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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
19.12  Mathbox for Brendan Leahy
 
Theoremsupaddc 26201* The supremum function distributes over addition in a sense similar to that in supmul1 9963. (Contributed by Brendan Leahy, 25-Sep-2017.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   &    |-  ( ph  ->  B  e.  RR )   &    |-  C  =  { z  |  E. v  e.  A  z  =  ( v  +  B ) }   =>    |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  B )  =  sup ( C ,  RR ,  <  ) )
 
Theoremsupadd 26202* The supremum function distributes over addition in a sense similar to that in supmul 9966. (Contributed by Brendan Leahy, 26-Sep-2017.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   &    |-  ( ph  ->  B 
 C_  RR )   &    |-  ( ph  ->  B  =/=  (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  B  y  <_  x )   &    |-  C  =  {
 z  |  E. v  e.  A  E. b  e.  B  z  =  ( v  +  b ) }   =>    |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  ) )
 
Theoremrabiun2 26203* Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017.)
 |-  { x  e.  U_ y  e.  A  B  |  ph }  =  U_ y  e.  A  { x  e.  B  |  ph
 }
 
Theoremltflcei 26204 Theorem to move the floor function across a strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( |_ `  A )  <  B  <->  A  <  -u ( |_ `  -u B ) ) )
 
Theoremleceifl 26205 Theorem to move the floor function across a non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( -u ( |_ `  -u A )  <_  B  <->  A  <_  ( |_ `  B ) ) )
 
Theoremlxflflp1 26206 Theorem to move floor function between strict and non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( |_ `  A )  <_  B  <->  A  <  ( ( |_ `  B )  +  1 ) ) )
 
Theoremmblfinlem 26207* Lemma for ismblfin 26210, effectively one direction of the same fact for open sets, made necessary by Viaclovsky's slightly different defintion of outer measure. Note that unlike the main theorem, this holds for sets of infinite measure. (Contributed by Brendan Leahy, 21-Feb-2018.)
 |-  (
 ( A  e.  ( topGen `
  ran  (,) )  /\  M  e.  RR  /\  M  <  ( vol * `  A ) )  ->  E. s  e.  ( Clsd `  ( topGen `  ran  (,) ) ) ( s 
 C_  A  /\  M  <  ( vol * `  s ) ) )
 
Theoremmblfinlem2 26208* The difference between two sets measurable by the criterion in ismblfin 26210 is itself measurable by the same. Proposition 0.3 of [Viaclovsky7] p. 3. (Contributed by Brendan Leahy, 25-Mar-2018.)
 |-  (
 ( ( A  C_  RR  /\  ( vol * `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol * `  B )  e.  RR )  /\  ( ( vol
 * `  A )  =  sup ( { y  |  E. b  e.  ( Clsd `  ( topGen `  ran  (,) ) ) ( b 
 C_  A  /\  y  =  ( vol `  b
 ) ) } ,  RR ,  <  )  /\  ( vol * `  B )  =  sup ( {
 y  |  E. b  e.  ( Clsd `  ( topGen `  ran  (,) ) ) ( b 
 C_  B  /\  y  =  ( vol `  b
 ) ) } ,  RR ,  <  ) ) )  ->  sup ( {
 y  |  E. b  e.  ( Clsd `  ( topGen `  ran  (,) ) ) ( b 
 C_  ( A  \  B )  /\  y  =  ( vol `  b
 ) ) } ,  RR ,  <  )  =  ( vol * `  ( A  \  B ) ) )
 
Theoremmblfinlem3 26209* Backward direction of ismblfin 26210. (Contributed by Brendan Leahy, 28-Mar-2018.)
 |-  (
 ( ( A  C_  RR  /\  ( vol * `  A )  e.  RR )  /\  A  e.  dom  vol )  ->  ( vol * `
  A )  = 
 sup ( { y  |  E. b  e.  ( Clsd `  ( topGen `  ran  (,) ) ) ( b 
 C_  A  /\  y  =  ( vol `  b
 ) ) } ,  RR ,  <  ) )
 
Theoremismblfin 26210* Measurability in terms of inner and outer measure. Proposition 7 of [Viaclovsky8] p. 3. (Contributed by Brendan Leahy, 4-Mar-2018.) (Revised by Brendan Leahy, 28-Mar-2018.)
 |-  (
 ( A  C_  RR  /\  ( vol * `  A )  e.  RR )  ->  ( A  e.  dom 
 vol 
 <->  ( vol * `  A )  =  sup ( { y  |  E. b  e.  ( Clsd `  ( topGen `  ran  (,) )
 ) ( b  C_  A  /\  y  =  ( vol `  b )
 ) } ,  RR ,  <  ) ) )
 
