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Theorem List for Intuitionistic Logic Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmopnex 12601* The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( *Met `  X )  ->  E. d  e.  ( Met `  X ) J  =  ( MetOpen `  d
 ) )
 
Theoremmetrest 12602 Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)
 |-  D  =  ( C  |`  ( Y  X.  Y ) )   &    |-  J  =  (
 MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   =>    |-  (
 ( C  e.  ( *Met `  X )  /\  Y  C_  X )  ->  ( Jt  Y )  =  K )
 
Theoremxmetxp 12603* The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)
 |-  P  =  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  sup ( { (
 ( 1st `  u ) M ( 1st `  v
 ) ) ,  (
 ( 2nd `  u ) N ( 2nd `  v
 ) ) } ,  RR*
 ,  <  ) )   &    |-  ( ph  ->  M  e.  ( *Met `  X )
 )   &    |-  ( ph  ->  N  e.  ( *Met `  Y ) )   =>    |-  ( ph  ->  P  e.  ( *Met `  ( X  X.  Y ) ) )
 
Theoremxmetxpbl 12604* The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point  C with radius  R. (Contributed by Jim Kingdon, 22-Oct-2023.)
 |-  P  =  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  sup ( { (
 ( 1st `  u ) M ( 1st `  v
 ) ) ,  (
 ( 2nd `  u ) N ( 2nd `  v
 ) ) } ,  RR*
 ,  <  ) )   &    |-  ( ph  ->  M  e.  ( *Met `  X )
 )   &    |-  ( ph  ->  N  e.  ( *Met `  Y ) )   &    |-  ( ph  ->  R  e.  RR* )   &    |-  ( ph  ->  C  e.  ( X  X.  Y ) )   =>    |-  ( ph  ->  ( C ( ball `  P ) R )  =  ( ( ( 1st `  C ) ( ball `  M ) R )  X.  (
 ( 2nd `  C )
 ( ball `  N ) R ) ) )
 
Theoremxmettxlem 12605* Lemma for xmettx 12606. (Contributed by Jim Kingdon, 15-Oct-2023.)
 |-  P  =  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  sup ( { (
 ( 1st `  u ) M ( 1st `  v
 ) ) ,  (
 ( 2nd `  u ) N ( 2nd `  v
 ) ) } ,  RR*
 ,  <  ) )   &    |-  ( ph  ->  M  e.  ( *Met `  X )
 )   &    |-  ( ph  ->  N  e.  ( *Met `  Y ) )   &    |-  J  =  (
 MetOpen `  M )   &    |-  K  =  ( MetOpen `  N )   &    |-  L  =  ( MetOpen `  P )   =>    |-  ( ph  ->  L  C_  ( J  tX  K ) )
 
Theoremxmettx 12606* The maximum metric (Chebyshev distance) on the product of two sets, expressed as a binary topological product. (Contributed by Jim Kingdon, 11-Oct-2023.)
 |-  P  =  ( u  e.  ( X  X.  Y ) ,  v  e.  ( X  X.  Y )  |->  sup ( { (
 ( 1st `  u ) M ( 1st `  v
 ) ) ,  (
 ( 2nd `  u ) N ( 2nd `  v
 ) ) } ,  RR*
 ,  <  ) )   &    |-  ( ph  ->  M  e.  ( *Met `  X )
 )   &    |-  ( ph  ->  N  e.  ( *Met `  Y ) )   &    |-  J  =  (
 MetOpen `  M )   &    |-  K  =  ( MetOpen `  N )   &    |-  L  =  ( MetOpen `  P )   =>    |-  ( ph  ->  L  =  ( J  tX  K )
 )
 
7.2.5  Continuity in metric spaces
 
Theoremmetcnp3 12607* Two ways to express that  F is continuous at  P for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
 ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  ( F " ( P ( ball `  C )
 z ) )  C_  ( ( F `  P ) ( ball `  D ) y ) ) ) )
 
Theoremmetcnp 12608* Two ways to say a mapping from metric  C to metric  D is continuous at point  P. (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
 ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( P C w )  <  z  ->  ( ( F `  P ) D ( F `  w ) )  <  y ) ) ) )
 
Theoremmetcnp2 12609* Two ways to say a mapping from metric  C to metric  D is continuous at point  P. The distance arguments are swapped compared to metcnp 12608 (and Munkres' metcn 12610) for compatibility with df-lm 12286. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
 ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( w C P )  <  z  ->  (
 ( F `  w ) D ( F `  P ) )  < 
 y ) ) ) )
 
