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Theorem List for Metamath Proof Explorer - 11901-12000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrlimuni 11901 A real function whose domain is unbounded above converges to at most one limit. (Contributed by Mario Carneiro, 8-May-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  sup ( A ,  RR*
 ,  <  )  =  +oo )   &    |-  ( ph  ->  F  ~~> r  B )   &    |-  ( ph  ->  F  ~~> r  C )   =>    |-  ( ph  ->  B  =  C )
 
Theoremrlimdm 11902 Two ways to express that a function has a limit. (The expression  (  ~~> r  `  F ) is sometimes useful as a shorthand for "the unique limit of the function  F"). (Contributed by Mario Carneiro, 8-May-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  sup ( A ,  RR*
 ,  <  )  =  +oo )   =>    |-  ( ph  ->  ( F  e.  dom  ~~> r  <->  F  ~~> r  (  ~~> r  `  F ) ) )
 
Theoremclimuni 11903 An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( F  ~~>  A  /\  F 
 ~~>  B )  ->  A  =  B )
 
Theoremfclim 11904 The limit relation is function-like, and with range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ~~>  : dom  ~~>  --> CC
 
Theoremclimdm 11905 Two ways to express that a function has a limit. (The expression  (  ~~>  `  F
) is sometimes useful as a shorthand for "the unique limit of the function  F"). (Contributed by Mario Carneiro, 18-Mar-2014.)
 |-  ( F  e.  dom  ~~>  <->  F  ~~>  ( 
 ~~>  `  F ) )
 
Theoremclimeu 11906* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
 |-  ( F  ~~>  A  ->  E! x  F  ~~>  x )
 
Theoremclimreu 11907* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
 |-  ( F  ~~>  A  ->  E! x  e.  CC  F  ~~>  x )
 
Theoremclimmo 11908* An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)
 |- 
 E* x  F  ~~>  x
 
Theoremrlimres 11909 The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( F  ~~> r  A  ->  ( F  |`  B )  ~~> r  A )
 
Theoremlo1res 11910 The restriction of an eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( F  e.  <_ O ( 1 )  ->  ( F  |`  A )  e.  <_ O ( 1 ) )
 
Theoremo1res 11911 The restriction of an eventually bounded function is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  ( F  e.  O ( 1 )  ->  ( F  |`  A )  e.  O ( 1 ) )
 
Theoremrlimres2 11912* The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  ( x  e.  B  |->  C )  ~~> r  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D )
 
Theoremlo1res2 11913* The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  ( x  e.  B  |->  C )  e.  <_ O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  e. 
 <_ O ( 1 ) )
 
Theoremo1res2 11914* The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  ( x  e.  B  |->  C )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )
 
Theoremlo1resb 11915 The restriction of a function to an unbounded-above interval is eventually upper bounded iff the original is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( F  e.  <_ O ( 1 )  <->  ( F  |`  ( B [,)  +oo ) )  e. 
 <_ O ( 1 ) ) )
 
Theoremrlimresb 11916 The restriction of a function to an unbounded-above interval converges iff the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( F 
 ~~> r  C  <->  ( F  |`  ( B [,)  +oo ) )  ~~> r  C ) )
 
Theoremo1resb 11917 The restriction of a function to an unbounded-above interval is eventually bounded iff the original is eventually bounded. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( F  e.  O (
 1 )  <->  ( F  |`  ( B [,)  +oo ) )  e.  O ( 1 ) ) )
 
Theoremclimeq 11918* Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  ( G `
  k ) )   =>    |-  ( ph  ->  ( F  ~~>  A 
 <->  G  ~~>  A ) )
 
Theoremlo1eq 11919* Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ( ph  /\  ( x  e.  A  /\  D  <_  x ) ) 
 ->  B  =  C )   =>    |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  <_ O ( 1 )  <->  ( x  e.  A  |->  C )  e. 
 <_ O ( 1 ) ) )
 
Theoremrlimeq 11920* Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ( ph  /\  ( x  e.  A  /\  D  <_  x ) ) 
 ->  B  =  C )   =>    |-  ( ph  ->  ( ( x  e.  A  |->  B )  ~~> r  E  <->  ( x  e.  A  |->  C )  ~~> r  E ) )
 
Theoremo1eq 11921* Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ( ph  /\  ( x  e.  A  /\  D  <_  x ) ) 
 ->  B  =  C )   =>    |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  O ( 1 )  <->  ( x  e.  A  |->  C )  e.  O ( 1 ) ) )
 