Theoremovoliunnfl 26211* ovoliun 19391 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 21-Nov-2017.)
 |-  (
 ( f  Fn  NN  /\ 
 A. n  e.  NN  ( ( f `  n )  C_  RR  /\  ( vol * `  (
 f `  n )
 )  e.  RR )
 )  ->  ( vol * `
  U_ m  e.  NN  ( f `  m ) )  <_  sup ( ran  seq  1 (  +  ,  ( m  e.  NN  |->  ( vol * `  (
 f `  m )
 ) ) ) , 
 RR* ,  <  ) )   =>    |-  ( ( A  ~<_  NN  /\  A. x  e.  A  x  ~<_  NN )  ->  U. A  =/=  RR )
 
Theoremex-ovoliunnfl 26212* Demonstration of ovoliunnfl 26211. (Contributed by Brendan Leahy, 21-Nov-2017.)
 |-  (
 ( A  ~<_  NN  /\  A. x  e.  A  x  ~<_  NN )  ->  U. A  =/=  RR )
 
Theoremvoliunnfl 26213* voliun 19438 is incompatible with the Feferman-Levy model; in that model, therefore, the Lebesgue measure as we've defined it isn't actually a measure. (Contributed by Brendan Leahy, 16-Dec-2017.)
 |-  S  =  seq  1 (  +  ,  G )   &    |-  G  =  ( n  e.  NN  |->  ( vol `  ( f `  n ) ) )   &    |-  ( ( A. n  e.  NN  ( ( f `
  n )  e. 
 dom  vol  /\  ( vol `  ( f `  n ) )  e.  RR )  /\ Disj  n  e.  NN (
 f `  n )
 )  ->  ( vol ` 
 U_ n  e.  NN  ( f `  n ) )  =  sup ( ran  S ,  RR* ,  <  ) )   =>    |-  ( ( A  ~<_  NN  /\  A. x  e.  A  x  ~<_  NN )  ->  U. A  =/=  RR )
 
Theoremvolsupnfl 26214* volsup 19440 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 2-Jan-2018.)
 |-  (
 ( f : NN --> dom  vol  /\  A. n  e. 
 NN  ( f `  n )  C_  ( f `
  ( n  +  1 ) ) ) 
 ->  ( vol `  U. ran  f )  =  sup ( ( vol " ran  f ) ,  RR* ,  <  ) )   =>    |-  ( ( A  ~<_  NN  /\  A. x  e.  A  x  ~<_  NN )  ->  U. A  =/=  RR )
 
Theorem0mbf 26215 The empty function is measurable. (Contributed by Brendan Leahy, 28-Mar-2018.)
 |-  (/)  e. MblFn
 
Theoremmbfresfi 26216* Measurability of a piecewise function across arbitrarily many subsets. (Contributed by Brendan Leahy, 31-Mar-2018.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  S  e.  Fin )   &    |-  ( ph  ->  A. s  e.  S  ( F  |`  s )  e. MblFn )   &    |-  ( ph  ->  U. S  =  A )   =>    |-  ( ph  ->  F  e. MblFn )
 
Theoremmbfposadd 26217* If the sum of two measurable functions is measurable, the sum of their nonnegative parts is measurable. (Contributed by Brendan Leahy, 2-Apr-2018.)
 |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn
 )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e. MblFn )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( if ( 0  <_  B ,  B ,  0 )  +  if ( 0 
 <_  C ,  C , 
 0 ) ) )  e. MblFn )
 
Theoremcnambfre 26218 A real-valued, a.e. continuous function is measurable. (Contributed by Brendan Leahy, 4-Apr-2018.)
 |-  (
 ( F : A --> RR  /\  A  e.  dom  vol  /\  ( vol * `  ( A  \  ( ( `' ( ( ( topGen `  ran  (,) )t  A )  CnP  ( topGen `
  ran  (,) ) )  o.  _E  ) " { F } ) ) )  =  0 ) 
 ->  F  e. MblFn )
 
