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Theorem List for Metamath Proof Explorer - 26401-26500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremmapfiOLD 26401 Set exponentiation of finite sets is finite. (Moved into main set.mm as mapfi 7152 and may be deleted by mathbox owner, JM. --NM 24-Sep-2013.) (Contributed by Jeff Madsen, 19-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  ^m  B )  e.  Fin )
 
TheoremixpfiOLD 26402* A cross product of finitely many finite sets is finite. (Moved into main set.mm as ixpfi 7153 and may be deleted by mathbox owner, JM. --NM 24-Sep-2013.) (Contributed by Jeff Madsen, 19-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  Fin  /\ 
 A. x  e.  A  B  e.  Fin )  ->  X_ x  e.  A  B  e.  Fin )
 
Theoremupixp 26403* Universal property of the indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  X  =  X_ b  e.  A  ( C `  b )   &    |-  P  =  ( w  e.  A  |->  ( x  e.  X  |->  ( x `  w ) ) )   =>    |-  ( ( A  e.  R  /\  B  e.  S  /\  A. a  e.  A  ( F `  a ) : B --> ( C `
  a ) ) 
 ->  E! h ( h : B --> X  /\  A. a  e.  A  ( F `  a )  =  ( ( P `
  a )  o.  h ) ) )
 
Theoremabrexex2gOLD 26404* Existence of an existentially restricted class abstraction. (Moved to abrexex2g 5768 in main set.mm and may be deleted by mathbox owner, JM. --NM 21-May-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  B  /\  A. x  e.  A  { y  |  ph }  e.  C )  ->  { y  |  E. x  e.  A  ph
 }  e.  _V )
 
Theoremabrexdom 26405* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 y  e.  A  ->  E* x ph )   =>    |-  ( A  e.  V  ->  { x  |  E. y  e.  A  ph
 }  ~<_  A )
 
Theoremabrexdom2 26406* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  V  ->  { x  |  E. y  e.  A  x  =  B } 
 ~<_  A )
 
Theoremfindcard2OLD 26407* Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Moved to findcard2 7098 in main set.mm and may be deleted by mathbox owner, JM. --NM 30-Nov-2012.) (Contributed by Jeff Madsen, 8-Jul-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  (/)  ->  ( ph 
 <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  Fin  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  Fin  ->  ta )
 
TheoremfimaxOLD 26408* A finite set has a maximum under a total order. (Moved to fimaxg 7104 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  R  Or  A   =>    |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  y R x ) )
 
TheoremfimaxgOLD 26409* A finite set has a maximum under a total order. (Moved to fimaxg 7104 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  y R x ) )
 
TheoremfisupgOLD 26410* Lemma showing existence and closure of supremum of a finite set. (Moved to fisupg 7105 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A. y  e.  A  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  A  y R z ) ) )
 
Theoremac6gf 26411* Axiom of Choice. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  F/ y ps   &    |-  ( y  =  ( f `  x )  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  C  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremindexa 26412* If for every element of an indexing set  A there exists a corresponding element of another set  B, then there exists a subset of  B consisting only of those elements which are indexed by  A. Used to avoid the Axiom of Choice in situations where only the range of the choice function is needed. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( B  e.  M  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. c
 ( c  C_  B  /\  A. x  e.  A  E. y  e.  c  ph 
 /\  A. y  e.  c  E. x  e.  A  ph ) )
 
Theoremindexdom 26413* If for every element of an indexing set  A there exists a corresponding element of another set  B, then there exists a subset of  B consisting only of those elements which are indexed by  A, and which is dominated by the set  A. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( A  e.  M  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. c
 ( ( c  ~<_  A 
 /\  c  C_  B )  /\  ( A. x  e.  A  E. y  e.  c  ph  /\  A. y  e.  c  E. x  e.  A  ph ) ) )
 
TheoremindexfiOLD 26414* If for every element of a finite indexing set  A there exists a corresponding element of another set  B, then there exists a finite subset of  B consisting only of those elements which are indexed by  A. Proven without the Axiom of Choice, unlike indexdom 26413. (Moved to indexfi 7163 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Apr-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  Fin  /\  B  e.  M  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. c  e.  Fin  ( c  C_  B  /\  A. x  e.  A  E. y  e.  c  ph  /\ 
 A. y  e.  c  E. x  e.  A  ph ) )
 
