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Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprtlem10 26101* Lemma for prter3 26118. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  (  .~  Er  A  ->  (
 z  e.  A  ->  ( z  .~  w  <->  E. v  e.  A  ( z  e.  [ v ]  .~  /\  w  e. 
 [ v ]  .~  ) ) ) )
 
Theoremprtlem11 26102 Lemma for prter2 26117. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( B  e.  D  ->  ( C  e.  A  ->  ( B  =  [ C ]  .~  ->  B  e.  ( A /.  .~  )
 ) ) )
 
Theoremprtlem12 26103* Lemma for prtex 26116 and prter3 26118. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  (  .~  =  { <. x ,  y >.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }  ->  Rel  .~  )
 
Theoremprtlem13 26104* Lemma for prter1 26115, prter2 26117, prter3 26118 and prtex 26116. (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 26105* Lemma for prtex 26116, prter2 26117 and prter3 26118. (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 26106* Lemma for prter2 26117 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 26107 Extend the definition of a wff to include the partition predicate.
 wff  Prt 
 A
 
Definitiondf-prt 26108* 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 26109 The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  (  .~  Er  X  ->  Prt  ( A /.  .~  ) )
 
Theoremprtlem14 26110* Lemma for prter1 26115, prter2 26117 and prtex 26116. (Contributed by Rodolfo Medina, 13-Oct-2010.)
 |-  ( Prt  A  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( w  e.  x  /\  w  e.  y )  ->  x  =  y ) ) )
 
Theoremprtlem15 26111* Lemma for prter1 26115 and prtex 26116. (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 26112* Lemma for prter2 26117. (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 26113* Lemma for prter2 26117. (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 26114* Lemma for prter2 26117. (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 26115* 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 26116* 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 26117* 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 26118* 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 )
 
18.16  Mathbox for Stefan O'Rear
 
18.16.1  Additional elementary logic and set theory
 
Theoremnelss 26119 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 26120* 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 26121 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 ) )
 
18.16.2  Additional theory of functions
 
Theoremfninfp 26122* 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 26123 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 26124* 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 26125 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 26126 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 26127* 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 26128* 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 26129* 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 26130 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 26131* 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 ) )
 
18.16.3  Extensions beyond function theory
 
Theoremgsumvsmul 26132* Pull a scalar multiplication out of a sum of vectors. EDITORIAL: properly generalizes gsummulc2 15354, 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 26133 The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.) . (Moved to grpinvnz 14502 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 26134 The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.) . (Moved to grpinvnzcl 14503 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 26135 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 26136 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 )
 
18.16.4  Additional topology
 
Theoremelrfi 26137* 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 26138* 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 26139* 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 26140* 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 )  =/=  (/) )
 
18.16.5  Characterization of closure operators. Kuratowski closure axioms
 
Theoremismrcd1 26141* Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 13482), isotone (satisfies mrcss 13481), and idempotent (satisfies mrcidm 13484) 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 26142 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 26142* Second half of ismrcd1 26141. (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 26143* A closure function which satisfies sscls 16756, clsidm 16767, cls0 16780, and clsun 25614 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 26144* A function is a Moore closure operator iff it satisfies mrcssid 13482, mrcss 13481, and mrcidm 13484. (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 ) ) ) ) )
 
18.16.6  Algebraic closure systems
 
Syntaxcnacs 26145 Class of Noetherian closure systems.
 class NoeACS
 
Definitiondf-nacs 26146* 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 26147* 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 26148* 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 26149 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 26150* 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 26151* 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 26152 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 26153* 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 26154* Transitivity induction of subsets, lemma for nacsfix 26155. (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 26155* 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 ) )
 
18.16.7  Miscellanea 1. Map utilities
 
Theoremconstmap 26156 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 26157 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 26158 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 26159 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 26160 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 26161 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 26162 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 26163 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 26164 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 )
 
18.16.8  Miscellanea for polynomials
 
Theoremofmpteq 26165* 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 26166* Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( t  e.  A  |->  B ) : A --> C  ->  ( t  e.  A  ->  B  e.  C ) )
 
18.16.9  Multivariate polynomials over the integers
 
Syntaxcmzpcl 26167 Extend class notation to include pre-polynomial rings.
 class mzPolyCld
 
Syntaxcmzp 26168 Extend class notation to include polynomial rings.
 class mzPoly
 
Definitiondf-mzpcl 26169* 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 26170. (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 26170 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 26171* 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 26172* 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 26173 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 26170 is well defined. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  ( ZZ  ^m  ( ZZ 
 ^m  V ) )  e.  (mzPolyCld `  V ) )
 
Theoremmzpcln0 26174 Corrolary of mzpclall 26173: Polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  (mzPolyCld `  V )  =/=  (/) )
 
Theoremmzpcl1 26175 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 26176* 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 26177 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 26178 Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  ( V  e.  _V  ->  (mzPoly `  V )  =  |^| (mzPolyCld `
  V ) )
 
Theoremdmmzp 26179 mzPoly is defined for all index sets which are sets. This is used with elfvdm 5488 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  dom mzPoly  =  _V
 
Theoremmzpincl 26180 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 26181 Constant functions are polynomial. See also mzpconstmpt 26186. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( V  e.  _V  /\  C  e.  ZZ )  ->  ( ( ZZ  ^m  V )  X.  { C } )  e.  (mzPoly `  V ) )
 
Theoremmzpf 26182 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 26183* 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 26184 The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 26187. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  (mzPoly `  V )  /\  B  e.  (mzPoly `  V )
 )  ->  ( A  o F  +  B )  e.  (mzPoly `  V ) )
 
Theoremmzpmul 26185 The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 26188. (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 26186* A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 26187, mzpmulmpt 26188, mzpnegmpt 26190, mzpsubmpt 26189, mzpexpmpt 26191) 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 26183 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 26187* Sum of polynomial functions is polynomial. Maps-to version of mzpadd 26184. (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 26188* Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 26188. (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 26189* 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 26190* 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 26191* 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 26192* "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 26193 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 26194* 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 26195* Simplified version of mzpsubst 26194 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 26196* 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 26197* Lemma for mzpcompact2 26198. (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 26198* Polynomials are finitary objects and can only reference a finite number of variables, even if the index set is infinite. Thus, every polynomial can be expressed as a (uniquely minimal, although we do not prove that) polynomial on a finite number of variables, which is then extended by adding an arbitrary set of ignored variables. (Contributed by Stefan O'Rear, 9-Oct-2014.)
 |-  ( A  e.  (mzPoly `  B )  ->  E. a  e.  Fin  E. b  e.  (mzPoly `  a
 ) ( a  C_  B  /\  A  =  ( c  e.  ( ZZ 
 ^m  B )  |->  ( b `  ( c  |`  a ) ) ) ) )
 
18.16.10  Miscellanea for Diophantine sets 1
 
Theoremcoeq0 26199 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 5161 and coundir 5162 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 26200 coeq0 26199 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 )  =  (/) )
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