Theoremitg2addnclem 26219* An alternate expression for the 
S.2 integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017.) (Revised by Brendan Leahy, 10-Mar-2018.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( E. y  e.  RR+  ( z  e.  RR  |->  if (
 ( g `  z
 )  =  0 ,  0 ,  ( ( g `  z )  +  y ) ) )  o R  <_  F 
 /\  x  =  (
 S.1 `  g )
 ) }   =>    |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  =  sup ( L ,  RR* ,  <  )
 )
 
Theoremitg2addnclem2 26220* Lemma for itg2addnc 26222. The function described is a simple function. (Contributed by Brendan Leahy, 29-Oct-2017.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   =>    |-  ( ( ( ph  /\  h  e.  dom  S.1 )  /\  v  e.  RR+ )  ->  ( x  e. 
 RR  |->  if ( ( ( ( ( |_ `  (
 ( F `  x )  /  ( v  / 
 3 ) ) )  -  1 )  x.  ( v  /  3
 ) )  <_  ( h `  x )  /\  ( h `  x )  =/=  0 ) ,  ( ( ( |_ `  ( ( F `  x )  /  (
 v  /  3 )
 ) )  -  1
 )  x.  ( v 
 /  3 ) ) ,  ( h `  x ) ) )  e.  dom  S.1 )
 
Theoremitg2addnclem3 26221* Lemma incomprehensible in isolation split off to shorten proof of itg2addnc 26222. (Contributed by Brendan Leahy, 11-Mar-2018.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   &    |-  ( ph  ->  G : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  ( S.2 `  G )  e.  RR )   =>    |-  ( ph  ->  ( E. h  e.  dom  S.1 ( E. y  e.  RR+  ( z  e. 
 RR  |->  if ( ( h `
  z )  =  0 ,  0 ,  ( ( h `  z )  +  y
 ) ) )  o R  <_  ( F  o F  +  G )  /\  s  =  (
 S.1 `  h )
 )  ->  E. t E. u ( E. f  e.  dom  S.1 E. g  e. 
 dom  S.1 ( ( E. c  e.  RR+  ( z  e.  RR  |->  if (
 ( f `  z
 )  =  0 ,  0 ,  ( ( f `  z )  +  c ) ) )  o R  <_  F 
 /\  t  =  (
 S.1 `  f )
 )  /\  ( E. d  e.  RR+  ( z  e.  RR  |->  if (
 ( g `  z
 )  =  0 ,  0 ,  ( ( g `  z )  +  d ) ) )  o R  <_  G 
 /\  u  =  (
 S.1 `  g )
 ) )  /\  s  =  ( t  +  u ) ) ) )
 
Theoremitg2addnc 26222 Alternate proof of itg2add 19641 using the "buffer zone" definition from the first lemma, in which every simple function in the set is divided into to by dividing its buffer by a third and finding the largest allowable function locked to a grid laid out in increments of the new, smaller buffer up to the original simple function. The measurability of this function follows from that of the augend, and subtracting it from the original simple function yields another simple function by i1fsub 19590, which is allowable by the fact that the grid must have a mark between one third and two thirds the original buffer. This has two advantages over the current approach: first, eliminating ax-cc 8305, and second, weakening the measurability hypothesis to only the augend. (Contributed by Brendan Leahy, 31-Oct-2017.) (Revised by Brendan Leahy, 13-Mar-2018.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   &    |-  ( ph  ->  G : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  ( S.2 `  G )  e.  RR )   =>    |-  ( ph  ->  ( S.2 `  ( F  o F  +  G ) )  =  (
 ( S.2 `  F )  +  ( S.2 `  G ) ) )
 
Theoremitg2gt0cn 26223* itg2gt0 19642 holds on functions continuous on an open interval in the absence of ax-cc 8305. The fourth hypothesis is made unnecessary by the continuity hypothesis. (Contributed by Brendan Leahy, 16-Nov-2017.)
 |-  ( ph  ->  X  <  Y )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ( ph  /\  x  e.  ( X (,) Y ) ) 
 ->  0  <  ( F `
  x ) )   &    |-  ( ph  ->  ( F  |`  ( X (,) Y ) )  e.  (
 ( X (,) Y ) -cn-> CC ) )   =>    |-  ( ph  ->  0  <  ( S.2 `  F ) )
 
