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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorema1i4 26201 Add an antecedent to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ta )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i14 26202 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ps  ->  ( ch  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i24 26203 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)
 |-  ( ph  ->  ( ch  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i34 26204 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)
 |-  ( ph  ->  ( ps  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremimp5gOLD 26205 An importation inference. (Moved into main set.mm as imp5g 583 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( ( ch  /\  th )  /\  ta )  ->  et )
 )
 
Theoremimp55OLD 26206 An importation inference. (Moved into main set.mm as imp55 584 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  /\  ta )  ->  et )
 
Theoremimp511OLD 26207 An importation inference. (Moved into main set.mm as imp511 585 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ( ph  /\  (
 ( ps  /\  ( ch  /\  th ) ) 
 /\  ta ) )  ->  et )
 
Theoremexp5d 26208 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  ->  ( ( th  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5g 26209 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ps )  ->  ( ( ( ch 
 /\  th )  /\  ta )  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5j 26210 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ( ps  /\  ch )  /\  th )  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5k 26211 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ps  /\  ( ch 
 /\  th ) )  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5l 26212 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ps  /\  ch )  /\  ( th  /\  ta ) )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp56 26213 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  ( th  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp58 26214 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ph  /\  ps )  /\  ( ( ch 
 /\  th )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp510 26215 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ( ps  /\  ch )  /\  th )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp511 26216 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ps  /\  ( ch 
 /\  th ) )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp512 26217 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ps  /\  ch )  /\  ( th  /\  ta ) ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
TheoremmtordOLD 26218 A modus tollens deduction involving disjunction. (Moved into main set.mm as mtord 641 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 15-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  ( ps  ->  ( ch  \/  th )
 ) )   =>    |-  ( ph  ->  -.  ps )
 
Theorem3com12d 26219 Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ph  ->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ( ch  /\  ps  /\  th ) )
 
Theoremimp5p 26220 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch  /\  th 
 /\  ta )  ->  et )
 ) )
 
Theoremimp5q 26221 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( ch 
 /\  th  /\  ta )  ->  et ) )
 
Theoremecase13d 26222 Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  ( ch  \/  ps 
 \/  th ) )   =>    |-  ( ph  ->  ps )
 
TheoremeqeuOLD 26223* A condition which implies existential uniqueness. (Moved into main set.mm as eqeu 2936 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 8-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( A  e.  B  /\  ps  /\ 
 A. x ( ph  ->  x  =  A ) )  ->  E! x ph )
 
Theoremsubtr 26224 Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x Y   &    |-  F/_ x Z   &    |-  ( x  =  A  ->  X  =  Y )   &    |-  ( x  =  B  ->  X  =  Z )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  Y  =  Z ) )
 
Theoremsubtr2 26225 Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/ x ps   &    |-  F/ x ch   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ch ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  ( ps  <->  ch ) ) )
 
TheoremcnvresimaOLD 26226 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) (Moved to cnvresima 5162 in main set.mm and may be deleted by mathbox owner, JGH. --NM 23-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( `' ( F  |`  A )
 " B )  =  ( ( `' F " B )  i^i  A )
 
Theoremtrer 26227* A relation intersected with its converse is an equivalence relation if the relation is transitive. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( A. a A. b A. c ( ( a 
 .<_  b  /\  b  .<_  c )  ->  a  .<_  c )  ->  (  .<_  i^i  `'  .<_  )  Er  dom  (  .<_  i^i  `'  .<_  ) )
 
Theoremelicc3 26228 An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) ) )
 
TheoremccidOLD 26229 A closed interval with identical lower and upper bounds is a singleton. (Moved into main set.mm as iccid 10701 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
 
TheoremioodisjOLD 26230 If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Moved into main set.mm as ioodisj 10765 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  /\  B  <_  C )  ->  (
 ( A (,) B )  i^i  ( C (,) D ) )  =  (/) )
 
