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Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprtlem12 25901* Lemma for prtex 25914 and prter3 25916. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  (  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }  ->  Rel  .~  )
 
Theoremprtlem13 25902* Lemma for prter1 25913, prter2 25915, prter3 25916 and prtex 25914. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( z  .~  w 
 <-> 
 E. v  e.  A  ( z  e.  v  /\  w  e.  v
 ) )
 
Theoremprtlem16 25903* Lemma for prtex 25914, prter2 25915 and prter3 25916. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  dom  .~  =  U. A
 
Theoremprtlem400 25904* Lemma for prter2 25915 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  -.  (/)  e.  ( U. A /.  .~  )
 
Syntaxwprt 25905 Extend the definition of a wff to include the partition predicate.
 wff  Prt 
 A
 
Definitiondf-prt 25906* Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  ( Prt  A  <->  A. x  e.  A  A. y  e.  A  ( x  =  y  \/  ( x  i^i  y
 )  =  (/) ) )
 
Theoremerprt 25907 The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  (  .~  Er  X  ->  Prt  ( A /.  .~  ) )
 
Theoremprtlem14 25908* Lemma for prter1 25913, prter2 25915 and prtex 25914. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  ( Prt  A  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( w  e.  x  /\  w  e.  y )  ->  x  =  y ) ) )
 
Theoremprtlem15 25909* Lemma for prter1 25913 and prtex 25914. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  ( Prt  A  ->  ( E. x  e.  A  E. y  e.  A  ( ( u  e.  x  /\  w  e.  x )  /\  ( w  e.  y  /\  v  e.  y )
 )  ->  E. z  e.  A  ( u  e.  z  /\  v  e.  z ) ) )
 
Theoremprtlem17 25910* Lemma for prter2 25915. (Contributed by Rodolfo Medina, 15-Oct-2010.)
 |-  ( Prt  A  ->  ( ( x  e.  A  /\  z  e.  x )  ->  ( E. y  e.  A  ( z  e.  y  /\  w  e.  y )  ->  w  e.  x ) ) )
 
Theoremprtlem18 25911* Lemma for prter2 25915. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( Prt  A  ->  ( ( v  e.  A  /\  z  e.  v )  ->  ( w  e.  v  <->  z  .~  w ) ) )
 
Theoremprtlem19 25912* Lemma for prter2 25915. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( Prt  A  ->  ( ( v  e.  A  /\  z  e.  v )  ->  v  =  [ z ]  .~  ) )
 
Theoremprter1 25913* Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( Prt  A  ->  .~  Er  U. A )
 
Theoremprtex 25914* The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( Prt  A  ->  (  .~  e.  _V  <->  A  e.  _V ) )
 
Theoremprter2 25915* The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( Prt  A  ->  ( U. A /.  .~  )  =  ( A 
 \  { (/) } )
 )
 
Theoremprter3 25916* For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
 |-  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }   =>    |-  ( ( S  Er  U. A  /\  ( U. A /. S )  =  ( A  \  { (/) } ) ) 
 ->  .~  =  S )
 
16.15  Mathbox for Stefan O'Rear
 
16.15.1  Additional elementary logic and set theory
 
Theoremnelss 25917 Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |-  (
 ( A  e.  B  /\  -.  A  e.  C )  ->  -.  B  C_  C )
 
Theoremmoxfr 25918* Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)
 |-  A  e.  _V   &    |-  E! y  x  =  A   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( E* x ph  <->  E* y ps )
 
Theoremraldifsni 25919 Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  ( A. x  e.  ( A  \  { B }
 )  -.  ph  <->  A. x  e.  A  ( ph  ->  x  =  B ) )
 
16.15.2  Additional theory of functions
 
Theoremfninfp 25920* Express the class of fixed points of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( F  Fn  A  ->  dom  (  F  i^i  _I  )  =  { x  e.  A  |  ( F `  x )  =  x }
 )
 
Theoremfnelfp 25921 Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  (
 ( F  Fn  A  /\  X  e.  A ) 
 ->  ( X  e.  dom  (  F  i^i  _I  )  <->  ( F `  X )  =  X ) )
 