Theoremclimmpt 11922* Exhibit a function  G with the same convergence properties as the not-quite-function  F. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  G  =  ( k  e.  Z  |->  ( F `  k ) )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( F  ~~>  A  <->  G  ~~>  A ) )
 
Theorem2clim 11923* If two sequences converge to each other, they converge to the same limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( abs `  ( ( F `  k )  -  ( G `  k ) ) )  <  x )   &    |-  ( ph  ->  F  ~~>  A )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremclimmpt2 11924* Relate an integer limit on a not-quite-function to a real limit. (Contributed by Mario Carneiro, 17-Sep-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  ( F 
 ~~>  A  <->  ( n  e.  Z  |->  ( F `  n ) )  ~~> r  A ) )
 
Theoremclimshftlem 11925 A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( M  e.  ZZ  ->  ( F  ~~>  A  ->  ( F  shift  M )  ~~>  A )
 )
 
Theoremclimres 11926 A function restricted to upper integers converges iff the original function converges. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( ( F  |`  ( ZZ>= `  M )
 )  ~~>  A  <->  F  ~~>  A ) )
 
Theoremclimshft 11927 A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( ( F 
 shift  M )  ~~>  A  <->  F  ~~>  A ) )
 
Theoremserclim0 11928 The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
 |-  ( M  e.  ZZ  ->  seq  M (  +  ,  ( ( ZZ>= `  M )  X.  { 0 } ) )  ~~>  0 )
 
Theoremrlimcld2 11929* If  D is a closed set in the topology of the complexes (stated here in basic form), and all the elements of the sequence lie in  D, then the limit of the sequence also lies in  D. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ph  ->  sup ( A ,  RR* ,  <  )  =  +oo )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  C )   &    |-  ( ph  ->  D  C_ 
 CC )   &    |-  ( ( ph  /\  y  e.  ( CC  \  D ) )  ->  R  e.  RR+ )   &    |-  (
 ( ( ph  /\  y  e.  ( CC  \  D ) )  /\  z  e.  D )  ->  R  <_  ( abs `  (
 z  -  y ) ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  D )   =>    |-  ( ph  ->  C  e.  D )
 
Theoremrlimrege0 11930* The limit of a sequence of complexes with nonnegative real part has nonnegative real part. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ph  ->  sup ( A ,  RR* ,  <  )  =  +oo )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  ( Re
 `  B ) )   =>    |-  ( ph  ->  0  <_  ( Re `  C ) )
 
Theoremrlimrecl 11931* The limit of a real sequence is real. (Contributed by Mario Carneiro, 9-May-2016.)
 |-  ( ph  ->  sup ( A ,  RR* ,  <  )  =  +oo )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   =>    |-  ( ph  ->  C  e.  RR )
 
Theoremrlimge0 11932* The limit of a sequence of nonnegative reals is nonnegative. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ph  ->  sup ( A ,  RR* ,  <  )  =  +oo )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  C )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  B )   =>    |-  ( ph  ->  0  <_  C )
 
Theoremclimshft2 11933* A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ph  ->  F  e.  W )   &    |-  ( ph  ->  G  e.  X )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  ( k  +  K ) )  =  ( F `  k
 ) )   =>    |-  ( ph  ->  ( F 
 ~~>  A  <->  G  ~~>  A ) )
 
Theoremclimrecl 11934* The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremclimge0 11935* A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  0  <_  ( F `
  k ) )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremclimabs0 11936* Convergence to zero of the absolute value is equivalent to convergence to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( abs `  ( F `  k ) ) )   =>    |-  ( ph  ->  ( F 
 ~~>  0  <->  G  ~~>  0 ) )
 
Theoremo1co 11937* Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  F  e.  O ( 1 ) )   &    |-  ( ph  ->  G : B
 --> A )   &    |-  ( ph  ->  B 
 C_  RR )   &    |-  ( ( ph  /\  m  e.  RR )  ->  E. x  e.  RR  A. y  e.  B  ( x  <_  y  ->  m 
 <_  ( G `  y
 ) ) )   =>    |-  ( ph  ->  ( F  o.  G )  e.  O ( 1 ) )
 
Theoremo1compt 11938* Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  F  e.  O ( 1 ) )   &    |-  ( ( ph  /\  y  e.  B )  ->  C  e.  A )   &    |-  ( ph  ->  B 
 C_  RR )   &    |-  ( ( ph  /\  m  e.  RR )  ->  E. x  e.  RR  A. y  e.  B  ( x  <_  y  ->  m 
 <_  C ) )   =>    |-  ( ph  ->  ( F  o.  ( y  e.  B  |->  C ) )  e.  O ( 1 ) )
 
