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Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivsmulf 13701* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  R  e.  Z )   &    |-  ( ph  ->  ( ( a 
 .~  p  /\  b  .~  q )  ->  (
 a  .x.  b )  .~  ( p  .x.  q
 ) ) )   &    |-  (
 ( ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( p  .x.  q )  e.  V )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  U )   =>    |-  ( ph  ->  .xb  : ( ( V /.  .~  )  X.  ( V /.  .~  ) ) --> ( V
 /.  .~  ) )
 
Theoremxpsc 13702 A short expression for the pair function mapping  0 to  A and  1 to  B. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  `' ( { A }  +c  { B } )  =  ( ( { (/) }  X.  { A } )  u.  ( { 1o }  X.  { B } )
 )
 
Theoremxpscg 13703 A short expression for the pair function mapping  0 to  A and  1 to  B. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  `' ( { A }  +c  { B } )  =  { <.
 (/) ,  A >. , 
 <. 1o ,  B >. } )
 
Theoremxpscfn 13704 The pair function is a function on 
2o  =  { (/) ,  1o }. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  `' ( { A }  +c  { B } )  Fn  2o )
 
Theoremxpsc0 13705 The pair function maps  0 to  A. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( A  e.  V  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
 
Theoremxpsc1 13706 The pair function maps  1 to  B. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B } ) `  1o )  =  B )
 
Theoremxpscfv 13707 The value of the pair function at an element of  2o. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  2o )  ->  ( `' ( { A }  +c  { B } ) `  C )  =  if ( C  =  (/) ,  A ,  B ) )
 
Theoremxpsfrnel 13708* Elementhood in the target space of the function  F appearing in xpsval 13717. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( G  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <->  ( G  Fn  2o  /\  ( G `  (/) )  e.  A  /\  ( G `  1o )  e.  B ) )
 
Theoremxpsfeq 13709 A function on  2o is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( G  Fn  2o  ->  `' ( { ( G `
  (/) ) }  +c  { ( G `  1o ) } )  =  G )
 
Theoremxpsfrnel2 13710* Elementhood in the target space of the function  F appearing in xpsval 13717. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  ( `' ( { X }  +c  { Y } )  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <->  ( X  e.  A  /\  Y  e.  B ) )
 
Theoremxpscf 13711 Equivalent condition for the pair function to be a proper function on  A. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( `' ( { X }  +c  { Y } ) : 2o --> A 
 <->  ( X  e.  A  /\  Y  e.  A ) )
 
Theoremxpsfval 13712* The value of the function appearing in xpsval 13717. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } )
 )   =>    |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( X F Y )  =  `' ( { X }  +c  { Y } ) )
 
Theoremxpsff1o 13713* The function appearing in xpsval 13717 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair  2o  =  { (/)
,  1o }. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } )
 )   =>    |-  F : ( A  X.  B ) -1-1-onto-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
 
Theoremxpsfrn 13714* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } )
 )   =>    |- 
 ran  F  =  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
 
Theoremxpsfrn2 13715* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } )
 )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  F  =  X_ k  e.  2o  ( `' ( { A }  +c  { B } ) `  k ) )
 
Theoremxpsff1o2 13716* The function appearing in xpsval 13717 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair  2o  =  { (/)
,  1o }. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } )
 )   =>    |-  F : ( A  X.  B ) -1-1-onto-> ran  F
 
Theoremxpsval 13717* Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
 y } ) )   &    |-  G  =  (Scalar `  R )   &    |-  U  =  ( G
 X_s `' ( { R }  +c  { S } )
 )   =>    |-  ( ph  ->  T  =  ( `' F  "s  U ) )
 
Theoremxpslem 13718* The indexed structure product that appears in xpsval 13717 has the same base as the target of the function  F. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
 y } ) )   &    |-  G  =  (Scalar `  R )   &    |-  U  =  ( G
 X_s `' ( { R }  +c  { S } )
 )   =>    |-  ( ph  ->  ran  F  =  ( Base `  U )
 )
 
Theoremxpsbas 13719 The base set of the binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   =>    |-  ( ph  ->  ( X  X.  Y )  =  ( Base `  T )
 )
 