Theoremfinminlem 26231* A useful lemma about finite sets. If a property holds for a finite set, it holds for a minimal set. (Contributed by Jeff Hankins, 4-Dec-2009.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  Fin  ph  ->  E. x ( ph  /\  A. y
 ( ( y  C_  x  /\  ps )  ->  x  =  y )
 ) )
 
Theoremdivcan7OLD 26232 Cancel equal divisors in a division. (Contributed by Jeff Hankins, 29-Sep-2013.) (Moved to divcan7 9469 in main set.mm and may be deleted by mathbox owner, JGH. --NM 21-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) 
 /\  ( C  e.  CC  /\  C  =/=  0
 ) )  ->  (
 ( A  /  C )  /  ( B  /  C ) )  =  ( A  /  B ) )
 
TheoreminfleOLD 26233* If a nonempty set of real numbers has a lower bound, its infimum is less than or equal to any of its elements. (Contributed by Jeff Hankins, 15-Sep-2013.) (Moved to infmrlb 9735 in main set.mm and may be deleted by mathbox owner, JGH. --NM 21-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( S  C_  RR  /\ 
 E. x  e.  RR  A. y  e.  S  x  <_  y  /\  A  e.  S )  ->  sup ( S ,  RR ,  `'  <  )  <_  A )
 
Theoremgtinf 26234* Any number greater than an infimum is greater than some element of the set. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  (
 ( ( S  C_  RR  /\  S  =/=  (/)  /\  E. x  e.  RR  A. y  e.  S  x  <_  y
 )  /\  ( A  e.  RR  /\  sup ( S ,  RR ,  `'  <  )  <  A ) )  ->  E. z  e.  S  z  <  A )
 
Theoremopnrebl 26235* A set is open in the standard topology of the reals precisely when every point can be enclosed in an open ball. (Contributed by Jeff Hankins, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
 |-  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. x  e.  A  E. y  e.  RR+  ( ( x  -  y ) (,) ( x  +  y )
 )  C_  A )
 )
 
Theoremopnrebl2 26236* A set is open in the standard topology of the reals precisely when every point can be enclosed in an arbitrarily small ball. (Contributed by Jeff Hankins, 22-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
 |-  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  ( z  <_  y  /\  ( ( x  -  z ) (,) ( x  +  z )
 )  C_  A )
 ) )
 
Theoremdivides2OLD 26237 One nonzero integer divides another integer if and only if their quotient is an integer. (Moved to cnvresima 5162 in main set.mm and may be deleted by mathbox owner, JGH. --NM 28-Feb-2014.) (Contributed by Jeff Hankins, 29-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )
 
Theoremnn0prpwlem 26238* Lemma for nn0prpw 26239. Use strong induction to show that every natural number has unique prime power divisors. (Contributed by Jeff Hankins, 28-Sep-2013.)
 |-  ( A  e.  NN  ->  A. k  e.  NN  (
 k  <  A  ->  E. p  e.  Prime  E. n  e.  NN  -.  ( ( p ^ n ) 
 ||  k  <->  ( p ^ n )  ||  A ) ) )
 
Theoremnn0prpw 26239* Two nonnegative integers are the same if and only if they are divisible by the same prime powers. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  =  B  <->  A. p  e.  Prime  A. n  e.  NN  ( ( p ^ n )  ||  A 
 <->  ( p ^ n )  ||  B ) ) )
 
TheoremqredeqOLD 26240 Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.) (Moved into main set.mm as qredeq 12785 and may be deleted by mathbox owner, JGH. --NM 13-Oct-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  /\  ( M  /  N )  =  ( P  /  Q ) )  ->  ( M  =  P  /\  N  =  Q ) )
 
TheoremqredeuOLD 26241* Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.) (Moved into main set.mm as qredeq 12785 and may be deleted by mathbox owner, JGH. --NM 13-Oct-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  QQ  ->  E! x  e.  ( ZZ 
 X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) )
 