Theoremmetcn 12610* Two ways to say a mapping from metric  C to metric  D is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon"  y there is a positive "delta"  z such that a distance less than delta in  C maps to a distance less than epsilon in  D. (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. x  e.  X  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( x C w )  <  z  ->  ( ( F `  x ) D ( F `  w ) )  <  y ) ) ) )
 
Theoremmetcnpi 12611* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 12608. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y ) ) 
 /\  ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  <  x  ->  ( ( F `  P ) D ( F `  y ) )  <  A ) )
 
Theoremmetcnpi2 12612* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 12609. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y ) ) 
 /\  ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  <  x  ->  ( ( F `  y ) D ( F `  P ) )  <  A ) )
 
Theoremmetcnpi3 12613* Epsilon-delta property of a metric space function continuous at  P. A variation of metcnpi2 12612 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y ) ) 
 /\  ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  <_  x  ->  ( ( F `  y ) D ( F `  P ) )  <_  A )
 )
 
Theoremtxmetcnp 12614* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  L  =  ( MetOpen `  E )   =>    |-  (
 ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) ) 
 /\  ( A  e.  X  /\  B  e.  Y ) )  ->  ( F  e.  ( ( ( J  tX  K )  CnP  L ) `  <. A ,  B >. )  <->  ( F :
 ( X  X.  Y )
 --> Z  /\  A. z  e.  RR+  E. w  e.  RR+  A. u  e.  X  A. v  e.  Y  ( ( ( A C u )  <  w  /\  ( B D v )  <  w )  ->  ( ( A F B ) E ( u F v ) )  <  z ) ) ) )
 
Theoremtxmetcn 12615* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  L  =  ( MetOpen `  E )   =>    |-  (
 ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y )  /\  E  e.  ( *Met `  Z ) ) 
 ->  ( F  e.  (
 ( J  tX  K )  Cn  L )  <->  ( F :
 ( X  X.  Y )
 --> Z  /\  A. x  e.  X  A. y  e.  Y  A. z  e.  RR+  E. w  e.  RR+  A. u  e.  X  A. v  e.  Y  (
 ( ( x C u )  <  w  /\  ( y D v )  <  w ) 
 ->  ( ( x F y ) E ( u F v ) )  <  z ) ) ) )
 
Theoremmetcnpd 12616* Two ways to say a mapping from metric  C to metric  D is continuous at point  P. (Contributed by Jim Kingdon, 14-Jun-2023.)
 |-  ( ph  ->  J  =  ( MetOpen `  C )
 )   &    |-  ( ph  ->  K  =  ( MetOpen `  D )
 )   &    |-  ( ph  ->  C  e.  ( *Met `  X ) )   &    |-  ( ph  ->  D  e.  ( *Met `  Y ) )   &    |-  ( ph  ->  P  e.  X )   =>    |-  ( ph  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <->  ( F : X
 --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( P C w )  <  z  ->  ( ( F `  P ) D ( F `  w ) )  <  y ) ) ) )
 
7.2.6  Topology on the reals
 
Theoremqtopbasss 12617* The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.)
 |-  S  C_  RR*   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  sup ( { x ,  y } ,  RR* ,  <  )  e.  S )   &    |-  ( ( x  e.  S  /\  y  e.  S )  -> inf ( { x ,  y } ,  RR* ,  <  )  e.  S )   =>    |-  ( (,) " ( S  X.  S ) )  e.  TopBases
 
Theoremqtopbas 12618 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
 |-  ( (,) " ( QQ  X.  QQ ) )  e.  TopBases
 
Theoremretopbas 12619 A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)
 |- 
 ran  (,)  e.  TopBases
 
Theoremretop 12620 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
 |-  ( topGen `  ran  (,) )  e.  Top
 
Theoremuniretop 12621 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
 |- 
 RR  =  U. ( topGen `
  ran  (,) )
 
Theoremretopon 12622 The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( topGen `  ran  (,) )  e.  (TopOn `  RR )
 
Theoremretps 12623 The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
 |-  K  =  { <. (
 Base `  ndx ) ,  RR >. ,  <. (TopSet `  ndx ) ,  ( topGen `  ran  (,) ) >. }   =>    |-  K  e.  TopSp
 
Theoremiooretopg 12624 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) (Revised by Jim Kingdon, 23-May-2023.)
 |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e.  ( topGen `  ran  (,) ) )
 
