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Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremipoval 14101* Value of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  I  =  (toInc `  F )   &    |-  .<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  F  /\  x  C_  y ) }   =>    |-  ( F  e.  V  ->  I  =  ( { <. ( Base `  ndx ) ,  F >. , 
 <. (TopSet `  ndx ) ,  (ordTop `  .<_  ) >. }  u.  { <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |->  U. {
 y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. } ) )
 
Theoremipobas 14102 Base set of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by Mario Carneiro, 25-Oct-2015.)
 |-  I  =  (toInc `  F )   =>    |-  ( F  e.  V  ->  F  =  ( Base `  I ) )
 
Theoremipolerval 14103* Relation of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  I  =  (toInc `  F )   =>    |-  ( F  e.  V  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  F  /\  x  C_  y ) }  =  ( le `  I ) )
 
Theoremipotset 14104 Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015.)
 |-  I  =  (toInc `  F )   &    |-  .<_  =  ( le `  I )   =>    |-  ( F  e.  V  ->  (ordTop `  .<_  )  =  (TopSet `  I )
 )
 
Theoremipole 14105 Weak order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  I  =  (toInc `  F )   &    |-  .<_  =  ( le `  I )   =>    |-  ( ( F  e.  V  /\  X  e.  F  /\  Y  e.  F ) 
 ->  ( X  .<_  Y  <->  X  C_  Y ) )
 
Theoremipolt 14106 Strict order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  I  =  (toInc `  F )   &    |-  .<  =  ( lt `  I )   =>    |-  ( ( F  e.  V  /\  X  e.  F  /\  Y  e.  F )  ->  ( X 
 .<  Y  <->  X  C.  Y ) )
 
Theoremipopos 14107 The inclusion poset on a family of sets is actually a poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  I  =  (toInc `  F )   =>    |-  I  e.  Poset
 
Theoremisipodrs 14108* Condition for a family of sets to be directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( (toInc `  A )  e. Dirset  <->  ( A  e.  _V 
 /\  A  =/=  (/)  /\  A. x  e.  A  A. y  e.  A  E. z  e.  A  ( x  u.  y )  C_  z ) )
 
Theoremipodrscl 14109 Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( (toInc `  A )  e. Dirset  ->  A  e.  _V )
 
Theoremipodrsfi 14110* Finite upper bound property for directed collections of sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( ( (toInc `  A )  e. Dirset  /\  X  C_  A  /\  X  e.  Fin )  ->  E. z  e.  A  U. X  C_  z )
 
Theoremfpwipodrs 14111 The finite subsets of any set are directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( A  e.  V  ->  (toInc `  ( ~P A  i^i  Fin ) )  e. Dirset )
 
Theoremipodrsima 14112* The monotone image of a directed set. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( ph  ->  F  Fn  ~P A )   &    |-  (
 ( ph  /\  ( u 
 C_  v  /\  v  C_  A ) )  ->  ( F `  u ) 
 C_  ( F `  v ) )   &    |-  ( ph  ->  (toInc `  B )  e. Dirset )   &    |-  ( ph  ->  B 
 C_  ~P A )   &    |-  ( ph  ->  ( F " B )  e.  V )   =>    |-  ( ph  ->  (toInc `  ( F " B ) )  e. Dirset )
 
Theoremisacs3lem 14113* An algebraic closure system satisfies isacs3 14121. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  ->  ( C  e.  (Moore `  X )  /\  A. s  e. 
 ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  C ) ) )
 
Theoremacsdrsel 14114 An algebraic closure system contains all directed unions of closed sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( ( C  e.  (ACS `  X )  /\  Y  C_  C  /\  (toInc `  Y )  e. Dirset )  ->  U. Y  e.  C )
 
Theoremisacs4lem 14115* In a closure system in which directed unions of closed sets are closed, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  C ) )  ->  ( C  e.  (Moore `  X )  /\  A. t  e. 
 ~P  ~P X ( (toInc `  t )  e. Dirset  ->  ( F `  U. t
 )  =  U. ( F " t ) ) ) )
 