Theoremibladdnclem 26224* Lemma for ibladdnc 26225; cf ibladdlem 19701, whose fifth hypothesis is rendered unnecessary by the weakened hypotheses of itg2addnc 26222. (Contributed by Brendan Leahy, 31-Oct-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  D  =  ( B  +  C ) )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ph  ->  (
 S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 ) ) )  e.  RR )   &    |-  ( ph  ->  ( S.2 `  ( x  e.  RR  |->  if (
 ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) ) )  e. 
 RR )   =>    |-  ( ph  ->  ( S.2 `  ( x  e. 
 RR  |->  if ( ( x  e.  A  /\  0  <_  D ) ,  D ,  0 ) ) )  e.  RR )
 
Theoremibladdnc 26225* Choice-free analogue of itgadd 19706. A measurability hypothesis is necessitated by the loss of mbfadd 19543; for large classes of functions, such as continuous functions, it should be relatively easy to show. (Contributed by Brendan Leahy, 1-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e.  L ^1 )
 
Theoremitgaddnclem1 26226* Lemma for itgaddnc 26228; cf. itgaddlem1 19704. (Contributed by Brendan Leahy, 7-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  0  <_  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  C )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgaddnclem2 26227* Lemma for itgaddnc 26228; cf. itgaddlem2 19705. (Contributed by Brendan Leahy, 10-Nov-2017.) (Revised by Brendan Leahy, 3-Apr-2018.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgaddnc 26228* Choice-free analogue of itgadd 19706. (Contributed by Brendan Leahy, 11-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremiblsubnc 26229* Choice-free analogue of iblsub 19703. (Contributed by Brendan Leahy, 11-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e.  L ^1 )
 
Theoremitgsubnc 26230* Choice-free analogue of itgsub 19707. (Contributed by Brendan Leahy, 11-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  S. A ( B  -  C )  _d x  =  ( S. A B  _d x  -  S. A C  _d x ) )
 
Theoremiblabsnclem 26231* Lemma for iblabsnc 26232; cf. iblabslem 19709. (Contributed by Brendan Leahy, 7-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( abs `  ( F `  B ) ) ,  0 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  ( F `  B ) )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  B )  e. 
 RR )   =>    |-  ( ph  ->  ( G  e. MblFn  /\  ( S.2 `  G )  e.  RR ) )
 
Theoremiblabsnc 26232* Choice-free analogue of iblabs 19710. As with ibladdnc 26225, a measurability hypothesis is needed. (Contributed by Brendan Leahy, 7-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B )
 )  e. MblFn )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B ) )  e.  L ^1 )
 
Theoremiblmulc2nc 26233* Choice-free analogue of iblmulc2 19712. (Contributed by Brendan Leahy, 17-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e.  L ^1 )
 
Theoremitgmulc2nclem1 26234* Lemma for itgmulc2nc 26236; cf. itgmulc2lem1 19713. (Contributed by Brendan Leahy, 17-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   &    |-  ( ph  ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  0 
 <_  C )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  B )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgmulc2nclem2 26235* Lemma for itgmulc2nc 26236; cf. itgmulc2lem2 19714. (Contributed by Brendan Leahy, 19-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   &    |-  ( ph  ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgmulc2nc 26236* Choice-free analogue of itgmulc2 19715. (Contributed by Brendan Leahy, 19-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgabsnc 26237* Choice-free analogue of itgabs 19716. (Contributed by Brendan Leahy, 19-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B )
 )  e. MblFn )   &    |-  ( ph  ->  ( x  e.  A  |->  ( ( * `  S. A B  _d x )  x.  B ) )  e. MblFn )   =>    |-  ( ph  ->  ( abs `  S. A B  _d x )  <_  S. A ( abs `  B )  _d x )
 
Theorembddiblnc 26238* Choice-free proof of bddibl 19721. (Contributed by Brendan Leahy, 2-Nov-2017.) (Revised by Brendan Leahy, 6-Nov-2017.)
 |-  (
 ( F  e. MblFn  /\  ( vol `  dom  F )  e.  RR  /\  E. x  e.  RR  A. y  e.  dom  F ( abs `  ( F `  y
 ) )  <_  x )  ->  F  e.  L ^1 )
 
Theoremcnicciblnc 26239 Choice-free proof of cniccibl 19722. (Contributed by Brendan Leahy, 2-Nov-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  F  e.  ( ( A [,] B ) -cn-> CC ) )  ->  F  e.  L ^1 )
 