Theoremcnmetdval 12625 Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  D  =  ( abs 
 o.  -  )   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A D B )  =  ( abs `  ( A  -  B ) ) )
 
Theoremcnmet 12626 The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)
 |-  ( abs  o.  -  )  e.  ( Met `  CC )
 
Theoremcnxmet 12627 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  ( abs  o.  -  )  e.  ( *Met `  CC )
 
Theoremcntoptopon 12628 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |-  J  e.  (TopOn `  CC )
 
Theoremcntoptop 12629 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |-  J  e.  Top
 
Theoremcnbl0 12630 Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  D  =  ( abs 
 o.  -  )   =>    |-  ( R  e.  RR* 
 ->  ( `' abs " (
 0 [,) R ) )  =  ( 0 (
 ball `  D ) R ) )
 
Theoremcnblcld 12631* Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  D  =  ( abs 
 o.  -  )   =>    |-  ( R  e.  RR* 
 ->  ( `' abs " (
 0 [,] R ) )  =  { x  e. 
 CC  |  ( 0 D x )  <_  R } )
 
Theoremunicntopcntop 12632 The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
 |- 
 CC  =  U. ( MetOpen `  ( abs  o.  -  ) )
 
Theoremcnopncntop 12633 The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
 |- 
 CC  e.  ( MetOpen `  ( abs  o.  -  )
 )
 
Theoremreopnap 12634* The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
 |-  ( A  e.  RR  ->  { w  e.  RR  |  w #  A }  e.  ( topGen `  ran  (,) )
 )
 
Theoremremetdval 12635 Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B ) ) )
 
Theoremremet 12636 The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  D  e.  ( Met `  RR )
 
Theoremrexmet 12637 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  D  e.  ( *Met `  RR )
 
Theorembl2ioo 12638 A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (
 ball `  D ) B )  =  ( ( A  -  B ) (,) ( A  +  B ) ) )
 
Theoremioo2bl 12639 An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (,) B )  =  ( ( ( A  +  B )  /  2 ) (
 ball `  D ) ( ( B  -  A )  /  2 ) ) )
 
Theoremioo2blex 12640 An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (,) B )  e.  ran  ( ball `  D ) )
 
Theoremblssioo 12641 The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |- 
 ran  ( ball `  D )  C_  ran  (,)
 
Theoremtgioo 12642 The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  D  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   &    |-  J  =  (
 MetOpen `  D )   =>    |-  ( topGen `  ran  (,) )  =  J
 
Theoremtgqioo 12643 The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)
 |-  Q  =  ( topGen `  ( (,) " ( QQ 
 X.  QQ ) ) )   =>    |-  ( topGen `  ran  (,) )  =  Q
 
Theoremresubmet 12644 The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)
 |-  R  =  ( topGen `  ran  (,) )   &    |-  J  =  (
 MetOpen `  ( ( abs 
 o.  -  )  |`  ( A  X.  A ) ) )   =>    |-  ( A  C_  RR  ->  J  =  ( Rt  A ) )
 
Theoremtgioo2cntop 12645 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |-  ( topGen `  ran  (,) )  =  ( Jt  RR )
 
Theoremrerestcntop 12646 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  R  =  ( topGen `  ran  (,) )   =>    |-  ( A  C_  RR  ->  ( Jt  A )  =  ( Rt  A ) )
 
Theoremaddcncntoplem 12647* Lemma for addcncntop 12648, subcncntop 12649, and mulcncntop 12650. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |- 
 .+  : ( CC 
 X.  CC ) --> CC   &    |-  (
 ( a  e.  RR+  /\  b  e.  CC  /\  c  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e. 
 CC  ( ( ( abs `  ( u  -  b ) )  < 
 y  /\  ( abs `  ( v  -  c
 ) )  <  z
 )  ->  ( abs `  ( ( u  .+  v )  -  (
 b  .+  c )
 ) )  <  a
 ) )   =>    |- 
 .+  e.  ( ( J  tX  J )  Cn  J )
 
Theoremaddcncntop 12648 Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |- 
 +  e.  ( ( J  tX  J )  Cn  J )
 
Theoremsubcncntop 12649 Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |- 
 -  e.  ( ( J  tX  J )  Cn  J )
 
Theoremmulcncntop 12650 Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |- 
 x.  e.  ( ( J  tX  J )  Cn  J )
 
