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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrlimim 12101* 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 12102* 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 12103 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 12104 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 12105 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 12106 Any function with a finite limit is eventually bounded. (Contributed by Mario Carneiro, 18-Sep-2014.)
 |-  ( F  ~~> r  A  ->  F  e.  O ( 1 ) )
 
Theoremrlimdmo1 12107 A convergent function is eventually bounded. (Contributed by Mario Carneiro, 12-May-2016.)
 |-  ( F  e.  dom  ~~> r 
 ->  F  e.  O ( 1 ) )
 
Theoremo1rlimmul 12108 The product of an 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 12109* 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 12110* 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 12111* 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 12112* 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 12113* 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 12114* 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 12115* 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 12116* 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 12117* 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 12118* 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 12119* 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 12120* 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 12116. (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 12121* 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 12122* 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 12123* 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 12124* 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 12125* 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 12126* 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 12127* 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 12128* 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 12129* 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 12130* 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 12131* 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 12132* 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 12133* 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 12134* 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 12135* 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 12136* 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 12137* 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 12138* Lemma for rlimsqz 12139 and rlimsqz2 12140. (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 12139* 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 12140* 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 12141* 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 12142* 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 12143* 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 12144* 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 12145* 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 12146* 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 12147* 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 12148* 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 12149* 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 12150* 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 12151* 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 )
 
Theoremclimserle 12152* The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  0  <_  ( F `  k
 ) )   =>    |-  ( ph  ->  (  seq  M (  +  ,  F ) `  N )  <_  A )
 
Theoremisershft 12153 Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  (  seq  M (  .+  ,  F )  ~~>  A  <->  seq  ( M  +  N ) (  .+  ,  ( F  shift  N ) )  ~~>  A ) )
 
Theoremisercolllem1 12154* Lemma for isercoll 12157. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : NN --> Z )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( G `  k
 )  <  ( G `  ( k  +  1 ) ) )   =>    |-  ( ( ph  /\  S  C_  NN )  ->  ( G  |`  S ) 
 Isom  <  ,  <  ( S ,  ( G " S ) ) )
 
Theoremisercolllem2 12155* Lemma for isercoll 12157. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : NN --> Z )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( G `  k
 )  <  ( G `  ( k  +  1 ) ) )   =>    |-  ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) )  ->  (
 1 ... ( # `  ( G " ( `' G " ( M ... N ) ) ) ) )  =  ( `' G " ( M
 ... N ) ) )
 
Theoremisercolllem3 12156* Lemma for isercoll 12157. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : NN --> Z )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( G `  k
 )  <  ( G `  ( k  +  1 ) ) )   &    |-  (
 ( ph  /\  n  e.  ( Z  \  ran  G ) )  ->  ( F `  n )  =  0 )   &    |-  ( ( ph  /\  n  e.  Z ) 
 ->  ( F `  n )  e.  CC )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( H `  k )  =  ( F `  ( G `  k ) ) )   =>    |-  ( ( ph  /\  N  e.  ( ZZ>= `  ( G `  1 ) ) ) 
 ->  (  seq  M (  +  ,  F ) `
  N )  =  (  seq  1 (  +  ,  H ) `
  ( # `  ( G " ( `' G " ( M ... N ) ) ) ) ) )
 
Theoremisercoll 12157* Rearrange an infinite series by spacing out the terms using an order isomorphism. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : NN --> Z )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( G `  k
 )  <  ( G `  ( k  +  1 ) ) )   &    |-  (
 ( ph  /\  n  e.  ( Z  \  ran  G ) )  ->  ( F `  n )  =  0 )   &    |-  ( ( ph  /\  n  e.  Z ) 
 ->  ( F `  n )  e.  CC )   &    |-  (
 ( ph  /\  k  e. 
 NN )  ->  ( H `  k )  =  ( F `  ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq  1 (  +  ,  H )  ~~>  A  <->  seq  M (  +  ,  F )  ~~>  A )
 )
 
Theoremisercoll2 12158* Generalize isercoll 12157 so that both sequences have arbitrary starting point. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  (
 ZZ>= `  N )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  G : Z --> W )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( G `  k )  < 
 ( G `  (
 k  +  1 ) ) )   &    |-  ( ( ph  /\  n  e.  ( W 
 \  ran  G )
 )  ->  ( F `  n )  =  0 )   &    |-  ( ( ph  /\  n  e.  W ) 
 ->  ( F `  n )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( F `  ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq  M (  +  ,  H )  ~~>  A  <->  seq  N (  +  ,  F )  ~~>  A )
 )
 
