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Theorem List for Metamath Proof Explorer - 11901-12000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrlim2lt 11901* Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( ph  ->  A. z  e.  A  B  e.  CC )   &    |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( ( z  e.  A  |->  B )  ~~> r  C  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  (
 y  <  z  ->  ( abs `  ( B  -  C ) )  < 
 x ) ) )
 
Theoremrlim3 11902* Restrict the range of the domain bound to reals greater than some  D  e.  RR. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( ph  ->  A. z  e.  A  B  e.  CC )   &    |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  RR )   =>    |-  ( ph  ->  (
 ( z  e.  A  |->  B )  ~~> r  C  <->  A. x  e.  RR+  E. y  e.  ( D [,)  +oo ) A. z  e.  A  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  <  x ) ) )
 
Theoremclimcl 11903 Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( F  ~~>  A  ->  A  e.  CC )
 
Theoremrlimpm 11904 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( F  ~~> r  A  ->  F  e.  ( CC 
 ^pm  RR ) )
 
Theoremrlimf 11905 Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( F  ~~> r  A  ->  F : dom  F --> CC )
 
Theoremrlimss 11906 Domain closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( F  ~~> r  A  ->  dom  F  C_  RR )
 
Theoremrlimcl 11907 Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( F  ~~> r  A  ->  A  e.  CC )
 
Theoremclim2 11908* Express the predicate: The limit of complex number sequence  F is  A, or  F converges to  A, with more general quantifier restrictions than clim 11898. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   =>    |-  ( ph  ->  ( F 
 ~~>  A  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  < 
 x ) ) ) )
 
Theoremclim2c 11909* Express the predicate  F converges to  A. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ph  ->  A  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  B  e.  CC )   =>    |-  ( ph  ->  ( F 
 ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( B  -  A ) )  <  x ) )
 
Theoremclim0 11910* Express the predicate  F converges to  0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   =>    |-  ( ph  ->  ( F 
 ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  B )  < 
 x ) ) )
 
Theoremclim0c 11911* Express the predicate  F converges to  0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  B )  <  x ) )
 
Theoremrlim0 11912* Express the predicate  B ( z ) converges to  0. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
 |-  ( ph  ->  A. z  e.  A  B  e.  CC )   &    |-  ( ph  ->  A  C_ 
 RR )   =>    |-  ( ph  ->  (
 ( z  e.  A  |->  B )  ~~> r  0  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  (
 y  <_  z  ->  ( abs `  B )  <  x ) ) )
 
Theoremrlim0lt 11913* Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
 |-  ( ph  ->  A. z  e.  A  B  e.  CC )   &    |-  ( ph  ->  A  C_ 
 RR )   =>    |-  ( ph  ->  (
 ( z  e.  A  |->  B )  ~~> r  0  <->  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  (
 y  <  z  ->  ( abs `  B )  <  x ) ) )
 
Theoremclimi 11914* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ph  ->  F  ~~>  A )   =>    |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  C ) )
 
Theoremclimi2 11915* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ph  ->  F  ~~>  A )   =>    |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( abs `  ( B  -  A ) )  <  C )
 
Theoremclimi0 11916* Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ph  ->  F  ~~>  0 )   =>    |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( abs `  B )  <  C )
 
Theoremrlimi 11917* Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 28-Feb-2015.)
 |-  ( ph  ->  A. z  e.  A  B  e.  V )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  ( z  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ph  ->  E. y  e.  RR  A. z  e.  A  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  <  R ) )
 
Theoremrlimi2 11918* Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  ( ph  ->  A. z  e.  A  B  e.  V )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  ( z  e.  A  |->  B )  ~~> r  C )   &    |-  ( ph  ->  D  e.  RR )   =>    |-  ( ph  ->  E. y  e.  ( D [,)  +oo ) A. z  e.  A  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  <  R ) )
 
Theoremello1 11919* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( F  e.  <_ O ( 1 )  <->  ( F  e.  ( RR  ^pm  RR )  /\  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  F  i^i  ( x [,)  +oo ) ) ( F `
  y )  <_  m ) )
 
Theoremello12 11920* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( F : A
 --> RR  /\  A  C_  RR )  ->  ( F  e.  <_ O ( 1 )  <->  E. x  e.  RR  E. m  e.  RR  A. y  e.  A  ( x  <_  y  ->  ( F `  y )  <_  m ) ) )
 
Theoremello12r 11921* Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ( F : A --> RR  /\  A  C_  RR )  /\  ( C  e.  RR  /\  M  e.  RR )  /\  A. x  e.  A  ( C  <_  x  ->  ( F `  x ) 
 <_  M ) )  ->  F  e.  <_ O ( 1 ) )
 
Theoremlo1f 11922 An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( F  e.  <_ O ( 1 )  ->  F : dom  F --> RR )
 