Theoremrlimcn1 11939* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 17-Sep-2014.)
 |-  ( ph  ->  G : A --> X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  G  ~~> r  C )   &    |-  ( ph  ->  F : X --> CC )   &    |-  (
 ( ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  X  ( ( abs `  (
 z  -  C ) )  <  y  ->  ( abs `  ( ( F `  z )  -  ( F `  C ) ) )  <  x ) )   =>    |-  ( ph  ->  ( F  o.  G )  ~~> r  ( F `  C ) )
 
Theoremrlimcn1b 11940* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   &    |-  ( ph  ->  F : X --> CC )   &    |-  (
 ( ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  X  ( ( abs `  (
 z  -  C ) )  <  y  ->  ( abs `  ( ( F `  z )  -  ( F `  C ) ) )  <  x ) )   =>    |-  ( ph  ->  (
 k  e.  A  |->  ( F `  B ) )  ~~> r  ( F `
  C ) )
 
Theoremrlimcn2 11941* Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014.)
 |-  ( ( ph  /\  z  e.  A )  ->  B  e.  X )   &    |-  ( ( ph  /\  z  e.  A ) 
 ->  C  e.  Y )   &    |-  ( ph  ->  R  e.  X )   &    |-  ( ph  ->  S  e.  Y )   &    |-  ( ph  ->  ( z  e.  A  |->  B )  ~~> r  R )   &    |-  ( ph  ->  (
 z  e.  A  |->  C )  ~~> r  S )   &    |-  ( ph  ->  F :
 ( X  X.  Y )
 --> CC )   &    |-  ( ( ph  /\  x  e.  RR+ )  ->  E. r  e.  RR+  E. s  e.  RR+  A. u  e.  X  A. v  e.  Y  ( ( ( abs `  ( u  -  R ) )  < 
 r  /\  ( abs `  ( v  -  S ) )  <  s ) 
 ->  ( abs `  (
 ( u F v )  -  ( R F S ) ) )  <  x ) )   =>    |-  ( ph  ->  (
 z  e.  A  |->  ( B F C ) )  ~~> r  ( R F S ) )
 
Theoremclimcn1 11942* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ( ph  /\  z  e.  B )  ->  ( F `  z )  e. 
 CC )   &    |-  ( ph  ->  G  ~~>  A )   &    |-  ( ph  ->  H  e.  W )   &    |-  (
 ( ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  B  ( ( abs `  (
 z  -  A ) )  <  y  ->  ( abs `  ( ( F `  z )  -  ( F `  A ) ) )  <  x ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  B )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( F `  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( F `  A ) )
 
Theoremclimcn2 11943* Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ( ph  /\  ( u  e.  C  /\  v  e.  D ) )  ->  ( u F v )  e. 
 CC )   &    |-  ( ph  ->  G  ~~>  A )   &    |-  ( ph  ->  H  ~~>  B )   &    |-  ( ph  ->  K  e.  W )   &    |-  (
 ( ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  C  A. v  e.  D  ( ( ( abs `  ( u  -  A ) )  <  y  /\  ( abs `  ( v  -  B ) )  < 
 z )  ->  ( abs `  ( ( u F v )  -  ( A F B ) ) )  <  x ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  C )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( H `  k )  e.  D )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( K `  k
 )  =  ( ( G `  k ) F ( H `  k ) ) )   =>    |-  ( ph  ->  K  ~~>  ( A F B ) )
 
Theoremaddcn2 11944* Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (We write out the definition directly because df-cn 16789 and df-cncf 18214 are not yet available to us. See addcn 18201 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e. 
 CC  ( ( ( abs `  ( u  -  B ) )  < 
 y  /\  ( abs `  ( v  -  C ) )  <  z ) 
 ->  ( abs `  (
 ( u  +  v
 )  -  ( B  +  C ) ) )  <  A ) )
 
Theoremsubcn2 11945* Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e. 
 CC  ( ( ( abs `  ( u  -  B ) )  < 
 y  /\  ( abs `  ( v  -  C ) )  <  z ) 
 ->  ( abs `  (
 ( u  -  v
 )  -  ( B  -  C ) ) )  <  A ) )
 
Theoremmulcn2 11946* Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e. 
 CC  ( ( ( abs `  ( u  -  B ) )  < 
 y  /\  ( abs `  ( v  -  C ) )  <  z ) 
 ->  ( abs `  (
 ( u  x.  v
 )  -  ( B  x.  C ) ) )  <  A ) )
 
