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Theorem List for Metamath Proof Explorer - 12001-12100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlo1res2 12001* 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 12002* 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 12003 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 12004 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 12005 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 12006* 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 12007* 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 12008* 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 12009* 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 12010* 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 12011* 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 12012* 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 12013 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 12014 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 12015 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 12016 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 12017* 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 12018* 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 12019* 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 12020* 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 12021* 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 12022* 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 12023* 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 12024* 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 12025* 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 12026* 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 12027* 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 12028* 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 12029* 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 12030* 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 12031* 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 12032* 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 16919 and df-cncf 18344 are not yet available to us. See addcn 18331 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 12033* 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 12034* 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 12035* The reciprocal function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) (Revised by Mario Carneiro, 22-Sep-2014.)
 |-  T  =  ( if ( 1  <_  (
 ( abs `  A )  x.  B ) ,  1 ,  ( ( abs `  A )  x.  B ) )  x.  (
 ( abs `  A )  /  2 ) )   =>    |-  ( ( A  e.  ( CC  \  { 0 } )  /\  B  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  ( CC  \  { 0 } ) ( ( abs `  ( z  -  A ) )  <  y  ->  ( abs `  ( (
 1  /  z )  -  ( 1  /  A ) ) )  <  B ) )
 
Theoremcn1lem 12036* A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  F : CC --> CC   &    |-  (
 ( z  e.  CC  /\  A  e.  CC )  ->  ( abs `  (
 ( F `  z
 )  -  ( F `
  A ) ) )  <_  ( abs `  ( z  -  A ) ) )   =>    |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  ( ( abs `  (
 z  -  A ) )  <  y  ->  ( abs `  ( ( F `  z )  -  ( F `  A ) ) )  <  x ) )
 
Theoremabscn2 12037* The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( abs `  z
 )  -  ( abs `  A ) ) )  <  x ) )
 
Theoremcjcn2 12038* The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( * `  z )  -  ( * `  A ) ) )  <  x ) )
 
Theoremrecn2 12039* The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( Re `  z )  -  ( Re `  A ) ) )  <  x ) )
 
Theoremimcn2 12040* The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( Im `  z )  -  ( Im `  A ) ) )  <  x ) )
 
Theoremclimcn1lem 12041* The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  H : CC --> CC   &    |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  (
 ( abs `  ( z  -  A ) )  < 
 y  ->  ( abs `  ( ( H `  z )  -  ( H `  A ) ) )  <  x ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( H `
  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( H `  A ) )
 
Theoremclimabs 12042* Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( abs `  ( F `  k
 ) ) )   =>    |-  ( ph  ->  G  ~~>  ( abs `  A )
 )
 
Theoremclimcj 12043* Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( * `
  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( * `  A ) )
 
Theoremclimre 12044* Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( Re
 `  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( Re `  A ) )
 
Theoremclimim 12045* Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  =  ( Im
 `  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( Im `  A ) )
 
Theoremrlimmptrcl 12046* Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )
 
Theoremrlimabs 12047* Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ph  ->  ( k  e.  A  |->  ( abs `  B ) )  ~~> r  ( abs `  C ) )
 
Theoremrlimcj 12048* Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ph  ->  ( k  e.  A  |->  ( * `  B ) )  ~~> r  ( * `  C ) )
 
Theoremrlimre 12049* Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ph  ->  ( k  e.  A  |->  ( Re `  B ) )  ~~> r  ( Re `  C ) )
 
Theoremrlimim 12050* Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ph  ->  ( k  e.  A  |->  ( Im `  B ) )  ~~> r  ( Im `  C ) )
 
Theoremo1of2 12051* Show that a binary operation preserves eventual boundedness. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( m  e. 
 RR  /\  n  e.  RR )  ->  M  e.  RR )   &    |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x R y )  e. 
 CC )   &    |-  ( ( ( m  e.  RR  /\  n  e.  RR )  /\  ( x  e.  CC  /\  y  e.  CC )
 )  ->  ( (
 ( abs `  x )  <_  m  /\  ( abs `  y )  <_  n )  ->  ( abs `  ( x R y ) ) 
 <_  M ) )   =>    |-  ( ( F  e.  O ( 1 )  /\  G  e.  O ( 1 ) )  ->  ( F  o F R G )  e.  O ( 1 ) )
 