Theoremlo1dm 11923 An eventually upper bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( F  e.  <_ O ( 1 )  ->  dom  F  C_  RR )
 
Theoremlo1bdd 11924* The defining property of an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( F  e.  <_ O ( 1 ) 
 /\  F : A --> RR )  ->  E. x  e.  RR  E. m  e. 
 RR  A. y  e.  A  ( x  <_  y  ->  ( F `  y ) 
 <_  m ) )
 
Theoremello1mpt 11925* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  <_ O ( 1 )  <->  E. y  e.  RR  E. m  e.  RR  A. x  e.  A  (
 y  <_  x  ->  B 
 <_  m ) ) )
 
Theoremello1mpt2 11926* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  <_ O ( 1 )  <->  E. y  e.  ( C [,)  +oo ) E. m  e.  RR  A. x  e.  A  ( y  <_  x  ->  B  <_  m ) ) )
 
Theoremello1d 11927* Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ( ph  /\  ( x  e.  A  /\  C  <_  x )
 )  ->  B  <_  M )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) )
 
Theoremlo1bdd2 11928* If an eventually bounded function is bounded on every interval  A  i^i  (  -oo ,  y ) by a function  M ( y ), then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) )   &    |-  ( ( ph  /\  (
 y  e.  RR  /\  C  <_  y ) ) 
 ->  M  e.  RR )   &    |-  (
 ( ( ph  /\  x  e.  A )  /\  (
 ( y  e.  RR  /\  C  <_  y )  /\  x  <  y ) )  ->  B  <_  M )   =>    |-  ( ph  ->  E. m  e.  RR  A. x  e.  A  B  <_  m )
 
Theoremlo1bddrp 11929* Refine o1bdd2 11945 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) )   &    |-  ( ( ph  /\  (
 y  e.  RR  /\  C  <_  y ) ) 
 ->  M  e.  RR )   &    |-  (
 ( ( ph  /\  x  e.  A )  /\  (
 ( y  e.  RR  /\  C  <_  y )  /\  x  <  y ) )  ->  B  <_  M )   =>    |-  ( ph  ->  E. m  e.  RR+  A. x  e.  A  B  <_  m )
 
Theoremelo1 11930* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( F  e.  O ( 1 )  <->  ( F  e.  ( CC  ^pm  RR )  /\  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  F  i^i  ( x [,)  +oo ) ) ( abs `  ( F `  y
 ) )  <_  m ) )
 
Theoremelo12 11931* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( F : A
 --> CC  /\  A  C_  RR )  ->  ( F  e.  O ( 1 )  <->  E. x  e.  RR  E. m  e.  RR  A. y  e.  A  ( x  <_  y  ->  ( abs `  ( F `  y ) )  <_  m ) ) )
 
Theoremelo12r 11932* Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( ( F : A --> CC  /\  A  C_  RR )  /\  ( C  e.  RR  /\  M  e.  RR )  /\  A. x  e.  A  ( C  <_  x  ->  ( abs `  ( F `  x ) )  <_  M ) )  ->  F  e.  O (
 1 ) )
 
Theoremo1f 11933 An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( F  e.  O ( 1 )  ->  F : dom  F --> CC )
 
Theoremo1dm 11934 An eventually bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( F  e.  O ( 1 )  ->  dom  F  C_  RR )
 
Theoremo1bdd 11935* The defining property of an eventually bounded function. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( F  e.  O ( 1 ) 
 /\  F : A --> CC )  ->  E. x  e.  RR  E. m  e. 
 RR  A. y  e.  A  ( x  <_  y  ->  ( abs `  ( F `  y ) )  <_  m ) )
 
Theoremlo1o1 11936 A function is eventually bounded iff its absolute value is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( F : A --> CC  ->  ( F  e.  O ( 1 )  <-> 
 ( abs  o.  F )  e.  <_ O ( 1 ) ) )
 
Theoremlo1o12 11937* A function is eventually bounded iff its absolute value is eventually upper bounded. (This function is useful for converting theorems about  <_ O ( 1 ) to  O ( 1 ).) (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  O ( 1 )  <->  ( x  e.  A  |->  ( abs `  B ) )  e.  <_ O ( 1 ) ) )
 
Theoremelo1mpt 11938* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  O ( 1 )  <->  E. y  e.  RR  E. m  e.  RR  A. x  e.  A  (
 y  <_  x  ->  ( abs `  B )  <_  m ) ) )
 
Theoremelo1mpt2 11939* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 12-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  O ( 1 )  <->  E. y  e.  ( C [,)  +oo ) E. m  e.  RR  A. x  e.  A  ( y  <_  x  ->  ( abs `  B )  <_  m ) ) )
 