Theoremisacs5lem 14116* If closure commutes with directed unions, then the closure of a set is the closure of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  A. t  e.  ~P  ~P X ( (toInc `  t )  e. Dirset  ->  ( F `  U. t )  =  U. ( F
 " t ) ) )  ->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( F `  s )  =  U. ( F " ( ~P s  i^i  Fin )
 ) ) )
 
Theoremacsdrscl 14117 In an algebraic closure system, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (ACS `  X )  /\  Y  C_  ~P X  /\  (toInc `  Y )  e. Dirset )  ->  ( F `  U. Y )  =  U. ( F " Y ) )
 
Theoremacsficl 14118 A closure in an algebraic closure system is the union of the closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  ( F `  S )  =  U. ( F
 " ( ~P S  i^i  Fin ) ) )
 
Theoremisacs5 14119* A closure system is algebraic iff the closure of a generating set is the union of the closures of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( F `  s )  =  U. ( F
 " ( ~P s  i^i  Fin ) ) ) )
 
Theoremisacs4 14120* A closure system is algebraic iff closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  ~P X ( (toInc `  s )  e. Dirset  ->  ( F `  U. s )  =  U. ( F
 " s ) ) ) )
 
Theoremisacs3 14121* A closure system is algebraic iff directed unions of closed sets are closed. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  C ) ) )
 
Theoremmrelatglb 14122 Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  I  =  (toInc `  C )   &    |-  G  =  ( glb `  I )   =>    |-  (
 ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  ( G `  U )  =  |^| U )
 
Theoremmrelatglb0 14123 The empty intersection in a Moore space is realized by the base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  I  =  (toInc `  C )   &    |-  G  =  ( glb `  I )   =>    |-  ( C  e.  (Moore `  X )  ->  ( G `  (/) )  =  X )
 
Theoremmrelatlub 14124 Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  I  =  (toInc `  C )   &    |-  F  =  (mrCls `  C )   &    |-  L  =  ( lub `  I )   =>    |-  (
 ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  ( L `  U )  =  ( F `  U. U ) )
 
Theoremmreclat 14125 A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  I  =  (toInc `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  I  e.  CLat )
 
9.2.5  Distributive lattices
 
Theoremlatmass 14126 Lattice meet is associative. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  ./\  Y )  ./\  Z )  =  ( X  ./\  ( Y  ./\  Z ) ) )
 
Theoremlatdisdlem 14127* Lemma for latdisd 14128. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( K  e.  Lat  ->  ( A. u  e.  B  A. v  e.  B  A. w  e.  B  ( u  .\/  ( v  ./\  w ) )  =  ( ( u  .\/  v
 )  ./\  ( u  .\/  w ) )  ->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y  .\/  z ) )  =  ( ( x  ./\  y )  .\/  ( x  ./\  z
 ) ) ) )
 
Theoremlatdisd 14128* In a lattice, joins distribute over meets if and only if meets distribute over joins; the distributive property is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( K  e.  Lat  ->  ( A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .\/  ( y  ./\  z ) )  =  ( ( x  .\/  y )  ./\  ( x 
 .\/  z ) )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y  .\/  z ) )  =  ( ( x  ./\  y )  .\/  ( x 
 ./\  z ) ) ) )
 
Syntaxcdlat 14129 The class of distributive lattices.
 class DLat
 
Definitiondf-dlat 14130* A distributive lattice is a lattice in which meets distribute over joins, or equivalently (latdisd 14128) joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |- DLat  =  { k  e.  Lat  | 
 [. ( Base `  k
 )  /  b ]. [. ( join `  k )  /  j ]. [. ( meet `  k )  /  m ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x m ( y j z ) )  =  ( ( x m y ) j ( x m z ) ) }
 