Theoremo1add 12052 The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
 |-  ( ( F  e.  O ( 1 ) 
 /\  G  e.  O ( 1 ) ) 
 ->  ( F  o F  +  G )  e.  O ( 1 ) )
 
Theoremo1mul 12053 The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
 |-  ( ( F  e.  O ( 1 ) 
 /\  G  e.  O ( 1 ) ) 
 ->  ( F  o F  x.  G )  e.  O ( 1 ) )
 
Theoremo1sub 12054 The difference of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
 |-  ( ( F  e.  O ( 1 ) 
 /\  G  e.  O ( 1 ) ) 
 ->  ( F  o F  -  G )  e.  O ( 1 ) )
 
Theoremrlimo1 12055 Any function with a finite limit is eventually bounded. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( F  ~~> r  A  ->  F  e.  O ( 1 ) )
 
Theoremrlimdmo1 12056 A convergent function is eventually bounded. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  ( F  e.  dom  ~~> r 
 ->  F  e.  O ( 1 ) )
 
Theoremo1rlimmul 12057 The product of a eventually bounded function and a function of limit zero has limit zero. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( ( F  e.  O ( 1 ) 
 /\  G  ~~> r  0 )  ->  ( F  o F  x.  G ) 
 ~~> r  0 )
 
Theoremo1const 12058* A constant function is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  ( ( A  C_  RR  /\  B  e.  CC )  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )
 
Theoremlo1const 12059* A constant function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( A  C_  RR  /\  B  e.  RR )  ->  ( x  e.  A  |->  B )  e. 
 <_ O ( 1 ) )
 
Theoremlo1mptrcl 12060* Reverse closure for an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) )   =>    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )
 
Theoremo1mptrcl 12061* Reverse closure for an eventually bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   =>    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  CC )
 
Theoremo1add2 12062* The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e.  O ( 1 ) )
 
Theoremo1mul2 12063* The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  C ) )  e.  O ( 1 ) )
 
Theoremo1sub2 12064* The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e.  O ( 1 ) )
 
Theoremlo1add 12065* The sum of two eventually upper bounded functions is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. 
 <_ O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  <_ O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. 
 <_ O ( 1 ) )
 
Theoremlo1mul 12066* The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. 
 <_ O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  <_ O ( 1 ) )   &    |-  ( ( ph  /\  x  e.  A )  ->  0  <_  B )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  C ) )  e.  <_ O ( 1 ) )
 
Theoremlo1mul2 12067* The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. 
 <_ O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  <_ O ( 1 ) )   &    |-  ( ( ph  /\  x  e.  A )  ->  0  <_  B )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e.  <_ O ( 1 ) )
 
Theoremo1dif 12068* If the difference of two functions is eventually bounded, eventual boundedness of either one implies the other. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  CC )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e.  O ( 1 ) )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  O ( 1 )  <->  ( x  e.  A  |->  C )  e.  O ( 1 ) ) )
 
Theoremlo1sub 12069* The difference of an eventually upper bounded function and an eventually bounded function is eventually upper bounded. The "correct" sharp result here takes the second function to be eventually lower bounded instead of just bounded, but our notation for this is simply  ( x  e.  A  |->  -u C
)  e.  <_ O
( 1 ), so it is just a special case of lo1add 12065. (Contributed by Mario Carneiro, 31-May-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. 
 <_ O ( 1 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e.  <_ O ( 1 ) )
 
Theoremclimadd 12070* Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  +  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  +  B ) )
 
Theoremclimmul 12071* Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  x.  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
 
Theoremclimsub 12072* Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  -  ( G `  k ) ) )   =>    |-  ( ph  ->  H  ~~>  ( A  -  B ) )
 
Theoremclimaddc1 12073* Limit of a constant  C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `  k )  +  C ) )   =>    |-  ( ph  ->  G  ~~>  ( A  +  C ) )
 
Theoremclimaddc2 12074* Limit of a constant  C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  +  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  +  A ) )
 
Theoremclimmulc2 12075* Limit of a sequence multiplied by a constant  C. Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  x.  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  x.  A ) )
 
Theoremclimsubc1 12076* Limit of a constant  C subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( ( F `  k )  -  C ) )   =>    |-  ( ph  ->  G  ~~>  ( A  -  C ) )
 
Theoremclimsubc2 12077* Limit of a constant  C minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  -  ( F `  k ) ) )   =>    |-  ( ph  ->  G  ~~>  ( C  -  A ) )
 
