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Theorem List for Metamath Proof Explorer - 11801-11900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabsvalsqd 11801 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A ) ^ 2 )  =  ( A  x.  ( * `  A ) ) )
 
Theoremabsvalsq2d 11802 Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( abs `  A ) ^ 2 )  =  ( ( ( Re
 `  A ) ^
 2 )  +  (
 ( Im `  A ) ^ 2 ) ) )
 
Theoremabsge0d 11803 Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( abs `  A ) )
 
Theoremabsval2d 11804 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  A )  =  ( sqr `  (
 ( ( Re `  A ) ^ 2
 )  +  ( ( Im `  A ) ^ 2 ) ) ) )
 
Theoremabs00d 11805 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( abs `  A )  =  0 )   =>    |-  ( ph  ->  A  =  0 )
 
Theoremabsne0d 11806 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( abs `  A )  =/=  0 )
 
Theoremabsrpcld 11807 The absolute value of a nonzero number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( abs `  A )  e.  RR+ )
 
Theoremabsnegd 11808 Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  -u A )  =  ( abs `  A ) )
 
Theoremabscjd 11809 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( * `  A ) )  =  ( abs `  A ) )
 
Theoremreleabsd 11810 The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  A )  <_  ( abs `  A )
 )
 
Theoremabsexpd 11811 Absolute value of natural number exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( abs `  ( A ^ N ) )  =  ( ( abs `  A ) ^ N ) )
 
Theoremabssubd 11812 Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  =  ( abs `  ( B  -  A ) ) )
 
Theoremabsmuld 11813 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  x.  B ) )  =  ( ( abs `  A )  x.  ( abs `  B ) ) )
 
Theoremabsdivd 11814 Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( abs `  ( A  /  B ) )  =  ( ( abs `  A )  /  ( abs `  B ) ) )
 
Theoremabstrid 11815 Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  +  B ) )  <_  ( ( abs `  A )  +  ( abs `  B ) ) )
 
Theoremabs2difd 11816 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( abs `  A )  -  ( abs `  B ) )  <_  ( abs `  ( A  -  B ) ) )
 
Theoremabs2dif2d 11817 Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <_  ( ( abs `  A )  +  ( abs `  B ) ) )
 
Theoremabs2difabsd 11818 Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( ( abs `  A )  -  ( abs `  B )
 ) )  <_  ( abs `  ( A  -  B ) ) )
 
Theoremabs3difd 11819 Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <_  ( ( abs `  ( A  -  C ) )  +  ( abs `  ( C  -  B ) ) ) )
 
Theoremabs3lemd 11820 Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  ( abs `  ( A  -  C ) )  < 
 ( D  /  2
 ) )   &    |-  ( ph  ->  ( abs `  ( C  -  B ) )  < 
 ( D  /  2
 ) )   =>    |-  ( ph  ->  ( abs `  ( A  -  B ) )  <  D )
 
5.8  Elementary limits and convergence
 
5.8.1  Superior limit (lim sup)
 
Syntaxclsp 11821 Extend class notation to include the limsup function.
 class  limsup
 
Definitiondf-limsup 11822* Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 11825 for its value. (Contributed by NM, 26-Oct-2005.)
 |-  limsup  =  ( x  e. 
 _V  |->  sup ( ran  (  k  e.  RR  |->  sup (
 ( ( x "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 ) ,  RR* ,  `'  <  ) )
 
Theoremlimsupgord 11823 Ordering property of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  sup ( ( ( F " ( B [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) 
 <_  sup ( ( ( F " ( A [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
 
Theoremlimsupcl 11824 Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  ( F  e.  V  ->  ( limsup `  F )  e.  RR* )
 
Theoremlimsupval 11825* The superior limit of an infinite sequence  F of extended real numbers, which is the infimum (indicated by  `'  <) of the set of suprema of all upper infinite subsequences of  F. Definition 12-4.1 of [Gleason] p. 175. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 5-Sep-2014.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( F  e.  V  ->  ( limsup `  F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
 
Theoremlimsupgf 11826* Closure of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  G : RR --> RR*
 
Theoremlimsupgval 11827* Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( M  e.  RR  ->  ( G `  M )  =  sup ( ( ( F " ( M [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
 
Theoremlimsupgle 11828* The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( ( ( B 
 C_  RR  /\  F : B
 --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( G `  C )  <_  A 
 <-> 
 A. j  e.  B  ( C  <_  j  ->  ( F `  j ) 
 <_  A ) ) )
 
Theoremlimsuple 11829* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  ->  ( A  <_  ( limsup `
  F )  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
 
Theoremlimsuplt 11830* The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   =>    |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  ->  ( ( limsup `  F )  <  A  <->  E. j  e.  RR  ( G `  j )  <  A ) )
 
Theoremlimsupval2 11831* The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 8-May-2016.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  A 
 C_  RR )   &    |-  ( ph  ->  sup ( A ,  RR* ,  <  )  =  +oo )   =>    |-  ( ph  ->  ( limsup `
  F )  = 
 sup ( ( G
 " A ) , 
 RR* ,  `'  <  ) )
 
Theoremlimsupgre 11832* If a sequence of real numbers has upper bounded limit supremum, then all the partial suprema are real. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  G  =  ( k  e.  RR  |->  sup (
 ( ( F "
 ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
 )   &    |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F : Z --> RR  /\  ( limsup `  F )  <  +oo )  ->  G : RR --> RR )
 