Theoremitggt0cn 26240* itggt0 19723 holds for continuous functions in the absence of ax-cc 8305. (Contributed by Brendan Leahy, 16-Nov-2017.)
 |-  ( ph  ->  X  <  Y )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  ( X (,) Y ) )  ->  B  e.  RR+ )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  ( ( X (,) Y )
 -cn-> CC ) )   =>    |-  ( ph  ->  0  <  S. ( X (,) Y ) B  _d x )
 
Theoremftc1cnnclem 26241* Lemma for ftc1cnnc 26242; cf. ftc1lem4 19913. The stronger assumptions of ftc1cn 19917 are exploited to make use of weaker theorems. (Contributed by Brendan Leahy, 19-Nov-2017.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `  t )  _d t
 )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  c  e.  ( A (,) B ) )   &    |-  H  =  ( z  e.  ( ( A [,] B ) 
 \  { c }
 )  |->  ( ( ( G `  z )  -  ( G `  c ) )  /  ( z  -  c
 ) ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  y  e.  ( A (,) B ) ) 
 ->  ( ( abs `  (
 y  -  c ) )  <  R  ->  ( abs `  ( ( F `  y )  -  ( F `  c ) ) )  <  E ) )   &    |-  ( ph  ->  X  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( X  -  c ) )  <  R )   &    |-  ( ph  ->  Y  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( Y  -  c ) )  <  R )   =>    |-  ( ( ph  /\  X  <  Y )  ->  ( abs `  ( ( ( ( G `  Y )  -  ( G `  X ) )  /  ( Y  -  X ) )  -  ( F `  c ) ) )  <  E )
 
Theoremftc1cnnc 26242* Choice-free proof of ftc1cn 19917. (Contributed by Brendan Leahy, 20-Nov-2017.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `  t )  _d t
 )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  F  e.  L ^1 )   =>    |-  ( ph  ->  ( RR  _D  G )  =  F )
 
Theoremdvreasin 26243 Real derivative of arcsine. (Contributed by Brendan Leahy, 3-Aug-2017.)
 |-  ( RR  _D  (arcsin  |`  ( -u 1 (,) 1 ) ) )  =  ( x  e.  ( -u 1 (,) 1 )  |->  ( 1 
 /  ( sqr `  (
 1  -  ( x ^ 2 ) ) ) ) )
 
Theoremdvreacos 26244 Real derivative of arccosine. (Contributed by Brendan Leahy, 3-Aug-2017.)
 |-  ( RR  _D  (arccos  |`  ( -u 1 (,) 1 ) ) )  =  ( x  e.  ( -u 1 (,) 1 )  |->  ( -u 1  /  ( sqr `  (
 1  -  ( x ^ 2 ) ) ) ) )
 
Theoremareacirclem2 26245* Antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.)
 |-  ( R  e.  RR+  ->  ( RR  _D  ( t  e.  ( -u R (,) R )  |->  ( ( R ^ 2 )  x.  ( (arcsin `  (
 t  /  R )
 )  +  ( ( t  /  R )  x.  ( sqr `  (
 1  -  ( ( t  /  R ) ^ 2 ) ) ) ) ) ) ) )  =  ( t  e.  ( -u R (,) R )  |->  ( 2  x.  ( sqr `  ( ( R ^
 2 )  -  (
 t ^ 2 ) ) ) ) ) )
 
Theoremareacirclem3 26246* Continuity of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.)
 |-  ( R  e.  RR+  ->  (
 t  e.  ( -u R (,) R )  |->  ( 2  x.  ( sqr `  ( ( R ^
 2 )  -  (
 t ^ 2 ) ) ) ) )  e.  ( ( -u R (,) R ) -cn-> CC ) )
 
Theoremareacirclem4 26247* Endpoint-inclusive continuity of Cartesian ordinate of circle. (Contributed by Brendan Leahy, 29-Aug-2017.)
 |-  (
 ( R  e.  RR  /\  0  <_  R )  ->  ( t  e.  ( -u R [,] R ) 
 |->  ( sqr `  (
 ( R ^ 2
 )  -  ( t ^ 2 ) ) ) )  e.  (
 ( -u R [,] R ) -cn-> CC ) )
 
Theoremareacirclem1 26248* Integrability of cross-section of circle. (Contributed by Brendan Leahy, 26-Aug-2017.)
 |-  (
 ( R  e.  RR  /\  0  <_  R )  ->  ( t  e.  ( -u R [,] R ) 
 |->  ( 2  x.  ( sqr `  ( ( R ^ 2 )  -  ( t ^ 2
 ) ) ) ) )  e.  L ^1 )
 