Theoremxpsaddlem 13720* Lemma for xpsadd 13721 and xpsmul 13722. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   &    |-  ( ph  ->  ( A  .x.  C )  e.  X )   &    |-  ( ph  ->  ( B  .X. 
 D )  e.  Y )   &    |- 
 .x.  =  ( E `  R )   &    |-  .X.  =  ( E `  S )   &    |-  .xb  =  ( E `  T )   &    |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } )
 )   &    |-  U  =  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } )
 )   &    |-  ( ( ph  /\  `' ( { A }  +c  { B } )  e. 
 ran  F  /\  `' ( { C }  +c  { D } )  e.  ran  F )  ->  ( ( `' F `  `' ( { A }  +c  { B } ) )  .xb  ( `' F `  `' ( { C }  +c  { D } ) ) )  =  ( `' F `  ( `' ( { A }  +c  { B } ) ( E `
  U ) `' ( { C }  +c  { D } )
 ) ) )   &    |-  (
 ( `' ( { R }  +c  { S } )  Fn  2o  /\  `' ( { A }  +c  { B } )  e.  ( Base `  U )  /\  `' ( { C }  +c  { D } )  e.  ( Base `  U )
 )  ->  ( `' ( { A }  +c  { B } ) ( E `  U ) `' ( { C }  +c  { D } )
 )  =  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `  k ) ( E `
  ( `' ( { R }  +c  { S } ) `  k
 ) ) ( `' ( { C }  +c  { D } ) `  k ) ) ) )   =>    |-  ( ph  ->  ( <. A ,  B >.  .xb  <. C ,  D >. )  =  <. ( A  .x.  C ) ,  ( B 
 .X.  D ) >. )
 
Theoremxpsadd 13721 Value of the addition operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   &    |-  ( ph  ->  ( A  .x.  C )  e.  X )   &    |-  ( ph  ->  ( B  .X. 
 D )  e.  Y )   &    |- 
 .x.  =  ( +g  `  R )   &    |-  .X.  =  ( +g  `  S )   &    |-  .xb  =  ( +g  `  T )   =>    |-  ( ph  ->  ( <. A ,  B >.  .xb  <. C ,  D >. )  =  <. ( A 
 .x.  C ) ,  ( B  .X.  D ) >. )
 
Theoremxpsmul 13722 Value of the multiplication operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   &    |-  ( ph  ->  ( A  .x.  C )  e.  X )   &    |-  ( ph  ->  ( B  .X. 
 D )  e.  Y )   &    |- 
 .x.  =  ( .r `  R )   &    |-  .X.  =  ( .r `  S )   &    |-  .xb  =  ( .r `  T )   =>    |-  ( ph  ->  ( <. A ,  B >.  .xb  <. C ,  D >. )  =  <. ( A  .x.  C ) ,  ( B  .X.  D ) >. )
 
Theoremxpssca 13723 Value of the scalar field of a binary structure product. For concreteness, we choose the scalar field to match the left argument, but in most cases where this slot is meaningful both factors will have the same scalar field, so that it doesn't matter which factor is chosen. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  G  =  (Scalar `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   =>    |-  ( ph  ->  G  =  (Scalar `  T )
 )
 
Theoremxpsvsca 13724 Value of the scalar multiplication function in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  G  =  (Scalar `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  X  =  ( Base `  R )   &    |-  Y  =  (
 Base `  S )   &    |-  K  =  ( Base `  G )   &    |-  .x.  =  ( .s `  R )   &    |-  .X. 
 =  ( .s `  S )   &    |-  .xb  =  ( .s `  T )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  Y )   &    |-  ( ph  ->  ( A  .x.  B )  e.  X )   &    |-  ( ph  ->  ( A  .X. 
 C )  e.  Y )   =>    |-  ( ph  ->  ( A  .xb  <. B ,  C >. )  =  <. ( A 
 .x.  B ) ,  ( A  .X.  C ) >. )
 
Theoremxpsless 13725 Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  .<_  =  ( le `  T )   =>    |-  ( ph  ->  .<_  C_  (
 ( X  X.  Y )  X.  ( X  X.  Y ) ) )
 
Theoremxpsle 13726 Value of the ordering in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  .<_  =  ( le `  T )   &    |-  M  =  ( le `  R )   &    |-  N  =  ( le `  S )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   =>    |-  ( ph  ->  (
 <. A ,  B >.  .<_  <. C ,  D >.  <->  ( A M C  /\  B N D ) ) )
 