Theoremisdlat 14131* Property of being a distributive lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( K  e. DLat  <->  ( K  e.  Lat  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y  .\/  z ) )  =  ( ( x  ./\  y )  .\/  ( x 
 ./\  z ) ) ) )
 
Theoremdlatmjdi 14132 In a distributive lattice, meets distribute over joins. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e. DLat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  (
 ( X  ./\  Y ) 
 .\/  ( X  ./\  Z ) ) )
 
Theoremdlatl 14133 A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( K  e. DLat  ->  K  e.  Lat )
 
Theoremodudlatb 14134 The dual of a distributive lattice is a distributive lattice and conversely. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  D  =  (ODual `  K )   =>    |-  ( K  e.  V  ->  ( K  e. DLat  <->  D  e. DLat ) )
 
Theoremdlatjmdi 14135 In a distributive lattice, joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e. DLat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .\/  ( Y  ./\  Z ) )  =  ( ( X  .\/  Y )  ./\  ( X  .\/  Z ) ) )
 
9.2.6  Posets and lattices as relations
 
Syntaxcps 14136 Extend class notation with the class of all posets.
 class  PosetRel
 
Syntaxctsr 14137 Extend class notation with the class of all totally ordered sets.
 class  TosetRel
 
Syntaxcspw 14138 Extend class notation with the supremum of an ordered set.
 class  sup w
 
Syntaxcinf 14139 Extend class notation with the infimum of an ordered set.
 class  inf w
 
Syntaxcla 14140 Extend class notation with the class of all lattices.
 class  LatRel
 
Definitiondf-ps 14141 Define the class of all posets (partially ordered sets) with weak ordering (e.g. "less than or equal to" instead of "less than"). A poset is a relation which is transitive, reflexive, and antisymmetric. (Contributed by NM, 11-May-2008.)
 |-  PosetRel 
 =  { r  |  ( Rel  r  /\  ( r  o.  r
 )  C_  r  /\  ( r  i^i  `' r
 )  =  (  _I  |`  U. U. r ) ) }
 
Definitiondf-tsr 14142 Define the class of all totally ordered sets. (Contributed by FL, 1-Nov-2009.)
 |-  TosetRel 
 =  { r  e.  PosetRel 
 |  ( dom  r  X.  dom  r )  C_  ( r  u.  `' r
 ) }
 
Definitiondf-spw 14143* Define suprema under weak orderings. Unlike df-sup 7078 for strong orderings,  sup w is evaluates to a member of the field of  R iff the supremum exists. Read 
R  sup w  A as the  R-supremum of set  A. (Contributed by NM, 13-May-2008.)
 |- 
 sup w  =  (
 r  e.  PosetRel ,  x  e.  _V  |->  ( iota_ y  e. 
 U. U. r ( A. z  e.  x  z
 r y  /\  A. z  e.  U. U. r
 ( A. w  e.  x  w r z  ->  y r z ) ) ) )
 
Definitiondf-nfw 14144* Define the class of all infima of a weak ordering relation. (Contributed by FL, 6-Sep-2009.)
 |- 
 inf w  =  (
 r  e.  _V ,  x  e.  _V  |->  ( `' r  sup w  x ) )
 
Definitiondf-lar 14145* Define the class of all lattices, which are posets in which every two elements have upper and lower bounds. (Contributed by NM, 12-Jun-2008.)
 |-  LatRel  =  { r  e.  PosetRel 
 |  A. x  e.  dom  r A. y  e.  dom  r ( ( r 
 sup w  { x ,  y } )  e. 
 dom  r  /\  (
 r  inf w  { x ,  y } )  e. 
 dom  r ) }
 
Theoremisps 14146 The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.)
 |-  ( R  e.  A  ->  ( R  e.  PosetRel  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  ( R  i^i  `' R )  =  (  _I  |`  U. U. R ) ) ) )
 