Theoremclimsup 12159* A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   &    |-  ( ph  ->  E. x  e.  RR  A. k  e.  Z  ( F `  k ) 
 <_  x )   =>    |-  ( ph  ->  F  ~~>  sup ( ran  F ,  RR ,  <  ) )
 
Theoremclimcau 12160* A converging sequence of complex numbers is a Cauchy sequence. Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  dom  ~~>  ) 
 ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x )
 
Theoremcaucvgrlem 12161* Lemma for caurcvgr 12162. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  sup ( A ,  RR*
 ,  <  )  =  +oo )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x ) )   &    |-  ( ph  ->  R  e.  RR+ )   =>    |-  ( ph  ->  E. j  e.  A  ( ( limsup `  F )  e.  RR  /\ 
 A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F )
 ) )  <  (
 3  x.  R ) ) ) )
 
Theoremcaurcvgr 12162* A Cauchy sequence of real numbers converges to its limit supremum. The third hypothesis specifies that  F is a Cauchy sequence. (Contributed by Mario Carneiro, 7-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  sup ( A ,  RR*
 ,  <  )  =  +oo )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x ) )   =>    |-  ( ph  ->  F  ~~> r  ( limsup `  F )
 )
 
Theoremcaucvgrlem2 12163* Lemma for caucvgr 12164. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Mario Carneiro, 8-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  sup ( A ,  RR*
 ,  <  )  =  +oo )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x ) )   &    |-  H : CC --> RR   &    |-  ( ( ( F `
  k )  e. 
 CC  /\  ( F `  j )  e.  CC )  ->  ( abs `  (
 ( H `  ( F `  k ) )  -  ( H `  ( F `  j ) ) ) )  <_  ( abs `  ( ( F `  k )  -  ( F `  j ) ) ) )   =>    |-  ( ph  ->  ( n  e.  A  |->  ( H `  ( F `
  n ) ) )  ~~> r  (  ~~> r  `  ( H  o.  F ) ) )
 
Theoremcaucvgr 12164* A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 8-May-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  sup ( A ,  RR*
 ,  <  )  =  +oo )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x ) )   =>    |-  ( ph  ->  F  e.  dom  ~~> r  )
 
Theoremcaurcvg 12165* A Cauchy sequence of real numbers converges to its limit supremum. The fourth hypothesis specifies that  F is a Cauchy sequence. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 8-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F : Z --> RR )   &    |-  ( ph  ->  A. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x )   =>    |-  ( ph  ->  F  ~~>  ( limsup `  F ) )
 
Theoremcaurcvg2 12166* A Cauchy sequence of real numbers converges, existence version. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 7-Sep-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  RR  /\  ( abs `  ( ( F `
  k )  -  ( F `  j ) ) )  <  x ) )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremcaucvg 12167* A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Proof shortened by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  ( ph  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x )   &    |-  ( ph  ->  F  e.  V )   =>    |-  ( ph  ->  F  e.  dom  ~~>  )
 
Theoremcaucvgb 12168* A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Mario Carneiro, 15-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( ( M  e.  ZZ  /\  F  e.  V )  ->  ( F  e.  dom  ~~>  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
 ( ( F `  k )  e.  CC  /\  ( abs `  (
 ( F `  k
 )  -  ( F `
  j ) ) )  <  x ) ) )
 
Theoremserf0 12169* If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  F  ~~>  0 )
 
Theoremiseraltlem1 12170* Lemma for iseralt 12173. A decreasing sequence with limit zero consists of positive terms. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : Z --> RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  (
 k  +  1 ) )  <_  ( G `  k ) )   &    |-  ( ph  ->  G  ~~>  0 )   =>    |-  (
 ( ph  /\  N  e.  Z )  ->  0  <_  ( G `  N ) )
 
Theoremiseraltlem2 12171* Lemma for iseralt 12173. The terms of an alternating series form a chain of inequalities in alternate terms, so that for example  S ( 1 )  <_  S (
3 )  <_  S
( 5 )  <_  ... and  ...  <_  S
( 4 )  <_  S ( 2 )  <_  S ( 0 ) (assuming  M  =  0 so that these terms are defined). (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : Z --> RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  (
 k  +  1 ) )  <_  ( G `  k ) )   &    |-  ( ph  ->  G  ~~>  0 )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( ( -u 1 ^ k )  x.  ( G `  k
 ) ) )   =>    |-  ( ( ph  /\  N  e.  Z  /\  K  e.  NN0 )  ->  ( ( -u 1 ^ N )  x.  (  seq  M (  +  ,  F ) `  ( N  +  ( 2  x.  K ) ) ) )  <_  ( ( -u 1 ^ N )  x.  (  seq  M (  +  ,  F ) `  N ) ) )
 