Theoremdivcnap 12651* Complex number division is a continuous function, when the second argument is apart from zero. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Jim Kingdon, 25-Oct-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  K  =  ( Jt  { x  e.  CC  |  x #  0 } )   =>    |-  ( y  e.  CC ,  z  e.  { x  e.  CC  |  x #  0 }  |->  ( y  /  z ) )  e.  ( ( J  tX  K )  Cn  J )
 
Theoremfsumcncntop 12652* A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for  B normally contains free variables  k and  x to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  ( x  e.  X  |->  B )  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  sum_ k  e.  A  B )  e.  ( J  Cn  K ) )
 
7.2.7  Topological definitions using the reals
 
Syntaxccncf 12653 Extend class notation to include the operation which returns a class of continuous complex functions.
 class  -cn->
 
Definitiondf-cncf 12654* Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)
 |- 
 -cn->  =  ( a  e. 
 ~P CC ,  b  e.  ~P CC  |->  { f  e.  ( b  ^m  a
 )  |  A. x  e.  a  A. e  e.  RR+  E. d  e.  RR+  A. y  e.  a  ( ( abs `  ( x  -  y ) )  <  d  ->  ( abs `  ( ( f `
  x )  -  ( f `  y
 ) ) )  < 
 e ) } )
 
Theoremcncfval 12655* The value of the continuous complex function operation is the set of continuous functions from  A to  B. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
 |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( A -cn-> B )  =  { f  e.  ( B  ^m  A )  |  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w ) )  <  z  ->  ( abs `  ( ( f `
  x )  -  ( f `  w ) ) )  < 
 y ) } )
 
Theoremelcncf 12656* Membership in the set of continuous complex functions from  A to  B. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
 |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w ) )  <  z  ->  ( abs `  ( ( F `
  x )  -  ( F `  w ) ) )  <  y
 ) ) ) )
 
Theoremelcncf2 12657* Version of elcncf 12656 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
 |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x ) )  <  z  ->  ( abs `  ( ( F `
  w )  -  ( F `  x ) ) )  <  y
 ) ) ) )
 
Theoremcncfrss 12658 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
 
Theoremcncfrss2 12659 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
 
Theoremcncff 12660 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  F : A --> B )
 
Theoremcncfi 12661* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( F  e.  ( A -cn-> B )  /\  C  e.  A  /\  R  e.  RR+ )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  C ) )  <  z  ->  ( abs `  ( ( F `  w )  -  ( F `  C ) ) )  <  R ) )
 
Theoremelcncf1di 12662* Membership in the set of continuous complex functions from  A to  B. (Contributed by Paul Chapman, 26-Nov-2007.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ ) )   &    |-  ( ph  ->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  ( ( abs `  ( x  -  w ) )  <  Z  ->  ( abs `  ( ( F `
  x )  -  ( F `  w ) ) )  <  y
 ) ) )   =>    |-  ( ph  ->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A
 -cn-> B ) ) )
 
Theoremelcncf1ii 12663* Membership in the set of continuous complex functions from  A to  B. (Contributed by Paul Chapman, 26-Nov-2007.)
 |-  F : A --> B   &    |-  (
 ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ )   &    |-  (
 ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
 ( abs `  ( x  -  w ) )  <  Z  ->  ( abs `  (
 ( F `  x )  -  ( F `  w ) ) )  <  y ) )   =>    |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) )
 
Theoremrescncf 12664 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( C  C_  A  ->  ( F  e.  ( A -cn-> B )  ->  ( F  |`  C )  e.  ( C -cn-> B ) ) )
 
Theoremcncffvrn 12665 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) ) 
 ->  ( F  e.  ( A -cn-> C )  <->  F : A --> C ) )
 
Theoremcncfss 12666 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
 |-  ( ( B  C_  C  /\  C  C_  CC )  ->  ( A -cn-> B )  C_  ( A -cn-> C ) )
 
Theoremclimcncf 12667 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  ( A -cn-> B ) )   &    |-  ( ph  ->  G : Z
 --> A )   &    |-  ( ph  ->  G  ~~>  D )   &    |-  ( ph  ->  D  e.  A )   =>    |-  ( ph  ->  ( F  o.  G )  ~~>  ( F `  D ) )
 
Theoremabscncf 12668 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremrecncf 12669 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Re  e.  ( CC
 -cn-> RR )
 
Theoremimcncf 12670 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Im  e.  ( CC
 -cn-> RR )
 
Theoremcjcncf 12671 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  *  e.  ( CC
 -cn-> CC )
 