Theoremelo1d 11940* Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ( ph  /\  ( x  e.  A  /\  C  <_  x )
 )  ->  ( abs `  B )  <_  M )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )
 
Theoremo1lo1 11941* A real function is eventually bounded iff it is eventually lower bounded and eventually upper bounded. (Contributed by Mario Carneiro, 25-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  O ( 1 )  <->  ( ( x  e.  A  |->  B )  e.  <_ O ( 1 )  /\  ( x  e.  A  |->  -u B )  e.  <_ O ( 1 ) ) ) )
 
Theoremo1lo12 11942* A lower bounded real function is eventually bounded iff it is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  M  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  M  <_  B )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  O ( 1 )  <->  ( x  e.  A  |->  B )  e. 
 <_ O ( 1 ) ) )
 
Theoremo1lo1d 11943* A real eventually bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) )
 
Theoremicco1 11944* Derive eventual boundedness from separate upper and lower eventual bounds. (Contributed by Mario Carneiro, 15-Apr-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  N  e.  RR )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  C  <_  x ) )  ->  B  e.  ( M [,] N ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )
 
Theoremo1bdd2 11945* If an eventually bounded function is bounded on every interval  A  i^i  (  -oo ,  y ) by a function  M ( y ), then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   &    |-  (
 ( ph  /\  ( y  e.  RR  /\  C  <_  y ) )  ->  M  e.  RR )   &    |-  (
 ( ( ph  /\  x  e.  A )  /\  (
 ( y  e.  RR  /\  C  <_  y )  /\  x  <  y ) )  ->  ( abs `  B )  <_  M )   =>    |-  ( ph  ->  E. m  e.  RR  A. x  e.  A  ( abs `  B )  <_  m )
 
Theoremo1bddrp 11946* Refine o1bdd2 11945 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   &    |-  (
 ( ph  /\  ( y  e.  RR  /\  C  <_  y ) )  ->  M  e.  RR )   &    |-  (
 ( ( ph  /\  x  e.  A )  /\  (
 ( y  e.  RR  /\  C  <_  y )  /\  x  <  y ) )  ->  ( abs `  B )  <_  M )   =>    |-  ( ph  ->  E. m  e.  RR+  A. x  e.  A  ( abs `  B )  <_  m )
 
Theoremclimconst 11947* An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  =  A )   =>    |-  ( ph  ->  F  ~~>  A )
 
Theoremrlimconst 11948* A constant sequence converges to its value. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ( ( A  C_  RR  /\  B  e.  CC )  ->  ( x  e.  A  |->  B )  ~~> r  B )
 
Theoremrlimclim1 11949 Forward direction of rlimclim 11950. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~> r  A )   &    |-  ( ph  ->  Z  C_ 
 dom  F )   =>    |-  ( ph  ->  F  ~~>  A )
 
Theoremrlimclim 11950 A sequence on an upper integer set converges in the real sense iff it converges in the integer sense. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> CC )   =>    |-  ( ph  ->  ( F 
 ~~> r  A  <->  F  ~~>  A ) )
 
Theoremclimrlim2 11951* Produce a real limit from an integer limit, where the real function is only dependent on the integer part of  x. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( n  =  ( |_ `  x )  ->  B  =  C )   &    |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  ( n  e.  Z  |->  B )  ~~>  D )   &    |-  (
 ( ph  /\  n  e.  Z )  ->  B  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  M  <_  x )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D )
 
Theoremclimconst2 11952 A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ZZ>= `  M )  C_  Z   &    |-  Z  e.  _V   =>    |-  (
 ( A  e.  CC  /\  M  e.  ZZ )  ->  ( Z  X.  { A } )  ~~>  A )
 
Theoremclimz 11953 The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ZZ  X.  {
 0 } )  ~~>  0
 
Theoremrlimuni 11954 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 11955 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 11956 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 11957 The limit relation is function-like, and with range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ~~>  : dom  ~~>  --> CC
 
Theoremclimdm 11958 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 11959* 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 11960* 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 11961* An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)
 |- 
 E* x  F  ~~>  x
 
Theoremrlimres 11962 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 11963 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 11964 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 11965* 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 11966* 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 11967* 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 11968 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 11969 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 11970 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 11971* 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 11972* 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 11973* 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 11974* 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 11975* 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 11976* 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 11977* 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 11978 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 11979 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 11980 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 11981 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 11982* 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 11983* 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 11984* 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 11985* 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 11986* 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 11987* 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 11988* 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 11989* 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 11990* 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 11991* 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 11992* 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 11993* 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 11994* 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 11995* 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 11996* 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 11997* 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 16884 and df-cncf 18309 are not yet available to us. See addcn 18296 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 11998* 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 11999* 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 12000* 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 ) )
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