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Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrlim2 13301* Rewrite rlim 13300 for a mapping operation. (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  ->  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 ) ) )
 
Theoremrlim2lt 13302* 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 13303* 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 13304 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 13305 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 13306 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 13307 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 13308 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 13309* Express the predicate: The limit of complex number sequence  F is  A, or  F converges to  A, with more general quantifier restrictions than clim 13299. (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 13310* 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 13311* 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 13312* 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 13313* 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 13314* 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 13315* 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 13316* 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 13317* 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 13318* 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 13319* 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 13320* 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 13321* 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 13322* 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 13323 An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( F  e.  <_O(1)  ->  F : dom  F --> RR )
 
Theoremlo1dm 13324 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 13325* 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 13326* 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 13327* 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 13328* 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 13329* 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 13330* Refine o1bdd2 13346 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 13331* 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 13332* 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 13333* 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 13334 An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( F  e.  O(1)  ->  F : dom  F --> CC )
 
Theoremo1dm 13335 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 13336* 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 13337 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 13338* 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 13339* 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 13340* 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 13341* 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 13342* 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 13343* 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 13344* 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 13345* 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 13346* 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 13347* Refine o1bdd2 13346 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 13348* 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 13349* 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 13350 Forward direction of rlimclim 13351. (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 13351 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 13352* 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 13353 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 13354 The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ZZ  X.  {
 0 } )  ~~>  0
 
Theoremrlimuni 13355 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 13356 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 13357 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 13358 The limit relation is function-like, and with range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
 |-  ~~>  : dom  ~~>  --> CC
 
Theoremclimdm 13359 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 13360* 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 13361* 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 13362* An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)
 |- 
 E* x  F  ~~>  x
 
Theoremrlimres 13363 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 13364 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 13365 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 13366* 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 13367* 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 13368* 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 13369 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 13370 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 13371 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 13372* 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 13373* 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 13374* 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 13375* 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 13376* 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 13377* 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 13378* 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 13379 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 13380 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 13381 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 13382 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 13383* If  D is a closed set in the topology of the complex numbers (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 13384* The limit of a sequence of complex numbers 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 13385* 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 13386* 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 13387* 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 13388* 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 13389* 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 13390* 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 13391* 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 13392* 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 13393* 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 13394* 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 13395* 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 13396* 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 13397* 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 13398* 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 19706 and df-cncf 21360 are not yet available to us. See addcn 21347 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 13399* 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 13400* 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 ) )
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