Theoremiseraltlem3 12172* Lemma for iseralt 12173. From iseraltlem2 12171, we have  ( -u 1 ^ n )  x.  S ( n  + 
2 k )  <_ 
( -u 1 ^ n
)  x.  S ( n ) and  ( -u 1 ^ n )  x.  S ( n  + 
1 )  <_  ( -u 1 ^ n )  x.  S ( n  +  2 k  +  1 ), and we also have  ( -u 1 ^ n )  x.  S
( n  +  1 )  =  ( -u 1 ^ n )  x.  S ( n )  -  G ( n  +  1 ) for each  n by the definition of the partial sum  S, so combining the inequalities we get  ( -u 1 ^ n )  x.  S ( n )  -  G ( n  +  1 )  =  ( -u 1 ^ n )  x.  S ( n  + 
1 )  <_  ( -u 1 ^ n )  x.  S ( n  + 
2 k  +  1 )  =  ( -u 1 ^ n )  x.  S ( n  + 
2 k )  -  G ( n  + 
2 k  +  1 )  <_  ( -u 1 ^ n )  x.  S ( n  + 
2 k )  <_ 
( -u 1 ^ n
)  x.  S ( n )  <_  ( -u 1 ^ n )  x.  S ( n )  +  G ( n  +  1 ), so  |  ( -u
1 ^ n )  x.  S ( n  +  2 k  +  1 )  -  ( -u 1 ^ n )  x.  S ( n )  |  =  |  S ( n  +  2 k  +  1 )  -  S ( n )  |  <_  G (
n  +  1 ) and  |  ( -u
1 ^ n )  x.  S ( n  +  2 k )  -  ( -u 1 ^ n )  x.  S ( n )  |  =  |  S ( n  +  2 k )  -  S ( n )  |  <_  G ( n  +  1 ). Thus, both even and odd partial sums are Cauchy if  G converges to  0. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : Z --> RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  (
 k  +  1 ) )  <_  ( G `  k ) )   &    |-  ( ph  ->  G  ~~>  0 )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( ( -u 1 ^ k )  x.  ( G `  k
 ) ) )   =>    |-  ( ( ph  /\  N  e.  Z  /\  K  e.  NN0 )  ->  ( ( abs `  (
 (  seq  M (  +  ,  F ) `  ( N  +  (
 2  x.  K ) ) )  -  (  seq  M (  +  ,  F ) `  N ) ) )  <_  ( G `  ( N  +  1 ) ) 
 /\  ( abs `  (
 (  seq  M (  +  ,  F ) `  ( ( N  +  ( 2  x.  K ) )  +  1
 ) )  -  (  seq  M (  +  ,  F ) `  N ) ) )  <_  ( G `  ( N  +  1 ) ) ) )
 
Theoremiseralt 12173* The alternating series test. If  G ( k ) is a decreasing sequence that converges to  0, then  sum_ k  e.  Z
( -u 1 ^ k
)  x.  G ( k ) is a convergent series. (Note that the first term is positive if  M is even, and negative if  M is odd. If the parity of your series does not match up with this, you will need to post-compose the series with multiplication by 
-u 1 using isermulc2 12147.) (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  G : Z --> RR )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  (
 k  +  1 ) )  <_  ( G `  k ) )   &    |-  ( ph  ->  G  ~~>  0 )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( ( -u 1 ^ k )  x.  ( G `  k
 ) ) )   =>    |-  ( ph  ->  seq 
 M (  +  ,  F )  e.  dom  ~~>  )
 
5.8.3  Finite and infinite sums
 
Syntaxcsu 12174 Extend class notation to include finite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is a commonly used word in comments.)
 class  sum_ k  e.  A  B
 
Definitiondf-sum 12175* Define the sum of a series with an index set of integers  A.  k is normally a free variable in  B, i.e.  B can be thought of as  B ( k ). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. These two methods of summation produce the same result on their common region of definition (i.e. finite subsets of the upper integers) by summo 12206. Examples:  sum_ k  e. 
{ 1 ,  2 ,  4 }  k means  1  +  2  + 
4  =  7, and  sum_ k  e.  NN  ( 1  / 
( 2 ^ k
) )  =  1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 12354). (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |- 
 sum_ k  e.  A  B  =  ( iota x ( E. m  e. 
 ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m )
 -1-1-onto-> A  /\  x  =  ( 
 seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) ) ) )
 