TheoremfipreimaOLD 26415* Given a finite subset  A of the range of a function, there exists a finite subset of the domain whose image is  A. (Moved to fipreima 7161 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Apr-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 1-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( F  Fn  B  /\  B  e.  M )  /\  ( A  C_  ran 
 F  /\  A  e.  Fin ) )  ->  E. c  e.  ( ~P B  i^i  Fin ) ( F "
 c )  =  A )
 
Theoremfrinfm 26416* A subset of a well founded set has an infimum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( R  Fr  A  /\  ( B  e.  C  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  A  ( A. y  e.  B  -.  x `' R y 
 /\  A. y  e.  A  ( y `' R x  ->  E. z  e.  B  y `' R z ) ) )
 
Theoremwelb 26417* A non-empty subset of a well ordered set has a lower bound. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( R  We  A  /\  ( B  e.  C  /\  B  C_  A  /\  B  =/=  (/) ) )  ->  ( `' R  Or  B  /\  E. x  e.  B  ( A. y  e.  B  -.  x `' R y 
 /\  A. y  e.  B  ( y `' R x  ->  E. z  e.  B  y `' R z ) ) ) )
 
Theoremsupeq2OLD 26418 Equality theorem for supremum. (Moved to supeq2 7201 in main set.mm and may be deleted by mathbox owner, JM. --NM 24-Sep-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( B  =  C  ->  sup ( A ,  B ,  R )  =  sup ( A ,  C ,  R ) )
 
Theoremsupex2g 26419 Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  C  ->  sup ( B ,  A ,  R )  e.  _V )
 
Theoremsupclt 26420* Closure of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( R  Or  A  /\  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )  ->  sup ( B ,  A ,  R )  e.  A )
 
Theoremsupubt 26421* Upper bound property of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( R  Or  A  /\  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )  ->  ( C  e.  B  ->  -.  sup ( B ,  A ,  R ) R C ) )
 
TheoreminfmrlbOLD 26422* A member of a non-empty bounded set of reals is greater than or equal to the set's lower bound. (Contributed by Jeff Madsen, 2-Feb-2011.) (Moved to infmrlb 9735 in main set.mm and may be deleted by mathbox owner, JM. --NM 21-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
 )  /\  B  e.  A )  ->  sup ( A ,  RR ,  `'  <  )  <_  B )
 
TheoremsupeutOLD 26423* A supremum is unique. Closed version of supeu 7205. (Moved to supeu 7205 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 9-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( R  Or  A  /\  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )  ->  E! x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
 
Theoremfisup2gOLD 26424 A nonempty finite set contains its supremum. (Moved to fisupcl 7218 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 9-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( R  Or  A  /\  ( B  e.  Fin  /\  B  =/=  (/)  /\  B  C_  A ) )  ->  sup ( B ,  A ,  R )  e.  B )
 
Theoremfimax2gOLD 26425* A finite set has a maximum under a total order. (Moved to fimax2g 7103 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
TheoremwofiOLD 26426 A total order on a finite set is a well order. (Moved to wofi 7106 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
 
TheoremfrfiOLD 26427 A partial order is founded on a finite set. (Moved to frfi 7102 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( R  Po  A  /\  A  e.  Fin )  ->  R  Fr  A )
 
TheorempofunOLD 26428* A function preserves a partial order relation. (Moved to pofun 4330 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  S  =  { <. x ,  y >.  |  X R Y }   &    |-  ( x  =  y 
 ->  X  =  Y )   =>    |-  ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  ->  S  Po  A )
 
TheoremfrminexOLD 26429* If an element of a founded set satisfies a property  ph, then there is a minimal element that satisfies  ph. (Moved to frminex 4373 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( R  Fr  A  ->  ( E. x  e.  A  ph  ->  E. x  e.  A  ( ph  /\  A. y  e.  A  ( ps  ->  -.  y R x ) ) ) )
 
18.15.2  Real and complex numbers; integers
 
TheoremfimaxreOLD 26430* A finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to fimaxre 9701 in main set.mm and may be deleted by mathbox owner, JM. --NM 22-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  RR  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  y  <_  x )
 
Theoremfimaxre2OLD 26431* A nonempty finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 27-May-2011.) (Moved to fimaxre2 9702 in main set.mm and may be deleted by mathbox owner, JM. --NM 22-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  RR  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )
 
