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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxpsval 13401* 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 13402* The indexed structure product that appears in xpsval 13401 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 13403 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 13404* Lemma for xpsadd 13405 and xpsmul 13406. (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 13405 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 13406 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 13407 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 13408 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 13409 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 13410 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 13411 The class of Moore systems.
 class Moore
 
Syntaxcmrc 13412 The class function generating Moore closures.
 class mrCls
 
Syntaxcmri 13413 mrInd is a class function which takes a Moore system to its set of independent sets.
 class mrInd
 
Syntaxcacs 13414 The class of algebraic closure (Moore) systems.
 class ACS
 
Definitiondf-mre 13415* 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 16742) and vector spaces (lssmre 15650) 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 13419, mresspw 13421, mre1cl 13423 and mreintcl 13424 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13429); as such the disjoint union of all Moore collections is sometimes considered as  U. ran Moore, justified by mreunirn 13430. (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 13416* 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 16743) and linear span (mrclsp 15673).

A Moore closure operation  N is (1) extensive, i.e.,  x  C_  ( N `  x ) for all subsets  x of the base set (mrcssid 13446), (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 13445), and (3) idempotent, i.e.,  ( N `  ( N `  x )
)  =  ( N `
 x ) for all subsets  x of the base set (mrcidm 13448.) 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 13417* 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 13418* 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 7277 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 13419* 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 13420 The Moore collection generator is a well-behaved function. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |- Moore  Fn  _V
 
Theoremmresspw 13421 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 13422 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 13423 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 13424 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 13425* 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 13426 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 13427* 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 13428 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 13429 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 13430 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 13431* 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 13432* 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 13433 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 13434 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 13435* 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 13436 Moore-closure is a well behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- mrCls  Fn  U. ran Moore
 
Theoremmrcfval 13437* 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 13438 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 13439* 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 13440 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 13441 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 13442 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 13443 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 13444 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 13445 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 13446 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 13447 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 13448 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 13449 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 13450 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 13451 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 13452 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 13443. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   =>    |-  ( ph  ->  ( N `  B )  C_  X )
 
Theoremmrcssd 13453 Moore closure preserves subset ordering. Deduction form of mrcss 13445. (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 13454 A set is contained in its Moore closure. Deduction form of mrcssid 13446. (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 13455 Moore closure is idempotent. Deduction form of mrcidm 13448. (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 13456 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 13457 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 13458 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 13467 and mrieqv2d 13468. 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 13459* 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 13460* 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 13461* 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 13462* 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 13463* 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 13464 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 13465 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 13466 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 13467* 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 13468* 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 13469 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 13456, and so are equal by mrieqv2d 13468.) (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 13470 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 13471* 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 13472* 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 13473* This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 13477. (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 13474* Used in mreexexlem4d 13476 to prove the induction step in mreexexd 13477. 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 13475* Base case of the induction in mreexexd 13477. (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 13476* Induction step of the induction in mreexexd 13477. (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 13477* 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 13475 for the base case and mreexexlem4d 13476 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 13478* 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 13477. (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 13479* 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 13478 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 13480* 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 13481 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  ->  C  e.  (Moore `  X )
 )
 
Theoremisacs2 13482* 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 13483* 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 13484* 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 )
 )
 
Theoremacsmred 13485 An algebraic closure system is also a Moore system. Deduction form of acsmre 13481. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   =>    |-  ( ph  ->  A  e.  (Moore `  X )
 )
 
Theoremisacs1i 13486* A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( X  e.  V  /\  F : ~P X
 --> ~P X )  ->  { s  e.  ~P X  |  U. ( F
 " ( ~P s  i^i  Fin ) )  C_  s }  e.  (ACS `  X ) )
 
Theoremmreacs 13487 Algebraicity is a composible property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  e.  V  ->  (ACS `  X )  e.  (Moore `  ~P X ) )
 
Theoremacsfn 13488* Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( ( X  e.  V  /\  K  e.  X )  /\  ( T  C_  X  /\  T  e.  Fin ) )  ->  { a  e.  ~P X  |  ( T  C_  a  ->  K  e.  a ) }  e.  (ACS `  X ) )
 
