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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremrespreimaOLD 25701 The preimage of a restricted function. (Moved to respreima 5553 in main set.mm and may be deleted by mathbox owner, JM. --NM 30-Mar-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Fun  F  ->  ( `' ( F  |`  B )
 " A )  =  ( ( `' F " A )  i^i  B ) )
 
TheoremfoelrnOLD 25702* Property of a surjective function. (Moved to foelrn 5578 in main set.mm and may be deleted by mathbox owner, JM. --NM 4-Feb-2013.) (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( F : A -onto-> B  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
 
Theoremfoco2OLD 25703 If a composition of two functions is surjective, then the function on the left is surjective. (Moved to foco2 5579 in main set.mm and may be deleted by mathbox owner, JM. --NM 18-Apr-2013.) (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( F : B --> C  /\  G : A --> B  /\  ( F  o.  G ) : A -onto-> C )  ->  F : B -onto-> C )
 
TheoremfnimaprOLD 25704 The image of a pair under a funtion. (Moved to fnimapr 5482 in main set.mm and may be deleted by mathbox owner, JM. --NM 19-Apr-2013.) (Contributed by Jeff Madsen, 6-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( F " { B ,  C }
 )  =  { ( F `  B ) ,  ( F `  C ) } )
 
Theoremopelopab3 25705* Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  ( ch  ->  A  e.  C )   =>    |-  ( B  e.  D  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  ch ) )
 
Theoremcocanfo 25706 Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  (
 ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B )  /\  ( G  o.  F )  =  ( H  o.  F ) ) 
 ->  G  =  H )
 
TheoremxpengOLD 25707 Equinumerosity law for cross product. (Moved to xpen 6957 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Mar-2013.) (Contributed by Jeff Madsen, 17-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  W  /\  B  e.  X )  /\  ( C  e.  Y  /\  D  e.  Z ) )  ->  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A  X.  C ) 
 ~~  ( B  X.  D ) ) )
 
TheoremfvifOLD 25708 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to fvif 5438 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( F `  if ( ph ,  A ,  B ) )  =  if ( ph ,  ( F `  A ) ,  ( F `  B ) )
 
Theoremifeq1daOLD 25709 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to ifeq1da 3531 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
 
Theoremifeq2daOLD 25710 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to ifeq2da 3532 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  -.  ps )  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
 
TheoremifcldaOLD 25711 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to ifclda 3533 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  A  e.  C )   &    |-  ( ( ph  /\  -.  ps )  ->  B  e.  C )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )
 
Theoremxpeq1dOLD 25712 Equality deduction for cross product. (Moved to xpeq1d 4665 in main set.mm and may be deleted by mathbox owner, JM. --NM 4-Feb-2013.) (Contributed by Jeff Madsen, 17-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  X.  C )  =  ( B  X.  C ) )
 
Theoremxpeq2dOLD 25713 Equality deduction for cross product. (Moved to xpeq1d 4665 in main set.mm and may be deleted by mathbox owner, JM. --NM 4-Feb-2013.) (Contributed by Jeff Madsen, 17-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  X.  A )  =  ( C  X.  B ) )
 
TheoremresexOLD 25714 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.) (Moved to resex 4948 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( A  |`  B )  e.  _V
 
Theorembrresi 25715 Restriction of a binary relation. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  B  e.  _V   =>    |-  ( A ( R  |`  C ) B  ->  A R B )
 
Theoremopabex3OLD 25716* Existence of an ordered pair abstraction. (Moved to opabex3 5668 in main set.mm and may be deleted by mathbox owner, JM. --NM 30-Jan-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  ( x  e.  A  ->  { y  |  ph }  e.  _V )   =>    |- 
 { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
 
Theoremfnopabeqd 25717* Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) } )
 
Theoremfvopabf4g 25718* Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
 |-  C  e.  _V   &    |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  ( R 
 ^m  D )  |->  B )   =>    |-  ( ( D  e.  X  /\  R  e.  Y  /\  A : D --> R ) 
 ->  ( F `  A )  =  C )
 