Theoremfilbcmb 26432* Combine a finite set of lower bounds. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( A  e.  Fin  /\  A  =/=  (/)  /\  B  C_ 
 RR )  ->  ( A. x  e.  A  E. y  e.  B  A. z  e.  B  ( y  <_  z  ->  ph )  ->  E. y  e.  B  A. z  e.  B  ( y  <_  z  ->  A. x  e.  A  ph ) ) )
 
Theoremadd20OLD 26433 Two nonnegative numbers are zero iff their sum is zero. (Moved to add20 9286 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( A  +  B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
 
Theoremaddsubeq4OLD 26434 Relation between sums and differences.. (Moved to addsubeq4 9066 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 17-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  =  ( C  +  D )  <->  ( C  -  A )  =  ( B  -  D ) ) )
 
Theoremrdr 26435 Two ways of expressing the remainder when  A is divided by 
B. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  -  ( A  mod  B ) )  /  B )  =  ( |_ `  ( A  /  B ) ) )
 
TheoremeluzaddOLD 26436 Membership in a later set of upper integers. (Moved to eluzadd 10256 in main set.mm and may be deleted by mathbox owner, JM. --NM 2-Sep-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( N  e.  ( ZZ>=
 `  M )  /\  K  e.  ZZ )  ->  ( N  +  K )  e.  ( ZZ>= `  ( M  +  K ) ) )
 
TheoremeluzsubOLD 26437 Membership in an earlier set of upper integers. (Moved to eluzsub 10257 in main set.mm and may be deleted by mathbox owner, JM. --NM 2-Sep-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( M  e.  ZZ  /\  K  e.  ZZ  /\  N  e.  ( ZZ>= `  ( M  +  K ) ) )  ->  ( N  -  K )  e.  ( ZZ>= `  M ) )
 
Theoremuzm1OLD 26438 Choices for an element of an upper interval of integers. (Moved to uzm1 10258 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-May-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( N  =  M  \/  ( N  -  1 )  e.  ( ZZ>= `  M )
 ) )
 
Theoremuzp1OLD 26439 Choices for an element of an upper interval of integers. (Moved to uzp1 10261 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-May-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( N  =  M  \/  N  e.  ( ZZ>= `  ( M  +  1 ) ) ) )
 
TheoremfzfiOLD 26440 A finite interval of integers is finite. (Moved to fzfi 11034 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( M
 ... N )  e. 
 Fin )
 
Theoremfzfi2OLD 26441 Variant of fzfi 11034 with hypothesis weakened. (Moved to fzfi 11034 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( N  e.  A  ->  ( M ... N )  e.  Fin )
 
Theoremfz10OLD 26442 There are no integers between 1 and 0. (Moved to fz10 10814 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 1 ... 0 )  =  (/)
 
Theoremfzmul 26443 Membership of a product in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  NN )  ->  ( J  e.  ( M ... N )  ->  ( K  x.  J )  e.  ( ( K  x.  M ) ... ( K  x.  N ) ) ) )
 
Theoremfzadd2 26444 Membership of a sum in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  (
 ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( O  e.  ZZ  /\  P  e.  ZZ ) )  ->  ( ( J  e.  ( M
 ... N )  /\  K  e.  ( O ... P ) )  ->  ( J  +  K )  e.  ( ( M  +  O ) ... ( N  +  P ) ) ) )
 
TheoremfzsplitOLD 26445 Split a finite interval of integers into two parts. (Moved to fzsplit 10816 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 17-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( N  e.  A  /\  K  e.  ( M
 ... N ) ) 
 ->  ( M ... N )  =  ( ( M ... K )  u.  ( ( K  +  1 ) ... N ) ) )
 
TheoremfzdisjOLD 26446 Condition for two finite intervals of integers to be disjoint. (Moved to fzdisj 10817 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 17-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( K  e.  A  /\  K  <  M ) 
 ->  ( ( J ... K )  i^i  ( M
 ... N ) )  =  (/) )
 
Theoremfzp1elp1OLD 26447 Add one to an element of a finite set of integers. (Moved to fzp1elp1 10839 in main set.mm and may be deleted by mathbox owner, JM. --NM 28-Feb-2014.) (Contributed by Jeff Madsen, 6-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( N  e.  A  /\  K  e.  ( M
 ... N ) ) 
 ->  ( K  +  1 )  e.  ( M
 ... ( N  +  1 ) ) )
 