7.2  Moore spaces
 
Syntaxcmre 13727 The class of Moore systems.
 class Moore
 
Syntaxcmrc 13728 The class function generating Moore closures.
 class mrCls
 
Syntaxcmri 13729 mrInd is a class function which takes a Moore system to its set of independent sets.
 class mrInd
 
Syntaxcacs 13730 The class of algebraic closure (Moore) systems.
 class ACS
 
Definitiondf-mre 13731* Define a Moore collection, which is a family of subsets of a base set which preserve arbitrary intersection. Elements of a Moore collection are termed closed; Moore collections generalize the notion of closedness from topologies (cldmre 17058) and vector spaces (lssmre 15962) to the most general setting in which such concepts make sense. Definition of Moore collection of sets in [Schechter] p. 78. A Moore collection may also be called a closure system (Section 0.6 in [Gratzer] p. 23.) The name Moore collection is after Eliakim Hastings Moore, who discussed these systems in Part I of [Moore] p. 53 to 76.

See ismre 13735, mresspw 13737, mre1cl 13739 and mreintcl 13740 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13745); as such the disjoint union of all Moore collections is sometimes considered as  U. ran Moore, justified by mreunirn 13746. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.)

 |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }
 )
 
Definitiondf-mrc 13732* Define the Moore closure of a generating set, which is the smallest closed set containing all generating elements. Definition of Moore closure in [Schechter] p. 79. This generalizes topological closure (mrccls 17059) and linear span (mrclsp 15985).

A Moore closure operation  N is (1) extensive, i.e.,  x  C_  ( N `  x ) for all subsets  x of the base set (mrcssid 13762), (2) isotone, i.e.,  x  C_  y implies that  ( N `
 x )  C_  ( N `  y ) for all subsets  x and  y of the base set (mrcss 13761), and (3) idempotent, i.e.,  ( N `  ( N `  x )
)  =  ( N `
 x ) for all subsets  x of the base set (mrcidm 13764.) Operators satisfying these three properties are in bijective correspondence with Moore collections, so these properties may be used to give an alternate characterization of a Moore collection by providing a closure operation  N on the set of subsets of a given base set which satisfies (1), (2), and (3); the closed sets can be recovered as those sets which equal their closures (Section 4.5 in [Schechter] p. 82.) (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by David Moews, 1-May-2017.)

 |- mrCls  =  ( c  e.  U. ran Moore 
 |->  ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) )
 
Definitiondf-mri 13733* In a Moore system, a set is independent if no element of the set is in the closure of the set with the element removed (Section 0.6 in [Gratzer] p. 27; Definition 4.1.1 in [FaureFrolicher] p. 83.) mrInd is a class function which takes a Moore system to its set of independent sets. (Contributed by David Moews, 1-May-2017.)
 |- mrInd  =  ( c  e.  U. ran Moore 
 |->  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c
 ) `  ( s  \  { x } )
 ) } )
 
Definitiondf-acs 13734* An important subclass of Moore systems are those which can be interpreted as closure under some collection of operators of finite arity (the collection itself is not required to be finite). These are termed algebraic closure systems; similar to definition (A) of an algebraic closure system in [Schechter] p. 84, but to avoid the complexity of an arbitrary mixed collection of functions of various arities (especially if the axiom of infinity omex 7524 is to be avoided), we consider a single function defined on finite sets instead. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |- ACS 
 =  ( x  e. 
 _V  |->  { c  e.  (Moore `  x )  |  E. f ( f : ~P x --> ~P x  /\  A. s  e.  ~P  x ( s  e.  c  <->  U. ( f "
 ( ~P s  i^i 
 Fin ) )  C_  s ) ) }
 )
 
Theoremismre 13735* Property of being a Moore collection on some base set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e. 
 ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
 
Theoremfnmre 13736 The Moore collection generator is a well-behaved function. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |- Moore  Fn  _V
 
Theoremmresspw 13737 A Moore collection is a subset of the power of the base set; each closed subset of the system is actually a subset of the base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  ->  C  C_ 
 ~P X )
 
Theoremmress 13738 A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  S  C_  X )
 
Theoremmre1cl 13739 In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  ->  X  e.  C )
 
Theoremmreintcl 13740 A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/= 
 (/) )  ->  |^| S  e.  C )
 
Theoremmreiincl 13741* A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  e.  C )
 