Theoremfndifnfp 25922* Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( F  Fn  A  ->  dom  (  F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
 
Theoremfnelnfp 25923 Property of a non-fixed point of a function. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  (
 ( F  Fn  A  /\  X  e.  A ) 
 ->  ( X  e.  dom  (  F  \  _I  )  <->  ( F `  X )  =/=  X ) )
 
Theoremfnnfpeq0 25924 A function is the identify iff it moves no points. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  ( F  Fn  A  ->  ( dom  (  F  \  _I  )  =  (/)  <->  F  =  (  _I  |`  A ) ) )
 
Theoremimaiinfv 25925* Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (
 ( F  Fn  A  /\  B  C_  A )  -> 
 |^|_ x  e.  B  ( F `  x )  =  |^| ( F " B ) )
 
Theoremralxpxfr2d 25926* Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  A  e.  _V   &    |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  E. z  e.  D  x  =  A )
 )   &    |-  ( ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  B  ps 
 <-> 
 A. y  e.  C  A. z  e.  D  ch ) )
 
Theoremralxpmap 25927* Quantification over functions in terms of quantification over values and punctured functions. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  (
 f  =  ( g  u.  { <. J ,  y >. } )  ->  ( ph  <->  ps ) )   =>    |-  ( J  e.  T  ->  ( A. f  e.  ( S  ^m  T ) ph  <->  A. y  e.  S  A. g  e.  ( S 
 ^m  ( T  \  { J } ) ) ps ) )
 
Theoremfunsnfsup 25928 Finite support for a function extended by a singleton. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  (
 ( `' ( F  u.  { <. X ,  Y >. } ) " Z )  e.  Fin  <->  ( `' F " Z )  e.  Fin )
 
Theoremfvtresfn 25929* Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( X  e.  B  ->  ( F `  X )  =  ( X  |`  V ) )
 
16.15.3  Extensions beyond function theory
 
Theoremgsumvsmul 25930* Pull a scalar multiplication out of a sum of vectors. EDITORIAL: properly generalizes gsummulc2 15226, since every ring is a left module over itself. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  S  =  (Scalar `  R )   &    |-  K  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .x.  =  ( .s `  R )   &    |-  ( ph  ->  R  e.  LMod )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  X  e.  K )   &    |-  (
 ( ph  /\  k  e.  A )  ->  Y  e.  B )   &    |-  ( ph  ->  ( `' ( k  e.  A  |->  Y ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   =>    |-  ( ph  ->  ( R  gsumg  ( k  e.  A  |->  ( X  .x.  Y ) ) )  =  ( X  .x.  ( R  gsumg  (
 k  e.  A  |->  Y ) ) ) )
 
TheoremgrpinvnzOLD 25931 The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.) . (Moved to grpinvnz 14374 in main set.mm and may be deleted by mathbox owner, SO. --NM 23-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  B  /\  X  =/=  .0.  )  ->  ( N `  X )  =/=  .0.  )
 
TheoremgrpinvnzclOLD 25932 The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.) . (Moved to grpinvnzcl 14375 in main set.mm and may be deleted by mathbox owner, SO. --NM 23-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  ( B 
 \  {  .0.  }
 ) )  ->  ( N `  X )  e.  ( B  \  {  .0.  } ) )
 
Theoremlcomf 25933 A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  B  =  ( Base `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G : I --> K )   &    |-  ( ph  ->  H : I
 --> B )   &    |-  ( ph  ->  I  e.  V )   =>    |-  ( ph  ->  ( G  o F  .x.  H ) : I --> B )
 
Theoremlcomfsup 25934 A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  B  =  ( Base `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G : I --> K )   &    |-  ( ph  ->  H : I
 --> B )   &    |-  ( ph  ->  I  e.  V )   &    |-  .0.  =  ( 0g `  W )   &    |-  Y  =  ( 0g
 `  F )   &    |-  ( ph  ->  ( `' G " ( _V  \  { Y } ) )  e. 
 Fin )   =>    |-  ( ph  ->  ( `' ( G  o F  .x.  H ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )
 
