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Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremposglbd 14201* Properties which determine a greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- 
 .<_  =  ( le `  K )   &    |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  G  =  ( glb `  K ) )   &    |-  ( ph  ->  K  e.  Poset )   &    |-  ( ph  ->  S 
 C_  B )   &    |-  ( ph  ->  T  e.  B )   &    |-  ( ( ph  /\  x  e.  S )  ->  T  .<_  x )   &    |-  ( ( ph  /\  y  e.  B  /\  A. x  e.  S  y 
 .<_  x )  ->  y  .<_  T )   =>    |-  ( ph  ->  ( G `  S )  =  T )
 
9.2.4  Subset order structures
 
Syntaxcipo 14202 Class function defining inclusion posets.
 class toInc
 
Definitiondf-ipo 14203* For any family of sets, define the poset of that family ordered by inclusion. See ipobas 14206, ipolerval 14207, and ipole 14209 for its contract.

EDITORIAL: I'm not thrilled with the name. Any suggestions? (Contributed by Stefan O'Rear, 30-Jan-2015.) (New usage is discouraged.)

 |- toInc  =  ( f  e.  _V  |->  [_
 { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  /  o ]_ ( { <. ( Base `  ndx ) ,  f >. , 
 <. (TopSet `  ndx ) ,  (ordTop `  o ) >. }  u.  { <. ( le `  ndx ) ,  o >. ,  <. ( oc
 `  ndx ) ,  ( x  e.  f  |->  U. { y  e.  f  |  ( y  i^i  x )  =  (/) } ) >. } ) )
 
Theoremipostr 14204 The structure of df-ipo 14203 is a structure defining indexes up to 11. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. (TopSet `  ndx ) ,  J >. }  u.  { <. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ._|_  >. } ) Struct  <. 1 , ; 1 1 >.
 
Theoremipoval 14205* 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 14206 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 14207* 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 14208 Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015.)
 |-  I  =  (toInc `  F )   &    |-  .<_  =  ( le `  I )   =>    |-  ( F  e.  V  ->  (ordTop `  .<_  )  =  (TopSet `  I )
 )
 
Theoremipole 14209 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 14210 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 14211 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 14212* 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 14213 Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( (toInc `  A )  e. Dirset  ->  A  e.  _V )
 
Theoremipodrsfi 14214* 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 14215 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 14216* 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 14217* An algebraic closure system satisfies isacs3 14225. (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 14218 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 14219* 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 14220* 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 14221 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 14222 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 14223* 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 14224* 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 14225* 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 ) ) )
 
Theoremacsficld 14226 In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 14222. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( N `  S )  =  U. ( N
 " ( ~P S  i^i  Fin ) ) )
 
Theoremacsficl2d 14227* In an algebraic closure system, an element is in the closure of a set if and only if it is in the closure of a finite subset. Alternate form of acsficl 14222. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( Y  e.  ( N `
  S )  <->  E. x  e.  ( ~P S  i^i  Fin ) Y  e.  ( N `  x ) ) )
 
Theoremacsfiindd 14228 In an algebraic closure system, a set is independent if and only if all its finite subsets are independent. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  ( ~P S  i^i  Fin )  C_  I
 ) )
 
Theoremacsmapd 14229* In an algebraic closure system, if 
T is contained in the closure of  S, there is a map  f from  T into the set of finite subsets of  S such that the closure of  U. ran  f contains  T. This is proven by applying acsficl2d 14227 to each element of  T. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  T  C_  ( N `  S ) )   =>    |-  ( ph  ->  E. f
 ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f
 ) ) )
 
Theoremacsmap2d 14230* In an algebraic closure system, if 
S and  T have the same closure and  S is independent, then there is a map  f from  T into the set of finite subsets of  S such that  S equals the union of  ran  f. This is proven by taking the map  f from acsmapd 14229 and observing that, since  S and  T have the same closure, the closure of  U. ran  f must contain  S. Since  S is independent, by mrissmrcd 13490,  U. ran  f must equal  S. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   =>    |-  ( ph  ->  E. f
 ( f : T --> ( ~P S  i^i  Fin )  /\  S  =  U. ran  f ) )
 
Theoremacsinfd 14231 In an algebraic closure system, if 
S and  T have the same closure and  S is infinite independent, then  T is infinite. This follows from applying unirnffid 7101 to the map given in acsmap2d 14230. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   &    |-  ( ph  ->  -.  S  e.  Fin )   =>    |-  ( ph  ->  -.  T  e.  Fin )
 