TheoremabszOLD 26448 A real number is an integer iff its absolute value is an integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to absz 11796 in main set.mm and may be deleted by mathbox owner, JM. --NM 10-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  ( A  e.  ZZ  <->  ( abs `  A )  e.  ZZ )
 )
 
Theoremmod0OLD 26449  A  mod  B is zero iff  A is evenly divisible by  B. (Moved to mod0 10978 in main set.mm and may be deleted by mathbox owner, JM. --NM 19-Apr-2014.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  mod  B )  =  0  <->  ( A  /  B )  e.  ZZ ) )
 
Theoremnegmod0OLD 26450  A is divisible by  B iff its negative is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to negmod0 10979 in main set.mm and may be deleted by mathbox owner, JM. --NM 10-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  mod  B )  =  0  <->  ( -u A  mod  B )  =  0 ) )
 
Theoremabsmod0OLD 26451  A is divisible by  B iff its absolute value is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to absmod0 11788 in main set.mm and may be deleted by mathbox owner, JM. --NM 10-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  mod  B )  =  0  <->  ( ( abs `  A )  mod  B )  =  0 )
 )
 
18.15.3  Sequences and sums
 
Theoremsdclem2 26452* Lemma for sdc 26454. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  (
 g  =  ( f  |`  ( M ... n ) )  ->  ( ps  <->  ch ) )   &    |-  ( n  =  M  ->  ( ps  <->  ta ) )   &    |-  ( n  =  k  ->  ( ps  <->  th ) )   &    |-  ( ( g  =  h  /\  n  =  ( k  +  1 ) )  ->  ( ps 
 <-> 
 si ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. g ( g : { M } --> A  /\  ta ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( ( g : ( M ... k
 ) --> A  /\  th )  ->  E. h ( h : ( M ... ( k  +  1
 ) ) --> A  /\  g  =  ( h  |`  ( M ... k
 ) )  /\  si ) ) )   &    |-  J  =  { g  |  E. n  e.  Z  (
 g : ( M
 ... n ) --> A  /\  ps ) }   &    |-  F  =  ( w  e.  Z ,  x  e.  J  |->  { h  |  E. k  e.  Z  ( h : ( M
 ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k
 ) )  /\  si ) } )   &    |-  F/ k ph   &    |-  ( ph  ->  G : Z --> J )   &    |-  ( ph  ->  ( G `  M ) : ( M ... M ) --> A )   &    |-  (
 ( ph  /\  w  e.  Z )  ->  ( G `  ( w  +  1 ) )  e.  ( w F ( G `  w ) ) )   =>    |-  ( ph  ->  E. f
 ( f : Z --> A  /\  A. n  e.  Z  ch ) )
 
Theoremsdclem1 26453* Lemma for sdc 26454. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  (
 g  =  ( f  |`  ( M ... n ) )  ->  ( ps  <->  ch ) )   &    |-  ( n  =  M  ->  ( ps  <->  ta ) )   &    |-  ( n  =  k  ->  ( ps  <->  th ) )   &    |-  ( ( g  =  h  /\  n  =  ( k  +  1 ) )  ->  ( ps 
 <-> 
 si ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. g ( g : { M } --> A  /\  ta ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( ( g : ( M ... k
 ) --> A  /\  th )  ->  E. h ( h : ( M ... ( k  +  1
 ) ) --> A  /\  g  =  ( h  |`  ( M ... k
 ) )  /\  si ) ) )   &    |-  J  =  { g  |  E. n  e.  Z  (
 g : ( M
 ... n ) --> A  /\  ps ) }   &    |-  F  =  ( w  e.  Z ,  x  e.  J  |->  { h  |  E. k  e.  Z  ( h : ( M
 ... ( k  +  1 ) ) --> A  /\  x  =  ( h  |`  ( M ... k
 ) )  /\  si ) } )   =>    |-  ( ph  ->  E. f
 ( f : Z --> A  /\  A. n  e.  Z  ch ) )
 