Theoremclimle 12078* Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  ~~>  B )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremclimsqz 12079* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  <_  A )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremclimsqz2 12080* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  <_  ( F `  k ) )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  A  <_  ( G `  k ) )   =>    |-  ( ph  ->  G  ~~>  A )
 
Theoremrlimadd 12081* Limit of the sum of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  ~~> r  ( D  +  E ) )
 
Theoremrlimsub 12082* Limit of the difference of two converging functions. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  ~~> r  ( D  -  E ) )
 
Theoremrlimmul 12083* Limit of the product of two converging functions. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  C ) )  ~~> r  ( D  x.  E ) )
 
Theoremrlimdiv 12084* Limit of the quotient of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )   &    |-  ( ph  ->  E  =/=  0 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  C  =/=  0 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B 
 /  C ) )  ~~> r  ( D  /  E ) )
 
Theoremrlimneg 12085* Limit of the negative of a sequence. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C )   =>    |-  ( ph  ->  ( k  e.  A  |->  -u B )  ~~> r  -u C )
 
Theoremrlimle 12086* Comparison of the limits of two sequences. (Contributed by Mario Carneiro, 10-May-2016.)
 |-  ( ph  ->  sup ( A ,  RR* ,  <  )  =  +oo )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  C  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  <_  C )   =>    |-  ( ph  ->  D  <_  E )
 
Theoremrlimsqzlem 12087* Lemma for rlimsqz 12088 and rlimsqz2 12089. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)
 |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  E  e.  CC )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  ( abs `  ( C  -  E ) )  <_  ( abs `  ( B  -  D ) ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E )
 
Theoremrlimsqz 12088* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)
 |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  B  <_  C )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  <_  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D )
 
Theoremrlimsqz2 12089* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 3-Feb-2014.) (Revised by Mario Carneiro, 20-May-2016.)
 |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  <_  B )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  D  <_  C )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  D )
 
Theoremlo1le 12090* Transfer eventual upper boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  <_ O ( 1 ) )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  C  <_  B )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  <_ O ( 1 ) )
 
Theoremo1le 12091* Transfer eventual boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 25-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  O ( 1 ) )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  M  <_  x ) )  ->  ( abs `  C )  <_  ( abs `  B ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  O ( 1 ) )
 
Theoremrlimno1 12092* A function whose inverse converges to zero is unbounded. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  sup ( A ,  RR* ,  <  )  =  +oo )   &    |-  ( ph  ->  ( x  e.  A  |->  ( 1  /  B ) )  ~~> r  0 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  CC )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  =/=  0 )   =>    |-  ( ph  ->  -.  ( x  e.  A  |->  B )  e.  O ( 1 ) )
 
Theoremclim2ser 12093* The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ph  ->  seq 
 M (  +  ,  F )  ~~>  A )   =>    |-  ( ph  ->  seq  ( N  +  1 ) (  +  ,  F )  ~~>  ( A  -  (  seq  M (  +  ,  F ) `  N ) ) )
 
Theoremclim2ser2 12094* The limit of an infinite series with an initial segment added. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ph  ->  seq  ( N  +  1 ) (  +  ,  F )  ~~>  A )   =>    |-  ( ph  ->  seq 
 M (  +  ,  F )  ~~>  ( A  +  (  seq  M (  +  ,  F ) `  N ) ) )
 
Theoremiserex 12095* An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  (  seq  M (  +  ,  F )  e.  dom  ~~>  <->  seq  N (  +  ,  F )  e.  dom  ~~>  ) )
 
Theoremisermulc2 12096* Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  seq  M (  +  ,  F ) 
 ~~>  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  x.  ( F `  k ) ) )   =>    |-  ( ph  ->  seq  M (  +  ,  G ) 
 ~~>  ( C  x.  A ) )
 
Theoremclimlec2 12097* Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  F  ~~>  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  A  <_  ( F `  k
 ) )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremiserle 12098* Comparison of the limits of two infinite series. (Contributed by Paul Chapman, 12-Nov-2007.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  A )   &    |-  ( ph  ->  seq 
 M (  +  ,  G )  ~~>  B )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k ) )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremiserge0 12099* The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  0  <_  ( F `  k
 ) )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremclimub 12100* The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   =>    |-  ( ph  ->  ( F `  N ) 
 <_  A )
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