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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-ibcg 26201* Incidence-Betweenness Geometry plus congruence axioms. (Here is an excerpt of Aitken's handout.)

Axiom (C-1). Segment congruence is an equivalence relation for line segments.

Axiom (C-2). Suppose that 
A B is a line segment and  C D is a ray. Then there is a unique point  E on  C D, distinct from  C, such that  A B  .~  s C E. The next axiom concerns copying dividing or intermediate points on a segment.

Axiom (C-3). Suppose that 
A C and  A' C' are congruent line segments. If B is a point such that  A * B * C, then there is a point  B' such that  A' * B' * C',  A B  .~  s A' B', and  B C  .~  s B' C'.

Axiom (C-4). Angle congruence is an equivalence relation for angles.

Axiom (C-5). (Copying an angle) Suppose  B A C is an angle, and  D E is a ray. Then on any given side of  D E, there is a unique ray  D F such that  B A C  .~  a E D F.

Axiom (C-6). (Copying a triangle) Suppose  A B C is a triangle, and  A' B' is a segment such that  A B  .~  s A' B'. Then on any given side of  A' B', there is a point  C' such that  A B C  .~  a A' B' C'.

(For my private use only. Don't use.) Axiom C-1 C-2, C-3, C-4, C-5, C-6 of [AitkenIBCG] p. 2 . (Contributed by FL, 1-Apr-2016.)

 |- Ibcg  =  {
 g  e. Ibg  |  [. (angc `  g )  /  o ]. [. (PPoints `  g
 )  /  p ]. [. (btw `  g )  /  q ]. [. (ray `  g
 )  /  r ]. [. (segc `  g )  /  s ]. [. ( seg `  g )  /  t ]. [. ( line `  g )  /  l ]. [. (angle `  g
 )  /  u ]. [. (triangle `  g )  /  v ]. [. ( pdWords 3
 )  /  w ]. [. (Halfplane `  g )  /  x ].
 [. (Segments `  g )  /  y ]. [. (trcng `  g )  /  z ]. ( s  Er  y  /\  o  Er  ( u " w )  /\  A. d  e.  p  A. e  e.  p  (
 d  =/=  e  ->  (
 A. a  e.  p  A. b  e.  p  ( ( a  =/=  b  ->  E! f  e.  (
 d r e ) ( f  =/=  d  /\  ( a t b ) s ( d t f ) ) )  /\  A. c  e.  p  ( (
 c  e.  ( a q b )  /\  ( a t b ) s ( d t e ) ) 
 ->  E. f  e.  p  ( f  e.  (
 d q e ) 
 /\  ( a t c ) s ( d t f ) 
 /\  ( c t b ) s ( f t e ) ) ) )  /\  A. a  e.  w  (
 A. b  e.  ( x `  ( d l e ) ) E! f  e.  b  ( u `  a ) o ( u `  <" e d f "> )  /\  ( ( ( a `
  1 ) t ( a `  2
 ) ) s ( d t e ) 
 ->  A. b  e.  ( x `  ( d l e ) ) E. f  e.  b  a
 z ( v `  <" d e f "> ) ) ) ) ) ) }
 