Theoremsdc 26454* Strong dependent choice. Suppose we may choose an element of  A such that property  ps holds, and suppose that if we have already chosen the first  k elements (represented here by a function from  1 ... k to  A), we may choose another element so that all  k  +  1 elements taken together have property  ps. Then there exists an infinite sequence of elements of  A such that the first  n terms of this sequence satisfy  ps for all  n. This theorem allows us to construct infinite seqeunces where each term depends on all the previous terms in the sequence. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 3-Jun-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  (
 g  =  ( f  |`  ( M ... n ) )  ->  ( ps  <->  ch ) )   &    |-  ( n  =  M  ->  ( ps  <->  ta ) )   &    |-  ( n  =  k  ->  ( ps  <->  th ) )   &    |-  ( ( g  =  h  /\  n  =  ( k  +  1 ) )  ->  ( ps 
 <-> 
 si ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  E. g ( g : { M } --> A  /\  ta ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( ( g : ( M ... k
 ) --> A  /\  th )  ->  E. h ( h : ( M ... ( k  +  1
 ) ) --> A  /\  g  =  ( h  |`  ( M ... k
 ) )  /\  si ) ) )   =>    |-  ( ph  ->  E. f ( f : Z --> A  /\  A. n  e.  Z  ch ) )
 
Theoremfdc 26455* Finite version of dependent choice. Construct a function whose value depends on the previous function value, except at a final point at which no new value can be chosen. The final hypothesis ensures that the process will terminate. The proof does not use the Axiom of Choice. (Contributed by Jeff Madsen, 18-Jun-2010.)
 |-  A  e.  _V   &    |-  M  e.  ZZ   &    |-  Z  =  ( ZZ>= `  M )   &    |-  N  =  ( M  +  1 )   &    |-  ( a  =  ( f `  (
 k  -  1 ) )  ->  ( ph  <->  ps ) )   &    |-  ( b  =  ( f `  k
 )  ->  ( ps  <->  ch ) )   &    |-  ( a  =  ( f `  n )  ->  ( th  <->  ta ) )   &    |-  ( et  ->  C  e.  A )   &    |-  ( et  ->  R  Fr  A )   &    |-  ( ( et 
 /\  a  e.  A )  ->  ( th  \/  E. b  e.  A  ph ) )   &    |-  ( ( ( et  /\  ph )  /\  ( a  e.  A  /\  b  e.  A ) )  ->  b R a )   =>    |-  ( et  ->  E. n  e.  Z  E. f ( f : ( M
 ... n ) --> A  /\  ( ( f `  M )  =  C  /\  ta )  /\  A. k  e.  ( N ... n ) ch )
 )
 
Theoremfdc1 26456* Variant of fdc 26455 with no specified base value. (Contributed by Jeff Madsen, 18-Jun-2010.)
 |-  A  e.  _V   &    |-  M  e.  ZZ   &    |-  Z  =  ( ZZ>= `  M )   &    |-  N  =  ( M  +  1 )   &    |-  ( a  =  ( f `  M )  ->  ( ze  <->  si ) )   &    |-  (
 a  =  ( f `
  ( k  -  1 ) )  ->  ( ph  <->  ps ) )   &    |-  (
 b  =  ( f `
  k )  ->  ( ps  <->  ch ) )   &    |-  (
 a  =  ( f `
  n )  ->  ( th  <->  ta ) )   &    |-  ( et  ->  E. a  e.  A  ze )   &    |-  ( et  ->  R  Fr  A )   &    |-  (
 ( et  /\  a  e.  A )  ->  ( th  \/  E. b  e.  A  ph ) )   &    |-  ( ( ( et 
 /\  ph )  /\  (
 a  e.  A  /\  b  e.  A )
 )  ->  b R a )   =>    |-  ( et  ->  E. n  e.  Z  E. f ( f : ( M
 ... n ) --> A  /\  ( si  /\  ta )  /\  A. k  e.  ( N ... n ) ch ) )
 
Theoremseqpo 26457* Two ways to say that a sequence respects a partial order. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( R  Po  A  /\  F : NN --> A ) 
 ->  ( A. s  e. 
 NN  ( F `  s ) R ( F `  ( s  +  1 ) )  <->  A. m  e.  NN  A. n  e.  ( ZZ>= `  ( m  +  1
 ) ) ( F `
  m ) R ( F `  n ) ) )
 
Theoremincsequz 26458* An increasing sequence of natural numbers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( F : NN --> NN  /\  A. m  e. 
 NN  ( F `  m )  <  ( F `
  ( m  +  1 ) )  /\  A  e.  NN )  ->  E. n  e.  NN  ( F `  n )  e.  ( ZZ>= `  A ) )
 