Theoremmrerintcl 13742 The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C )  ->  ( X  i^i  |^| S )  e.  C )
 
Theoremmreriincl 13743* The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C )
 
Theoremmreincl 13744 Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  ( A  i^i  B )  e.  C )
 
Theoremmreuni 13745 Since the entire base set of a Moore collection is the greatest element of it, the base set can be recovered from a Moore collection by set union. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  ->  U. C  =  X )
 
Theoremmreunirn 13746 Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  U. ran Moore  <->  C  e.  (Moore `  U. C ) )
 
Theoremismred 13747* Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( ph  ->  C  C_ 
 ~P X )   &    |-  ( ph  ->  X  e.  C )   &    |-  ( ( ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C )   =>    |-  ( ph  ->  C  e.  (Moore `  X )
 )
 
Theoremismred2 13748* Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ph  ->  C  C_ 
 ~P X )   &    |-  (
 ( ph  /\  s  C_  C )  ->  ( X  i^i  |^| s )  e.  C )   =>    |-  ( ph  ->  C  e.  (Moore `  X )
 )
 
Theoremmremre 13749 The Moore collections of subsets of a space, viewed as a kind of subset of the power set, form a Moore collection in their own right on the power set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( X  e.  V  ->  (Moore `  X )  e.  (Moore `  ~P X ) )
 
Theoremsubmre 13750 The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  ( C  i^i  ~P A )  e.  (Moore `  A ) )
 
7.2.1  Moore closures
 
Theoremmrcflem 13751* The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  ->  ( x  e.  ~P X  |->  |^|
 { s  e.  C  |  x  C_  s }
 ) : ~P X --> C )
 
Theoremfnmrc 13752 Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- mrCls  Fn  U. ran Moore
 
Theoremmrcfval 13753* Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
 
Theoremmrcf 13754 The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  F : ~P X --> C )
 
Theoremmrcval 13755* Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
 
Theoremmrccl 13756 The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  e.  C )
 
Theoremmrcsncl 13757 The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  e.  X )  ->  ( F `  { U } )  e.  C )
 
Theoremmrcid 13758 The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  ( F `  U )  =  U )
 
Theoremmrcssv 13759 The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  ( F `  U )  C_  X )
 
Theoremmrcidb 13760 A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  ( U  e.  C  <->  ( F `  U )  =  U ) )
 
Theoremmrcss 13761 Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  C_  ( F `  V ) )
 
Theoremmrcssid 13762 The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  ( F `  U ) )
 
Theoremmrcidb2 13763 A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( U  e.  C  <->  ( F `  U ) 
 C_  U ) )
 
Theoremmrcidm 13764 The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  ( F `
  U ) )  =  ( F `  U ) )
 
Theoremmrcsscl 13765 The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  U )  C_  V )
 
Theoremmrcuni 13766 Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  ~P X ) 
 ->  ( F `  U. U )  =  ( F ` 
 U. ( F " U ) ) )
 
Theoremmrcun 13767 Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  ( U  u.  V ) )  =  ( F `  (
 ( F `  U )  u.  ( F `  V ) ) ) )
 
Theoremmrcssvd 13768 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 13759. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   =>    |-  ( ph  ->  ( N `  B )  C_  X )
 
Theoremmrcssd 13769 Moore closure preserves subset ordering. Deduction form of mrcss 13761. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  V )   &    |-  ( ph  ->  V  C_  X )   =>    |-  ( ph  ->  ( N `  U )  C_  ( N `  V ) )
 
Theoremmrcssidd 13770 A set is contained in its Moore closure. Deduction form of mrcssid 13762. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  X )   =>    |-  ( ph  ->  U 
 C_  ( N `  U ) )
 
Theoremmrcidmd 13771 Moore closure is idempotent. Deduction form of mrcidm 13764. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  X )   =>    |-  ( ph  ->  ( N `  ( N `
  U ) )  =  ( N `  U ) )
 
Theoremmressmrcd 13772 In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  ( N `  T ) )   &    |-  ( ph  ->  T  C_  S )   =>    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )
 
Theoremsubmrc 13773 In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  F  =  (mrCls `  C )   &    |-  G  =  (mrCls `  ( C  i^i  ~P D ) )   =>    |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  =  ( F `  U ) )
 