Theoremacsdomd 14232 In an algebraic closure system, if 
S and  T have the same closure and  S is infinite independent, then  T dominates  S. This follows from applying acsinfd 14231 and then applying unirnfdomd 8143 to the map given in acsmap2d 14230. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   &    |-  ( ph  ->  -.  S  e.  Fin )   =>    |-  ( ph  ->  S  ~<_  T )
 
Theoremacsinfdimd 14233 In an algebraic closure system, if two independent sets have equal closure and one is infinite, then they are equinumerous. This is proven by using acsdomd 14232 twice with acsinfd 14231. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  e.  I
 )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   &    |-  ( ph  ->  -.  S  e.  Fin )   =>    |-  ( ph  ->  S  ~~  T )
 
Theoremacsexdimd 14234* In an algebraic closure system whose closure operator has the exchange property, if two independent sets have equal closure, they are equinumerous. See mreexfidimd 13500 for the finite case and acsinfdimd 14233 for the infinite case. This is a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  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  ->  ( N `  S )  =  ( N `  T ) )   =>    |-  ( ph  ->  S  ~~  T )
 
Theoremmrelatglb 14235 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 14236 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 14237 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 14238 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 14239 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 14240* Lemma for latdisd 14241. (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 14241* 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 14242 The class of distributive lattices.
 class DLat
 
Definitiondf-dlat 14243* A distributive lattice is a lattice in which meets distribute over joins, or equivalently (latdisd 14241) 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 14244* 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 14245 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 14246 A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( K  e. DLat  ->  K  e.  Lat )
 
Theoremodudlatb 14247 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 14248 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 14249 Extend class notation with the class of all posets.
 class  PosetRel
 
Syntaxctsr 14250 Extend class notation with the class of all totally ordered sets.
 class  TosetRel
 
Syntaxcspw 14251 Extend class notation with the supremum of an ordered set.
 class  sup w
 
Syntaxcinf 14252 Extend class notation with the infimum of an ordered set.
 class  inf w
 
Syntaxcla 14253 Extend class notation with the class of all lattices.
 class  LatRel
 
Definitiondf-ps 14254 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 14255 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 14256* Define suprema under weak orderings. Unlike df-sup 7148 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 14257* 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 14258* 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 14259 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 14260 A poset is a relation. (Contributed by NM, 12-May-2008.)
 |-  ( A  e.  PosetRel  ->  Rel 
 A )
 
Theorempsref2 14261 A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)
 |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |` 
 U. U. R ) )
 
Theorempstr2 14262 A poset is transitive. (Contributed by FL, 3-Aug-2009.)
 |-  ( R  e.  PosetRel  ->  ( R  o.  R ) 
 C_  R )
 
Theorempslem 14263 Lemma for psref 14265 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 14264 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 14265 A poset is reflexive. (Contributed by NM, 13-May-2008.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  PosetRel  /\  A  e.  X )  ->  A R A )
 
Theorempsrn 14266 The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)
 |-  X  =  dom  R   =>    |-  ( R  e.  PosetRel  ->  X  =  ran  R )
 
Theorempsasym 14267 A poset is antisymmetric. (Contributed by NM, 12-May-2008.)
 |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R A )  ->  A  =  B )
 
Theorempstr 14268 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 14269 The converse of a poset is a poset. In the general case  ( `' R  e.  PosetRel  ->  R  e.  PosetRel ) is not true. See cnvpsb 14270 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 14270 The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)
 |-  ( Rel  R  ->  ( R  e.  PosetRel  <->  `' R  e.  PosetRel ) )
 
Theorempsss 14271 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 14272 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 14273 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 14274 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 14275* 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 14276 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 14277 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 14278 A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
 
Theoremcnvtsr 14279 The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  TosetRel  ->  `' R  e.  TosetRel  )
 
Theoremtsrss 14280 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 14281* 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 5171. Unlike df-sup 7148 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 14282* 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 5171. Unlike df-sup 7148 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 14283* A poset has at most one supremum. (Contributed by NM, 13-May-2008.) (Revised by NM, 16-Jun-2017.)
 |-  ( 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 )
 
Theoremspweu 14284* 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 14285* 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 14286* 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 14287* 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 14288* 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 14289 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 14290* 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 14291 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 14292 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 14293 A lattice is a poset. (Contributed by NM, 12-Jun-2008.)
 |-  ( R  e.  LatRel  ->  R  e.  PosetRel )
 
Theoremledm 14294 domain of  <_ is  RR*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  RR*  =  dom  <_
 
Theoremlern 14295 The range of  <_ is  RR*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  RR*  =  ran  <_
 
Theoremlefld 14296 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 14297 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 14298 Extend class notation with the class of all directed sets.
 class  DirRel
 
Syntaxctail 14299 Extend class notation with the tail function.
 class  tail
 
Definitiondf-dir 14300 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 ) ) ) }
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