Theoremreccn2 11947* The reciprocal function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) (Revised by Mario Carneiro, 22-Sep-2014.)
 |-  T  =  ( if ( 1  <_  (
 ( abs `  A )  x.  B ) ,  1 ,  ( ( abs `  A )  x.  B ) )  x.  (
 ( abs `  A )  /  2 ) )   =>    |-  ( ( A  e.  ( CC  \  { 0 } )  /\  B  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  ( CC  \  { 0 } ) ( ( abs `  ( z  -  A ) )  <  y  ->  ( abs `  ( (
 1  /  z )  -  ( 1  /  A ) ) )  <  B ) )
 
Theoremcn1lem 11948* A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  F : CC --> CC   &    |-  (
 ( z  e.  CC  /\  A  e.  CC )  ->  ( abs `  (
 ( F `  z
 )  -  ( F `
  A ) ) )  <_  ( abs `  ( z  -  A ) ) )   =>    |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  ( ( abs `  (
 z  -  A ) )  <  y  ->  ( abs `  ( ( F `  z )  -  ( F `  A ) ) )  <  x ) )
 
Theoremabscn2 11949* The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( abs `  z
 )  -  ( abs `  A ) ) )  <  x ) )
 
Theoremcjcn2 11950* The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( * `  z )  -  ( * `  A ) ) )  <  x ) )
 
Theoremrecn2 11951* The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( Re `  z )  -  ( Re `  A ) ) )  <  x ) )
 
Theoremimcn2 11952* The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( Im `  z )  -  ( Im `  A ) ) )  <  x ) )
 
Theoremclimcn1lem 11953* The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  H : CC --> CC   &    |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( H `  z )  -  ( H `  A ) ) )  <  x ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( H `
  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( H `  A ) )
 
Theoremclimabs 11954* Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( abs `  ( F `  k
 ) ) )   =>    |-  ( ph  ->  G  ~~>  ( abs `  A )
 )
 
Theoremclimcj 11955* Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( * `
  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( * `  A ) )
 
Theoremclimre 11956* Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( Re
 `  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( Re `  A ) )
 
Theoremclimim 11957* Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( Im
 `  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( Im `  A ) )
 
Theoremrlimmptrcl 11958* Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )
 
Theoremrlimabs 11959* Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ph  ->  ( k  e.  A  |->  ( abs `  B ) )  ~~> r  ( abs `  C ) )
 
Theoremrlimcj 11960* Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ph  ->  ( k  e.  A  |->  ( * `  B ) )  ~~> r  ( * `  C ) )
 
Theoremrlimre 11961* Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ph  ->  ( k  e.  A  |->  ( Re `  B ) )  ~~> r  ( Re `  C ) )
 
Theoremrlimim 11962* Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ph  ->  ( k  e.  A  |->  ( Im `  B ) )  ~~> r  ( Im `  C ) )
 
Theoremo1of2 11963* Show that a binary operation preserves eventual boundedness. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( m  e. 
 RR  /\  n  e.  RR )  ->  M  e.  RR )   &    |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x R y )  e. 
 CC )   &    |-  ( ( ( m  e.  RR  /\  n  e.  RR )  /\  ( x  e.  CC  /\  y  e.  CC )
 )  ->  ( (
 ( abs `  x )  <_  m  /\  ( abs `  y )  <_  n )  ->  ( abs `  ( x R y ) ) 
 <_  M ) )   =>    |-  ( ( F  e.  O ( 1 )  /\  G  e.  O ( 1 ) )  ->  ( F  o F R G )  e.  O ( 1 ) )
 
Theoremo1add 11964 The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
 |-  ( ( F  e.  O ( 1 ) 
 /\  G  e.  O ( 1 ) ) 
 ->  ( F  o F  +  G )  e.  O ( 1 ) )
 
Theoremo1mul 11965 The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
 |-  ( ( F  e.  O ( 1 ) 
 /\  G  e.  O ( 1 ) ) 
 ->  ( F  o F  x.  G )  e.  O ( 1 ) )
 
Theoremo1sub 11966 The difference of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
 |-  ( ( F  e.  O ( 1 ) 
 /\  G  e.  O ( 1 ) ) 
 ->  ( F  o F  -  G )  e.  O ( 1 ) )
 
Theoremrlimo1 11967 Any function with a finite limit is eventually bounded. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( F  ~~> r  A  ->  F  e.  O ( 1 ) )
 
Theoremrlimdmo1 11968 A convergent function is eventually bounded. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  ( F  e.  dom  ~~> r 
 ->  F  e.  O ( 1 ) )
 