Theoremisibcg 26202* The predicate "is an incidence betwenness geometry with congruences". (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  .~  a  =  (angc `  G )   &    |-  P  =  (PPoints `  G )   &    |-  U  =  (Segments `  G )   &    |-  B  =  (btw `  G )   &    |-  R  =  (ray `  G )   &    |-  .~  s  =  (segc `  G )   &    |-  S  =  ( seg `  G )   &    |-  L  =  ( line `  G )   &    |-  H  =  (Halfplane `  G )   &    |-  T  =  (triangle `  G )   &    |-  A  =  (angle `  G )   &    |-  .~  t  =  (trcng `  G )   &    |-  W  =  ( PdWords 3 )   =>    |-  ( G  e. Ibcg  <->  ( G  e. Ibg  /\  (  .~  s  Er  U  /\  .~  a  Er  ( A " W ) )  /\  A. d  e.  P  A. e  e.  P  ( d  =/=  e  ->  ( A. a  e.  P  A. b  e.  P  ( ( a  =/=  b  ->  E! f  e.  ( d R e ) ( f  =/=  d  /\  ( a S b )  .~  s ( d S f ) ) )  /\  A. c  e.  P  (
 ( c  e.  (
 a B b ) 
 /\  ( a S b )  .~  s
 ( d S e ) )  ->  E. f  e.  P  ( f  e.  ( d B e )  /\  ( a S c )  .~  s ( d S f )  /\  (
 c S b ) 
 .~  s ( f S e ) ) ) )  /\  A. a  e.  W  ( A. b  e.  ( H `  ( d L e ) ) E! f  e.  b  ( A `  a ) 
 .~  a ( A `
  <" e d f "> )  /\  ( ( ( a `
  1 ) S ( a `  2
 ) )  .~  s
 ( d S e )  ->  A. b  e.  ( H `  (
 d L e ) ) E. f  e.  b  a  .~  t
 ( T `  <" d
 e f "> ) ) ) ) ) ) )
 
Syntaxcslices 26203 Extend class notation with the slices symbol.
 class slices
 
Definitiondf-slices 26204* Return the slices generated by the Dedekind cut of a set of points. Definition 1 of [AitkenNG] p. 2. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
 |- slices  =  ( f  e. Ibg  |->  ( x  e.  ~P (PPoints `  f
 )  |->  { <. a ,  b >.  |  ( ( ( a  u.  b )  =  x  /\  (
 a  i^i  b )  =  (/) )  /\  (
 a  =/=  (/)  /\  b  =/= 
 (/) )  /\  (
 a  e.  (convex `  f
 )  /\  b  e.  (convex `  f ) ) ) } ) )
 
Syntaxccut 26205 Extend class notation with the cutpoint symbol.
 class Cut
 
Definitiondf-Cut 26206* Return the cutpoints of a set of points  ( x  u.  y
) where  x and  y are the slices of a Dedekind cut of  ( x  u.  y
). Definition 2 of [AitkenNG] p. 2. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
 |- Cut  =  ( f  e. Ibg  |->  ( x  e.  ~P (PPoints `  f
 ) ,  y  e. 
 ~P (PPoints `  f )  |->  { c  e.  ( x  u.  y )  | 
 A. u  e.  ( x  u.  y ) A. v  e.  ( x  u.  y ) ( u  e.  ( c (btw `  f ) v ) 
 ->  ( ( u  e.  x  /\  v  e.  x )  \/  ( u  e.  y  /\  v  e.  y )
 ) ) } )
 )
 
Syntaxcneug 26207 Extend class notation with the neutral geometry symbol.
 class Neug
 
Definitiondf-neug 26208* Definition of a neutral geometry. Every Dedekind cut of a line has a cut point. (Axiom of Dedekind in [AitkenNG] p. 3.) (For my private use only. Don't use.) (Contributed by FL, 1-Apr-2016.)
 |- Neug  =  {
 g  e. Ibcg  |  A. l  e.  (PLines `  g ) A. u  e.  (
 (slices `  g ) `  l ) ( ( 1st `  u )
 (Cut `  g )
 ( 2nd `  u )
 )  =/=  (/) }
 
Syntaxccircle 26209 Extend class notation with the Circle symbol.
 class Circle
 
Definitiondf-crcl 26210* Definition of a circle (degenerated or not). Definition 5 of [AitkenNG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 5-Mar-2016.)
 |- Circle  =  ( g  e. Ibcg  |->  ( x  e.  (PPoints `  g
 ) ,  y  e.  (PPoints `  g )  |->  { u  e.  (PPoints `  g )  |  ( x ( seg `  g
 ) u ) (segc `  g ) ( x ( seg `  g
 ) y ) }
 ) )
 