Theoremmulc1cncf 12672* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  CC  |->  ( A  x.  x ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremdivccncfap 12673* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Jim Kingdon, 9-Jan-2023.)
 |-  F  =  ( x  e.  CC  |->  ( x 
 /  A ) )   =>    |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremcncfco 12674 The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( ph  ->  F  e.  ( A -cn-> B ) )   &    |-  ( ph  ->  G  e.  ( B -cn-> C ) )   =>    |-  ( ph  ->  ( G  o.  F )  e.  ( A -cn-> C ) )
 
Theoremcncfmet 12675 Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
 |-  C  =  ( ( abs  o.  -  )  |`  ( A  X.  A ) )   &    |-  D  =  ( ( abs  o.  -  )  |`  ( B  X.  B ) )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   =>    |-  (
 ( A  C_  CC  /\  B  C_  CC )  ->  ( A -cn-> B )  =  ( J  Cn  K ) )
 
Theoremcncfcncntop 12676 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  K  =  ( Jt  A )   &    |-  L  =  ( Jt  B )   =>    |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( A -cn-> B )  =  ( K  Cn  L ) )
 
Theoremcncfcn1cntop 12677 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   =>    |-  ( CC -cn-> CC )  =  ( J  Cn  J )
 
Theoremcncfmptc 12678* A constant function is a continuous function on  CC. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
 |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  ( x  e.  S  |->  A )  e.  ( S -cn-> T ) )
 
Theoremcncfmptid 12679* The identity function is a continuous function on  CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
 |-  ( ( S  C_  T  /\  T  C_  CC )  ->  ( x  e.  S  |->  x )  e.  ( S -cn-> T ) )
 
Theoremcncfmpt1f 12680* Composition of continuous functions.  -cn-> analogue of cnmpt11f 12380. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  ( ph  ->  F  e.  ( CC -cn-> CC )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> CC ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( F `
  A ) )  e.  ( X -cn-> CC ) )
 
Theoremcncfmpt2fcntop 12681* Composition of continuous functions.  -cn-> analogue of cnmpt12f 12382. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  J  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  ( ph  ->  F  e.  ( ( J  tX  J )  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X
 -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( X -cn-> CC )
 )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( X -cn-> CC ) )
 
Theoremaddccncf 12682* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  F  =  ( x  e.  CC  |->  ( x  +  A ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremcdivcncfap 12683* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.)
 |-  F  =  ( x  e.  { y  e. 
 CC  |  y #  0 }  |->  ( A  /  x ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( { y  e.  CC  |  y #  0 } -cn->
 CC ) )
 
Theoremnegcncf 12684* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  F  =  ( x  e.  A  |->  -u x )   =>    |-  ( A  C_  CC  ->  F  e.  ( A
 -cn-> CC ) )
 
Theoremnegfcncf 12685* The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  G  =  ( x  e.  A  |->  -u ( F `  x ) )   =>    |-  ( F  e.  ( A -cn-> CC )  ->  G  e.  ( A -cn-> CC )
 )
 
Theoremmulcncflem 12686* Lemma for mulcncf 12687. (Contributed by Jim Kingdon, 29-May-2023.)
 |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( X
 -cn-> CC ) )   &    |-  ( ph  ->  V  e.  X )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  F  e.  RR+ )   &    |-  ( ph  ->  G  e.  RR+ )   &    |-  ( ph  ->  S  e.  RR+ )   &    |-  ( ph  ->  T  e.  RR+ )   &    |-  ( ph  ->  A. u  e.  X  ( ( abs `  ( u  -  V ) )  <  S  ->  ( abs `  ( ( ( x  e.  X  |->  A ) `  u )  -  ( ( x  e.  X  |->  A ) `
  V ) ) )  <  F ) )   &    |-  ( ph  ->  A. u  e.  X  ( ( abs `  ( u  -  V ) )  <  T  ->  ( abs `  ( ( ( x  e.  X  |->  B ) `  u )  -  ( ( x  e.  X  |->  B ) `
  V ) ) )  <  G ) )   &    |-  ( ph  ->  A. u  e.  X  ( ( ( abs `  ( [_ u  /  x ]_ A  -  [_ V  /  x ]_ A ) )  <  F  /\  ( abs `  ( [_ u  /  x ]_ B  -  [_ V  /  x ]_ B ) )  <  G )  ->  ( abs `  ( ( [_ u  /  x ]_ A  x.  [_ u  /  x ]_ B )  -  ( [_ V  /  x ]_ A  x.  [_ V  /  x ]_ B ) ) )  <  E ) )   =>    |-  ( ph  ->  E. d  e.  RR+  A. u  e.  X  ( ( abs `  ( u  -  V ) )  <  d  ->  ( abs `  ( ( ( x  e.  X  |->  ( A  x.  B ) ) `  u )  -  ( ( x  e.  X  |->  ( A  x.  B ) ) `
  V ) ) )  <  E ) )
 