Theoremmrieqvlemd 13774 In a Moore system, if  Y is a member of  S,  ( S 
\  { Y }
) and  S have the same closure if and only if  Y is in the closure of  ( S  \  { Y } ). Used in the proof of mrieqvd 13783 and mrieqv2d 13784. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  ( Y  e.  ( N `  ( S  \  { Y } ) )  <->  ( N `  ( S  \  { Y } ) )  =  ( N `  S ) ) )
 
7.2.2  Independent sets in a Moore system
 
Theoremmrisval 13775* Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   =>    |-  ( A  e.  (Moore `  X )  ->  I  =  { s  e.  ~P X  |  A. x  e.  s  -.  x  e.  ( N `  (
 s  \  { x } ) ) }
 )
 
Theoremismri 13776* Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   =>    |-  ( A  e.  (Moore `  X )  ->  ( S  e.  I  <->  ( S  C_  X  /\  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x }
 ) ) ) ) )
 
Theoremismri2 13777* Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   =>    |-  ( ( A  e.  (Moore `  X )  /\  S  C_  X )  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) ) )
 
Theoremismri2d 13778* Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) ) )
 
Theoremismri2dd 13779* Definition of independence of a subset of the base set in a Moore system. One-way deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  A. x  e.  S  -.  x  e.  ( N `  ( S  \  { x } ) ) )   =>    |-  ( ph  ->  S  e.  I )
 
Theoremmriss 13780 An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.)
 |-  I  =  (mrInd `  A )   =>    |-  ( ( A  e.  (Moore `  X )  /\  S  e.  I )  ->  S  C_  X )
 
Theoremmrissd 13781 An independent set of a Moore system is a subset of the base set. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S  e.  I )   =>    |-  ( ph  ->  S 
 C_  X )
 
Theoremismri2dad 13782 Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A  e.  (Moore `  X ) )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  -.  Y  e.  ( N `  ( S  \  { Y }
 ) ) )
 
Theoremmrieqvd 13783* In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  A. x  e.  S  ( N `  ( S 
 \  { x }
 ) )  =/=  ( N `  S ) ) )
 
Theoremmrieqv2d 13784* In a Moore system, a set is independent if and only if all its proper subsets have closure properly contained in the closure of the set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  A. s ( s 
 C.  S  ->  ( N `  s )  C.  ( N `  S ) ) ) )
 
Theoremmrissmrcd 13785 In a Moore system, if an independent set is between a set and its closure, the two sets are equal (since the two sets must have equal closures by mressmrcd 13772, and so are equal by mrieqv2d 13784.) (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  ( N `  T ) )   &    |-  ( ph  ->  T  C_  S )   &    |-  ( ph  ->  S  e.  I )   =>    |-  ( ph  ->  S  =  T )
 
Theoremmrissmrid 13786 In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  S )   =>    |-  ( ph  ->  T  e.  I )
 
Theoremmreexd 13787* In a Moore system, the closure operator is said to have the exchange property if, for all elements  y and  z of the base set and subsets  S of the base set such that  z is in the closure of  ( S  u.  { y } ) but not in the closure of  S,  y is in the closure of  ( S  u.  { z } ) (Definition 3.1.9 in [FaureFrolicher] p. 57 to 58.) This theorem allows us to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  Z  e.  ( N `  ( S  u.  { Y }
 ) ) )   &    |-  ( ph  ->  -.  Z  e.  ( N `  S ) )   =>    |-  ( ph  ->  Y  e.  ( N `  ( S  u.  { Z }
 ) ) )
 
Theoremmreexmrid 13788* In a Moore system whose closure operator has the exchange property, if a set is independent and an element is not in its closure, then adding the element to the set gives another independent set. Lemma 4.1.5 in [FaureFrolicher] p. 84. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  -.  Y  e.  ( N `  S ) )   =>    |-  ( ph  ->  ( S  u.  { Y }
 )  e.  I )
 
Theoremmreexexlemd 13789* This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 13793. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  X  e.  J )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  ( F  ~~  K  \/  G  ~~  K ) )   &    |-  ( ph  ->  A. t A. u  e.  ~P  ( X  \  t ) A. v  e.  ~P  ( X  \  t ) ( ( ( u  ~~  K  \/  v  ~~  K ) 
 /\  u  C_  ( N `  ( v  u.  t ) )  /\  ( u  u.  t
 )  e.  I ) 
 ->  E. i  e.  ~P  v ( u  ~~  i  /\  ( i  u.  t )  e.  I
 ) ) )   =>    |-  ( ph  ->  E. j  e.  ~P  G ( F  ~~  j  /\  ( j  u.  H )  e.  I )
 )
 