Theoremincsequz2 26459* An increasing sequence of natural numbers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( F : NN --> NN  /\  A. m  e. 
 NN  ( F `  m )  <  ( F `
  ( m  +  1 ) )  /\  A  e.  NN )  ->  E. n  e.  NN  A. k  e.  ( ZZ>= `  n ) ( F `
  k )  e.  ( ZZ>= `  A )
 )
 
Theoremnnubfi 26460* A bounded above set of natural numbers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)
 |-  (
 ( A  C_  NN  /\  B  e.  NN )  ->  { x  e.  A  |  x  <  B }  e.  Fin )
 
Theoremnninfnub 26461* An infinite set of natural numbers is unbounded above. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)
 |-  (
 ( A  C_  NN  /\ 
 -.  A  e.  Fin  /\  B  e.  NN )  ->  { x  e.  A  |  B  <  x }  =/= 
 (/) )
 
Theoremcsbrn 26462* Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  RR )   =>    |-  ( ph  ->  ( sum_ k  e.  A  ( B  x.  C ) ^ 2
 )  <_  ( sum_ k  e.  A  ( B ^ 2 )  x. 
 sum_ k  e.  A  ( C ^ 2 ) ) )
 
Theoremtrirn 26463* Triangle inequality in R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  e.  RR )   =>    |-  ( ph  ->  ( sqr `  sum_ k  e.  A  ( ( B  +  C ) ^
 2 ) )  <_  ( ( sqr `  sum_ k  e.  A  ( B ^
 2 ) )  +  ( sqr `  sum_ k  e.  A  ( C ^
 2 ) ) ) )
 
18.15.4  Topology
 
TheoremunopnOLD 26464 The union of two open sets is open. (Moved to unopn 16649 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  u.  B )  e.  J )
 
TheoremincldOLD 26465 The intersection of two closed sets is closed. (Moved to incld 16780 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( J  e.  Top  /\  A  e.  ( Clsd `  J )  /\  B  e.  ( Clsd `  J )
 )  ->  ( A  i^i  B )  e.  ( Clsd `  J ) )
 
Theoremsubspopn 26466 An open set is open in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  (
 ( ( J  e.  Top  /\  A  e.  V ) 
 /\  ( B  e.  J  /\  B  C_  A ) )  ->  B  e.  ( Jt  A ) )
 
Theoremneificl 26467 Neighborhoods are closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Nov-2013.)
 |-  (
 ( ( J  e.  Top  /\  N  C_  ( ( nei `  J ) `  S ) )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) ) 
 ->  |^| N  e.  (
 ( nei `  J ) `  S ) )
 
Theoremlpss2 26468 Limit points of a subset are limit points of the larger set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  ( ( limPt `  J ) `  B )  C_  ( ( limPt `  J ) `  A ) )
 
18.15.5  Metric spaces
 
Theoremmetf1o 26469* Use a bijection with a metric space to construct a metric on a set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  N  =  ( x  e.  Y ,  y  e.  Y  |->  ( ( F `  x ) M ( F `  y ) ) )   =>    |-  ( ( Y  e.  A  /\  M  e.  ( Met `  X )  /\  F : Y -1-1-onto-> X )  ->  N  e.  ( Met `  Y ) )
 
Theoremblssp 26470 A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jan-2014.)
 |-  N  =  ( M  |`  ( S  X.  S ) )   =>    |-  ( ( ( M  e.  ( Met `  X )  /\  S  C_  X )  /\  ( Y  e.  S  /\  R  e.  RR+ ) )  ->  ( Y ( ball `  N ) R )  =  (
 ( Y ( ball `  M ) R )  i^i  S ) )
 
TheoremstiooOLD 26471 Two ways of expressing a subspace of  RR. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to resubmet 18308 in main set.mm and may be deleted by mathbox owner, SF. --MC 23-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  RR  ->  (
 ( topGen `  ran  (,) )t  A )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( A  X.  A ) ) ) )
 
TheoremblhalfOLD 26472 A ball of radius  R  /  2 is contained in a ball of radius  R centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to blhalf 17960 in main set.mm and may be deleted by mathbox owner, JM. --NM 21-May-2014.) (Revised by Mario Carneiro, 14-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( M  e.  ( Met `  X )  /\  Y  e.  X ) 
 /\  ( R  e.  RR+  /\  Z  e.  ( Y ( ball `  M )
 ( R  /  2
 ) ) ) ) 
 ->  ( Y ( ball `  M ) ( R 
 /  2 ) ) 
 C_  ( Z (
 ball `  M ) R ) )
 