Theoremeqfnun 25719 Two functions on  A  u.  B are equal if and only if they have equal restrictions to both  A and  B. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  (
 ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  ->  ( F  =  G  <->  ( ( F  |`  A )  =  ( G  |`  A ) 
 /\  ( F  |`  B )  =  ( G  |`  B ) ) ) )
 
Theoremfnopabco 25720* Composition of a function with a function abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  ( x  e.  A  ->  B  e.  C )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }   &    |-  G  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( H `  B ) ) }   =>    |-  ( H  Fn  C  ->  G  =  ( H  o.  F ) )
 
Theoremopropabco 25721* Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
 |-  ( x  e.  A  ->  B  e.  R )   &    |-  ( x  e.  A  ->  C  e.  S )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  = 
 <. B ,  C >. ) }   &    |-  G  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( B M C ) ) }   =>    |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
 
TheoremoprabexdOLD 25722* Existence of an operator abstraction. (Moved to oprabexd 5859 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.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  E* z ps )   &    |-  ( ph  ->  F  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) } )   =>    |-  ( ph  ->  F  e.  _V )
 
Theoremf1opr 25723* Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( F : ( A  X.  B ) -1-1-> C  <->  ( F :
 ( A  X.  B )
 --> C  /\  A. r  e.  A  A. s  e.  B  A. t  e.  A  A. u  e.  B  ( ( r F s )  =  ( t F u )  ->  ( r  =  t  /\  s  =  u ) ) ) )
 
TheoremcnvcanOLD 25724 Composition with the converse. (Moved to funcocnv2 5401 in main set.mm and may be deleted by mathbox owner, JM. --NM 28-Feb-2015.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Fun  F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
 
Theoremcocnv 25725 Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( Fun  F  /\  Fun 
 G )  ->  (
 ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
 
Theoremf1ocan1fv 25726 Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  (
 ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  B )  ->  ( ( F  o.  G ) `
  ( `' G `  X ) )  =  ( F `  X ) )
 
Theoremf1ocan2fv 25727 Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  (
 ( Fun  F  /\  G : A -1-1-onto-> B  /\  X  e.  A )  ->  ( ( F  o.  `' G ) `  ( G `  X ) )  =  ( F `  X ) )
 
Theoremf1elimaOLD 25728 Membership in the image of a 1-1 map. (Moved to f1elima 5686 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.)
 |-  (
 ( F : A -1-1-> B 
 /\  X  e.  A  /\  Y  C_  A )  ->  ( ( F `  X )  e.  ( F " Y )  <->  X  e.  Y ) )
 
Theoremenf1f1oOLD 25729 A one-to-one mapping of finite sets with the same cardinality is bijective. (Moved to f1finf1o 7019 in main set.mm and may be deleted by mathbox owner, JM. --NM 24-Sep-2013.) (Contributed by Jeff Madsen, 5-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  Fin  /\  B  ~~  A ) 
 ->  ( F : A -1-1-> B 
 ->  F : A -1-1-onto-> B ) )
 
Theoremeqfnfv3OLD 25730* Derive equality of functions from equality of their values. (Moved to eqfnfv3 5523 in main set.mm and may be deleted by mathbox owner, JM. --NM 22-Nov-2011.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( F  Fn  A  /\  G  Fn  B ) 
 ->  ( F  =  G  <->  ( B  C_  A  /\  A. x  e.  A  ( x  e.  B  /\  ( F `  x )  =  ( G `  x ) ) ) ) )
 
Theoreminixp 25731* Intersection of Cartesian products over the same base set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( X_ x  e.  A  B  i^i  X_ x  e.  A  C )  =  X_ x  e.  A  ( B  i^i  C )
 
TheoremixpssmapgOLD 25732* An infinite Cartesian product is a subset of set exponentiation. (Moved into main set.mm as ixpssmapg 6779 and may be deleted by mathbox owner, JM. --NM 18-Jun-2013.) (Contributed by Jeff Madsen, 19-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  C  /\  A. x  e.  A  B  e.  D )  -> 
 X_ x  e.  A  B  C_  ( U_ x  e.  A  B  ^m  A ) )
 