18.14.2  Basic topological facts
 
Theoremtopbnd 26242 Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( ( ( cls `  J ) `  A )  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J ) `  A )  \  ( ( int `  J ) `  A ) ) )
 
Theoremopnbnd 26243 A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( A  e.  J  <->  ( A  i^i  ( ( ( cls `  J ) `  A )  i^i  ( ( cls `  J ) `  ( X  \  A ) ) ) )  =  (/) ) )
 
Theoremcldbnd 26244 A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  ( ( ( cls `  J ) `  A )  i^i  (
 ( cls `  J ) `  ( X  \  A ) ) )  C_  A ) )
 
Theoremntruni 26245* A union of interiors is a subset of the interior of the union. The reverse inclusion may not hold. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  O  C_  ~P X )  ->  U_ o  e.  O  ( ( int `  J ) `  o )  C_  ( ( int `  J ) `  U. O ) )
 
Theoremclsun 26246 A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  ( ( cls `  J ) `  ( A  u.  B ) )  =  ( ( ( cls `  J ) `  A )  u.  ( ( cls `  J ) `  B ) ) )
 
Theoremclsint2 26247* The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  ( ( cls `  J ) `  |^| C )  C_  |^|_ c  e.  C  ( ( cls `  J ) `  c ) )
 
Theoremopnregcld 26248* A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( ( ( cls `  J ) `  (
 ( int `  J ) `  A ) )  =  A  <->  E. o  e.  J  A  =  ( ( cls `  J ) `  o ) ) )
 
Theoremcldregopn 26249* A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( ( ( int `  J ) `  (
 ( cls `  J ) `  A ) )  =  A  <->  E. c  e.  ( Clsd `  J ) A  =  ( ( int `  J ) `  c
 ) ) )
 
Theoremneiin 26250 Two neighborhoods intersect to form a neighborhood of the intersection. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  (
 ( J  e.  Top  /\  M  e.  ( ( nei `  J ) `  A )  /\  N  e.  ( ( nei `  J ) `  B ) ) 
 ->  ( M  i^i  N )  e.  ( ( nei `  J ) `  ( A  i^i  B ) ) )
 
Theoremhmeoclda 26251 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
 |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Homeo  K ) )  /\  S  e.  ( Clsd `  J ) )  ->  ( F " S )  e.  ( Clsd `  K ) )
 
Theoremhmeocldb 26252 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.)
 |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Homeo  K ) )  /\  S  e.  ( Clsd `  K ) )  ->  ( `' F " S )  e.  ( Clsd `  J ) )
 
Theoremdfcon2OLD 26253* An alternate definition of connectedness. (Moved into main set.mm as dfcon2 17145 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 8-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( J  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( ( x  =/=  (/)  /\  y  =/=  (/)  /\  ( x  i^i  y
 )  =  (/) )  ->  X  =/=  ( x  u.  y ) ) ) )
 
TheoremconnsubOLD 26254* Two equivalent ways of saying that a subspace topology is connected. (Moved into main set.mm as connsub 17147 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  S  C_  X )  ->  ( ( Jt  S )  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( ( ( x  i^i  S )  =/=  (/)  /\  (
 y  i^i  S )  =/= 
 (/)  /\  ( x  i^i  y )  C_  ( X 
 \  S ) ) 
 ->  -.  S  C_  ( x  u.  y ) ) ) )
 
18.14.3  Topology of the real numbers
 
TheoremreconnOLD 26255* A subset of the reals is connected iff it has the interval property. (Moved into main set.mm as reconn 18333 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 15-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  RR  ->  (
 ( ( topGen `  ran  (,) )t  A )  e.  Con  <->  A. x  e.  A  A. y  e.  A  ( x [,] y )  C_  A ) )
 
TheoremretopconOLD 26256 Corollary of reconn 18333. The set of real numbers is connected. (Moved into main set.mm as retopcon 18334 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( topGen `
  ran  (,) )  e. 
 Con
 