Theorempsrel 14147 A poset is a relation. (Contributed by NM, 12-May-2008.)
 |-  ( A  e.  PosetRel  ->  Rel 
 A )
 
Theorempsref2 14148 A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)
 |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |` 
 U. U. R ) )
 
Theorempstr2 14149 A poset is transitive. (Contributed by FL, 3-Aug-2009.)
 |-  ( R  e.  PosetRel  ->  ( R  o.  R ) 
 C_  R )
 
Theorempslem 14150 Lemma for psref 14152 and others. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( R  e.  PosetRel  ->  ( ( ( A R B  /\  B R C )  ->  A R C )  /\  ( A  e.  U.
 U. R  ->  A R A )  /\  (
 ( A R B  /\  B R A ) 
 ->  A  =  B ) ) )
 
Theorempsdmrn 14151 The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.)
 |-  ( R  e.  PosetRel  ->  ( dom  R  =  U. U. R  /\  ran  R  =  U. U. R ) )
 
Theorempsref 14152 A poset is reflexive. (Contributed by NM, 13-May-2008.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  PosetRel  /\  A  e.  X )  ->  A R A )
 
Theorempsrn 14153 The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)
 |-  X  =  dom  R   =>    |-  ( R  e.  PosetRel  ->  X  =  ran  R )
 
Theorempsasym 14154 A poset is antisymmetric. (Contributed by NM, 12-May-2008.)
 |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R A )  ->  A  =  B )
 
Theorempstr 14155 A poset is transitive. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R C )  ->  A R C )
 
Theoremcnvps 14156 The converse of a poset is a poset. In the general case  ( `' R  e.  PosetRel  ->  R  e.  PosetRel ) is not true. See cnvpsb 14157 for a special case where the property holds. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
 
Theoremcnvpsb 14157 The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)
 |-  ( Rel  R  ->  ( R  e.  PosetRel  <->  `' R  e.  PosetRel ) )
 
Theorempsss 14158 Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.)
 |-  ( R  e.  PosetRel  ->  ( R  i^i  ( A  X.  A ) )  e.  PosetRel )
 
Theorempsssdm2 14159 Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   =>    |-  ( R  e.  PosetRel  ->  dom  (  R  i^i  ( A  X.  A ) )  =  ( X  i^i  A ) )
 
Theorempsssdm 14160 Field of a subposet. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  PosetRel  /\  A  C_  X )  ->  dom  (  R  i^i  ( A  X.  A ) )  =  A )
 
Theoremistsr 14161 The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  X  =  dom  R   =>    |-  ( R  e.  TosetRel  <->  ( R  e.  PosetRel  /\  ( X  X.  X )  C_  ( R  u.  `' R ) ) )
 
Theoremistsr2 14162* The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  X  =  dom  R   =>    |-  ( R  e.  TosetRel  <->  ( R  e.  PosetRel  /\ 
 A. x  e.  X  A. y  e.  X  ( x R y  \/  y R x ) ) )
 
Theoremtsrlin 14163 A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  TosetRel  /\  A  e.  X  /\  B  e.  X )  ->  ( A R B  \/  B R A ) )
 
Theoremtsrlemax 14164 Two ways of saying a number is less than or equal to the maximum of two others. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A R if ( B R C ,  C ,  B )  <->  ( A R B  \/  A R C ) ) )
 
Theoremtsrps 14165 A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
 
Theoremcnvtsr 14166 The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  TosetRel  ->  `' R  e.  TosetRel  )
 
Theoremtsrss 14167 Any subset of a totally ordered set is totally ordered. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Nov-2013.)
 |-  ( R  e.  TosetRel  ->  ( R  i^i  ( A  X.  A ) )  e.  TosetRel  )
 