16.15.4  Additional topology
 
Theoremelrfi 25935* 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 25936* 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 25937* 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 25938* 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 )  =/=  (/) )
 
16.15.5  Characterization of closure operators. Kuratowski closure axioms
 
Theoremismrcd1 25939* Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 13391), isotone (satisfies mrcss 13390), and idempotent (satisfies mrcidm 13393) 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 25940 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 25940* Second half of ismrcd1 25939. (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 25941* A closure function which satisfies sscls 16625, clsidm 16636, cls0 16649, and clsun 25412 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 25942* A function is a Moore closure operator iff it satisfies mrcssid 13391, mrcss 13390, and mrcidm 13393. (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 ) ) ) ) )
 
16.15.6  Algebraic closure systems
 
Syntaxcnacs 25943 Class of Noetherian closure systems.
 class NoeACS
 
Definitiondf-nacs 25944* 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 25945* 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 25946* 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 25947 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 25948* 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 25949* 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 25950 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 25951* 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 25952* Transitivity induction of subsets, lemma for nacsfix 25953. (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 25953* 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 ) )
 
16.15.7  Miscellanea 1. Map utilities
 
Theoremconstmap 25954 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 25955 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 25956 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 25957 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 25958 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 25959 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 25960 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 25961 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 25962 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 )
 
16.15.8  Miscellanea for polynomials
 
Theoremofmpteq 25963* 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 25964* 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 ) )
 
16.15.9  Multivariate polynomials over the integers
 
Syntaxcmzpcl 25965 Extend class notation to include pre-polynomial rings.
 class mzPolyCld
 
Syntaxcmzp 25966 Extend class notation to include polynomial rings.
 class mzPoly
 
Definitiondf-mzpcl 25967* 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 25968. (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 25968 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 25969* 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 25970* 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 25971 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 25968 is well defined. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  ( ZZ  ^m  ( ZZ 
 ^m  V ) )  e.  (mzPolyCld `  V ) )
 
Theoremmzpcln0 25972 Corrolary of mzpclall 25971: Polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  (mzPolyCld `  V )  =/=  (/) )
 
Theoremmzpcl1 25973 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 25974* 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 25975 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 25976 Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  (mzPoly `  V )  =  |^| (mzPolyCld `
  V ) )
 
Theoremdmmzp 25977 mzPoly is defined for all index sets which are sets. This is used with elfvdm 5407 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  dom mzPoly  =  _V
 
Theoremmzpincl 25978 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 25979 Constant functions are polynomial. See also mzpconstmpt 25984. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( V  e.  _V  /\  C  e.  ZZ )  ->  ( ( ZZ  ^m  V )  X.  { C } )  e.  (mzPoly `  V ) )
 
Theoremmzpf 25980 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 25981* 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 25982 The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 25985. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  (mzPoly `  V )  /\  B  e.  (mzPoly `  V )
 )  ->  ( A  o F  +  B )  e.  (mzPoly `  V ) )
 
Theoremmzpmul 25983 The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 25986. (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 25984* A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 25985, mzpmulmpt 25986, mzpnegmpt 25988, mzpsubmpt 25987, mzpexpmpt 25989) 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 25981 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 25985* Sum of polynomial functions is polynomial. Maps-to version of mzpadd 25982. (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 25986* Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 25986. (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 25987* 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 25988* 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 25989* 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 25990* "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 25991 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 25992* 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 25993* Simplified version of mzpsubst 25992 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 25994* 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 25995* Lemma for mzpcompact2 25996. (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 25996* 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 ) ) ) ) )
 
16.15.10  Miscellanea for Diophantine sets 1
 
Theoremcoeq0 25997 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 5080 and coundir 5081 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 25998 coeq0 25997 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 25999 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 ) ) )
 
16.15.11  Diophantine sets 1: definitions
 
Syntaxcdioph 26000 Extend class notation to include the family of Diophantine sets.
 class Dioph
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