Theoremmettrifi 26473* Generalized triangle inequality for arbitrary finite sums. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e.  X )   =>    |-  ( ph  ->  (
 ( F `  M ) D ( F `  N ) )  <_  sum_ k  e.  ( M
 ... ( N  -  1 ) ) ( ( F `  k
 ) D ( F `
  ( k  +  1 ) ) ) )
 
Theoremlmclim2 26474* A sequence in a metric space converges to a point iff the distance between the point and the elements of the sequence converges to 0. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
 |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  F : NN
 --> X )   &    |-  J  =  (
 MetOpen `  D )   &    |-  G  =  ( x  e.  NN  |->  ( ( F `  x ) D Y ) )   &    |-  ( ph  ->  Y  e.  X )   =>    |-  ( ph  ->  ( F ( ~~> t `  J ) Y  <->  G  ~~>  0 ) )
 
Theoremgeomcau 26475* If the distance between consecutive points in a sequence is bounded by a geometric sequence, then the sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
 |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  F : NN
 --> X )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  B  <  1 )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( ( F `  k ) D ( F `  ( k  +  1 ) ) )  <_  ( A  x.  ( B ^ k
 ) ) )   =>    |-  ( ph  ->  F  e.  ( Cau `  D ) )
 
Theoremcaures 26476 The restriction of a Cauchy sequence to a set of upper integers is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  F  e.  ( X  ^pm  CC ) )   =>    |-  ( ph  ->  ( F  e.  ( Cau `  D )  <->  ( F  |`  Z )  e.  ( Cau `  D ) ) )
 
Theoremcaushft 26477* A shifted Cauchy sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  W  =  (
 ZZ>= `  ( M  +  N ) )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( G `  (
 k  +  N ) ) )   &    |-  ( ph  ->  F  e.  ( Cau `  D ) )   &    |-  ( ph  ->  G : W --> X )   =>    |-  ( ph  ->  G  e.  ( Cau `  D )
 )
 
18.15.6  Continuous maps and homeomorphisms
 
Theoremconstcncf 26478* A constant function is a continuous function on  CC. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved into main set.mm as cncfmptc 18415 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  CC  |->  A )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremaddccncf 26479* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  CC  |->  ( x  +  A ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremidcncf 26480 The identity function is a continuous function on  CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 18416 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  CC  |->  x )   =>    |-  F  e.  ( CC
 -cn-> CC )
 
Theoremsub1cncf 26481* Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  CC  |->  ( x  -  A ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremsub2cncf 26482* Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  CC  |->  ( A  -  x ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremcnres2 26483* The restriction of a continuous function to a subset is continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  X  =  U. J   &    |-  Y  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A 
 C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K )  /\  A. x  e.  A  ( F `  x )  e.  B ) )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  B ) ) )
 
Theoremcnresima 26484 A continuous function is continuous onto its image. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Cn  K ) )  ->  F  e.  ( J  Cn  ( Kt  ran  F ) ) )
 
Theoremcncfres 26485* A continuous function on complex numbers restricted to a subset. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  A  C_ 
 CC   &    |-  B  C_  CC   &    |-  F  =  ( x  e.  CC  |->  C )   &    |-  G  =  ( x  e.  A  |->  C )   &    |-  ( x  e.  A  ->  C  e.  B )   &    |-  F  e.  ( CC -cn-> CC )   &    |-  J  =  (
 MetOpen `  ( ( abs 
 o.  -  )  |`  ( A  X.  A ) ) )   &    |-  K  =  (
 MetOpen `  ( ( abs 
 o.  -  )  |`  ( B  X.  B ) ) )   =>    |-  G  e.  ( J  Cn  K )
 
18.15.7  Product topologies
 
TheoremtxtopiOLD 26486 The product of two topologies is a topology. (Moved to txtopi 17285 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 15-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  R  e.  Top   &    |-  S  e.  Top   =>    |-  ( R  tX  S )  e.  Top
 
TheoremtxuniiOLD 26487 The underlying set of the product of two topologies. (Moved to txunii 17288 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 15-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  R  e.  Top   &    |-  S  e.  Top   &    |-  X  =  U. R   &    |-  Y  =  U. S   =>    |-  ( X  X.  Y )  =  U. ( R 
 tX  S )
 