Theoremspwval2 14168* Value of supremum under a weak ordering. Read  R  sup w  A as "the  R-supremum of  A."  U. U. R is the field of a relation  R by relfld 5104. Unlike df-sup 7078 for strong orderings, the supremum exists iff  R  sup w  A belongs to the field. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
 |-  X  =  U. U. R   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  V ) 
 ->  ( R  sup w  A )  =  ( iota_ x  e.  X ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) ) )
 
Theoremspwval 14169* Value of supremum under a weak ordering. Read  R  sup w  A as "the  R-supremum of  A."  U. U. R is the field of a relation  R by relfld 5104. Unlike df-sup 7078 for strong orderings, the supremum exists iff  R  sup w  A belongs to the field. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  PosetRel  /\  A  e.  V )  ->  ( R  sup w  A )  =  ( iota_ x  e.  X ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) ) )
 
Theoremspwmo 14170* A poset has at most one supremum. (Contributed by NM, 13-May-2008.)
 |-  ( ph  <->  ( A. y  e.  A  y R x 
 /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )   =>    |-  ( R  e.  PosetRel  ->  E* x ( x  e.  X  /\  ph )
 )
 
Theoremspweu 14171* A supremum is unique. (Contributed by NM, 15-May-2008.)
 |-  ( ph  <->  ( A. y  e.  A  y R x 
 /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )   =>    |-  ( ( R  e.  PosetRel  /\ 
 E. x  e.  X  ph )  ->  E! x  e.  X  ph )
 
Theoremspwpr2 14172* Property of supremum defining condition for an unordered pair. (Contributed by NM, 24-Jun-2008.)
 |-  ( ph  <->  ( A. y  e.  A  y R x 
 /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )   =>    |-  ( ( ( R  e.  T  /\  A  =  { B ,  C } )  /\  ( B  e.  U  /\  C  e.  W ) )  ->  ( ph  <->  ( ( B R x  /\  C R x )  /\  A. y  e.  X  (
 ( B R y 
 /\  C R y )  ->  x R y ) ) ) )
 
Theoremspwex 14173* A supremum exists iff  R  sup w  A belongs to the domain of  R. (Contributed by NM, 15-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
 |-  X  =  dom  R   &    |-  ( ph 
 <->  ( A. y  e.  A  y R x 
 /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  V ) 
 ->  ( E. x  e.  X  ph  <->  ( R  sup w  A )  e.  X ) )
 
Theoremspwcl 14174* Closure of a supremum. (Contributed by NM, 15-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
 |-  X  =  dom  R   &    |-  ( ph 
 <->  ( A. y  e.  A  y R x 
 /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  V  /\  E. x  e.  X  ph )  ->  ( R  sup w  A )  e.  X )
 
Theoremspwpr4 14175* Supremum of an unordered pair. (Contributed by NM, 7-Jul-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  PosetRel  /\  ( A R C  /\  B R C )  /\  A. x  e.  X  ( ( A R x 
 /\  B R x )  ->  C R x ) )  ->  ( R  sup w  { A ,  B }
 )  =  C )
 
Theoremspwpr4c 14176 Supremum of an unordered pair of comparable elements. (Contributed by NM, 7-Jul-2008.)
 |-  ( ( R  e.  PosetRel  /\  A R B ) 
 ->  ( R  sup w  { A ,  B }
 )  =  B )
 
Theoremisla 14177* The predicate "is a lattice" i.e. a poset in which any two elements have upper and lower bounds. (Contributed by NM, 12-Jun-2008.)
 |-  X  =  dom  R   =>    |-  ( R  e.  LatRel  <->  ( R  e.  PosetRel  /\ 
 A. x  e.  X  A. y  e.  X  ( ( R  sup w  { x ,  y }
 )  e.  X  /\  ( R  inf w  { x ,  y }
 )  e.  X ) ) )
 
Theoremlaspwcl 14178 Closure of the supremum (join) of two lattice elements. (Contributed by NM, 12-Jun-2008.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  LatRel  /\  A  e.  X  /\  B  e.  X )  ->  ( R  sup w  { A ,  B }
 )  e.  X )
 