TheoremiccconnOLD 26257 A closed interval is connected. (Moved into main set.mm as iccconn 18335 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B ) )  e.  Con )
 
TheoremivthALT 26258* An alternate proof of the Intermediate Value Theorem ivth 18814 using topology. (Contributed by Jeff Hankins, 17-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn->
 CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A ) (,) ( F `  B ) ) ) ) )  ->  E. x  e.  ( A (,) B ) ( F `  x )  =  U )
 
18.14.4  Refinements
 
Syntaxcfne 26259 Extend class definition to include the "finer than" relation.
 class  Fne
 
Syntaxcref 26260 Extend class definition to include the refinement relation.
 class  Ref
 
Syntaxcptfin 26261 Extend class definition to include the class of point-finite covers.
 class  PtFin
 
Syntaxclocfin 26262 Extend class definition to include the class of locally finite covers.
 class  LocFin
 
Definitiondf-fne 26263* Define the fineness relation for covers. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  Fne  =  { <. x ,  y >.  |  ( U. x  =  U. y  /\  A. z  e.  x  z  C_ 
 U. ( y  i^i 
 ~P z ) ) }
 
Definitiondf-ref 26264* Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  Ref  =  { <. x ,  y >.  |  ( U. x  =  U. y  /\  A. z  e.  y  E. w  e.  x  z  C_  w ) }
 
Definitiondf-ptfin 26265* Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  PtFin  =  { x  |  A. y  e. 
 U. x { z  e.  x  |  y  e.  z }  e.  Fin }
 
Definitiondf-locfin 26266* Define "locally finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  LocFin  =  ( x  e.  Top  |->  { y  |  ( U. x  = 
 U. y  /\  A. p  e.  U. x E. n  e.  x  ( p  e.  n  /\  { s  e.  y  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) } )
 
Theoremfnerel 26267 Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  Rel  Fne
 
Theoremisfne 26268* The predicate " B is finer than  A." This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x  C_ 
 U. ( B  i^i  ~P x ) ) ) )
 
Theoremisfne4 26269 The predicate " B is finer than  A " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B ) ) )
 
Theoremisfne4b 26270 A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  V  ->  ( A Fne B  <->  ( X  =  Y  /\  ( topGen `  A )  C_  ( topGen `  B )
 ) ) )
 
Theoremisfne2 26271* The predicate " B is finer than  A." (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  A. y  e.  x  E. z  e.  B  (
 y  e.  z  /\  z  C_  x ) ) ) )
 
Theoremisfne3 26272* The predicate " B is finer than  A." (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  E. y ( y  C_  B  /\  x  =  U. y ) ) ) )
 
Theoremfnebas 26273 A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( A Fne B  ->  X  =  Y )
 
Theoremfnetg 26274 A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( A Fne B  ->  A  C_  ( topGen `  B )
 )
 
Theoremfnessex 26275* If  B is finer than  A and  S is an element of  A, every point in  S is an element of a subset of  S which is in  B. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  (
 ( A Fne B  /\  S  e.  A  /\  P  e.  S )  ->  E. x  e.  B  ( P  e.  x  /\  x  C_  S ) )
 
Theoremfneuni 26276* If  B is finer than  A, every element of  A is a union of elements of  B. (Contributed by Jeff Hankins, 11-Oct-2009.)
 |-  (
 ( A Fne B  /\  S  e.  A ) 
 ->  E. x ( x 
 C_  B  /\  S  =  U. x ) )
 
Theoremfneint 26277* If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)
 |-  ( A Fne B  ->  |^| { x  e.  B  |  P  e.  x }  C_  |^| { x  e.  A  |  P  e.  x } )
 
Theoremrefrel 26278 Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  Rel  Ref
 