Theoremlimsupbnd1 11833* If a sequence is eventually at most 
A, then the limsup is also at most  A. (The converse is only true if the less or equal is replaced by strictly less than; consider the sequence  1  /  n which is never less or equal to zero even though the limsup is.) (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  ( ph  ->  B  C_ 
 RR )   &    |-  ( ph  ->  F : B --> RR* )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  E. k  e.  RR  A. j  e.  B  ( k  <_  j  ->  ( F `  j )  <_  A ) )   =>    |-  ( ph  ->  ( limsup `
  F )  <_  A )
 
Theoremlimsupbnd2 11834* If a sequence is eventually greater than  A, then the limsup is also greater than  A. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
 |-  ( ph  ->  B  C_ 
 RR )   &    |-  ( ph  ->  F : B --> RR* )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  sup ( B ,  RR* ,  <  )  =  +oo )   &    |-  ( ph  ->  E. k  e.  RR  A. j  e.  B  ( k  <_  j  ->  A 
 <_  ( F `  j
 ) ) )   =>    |-  ( ph  ->  A 
 <_  ( limsup `  F )
 )
 
5.8.2  Limits
 
Syntaxcli 11835 Extend class notation with convergence relation for limits.
 class  ~~>
 
Syntaxcrli 11836 Extend class notation with real convergence relation for limits.
 class  ~~> r
 
Syntaxco1 11837 Extend class notation with the set of all eventually bounded functions.
 class  O ( 1 )
 
Syntaxclo1 11838 Extend class notation with the set of all eventually upper bounded functions.
 class  <_ O ( 1 )
 
Definitiondf-clim 11839* Define the limit relation for complex number sequences. See clim 11845 for its relational expression. (Contributed by NM, 28-Aug-2005.)
 |-  ~~>  =  { <. f ,  y >.  |  ( y  e. 
 CC  /\  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ( ( f `  k )  e.  CC  /\  ( abs `  ( ( f `
  k )  -  y ) )  < 
 x ) ) }
 
Definitiondf-rlim 11840* Define the limit relation for partial functions on the reals. See rlim 11846 for its relational expression. (Contributed by Mario Carneiro, 16-Sep-2014.)
 |-  ~~> r  =  { <. f ,  x >.  |  (
 ( f  e.  ( CC  ^pm  RR )  /\  x  e.  CC )  /\  A. y  e.  RR+  E. z  e.  RR  A. w  e.  dom  f ( z  <_  w  ->  ( abs `  ( (
 f `  w )  -  x ) )  < 
 y ) ) }
 
Definitiondf-o1 11841* Define the set of eventually bounded functions. We don't bother to build the full conception of big-O notation, because we can represent any big-O in terms of  O ( 1 ) and division, and any little-O in terms of a limit and division. We could also use limsup for this, but it only works on integer sequences, while this will work for real sequences or integer sequences. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  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 }
 
Definitiondf-lo1 11842* Define the set of eventually upper bounded real functions. This fills a gap in  O ( 1 ) coverage, to express statements like  f (
x )  <_  g
( x )  +  O ( x ) via  ( x  e.  RR+  |->  ( f ( x )  -  g
( x ) )  /  x )  e. 
<_ O ( 1 ). (Contributed by Mario Carneiro, 25-May-2016.)
 |- 
 <_ 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 }
 
Theoremclimrel 11843 The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |- 
 Rel 
 ~~>
 
Theoremrlimrel 11844 The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)
 |- 
 Rel 
 ~~> r
 
Theoremclim 11845* Express the predicate: The limit of complex number sequence  F is  A, or  F converges to  A. This means that for any real  x, no matter how small, there always exists an integer 
j such that the absolute difference of any later complex number in the sequence and the limit is less than  x. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  ZZ )  ->  ( F `  k
 )  =  B )   =>    |-  ( ph  ->  ( F  ~~>  A 
 <->  ( A  e.  CC  /\ 
 A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  < 
 x ) ) ) )
 
Theoremrlim 11846* Express the predicate: The limit of complex number function  F is  C, or  F converges to  C, in the real sense. This means that for any real  x, no matter how small, there always exists a number  y such that the absolute difference of any number in the function beyond  y and the limit is less than  x. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ( ph  /\  z  e.  A )  ->  ( F `  z )  =  B )   =>    |-  ( ph  ->  ( F 
 ~~> r  C  <->  ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR  A. z  e.  A  ( y  <_  z  ->  ( abs `  ( B  -  C ) )  < 
 x ) ) ) )
 
Theoremrlim2 11847* Rewrite rlim 11846 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 11848* 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 11849* 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 11850 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 11851 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 11852 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 11853 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 11854 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 11855* Express the predicate: The limit of complex number sequence  F is  A, or  F converges to  A, with more general quantifier restrictions than clim 11845. (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 11856* 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 11857* 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 11858* 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 11859* 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 11860* 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 11861* 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 11862* 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 11863* 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 11864* 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 11865* 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 11866* 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 11867* 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 11868* 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 11869 An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( F  e.  <_ O ( 1 )  ->  F : dom  F --> RR )
 
Theoremlo1dm 11870 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 11871* 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 11872* 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 11873* 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 11874* 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 11875* 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 11876* Refine o1bdd2 11892 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 11877* 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 11878* 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 11879* 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 11880 An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( F  e.  O ( 1 )  ->  F : dom  F --> CC )
 
Theoremo1dm 11881 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 11882* 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 11883 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 11884* 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 11885* 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 11886* 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 11887* 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 11888* 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 11889* 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 11890* 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 11891* 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 11892* 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 11893* Refine o1bdd2 11892 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 11894* 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 11895* 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 11896 Forward direction of rlimclim 11897. (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 11897 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 11898* 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 11899 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 11900 The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ZZ  X.  {
 0 } )  ~~>  0
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