Theoremareacirclem5 26249* Endpoint-inclusive continuity of antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 31-Aug-2017.)
 |-  ( R  e.  RR+  ->  (
 t  e.  ( -u R [,] R )  |->  ( ( R ^ 2
 )  x.  ( (arcsin `  ( t  /  R ) )  +  (
 ( t  /  R )  x.  ( sqr `  (
 1  -  ( ( t  /  R ) ^ 2 ) ) ) ) ) ) )  e.  ( (
 -u R [,] R ) -cn-> CC ) )
 
Theoremareacirclem6 26250* Finding the cross-section of a circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.)
 |-  S  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  (
 ( x ^ 2
 )  +  ( y ^ 2 ) ) 
 <_  ( R ^ 2
 ) ) }   =>    |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e. 
 RR )  ->  ( S " { t }
 )  =  if (
 ( abs `  t )  <_  R ,  ( -u ( sqr `  ( ( R ^ 2 )  -  ( t ^ 2
 ) ) ) [,] ( sqr `  (
 ( R ^ 2
 )  -  ( t ^ 2 ) ) ) ) ,  (/) ) )
 
Theoremareacirc 26251* The area of a circle of radius  R is  pi  x.  R ^ 2. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.)
 |-  S  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  (
 ( x ^ 2
 )  +  ( y ^ 2 ) ) 
 <_  ( R ^ 2
 ) ) }   =>    |-  ( ( R  e.  RR  /\  0  <_  R )  ->  (area `  S )  =  ( pi  x.  ( R ^ 2 ) ) )
 
19.13  Mathbox for Jeff Hankins
 
19.13.1  Miscellany
 
Theorema1i13 26252 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ps  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theorema1i4 26253 Add an antecedent to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ta )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i14 26254 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ps  ->  ( ch  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i24 26255 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)
 |-  ( ph  ->  ( ch  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i34 26256 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)
 |-  ( ph  ->  ( ps  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp5d 26257 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  ->  ( ( th  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5g 26258 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ps )  ->  ( ( ( ch 
 /\  th )  /\  ta )  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5j 26259 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ( ps  /\  ch )  /\  th )  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5k 26260 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ps  /\  ( ch 
 /\  th ) )  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5l 26261 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ps  /\  ch )  /\  ( th  /\  ta ) )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp56 26262 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  ( th  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp58 26263 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ph  /\  ps )  /\  ( ( ch 
 /\  th )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp510 26264 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ( ps  /\  ch )  /\  th )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp511 26265 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ps  /\  ( ch 
 /\  th ) )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp512 26266 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ps  /\  ch )  /\  ( th  /\  ta ) ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theorem3com12d 26267 Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ph  ->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ( ch  /\  ps  /\  th ) )
 
Theoremimp5p 26268 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch  /\  th 
 /\  ta )  ->  et )
 ) )
 
Theoremimp5q 26269 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( ch 
 /\  th  /\  ta )  ->  et ) )
 
Theoremecase13d 26270 Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  ( ch  \/  ps 
 \/  th ) )   =>    |-  ( ph  ->  ps )
 
Theoremsubtr 26271 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 26272 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 ) ) )
 
Theoremtrer 26273* 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 26274 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 ) ) ) )
 
Theoremfinminlem 26275* 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 )
 ) )
 
Theoremgtinf 26276* 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 26277* 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 26278* 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 )
 ) )
 
Theoremnn0prpwlem 26279* Lemma for nn0prpw 26280. 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 26280* 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 ) ) )
 
19.13.2  Basic topological facts
 
Theoremtopbnd 26281 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 26282 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 26283 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 26284* 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 26285 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 26286* 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 26287* 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 26288* 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 26289 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 26290 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 26291 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 ) )
 
19.13.3  Topology of the real numbers
 
TheoremivthALT 26292* An alternate proof of the Intermediate Value Theorem ivth 19341 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 )
 
19.13.4  Refinements
 
Syntaxcfne 26293 Extend class definition to include the "finer than" relation.
 class  Fne
 
Syntaxcref 26294 Extend class definition to include the refinement relation.
 class  Ref
 
Syntaxcptfin 26295 Extend class definition to include the class of point-finite covers.
 class  PtFin
 
Syntaxclocfin 26296 Extend class definition to include the class of locally finite covers.
 class  LocFin
 
Definitiondf-fne 26297* 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 26298* 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 26299* 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 26300* 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 ) ) } )
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