TheoremtxopnOLD 26488 The product of two open sets is open in the product topology. (Moved to txopn 17297 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  T  =  ( R  tX  S )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( A  e.  R  /\  B  e.  S ) )  ->  ( A  X.  B )  e.  T )
 
TheoremtxcldOLD 26489 The product of two closed sets is closed in the product topology. (Moved to txcld 17298 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  T  =  ( R  tX  S )   =>    |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( A  e.  ( Clsd `  R )  /\  B  e.  ( Clsd `  S ) ) )  ->  ( A  X.  B )  e.  ( Clsd `  T ) )
 
18.15.8  Boundedness
 
Syntaxctotbnd 26490 Extend class notation with the class of totally bounded metric spaces.
 class  TotBnd
 
Syntaxcbnd 26491 Extend class notation with the class of bounded metric spaces.
 class  Bnd
 
Definitiondf-totbnd 26492* Define the class of totally bounded metrics. A metric space is totally bounded iff it can be covered by a finite number of balls of any given radius. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  TotBnd  =  ( x  e.  _V  |->  { m  e.  ( Met `  x )  |  A. d  e.  RR+  E. v  e.  Fin  ( U. v  =  x  /\  A. b  e.  v  E. y  e.  x  b  =  ( y ( ball `  m ) d ) ) } )
 
Theoremistotbnd 26493* The predicate "is a totally bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( M  e.  ( TotBnd `  X )  <->  ( M  e.  ( Met `  X )  /\  A. d  e.  RR+  E. v  e.  Fin  ( U. v  =  X  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M )
 d ) ) ) )
 
Theoremistotbnd2 26494* The predicate "is a totally bounded metric space." (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( M  e.  ( Met `  X )  ->  ( M  e.  ( TotBnd `  X )  <->  A. d  e.  RR+  E. v  e.  Fin  ( U. v  =  X  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M )
 d ) ) ) )
 
Theoremistotbnd3 26495* A metric space is totally bounded iff there is a finite ε-net for every positive ε. This differs from the definition in providing a finite set of ball centers rather than a finite set of balls. (Contributed by Mario Carneiro, 12-Sep-2015.)
 |-  ( M  e.  ( TotBnd `  X )  <->  ( M  e.  ( Met `  X )  /\  A. d  e.  RR+  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) d )  =  X ) )
 
Theoremtotbndmet 26496 The predicate "totally bounded" implies  M is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( M  e.  ( TotBnd `  X )  ->  M  e.  ( Met `  X )
 )
 
Theorem0totbnd 26497 The metric (there is only one) on the empty set is totally bounded. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( X  =  (/)  ->  ( M  e.  ( TotBnd `  X )  <->  M  e.  ( Met `  X ) ) )
 
Theoremsstotbnd2 26498* Condition for a subset of a metric space to be totally bounded. (Contributed by Mario Carneiro, 12-Sep-2015.)
 |-  N  =  ( M  |`  ( Y  X.  Y ) )   =>    |-  ( ( M  e.  ( Met `  X )  /\  Y  C_  X )  ->  ( N  e.  ( TotBnd `
  Y )  <->  A. d  e.  RR+  E. v  e.  ( ~P X  i^i  Fin ) Y  C_  U_ x  e.  v  ( x ( ball `  M ) d ) ) )
 
Theoremsstotbnd 26499* Condition for a subset of a metric space to be totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  N  =  ( M  |`  ( Y  X.  Y ) )   =>    |-  ( ( M  e.  ( Met `  X )  /\  Y  C_  X )  ->  ( N  e.  ( TotBnd `
  Y )  <->  A. d  e.  RR+  E. v  e.  Fin  ( Y  C_  U. v  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M ) d ) ) ) )
 
Theoremsstotbnd3 26500* Use a net that is not necessarily finite, but for which only finitely many balls meet the subset. (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  N  =  ( M  |`  ( Y  X.  Y ) )   =>    |-  ( ( M  e.  ( Met `  X )  /\  Y  C_  X )  ->  ( N  e.  ( TotBnd `
  Y )  <->  A. d  e.  RR+  E. v  e.  ~P  X ( Y  C_  U_ x  e.  v  ( x ( ball `  M )
 d )  /\  { x  e.  v  |  ( ( x (
 ball `  M ) d )  i^i  Y )  =/=  (/) }  e.  Fin ) ) )
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