162. (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 26162* 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 ) ) ) ) )
 
16.16.10  Miscellanea for Diophantine sets 1
 
Theoremcoeq0 26163 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 5126 and coundir 5127 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 26164 coeq0 26163 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 26165 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 ) ) )
 
16.16.11  Diophantine sets 1: definitions
 
Syntaxcdioph 26166 Extend class notation to include the family of Diophantine sets.
 class Dioph
 
Definitiondf-dioph 26167* 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 12938 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 26174. (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 26168* 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 26169* 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 26170* 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 26171* Lemma for eldioph2 26173. 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 26172* Lemma for eldioph2 26173. 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 26173* Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 26162. (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 26174* 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 26184 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 26175 Remove antecedent on  B from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  (Dioph `  B )  ->  B  e.  NN0 )
 
Theoremeldioph3b 26176* 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 26168 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 26177* Inference version of eldioph3b 26176 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 ) )
 
16.16.12  Diophantine sets 2 miscellanea
 
Theoremellz1 26178 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 26179 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 26180 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 26181 Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( N  e.  ZZ  ->  ( ZZ  \  ( ZZ>= `  ( N  +  1
 ) ) )  ~~  om )
 
Theoremelmapresaun 26182 fresaun 5315 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 26183 fresaunres2 5316 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 )
 
16.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra
 
Theoremdiophin 26184 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 26185 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 26186 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 ) ) )
 
16.16.14  Diophantine sets 3: construction
 
Theoremdiophrex 26187* 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 26188* 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 26189* 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 26190 The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  NN0  ->  (/)  e.  (Dioph `  A ) )
 
Theoremvdioph 26191 The "universal" set (as large as possible given eldiophss 26186) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  NN0  ->  ( NN0  ^m  ( 1 ...
 A ) )  e.  (Dioph `  A )
 )
 
Theoremanrabdioph 26192* 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 26193* 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 26194* 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 26195* 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 ) )
 
16.16.15  Diophantine sets 4 miscellanea
 
Theorem2sbcrex 26196* 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 26197* 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 26198* 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 26199 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 26200* 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 )
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