Theoremmulcncf 12687* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( X
 -cn-> CC ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  x.  B ) )  e.  ( X
 -cn-> CC ) )
 
Theoremexpcncf 12688* The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( N  e.  NN0  ->  ( x  e.  CC  |->  ( x ^ N ) )  e.  ( CC
 -cn-> CC ) )
 
Theoremcnrehmeocntop 12689* The canonical bijection from  ( RR  X.  RR ) to  CC described in cnref1o 9408 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if  ( RR  X.  RR ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y
 ) ) )   &    |-  J  =  ( topGen `  ran  (,) )   &    |-  K  =  ( MetOpen `  ( abs  o. 
 -  ) )   =>    |-  F  e.  (
 ( J  tX  J ) Homeo K )
 
Theoremcnopnap 12690* The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.)
 |-  ( A  e.  CC  ->  { w  e.  CC  |  w #  A }  e.  ( MetOpen `  ( abs  o. 
 -  ) ) )
 
PART 8  BASIC REAL AND COMPLEX ANALYSIS
 
8.0.1  Dedekind cuts
 
Theoremdedekindeulemuub 12691* Lemma for dedekindeu 12697. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  A. z  e.  L  z  <  A )
 
Theoremdedekindeulemub 12692* Lemma for dedekindeu 12697. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  E. x  e.  RR  A. y  e.  L  y  <  x )
 
Theoremdedekindeulemloc 12693* Lemma for dedekindeu 12697. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  A. x  e. 
 RR  A. y  e.  RR  ( x  <  y  ->  ( E. z  e.  L  x  <  z  \/  A. z  e.  L  z  <  y ) ) )
 
Theoremdedekindeulemlub 12694* Lemma for dedekindeu 12697. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  L  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  L  y  <  z
 ) ) )
 
Theoremdedekindeulemlu 12695* Lemma for dedekindeu 12697. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  E. x  e.  RR  ( A. q  e.  L  q  <  x  /\  A. r  e.  U  x  <  r ) )
 
Theoremdedekindeulemeu 12696* Lemma for dedekindeu 12697. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  (
 A. q  e.  L  q  <  A  /\  A. r  e.  U  A  <  r ) )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( A. q  e.  L  q  <  B  /\  A. r  e.  U  B  <  r ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  -> F.  )
 
Theoremdedekindeu 12697* A Dedekind cut identifies a unique real number. Similar to df-inp 7242 except that the the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 5-Jan-2024.)
 |-  ( ph  ->  L  C_ 
 RR )   &    |-  ( ph  ->  U 
 C_  RR )   &    |-  ( ph  ->  E. q  e.  RR  q  e.  L )   &    |-  ( ph  ->  E. r  e.  RR  r  e.  U )   &    |-  ( ph  ->  A. q  e.  RR  (
 q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e. 
 RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  RR  A. r  e. 
 RR  ( q  < 
 r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  E! x  e.  RR  ( A. q  e.  L  q  <  x  /\  A. r  e.  U  x  <  r ) )
 
Theoremsuplociccreex 12698* An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7805 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <  C )   &    |-  ( ph  ->  A  C_  ( B [,] C ) )   &    |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  A. x  e.  ( B [,] C ) A. y  e.  ( B [,] C ) ( x  <  y  ->  ( E. z  e.  A  x  <  z  \/  A. z  e.  A  z  <  y ) ) )   =>    |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )
 
Theoremsuplociccex 12699* An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7805 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <  C )   &    |-  ( ph  ->  A  C_  ( B [,] C ) )   &    |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  A. x  e.  ( B [,] C ) A. y  e.  ( B [,] C ) ( x  <  y  ->  ( E. z  e.  A  x  <  z  \/  A. z  e.  A  z  <  y ) ) )   =>    |-  ( ph  ->  E. x  e.  ( B [,] C ) ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e.  ( B [,] C ) ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )
 
Theoremdedekindicclemuub 12700* Lemma for dedekindicc 12707. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  C  e.  U )   =>    |-  ( ph  ->  A. z  e.  L  z  <  C )
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