Theoremacsfn0 13489* Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( X  e.  V  /\  K  e.  X )  ->  { a  e. 
 ~P X  |  K  e.  a }  e.  (ACS `  X ) )
 
Theoremacsfn1 13490* Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( X  e.  V  /\  A. b  e.  X  E  e.  X )  ->  { a  e. 
 ~P X  |  A. b  e.  a  E  e.  a }  e.  (ACS `  X ) )
 
Theoremacsfn1c 13491* Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( X  e.  V  /\  A. b  e.  K  A. c  e.  X  E  e.  X )  ->  { a  e. 
 ~P X  |  A. b  e.  K  A. c  e.  a  E  e.  a }  e.  (ACS `  X ) )
 
Theoremacsfn2 13492* Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( X  e.  V  /\  A. b  e.  X  A. c  e.  X  E  e.  X )  ->  { a  e. 
 ~P X  |  A. b  e.  a  A. c  e.  a  E  e.  a }  e.  (ACS `  X ) )
 
PART 8  BASIC CATEGORY THEORY
 
8.1  Categories
 
8.1.1  Categories
 
Syntaxccat 13493 Extend class notation with the class of categories.
 class  Cat
 
Syntaxccid 13494 Extend class notation with the identity arrow of a category.
 class  Id
 
Syntaxchomf 13495 Extend class notation to include functionalized Hom-set extractor.
 class  Homf
 
Syntaxccomf 13496 Extend class notation to include functionalized composition operation.
 class compf
 
Definitiondf-cat 13497* A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated to those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
 |- 
 Cat  =  { c  |  [. ( Base `  c
 )  /  b ]. [. (  Hom  `  c
 )  /  h ]. [. (comp `  c )  /  o ]. A. x  e.  b  ( E. g  e.  ( x h x ) A. y  e.  b  ( A. f  e.  (
 y h x ) ( g ( <. y ,  x >. o x ) f )  =  f  /\  A. f  e.  ( x h y ) ( f (
 <. x ,  x >. o y ) g )  =  f )  /\  A. y  e.  b  A. z  e.  b  A. f  e.  ( x h y ) A. g  e.  ( y h z ) ( ( g ( <. x ,  y >. o z ) f )  e.  ( x h z )  /\  A. w  e.  b  A. k  e.  ( z h w ) ( ( k ( <. y ,  z >. o w ) g ) ( <. x ,  y >. o w ) f )  =  ( k ( <. x ,  z >. o w ) ( g ( <. x ,  y >. o z ) f ) ) ) ) }
 
Definitiondf-cid 13498* Define the category identity arrow. Since it is uniquely defined when it exists, we do not need to add it to the data of the category, and instead extract it by uniqueness. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- 
 Id  =  ( c  e.  Cat  |->  [_ ( Base `  c )  /  b ]_ [_ (  Hom  `  c )  /  h ]_
 [_ (comp `  c
 )  /  o ]_ ( x  e.  b  |->  ( iota_ g  e.  ( x h x ) A. y  e.  b  ( A. f  e.  (
 y h x ) ( g ( <. y ,  x >. o x ) f )  =  f  /\  A. f  e.  ( x h y ) ( f (
 <. x ,  x >. o y ) g )  =  f ) ) ) )
 
Definitiondf-homf 13499* Define the functionalized Hom-set operator, which is exactly like  Hom but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |- 
 Homf  =  ( c  e.  _V  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c )  |->  ( x (  Hom  `  c
 ) y ) ) )
 
Definitiondf-comf 13500* Define the functionalized composition operator, which is exactly like comp but is guaranteed to be a function of the proper type. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |- compf  =  ( c  e.  _V  |->  ( x  e.  (
 ( Base `  c )  X.  ( Base `  c )
 ) ,  y  e.  ( Base `  c )  |->  ( g  e.  (
 ( 2nd `  x )
 (  Hom  `  c ) y ) ,  f  e.  ( (  Hom  `  c
 ) `  x )  |->  ( g ( x (comp `  c )
 y ) f ) ) ) )
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