Theoremo1rlimmul 11969 The product of a eventually bounded function and a function of limit zero has limit zero. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( ( F  e.  O ( 1 ) 
 /\  G  ~~> r  0 )  ->  ( F  o F  x.  G ) 
 ~~> r  0 )
 
Theoremo1const 11970* A constant function is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  ( ( A  C_  RR  /\  B  e.  CC )  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )
 
Theoremlo1const 11971* A constant function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( A  C_  RR  /\  B  e.  RR )  ->  ( x  e.  A  |->  B )  e. 
 <_ O ( 1 ) )
 
Theoremlo1mptrcl 11972* Reverse closure for an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) )   =>    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )
 
Theoremo1mptrcl 11973* Reverse closure for an eventually bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   =>    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  CC )
 
Theoremo1add2 11974* The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e.  O ( 1 ) )
 
Theoremo1mul2 11975* The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  C ) )  e.  O ( 1 ) )
 
Theoremo1sub2 11976* The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e.  O ( 1 ) )
 
Theoremlo1add 11977* The sum of two eventually upper bounded functions is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. 
 <_ O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  <_ O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. 
 <_ O ( 1 ) )
 
Theoremlo1mul 11978* The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. 
 <_ O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  <_ O ( 1 ) )   &    |-  ( ( ph  /\  x  e.  A )  ->  0  <_  B )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  C ) )  e.  <_ O ( 1 ) )
 
Theoremlo1mul2 11979* The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. 
 <_ O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  <_ O ( 1 ) )   &    |-  ( ( ph  /\  x  e.  A )  ->  0  <_  B )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e.  <_ O ( 1 ) )
 
Theoremo1dif 11980* If the difference of two functions is eventually bounded, eventual boundedness of either one implies the other. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  CC )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e.  O ( 1 ) )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  O ( 1 )  <->  ( x  e.  A  |->  C )  e.  O ( 1 ) ) )
 
Theoremlo1sub 11981* The difference of an eventually upper bounded function and an eventually bounded function is eventually upper bounded. The "correct" sharp result here takes the second function to be eventually lower bounded instead of just bounded, but our notation for this is simply  ( x  e.  A  |->  -u C
)  e.  <_ O
( 1 ), so it is just a special case of lo1add 11977. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. 
 <_ O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e.  <_ O ( 1 ) )
 
Theoremclimadd 11982* Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  +  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  +  B ) )
 
Theoremclimmul 11983* Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  x.  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
 
Theoremclimsub 11984* Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  -  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  -  B ) )
 
Theoremclimaddc1 11985* Limit of a constant  C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `  k )  +  C ) )   =>    |-  ( ph  ->  G  ~~>  ( A  +  C ) )
 
Theoremclimaddc2 11986* Limit of a constant  C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  +  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  +  A ) )
 
Theoremclimmulc2 11987* Limit of a sequence multiplied by a constant  C. Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  x.  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  x.  A ) )
 
Theoremclimsubc1 11988* Limit of a constant  C subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `  k )  -  C ) )   =>    |-  ( ph  ->  G  ~~>  ( A  -  C ) )
 
Theoremclimsubc2 11989* Limit of a constant  C minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  -  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  -  A ) )
 
Theoremclimle 11990* Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremclimsqz 11991* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  <_  A )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremclimsqz2 11992* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  <_  ( F `  k ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  <_  ( G `  k ) )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremrlimadd 11993* Limit of the sum of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  ~~> r  ( D  +  E ) )
 
Theoremrlimsub 11994* Limit of the difference of two converging functions. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  ~~> r  ( D  -  E ) )
 
Theoremrlimmul 11995* Limit of the product of two converging functions. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  C ) )  ~~> r  ( D  x.  E ) )
 
Theoremrlimdiv 11996* Limit of the quotient of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )   &    |-  ( ph  ->  E  =/=  0 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B 
 /  C ) )  ~~> r  ( D  /  E ) )
 
Theoremrlimneg 11997* Limit of the negative of a sequence. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ph  ->  ( k  e.  A  |->  -u B )  ~~> r  -u C )
 
Theoremrlimle 11998* Comparison of the limits of two sequences. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ph  ->  sup ( A ,  RR* ,  <  )  =  +oo )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  C  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  <_  C )   =>    |-  ( ph  ->  D  <_  E )
 
Theoremrlimsqzlem 11999* Lemma for rlimsqz 12000 and rlimsqz2 12001. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)
 |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  E  e.  CC )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  ( abs `  ( C  -  E ) )  <_  ( abs `  ( B  -  D ) ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )
 
Theoremrlimsqz 12000* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)
 |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  B  <_  C )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  <_  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D )
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