Theoremsumex 12176 A sum is a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |- 
 sum_ k  e.  A  B  e.  _V
 
Theoremsumeq1f 12177 Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  F/_ k A   &    |-  F/_ k B   =>    |-  ( A  =  B  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremsumeq1 12178* Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( A  =  B  -> 
 sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremnfsum1 12179* Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  F/_ k A   =>    |-  F/_ k sum_ k  e.  A  B
 
Theoremnfsum 12180* Bound-variable hypothesis builder for sum: if  x is (effectively) not free in  A and  B, it is not free in  sum_ k  e.  A B. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x sum_ k  e.  A  B
 
Theoremsumeq2w 12181* Equality theorem for sum, when the class expressions  B and  C are equal everywhere. Proved using only Extensionality. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( A. k  B  =  C  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2ii 12182* Equality theorem for sum, with the class expressions  B and  C guarded by  _I to be always sets. (Contributed by Mario Carneiro, 29-Mar-2014.)
 |-  ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2 12183* Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( A. k  e.  A  B  =  C  -> 
 sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremcbvsum 12184* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( j  =  k 
 ->  B  =  C )   &    |-  F/_ k A   &    |-  F/_ j A   &    |-  F/_ k B   &    |-  F/_ j C   =>    |- 
 sum_ j  e.  A  B  =  sum_ k  e.  A  C
 
Theoremcbvsumv 12185* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( j  =  k 
 ->  B  =  C )   =>    |-  sum_
 j  e.  A  B  =  sum_ k  e.  A  C
 
Theoremcbvsumi 12186* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
 |-  F/_ k B   &    |-  F/_ j C   &    |-  (
 j  =  k  ->  B  =  C )   =>    |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
 
Theoremsumeq1i 12187* Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
 |-  A  =  B   =>    |-  sum_ k  e.  A  C  =  sum_ k  e.  B  C
 
Theoremsumeq2i 12188* Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
 |-  ( k  e.  A  ->  B  =  C )   =>    |-  sum_
 k  e.  A  B  =  sum_ k  e.  A  C
 
Theoremsumeq12i 12189* Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
 |-  A  =  B   &    |-  (
 k  e.  A  ->  C  =  D )   =>    |-  sum_ k  e.  A  C  =  sum_ k  e.  B  D
 
Theoremsumeq1d 12190* Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  C )
 
Theoremsumeq2d 12191* Equality deduction for sum. Note that unlike sumeq2dv 12192, 
k may occur in  ph. (Contributed by NM, 1-Nov-2005.)
 |-  ( ph  ->  A. k  e.  A  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2dv 12192* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( ph  /\  k  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremsumeq2sdv 12193* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theorem2sumeq2dv 12194* Equality deduction for double sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
 |-  ( ( ph  /\  j  e.  A  /\  k  e.  B )  ->  C  =  D )   =>    |-  ( ph  ->  sum_ j  e.  A  sum_ k  e.  B  C  =  sum_ j  e.  A  sum_ k  e.  B  D )
 
Theoremsumeq12dv 12195* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  C  =  D )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  D )
 
Theoremsumeq12rdv 12196* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ( ph  /\  k  e.  B ) 
 ->  C  =  D )   =>    |-  ( ph  ->  sum_ k  e.  A  C  =  sum_ k  e.  B  D )
 
Theoremsum2id 12197* The second class argument to a sum can be chosen so that it is always a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |- 
 sum_ k  e.  A  B  =  sum_ k  e.  A  (  _I  `  B )
 
Theoremsumfc 12198* A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
 |- 
 sum_ j  e.  A  ( ( k  e.  A  |->  B ) `  j )  =  sum_ k  e.  A  B
 
Theoremfz1f1o 12199* A lemma for working with finite sums. (Contributed by Mario Carneiro, 22-Apr-2014.)
 |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
 ( # `  A )  e.  NN  /\  E. f  f : ( 1
 ... ( # `  A ) ) -1-1-onto-> A ) ) )
 
Theoremsumrblem 12200* Lemma for sumrb 12202. (Contributed by Mario Carneiro, 12-Aug-2013.)
 |-  F  =  ( k  e.  ZZ  |->  if (
 k  e.  A ,  B ,  0 )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   =>    |-  ( ( ph  /\  A  C_  ( ZZ>= `  N )
 )  ->  (  seq  M (  +  ,  F )  |`  ( ZZ>= `  N ) )  =  seq  N (  +  ,  F ) )
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