Theoremisref 26279* The property of being a refinement of a cover. Dr. Nyikos once commented in class that the term "refinement" is actually misleading and that people are inclined to confuse it with the notion defined in isfne 26268. On the other hand, the two concepts do seem to have a dual relationship. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Ref B  <->  ( X  =  Y  /\  A. x  e.  B  E. y  e.  A  x  C_  y ) ) )
 
Theoremrefbas 26280 A refinement covers the same set. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( A Ref B  ->  X  =  Y )
 
Theoremrefssex 26281* Every set in a refinement has a superset in the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  (
 ( A Ref B  /\  S  e.  B ) 
 ->  E. x  e.  A  S  C_  x )
 
Theoremfness 26282 A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( ( B  e.  C  /\  A  C_  B  /\  X  =  Y ) 
 ->  A Fne B )
 
Theoremssref 26283 A subcover is a refinement of the original cover. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( ( A  e.  C  /\  A  C_  B  /\  X  =  Y ) 
 ->  B Ref A )
 
Theoremfneref 26284 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
 |-  ( A  e.  V  ->  A Fne A )
 
Theoremrefref 26285 Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  ( A  e.  V  ->  A Ref A )
 
Theoremfnetr 26286 Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  (
 ( A Fne B  /\  B Fne C ) 
 ->  A Fne C )
 
Theoremfneval 26287 Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  .~  B 
 <->  ( topGen `  A )  =  ( topGen `  B )
 ) )
 
Theoremfneer 26288 Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |- 
 .~  Er  _V
 
Theoremreftr 26289 Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  (
 ( A Ref B  /\  B Ref C ) 
 ->  A Ref C )
 
Theoremtopfne 26290 Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( K  e.  Top  /\  X  =  Y ) 
 ->  ( J  C_  K  <->  J Fne K ) )
 
Theoremtopfneec 26291 A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |-  ( J  e.  Top  ->  ( A  e.  [ J ]  .~  <->  ( topGen `  A )  =  J )
 )
 
Theoremtopfneec2 26292 A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.)
 |-  .~  =  ( Fne  i^i  `' Fne )   =>    |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( [ J ]  .~  =  [ K ]  .~  <->  J  =  K ) )
 
Theoremfnessref 26293* A cover is finer iff it has a subcover which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( X  =  Y  ->  ( A Fne B  <->  E. c ( c  C_  B  /\  A ( Fne 
 i^i  Ref ) c ) ) )
 
Theoremrefssfne 26294* A cover is a refinement iff it is a subcover of something which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( X  =  Y  ->  ( A Ref B  <->  E. c ( B  C_  c  /\  A ( Fne 
 i^i  Ref ) c ) ) )
 
Theoremisptfin 26295* The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. A   =>    |-  ( A  e.  B  ->  ( A  e.  PtFin  <->  A. x  e.  X  { y  e.  A  |  x  e.  y }  e.  Fin ) )
 
Theoremislocfin 26296* The statement "is a locally finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. J   &    |-  Y  =  U. A   =>    |-  ( A  e.  ( LocFin `
  J )  <->  ( J  e.  Top  /\  X  =  Y  /\  A. x  e.  X  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) )
 
Theoremfinptfin 26297 A finite cover is a point-finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  ( A  e.  Fin  ->  A  e.  PtFin )
 
Theoremptfinfin 26298* A point covered by a point-finite cover is only covered by finitely many elements. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. A   =>    |-  ( ( A  e.  PtFin  /\  P  e.  X ) 
 ->  { x  e.  A  |  P  e.  x }  e.  Fin )
 
Theoremfinlocfin 26299 A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  X  =  U. J   &    |-  Y  =  U. A   =>    |-  ( ( J  e.  Top  /\  A  e.  Fin  /\  X  =  Y )  ->  A  e.  ( LocFin `  J ) )
 
Theoremlocfintop 26300 A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  ( A  e.  ( LocFin `  J )  ->  J  e.  Top )
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