18.14  Mathbox for Jeff Hankins
 
18.14.1  Miscellany
 
Theorema1i13 26211 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ps  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theorema1i4 26212 Add an antecedent to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ta )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i14 26213 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ps  ->  ( ch  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i24 26214 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)
 |-  ( ph  ->  ( ch  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i34 26215 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)
 |-  ( ph  ->  ( ps  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremimp5gOLD 26216 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 26217 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 26218 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 26219 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  ->  ( ( th  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5g 26220 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ps )  ->  ( ( ( ch 
 /\  th )  /\  ta )  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5j 26221 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ( ps  /\  ch )  /\  th )  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5k 26222 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ps  /\  ( ch 
 /\  th ) )  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5l 26223 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ps  /\  ch )  /\  ( th  /\  ta ) )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp56 26224 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  ( th  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp58 26225 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ph  /\  ps )  /\  ( ( ch 
 /\  th )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp510 26226 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ( ps  /\  ch )  /\  th )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp511 26227 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ps  /\  ( ch 
 /\  th ) )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp512 26228 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ps  /\  ch )  /\  ( th  /\  ta ) ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
TheoremmtordOLD 26229 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 26230 Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ph  ->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ( ch  /\  ps  /\  th ) )
 
Theoremimp5p 26231 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch  /\  th 
 /\  ta )  ->  et )
 ) )
 
Theoremimp5q 26232 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( ch 
 /\  th  /\  ta )  ->  et ) )
 
Theoremecase13d 26233 Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  ( ch  \/  ps 
 \/  th ) )   =>    |-  ( ph  ->  ps )
 
TheoremeqeuOLD 26234* A condition which implies existential uniqueness. (Moved into main set.mm as eqeu 2938 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 26235 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 26236 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 26237 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) (Moved to cnvresima 5164 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 26238* 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 26239 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 26240 A closed interval with identical lower and upper bounds is a singleton. (Moved into main set.mm as iccid 10703 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 26241 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 10767 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 26242* 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 26243 Cancel equal divisors in a division. (Contributed by Jeff Hankins, 29-Sep-2013.) (Moved to divcan7 9471 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 26244* 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 9737 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 26245* 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 26246* 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 26247* 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 26248 One nonzero integer divides another integer if and only if their quotient is an integer. (Moved to cnvresima 5164 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 26249* Lemma for nn0prpw 26250. 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 26250* 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 26251 Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.) (Moved into main set.mm as qredeq 12787 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 26252* Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.) (Moved into main set.mm as qredeq 12787 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 26253 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 26254 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 26255 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 26256* 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 26257 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 26258* 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 26259* 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 26260* 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 26261 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 26262 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 26263 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 26264* An alternate definition of connectedness. (Moved into main set.mm as dfcon2 17147 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 26265* Two equivalent ways of saying that a subspace topology is connected. (Moved into main set.mm as connsub 17149 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 26266* A subset of the reals is connected iff it has the interval property. (Moved into main set.mm as reconn 18335 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 26267 Corollary of reconn 18335. The set of real numbers is connected. (Moved into main set.mm as retopcon 18336 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 26268 A closed interval is connected. (Moved into main set.mm as iccconn 18337 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 26269* An alternate proof of the Intermediate Value Theorem ivth 18816 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 26270 Extend class definition to include the "finer than" relation.
 class  Fne
 
Syntaxcref 26271 Extend class definition to include the refinement relation.
 class  Ref
 
Syntaxcptfin 26272 Extend class definition to include the class of point-finite covers.
 class  PtFin
 
Syntaxclocfin 26273 Extend class definition to include the class of locally finite covers.
 class  LocFin
 
Definitiondf-fne 26274* 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 26275* 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 26276* 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 26277* 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 26278 Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  Rel  Fne
 
Theoremisfne 26279* 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 26280 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 26281 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 26282* 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 26283* 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 26284 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 26285 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 26286* 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 26287* 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 26288* 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 26289 Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  Rel  Ref
 
Theoremisref 26290* 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 26279. 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 26291 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 26292* 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 26293 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 26294 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 26295 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
 |-  ( A  e.  V  ->  A Fne A )
 
Theoremrefref 26296 Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  ( A  e.  V  ->  A Ref A )
 
Theoremfnetr 26297 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 26298 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 26299 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 26300 Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  (
 ( A Ref B  /\  B Ref C ) 
 ->  A Ref C )
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