Theoremmreexexlem2d 13790* Used in mreexexlem4d 13792 to prove the induction step in mreexexd 13793. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  Y  e.  F )   =>    |-  ( ph  ->  E. g  e.  G  ( -.  g  e.  ( F  \  { Y } )  /\  (
 ( F  \  { Y } )  u.  ( H  u.  { g }
 ) )  e.  I
 ) )
 
Theoremmreexexlem3d 13791* Base case of the induction in mreexexd 13793. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  ( F  =  (/)  \/  G  =  (/) ) )   =>    |-  ( ph  ->  E. i  e.  ~P  G ( F  ~~  i  /\  ( i  u.  H )  e.  I )
 )
 
Theoremmreexexlem4d 13792* Induction step of the induction in mreexexd 13793. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  L  e.  om )   &    |-  ( ph  ->  A. h A. f  e. 
 ~P  ( X  \  h ) A. g  e.  ~P  ( X  \  h ) ( ( ( f  ~~  L  \/  g  ~~  L ) 
 /\  f  C_  ( N `  ( g  u.  h ) )  /\  ( f  u.  h )  e.  I )  ->  E. j  e.  ~P  g ( f  ~~  j  /\  ( j  u.  h )  e.  I
 ) ) )   &    |-  ( ph  ->  ( F  ~~  suc 
 L  \/  G  ~~  suc 
 L ) )   =>    |-  ( ph  ->  E. j  e.  ~P  G ( F  ~~  j  /\  ( j  u.  H )  e.  I )
 )
 
Theoremmreexexd 13793* Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if  F and  G are disjoint from  H,  ( F  u.  H ) is independent,  F is contained in the closure of  ( G  u.  H ), and either  F or  G is finite, then there is a subset  q of  G equinumerous to  F such that  ( q  u.  H ) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either  ( A  \  B ) or  ( B  \  A ) is finite. The theorem is proven by induction using mreexexlem3d 13791 for the base case and mreexexlem4d 13792 for the induction step. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  F 
 C_  ( X  \  H ) )   &    |-  ( ph  ->  G  C_  ( X  \  H ) )   &    |-  ( ph  ->  F  C_  ( N `  ( G  u.  H ) ) )   &    |-  ( ph  ->  ( F  u.  H )  e.  I
 )   &    |-  ( ph  ->  ( F  e.  Fin  \/  G  e.  Fin ) )   =>    |-  ( ph  ->  E. q  e.  ~P  G ( F  ~~  q  /\  ( q  u.  H )  e.  I )
 )
 
Theoremmreexdomd 13794* In a Moore system whose closure operator has the exchange property, if  S is independent and contained in the closure of  T, and either  S or  T is finite, then  T dominates  S. This is an immediate consequence of mreexexd 13793. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S 
 C_  ( N `  T ) )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( S  e.  Fin  \/  T  e.  Fin ) )   &    |-  ( ph  ->  S  e.  I
 )   =>    |-  ( ph  ->  S  ~<_  T )
 
Theoremmreexfidimd 13795* In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd 13794 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  e.  I
 )   &    |-  ( ph  ->  S  e.  Fin )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   =>    |-  ( ph  ->  S 
 ~~  T )
 
7.2.3  Algebraic closure systems
 
Theoremisacs 13796* A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  E. f ( f : ~P X --> ~P X  /\  A. s  e.  ~P  X ( s  e.  C  <->  U. ( f "
 ( ~P s  i^i 
 Fin ) )  C_  s ) ) ) )
 
Theoremacsmre 13797 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  ->  C  e.  (Moore `  X )
 )
 
Theoremisacs2 13798* In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( s  e.  C  <->  A. y  e.  ( ~P s  i^i  Fin )
 ( F `  y
 )  C_  s )
 ) )
 
Theoremacsfiel 13799* A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (ACS `  X )  ->  ( S  e.  C  <->  ( S  C_  X  /\  A. y  e.  ( ~P S  i^i  Fin ) ( F `  y )  C_  S ) ) )
 
Theoremacsfiel2 13800* A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  ( S  e.  C  <->  A. y  e.  ( ~P S  i^i  Fin )
 ( F `  y
 )  C_  S )
 )
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