TheoremmapfiOLD 25733 Set exponentiation of finite sets is finite. (Moved into main set.mm as mapfi 7085 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 25734* A cross product of finitely many finite sets is finite. (Moved into main set.mm as ixpfi 7086 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 25735* 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 25736* Existence of an existentially restricted class abstraction. (Moved to abrexex2g 5667 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 25737* 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 25738* 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 25739* 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 7031 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 25740* A finite set has a maximum under a total order. (Moved to fimaxg 7037 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 25741* A finite set has a maximum under a total order. (Moved to fimaxg 7037 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 25742* Lemma showing existence and closure of supremum of a finite set. (Moved to fisupg 7038 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 25743* 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 25744* 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 25745* 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 25746* 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 25745. (Moved to indexfi 7096 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 25747* 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 7094 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 25748* 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 25749* 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 25750 Equality theorem for supremum. (Moved to supeq2 7134 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 25751 Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  C  ->  sup ( B ,  A ,  R )  e.  _V )
 
Theoremsupclt 25752* 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 25753* 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 25754* 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 9668 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 25755* A supremum is unique. Closed version of supeu 7138. (Moved to supeu 7138 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 25756 A nonempty finite set contains its supremum. (Moved to fisupcl 7151 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 25757* A finite set has a maximum under a total order. (Moved to fimax2g 7036 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 25758 A total order on a finite set is a well order. (Moved to wofi 7039 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 25759 A partial order is founded on a finite set. (Moved to frfi 7035 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 25760* A function preserves a partial order relation. (Moved to pofun 4267 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 25761* If an element of a founded set satisfies a property  ph, then there is a minimal element that satisfies  ph. (Moved to frminex 4310 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 ) ) ) )
 
16.14.2  Real and complex numbers; integers
 
TheoremfimaxreOLD 25762* A finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to fimaxre 9634 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 25763* A nonempty finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 27-May-2011.) (Moved to fimaxre2 9635 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 25764* 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 25765 Two nonnegative numbers are zero iff their sum is zero. (Moved to add20 9219 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 25766 Relation between sums and differences.. (Moved to addsubeq4 8999 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 25767 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 25768 Membership in a later set of upper integers. (Moved to eluzadd 10188 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 25769 Membership in an earlier set of upper integers. (Moved to eluzsub 10189 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 25770 Choices for an element of an upper interval of integers. (Moved to uzm1 10190 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 25771 Choices for an element of an upper interval of integers. (Moved to uzp1 10193 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 25772 A finite interval of integers is finite. (Moved to fzfi 10965 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 25773 Variant of fzfi 10965 with hypothesis weakened. (Moved to fzfi 10965 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 25774 There are no integers between 1 and 0. (Moved to fz10 10745 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 25775 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 25776 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 25777 Split a finite interval of integers into two parts. (Moved to fzsplit 10747 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 25778 Condition for two finite intervals of integers to be disjoint. (Moved to fzdisj 10748 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 25779 Add one to an element of a finite set of integers. (Moved to fzp1elp1 10770 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 25780 A real number is an integer iff its absolute value is an integer. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to absz 11726 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 25781  A  mod  B is zero iff  A is evenly divisible by  B. (Moved to mod0 10909 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 25782  A is divisible by  B iff its negative is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to negmod0 10910 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 25783  A is divisible by  B iff its absolute value is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved to absmod0 11718 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 )
 )
 
16.14.3  Sequences and sums
 
Theoremsdclem2 25784* Lemma for sdc 25786. (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 25785* Lemma for sdc 25786. (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 25786* 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 25787* 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 25788* Variant of fdc 25787 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 25789* 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 25790* 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 25791* 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 25792* 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 25793* 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 25794* 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 25795* 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 ) ) ) )
 
16.14.4  Topology
 
TheoremunopnOLD 25796 The union of two open sets is open. (Moved to unopn 16576 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 25797 The intersection of two closed sets is closed. (Moved to incld 16707 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 25798 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 25799 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 25800 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 ) )
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