Theoremlanfwcl 14179 Closure of the infimum (meet) of two lattice elements. (Contributed by NM, 20-Jun-2008.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  LatRel  /\  A  e.  X  /\  B  e.  X )  ->  ( R  inf w  { A ,  B }
 )  e.  X )
 
Theoremlaps 14180 A lattice is a poset. (Contributed by NM, 12-Jun-2008.)
 |-  ( R  e.  LatRel  ->  R  e.  PosetRel )
 
Theoremledm 14181 domain of  <_ is  RR*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  RR*  =  dom  <_
 
Theoremlern 14182 The range of  <_ is  RR*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  RR*  =  ran  <_
 
Theoremlefld 14183 The field of the 'less or equal to' relationship on the extended real. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  RR*  =  U. U.  <_
 
Theoremletsr 14184 The "less than or equal to" relationship on the extended reals is a toset. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |- 
 <_  e.  TosetRel
 
9.2.7  Directed sets, nets
 
Syntaxcdir 14185 Extend class notation with the class of all directed sets.
 class  DirRel
 
Syntaxctail 14186 Extend class notation with the tail function.
 class  tail
 
Definitiondf-dir 14187 Define the class of all directed sets/directions. (Contributed by Jeff Hankins, 25-Nov-2009.)
 |-  DirRel  =  { r  |  ( ( Rel  r  /\  (  _I  |`  U. U. r )  C_  r ) 
 /\  ( ( r  o.  r )  C_  r  /\  ( U. U. r  X.  U. U. r
 )  C_  ( `' r  o.  r ) ) ) }
 
Definitiondf-tail 14188* Define the tail function for directed sets. (Contributed by Jeff Hankins, 25-Nov-2009.)
 |- 
 tail  =  ( r  e.  DirRel  |->  ( x  e. 
 U. U. r  |->  ( r
 " { x }
 ) ) )
 
Theoremisdir 14189 A condition for a relation to be a direction. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  A  =  U. U. R   =>    |-  ( R  e.  V  ->  ( R  e.  DirRel  <->  (
 ( Rel  R  /\  (  _I  |`  A )  C_  R )  /\  (
 ( R  o.  R )  C_  R  /\  ( A  X.  A )  C_  ( `' R  o.  R ) ) ) ) )
 
Theoremreldir 14190 A direction is a relation. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  ( R  e.  DirRel  ->  Rel  R )
 
Theoremdirdm 14191 A direction's domain is equal to its field. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  ( R  e.  DirRel  ->  dom  R  =  U. U. R )
 
Theoremdirref 14192 A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  DirRel  /\  A  e.  X ) 
 ->  A R A )
 
Theoremdirtr 14193 A direction is transitive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  ( ( ( R  e.  DirRel  /\  C  e.  V )  /\  ( A R B  /\  B R C ) )  ->  A R C )
 
Theoremdirge 14194* For any two elements of a directed set, there exists a third element greater than or equal to both. (Note that this does not say that the two elements have a least upper bound.) (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  DirRel  /\  A  e.  X  /\  B  e.  X )  ->  E. x  e.  X  ( A R x  /\  B R x ) )
 
Theoremtsrdir 14195 A totally ordered set is a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  ( A  e.  TosetRel  ->  A  e.  DirRel )
 
PART 10  BASIC ALGEBRAIC STRUCTURES
 
10.1  Monoids
 
10.1.1  Definition and basic properties
 
Syntaxcmnd 14196 Extend class notation with class of all monoids.
 class  Mnd
 
Syntaxcgrp 14197 Extend class notation with class of all groups.
 class  Grp
 
Syntaxcminusg 14198 Extend class notation with inverse of group element.
 class  inv g
 
Syntaxcplusf 14199 Extend class notation with group addition as a function.
 class  + f
 
Syntaxcsg 14200 Extend class notation with group subtraction (or division) operation.
 class  -g
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