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Theorem List for Metamath Proof Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuzrdgsuci 10901* Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 10897. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   &    |-  A  e.  _V   &    |-  R  =  ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om )   &    |-  S  =  ran  R   =>    |-  ( B  e.  ( ZZ>=
 `  C )  ->  ( S `  ( B  +  1 ) )  =  ( B F ( S `  B ) ) )
 
Theoremltweuz 10902  < is a well-founded relation on any sequence of upper integers. (Contributed by Andrew Salmon, 13-Nov-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |- 
 <  We  ( ZZ>= `  A )
 
Theoremltwenn 10903 Less than well orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
 |- 
 <  We  NN
 
Theoremltwefz 10904 Less than well orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |- 
 <  We  ( M ... N )
 
Theoremuzenom 10905 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  Z  ~~  om )
 
Theoremuzinf 10906 An upper integer set is infinite. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  -.  Z  e.  Fin )
 
Theoremuzrdgxfr 10907* Transfer the value of the recursive sequence builder from one base to another. (Contributed by Mario Carneiro, 1-Apr-2014.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  A )  |`  om )   &    |-  H  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  B )  |`  om )   &    |-  A  e.  ZZ   &    |-  B  e.  ZZ   =>    |-  ( N  e.  om  ->  ( G `  N )  =  ( ( H `  N )  +  ( A  -  B ) ) )
 
Theoremfzennn 10908 The cardinality of a finite set of sequential integers. (See om2uz0i 10888 for a description of the hypothesis.) (Contributed by Mario Carneiro, 12-Feb-2013.) (Revised by Mario Carneiro, 7-Mar-2014.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   =>    |-  ( N  e.  NN0  ->  (
 1 ... N )  ~~  ( `' G `  N ) )
 
Theoremfzen2 10909 The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Mario Carneiro, 13-Feb-2014.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   =>    |-  ( N  e.  ( ZZ>= `  M )  ->  ( M
 ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
 
Theoremcardfz 10910 The cardinality of a finite set of sequential integers. (See om2uz0i 10888 for a description of the hypothesis.) (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   =>    |-  ( N  e.  NN0  ->  ( card `  ( 1 ...
 N ) )  =  ( `' G `  N ) )
 
Theoremhashgf1o 10911  G maps  om one-to-one onto  NN0. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   =>    |-  G : om
 -1-1-onto-> NN0
 
Theoremfzfi 10912 A finite interval of integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
 |-  ( M ... N )  e.  Fin
 
Theoremfzfid 10913 Commonly used special case of fzfi 10912. (Contributed by Mario Carneiro, 25-May-2014.)
 |-  ( ph  ->  ( M ... N )  e. 
 Fin )
 
Theoremfzofi 10914 Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( M..^ N )  e.  Fin
 
Theoremfsequb 10915* The values of a finite real sequence have an upper bound. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( A. k  e.  ( M ... N ) ( F `  k )  e.  RR  ->  E. x  e.  RR  A. k  e.  ( M
 ... N ) ( F `  k )  <  x )
 
Theoremfsequb2 10916* The values of a finite real sequence have an upper bound. (Contributed by NM, 20-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( F : ( M ... N ) --> RR  ->  E. x  e.  RR  A. y  e. 
 ran  F  y  <_  x )
 
Theoremfseqsupcl 10917 The values of a finite real sequence have a supremum. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M
 ... N ) --> RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
 
Theoremfseqsupubi 10918 The values of a finite real sequence are bounded by their supremum. (Contributed by NM, 20-Sep-2005.)
 |-  ( ( K  e.  ( M ... N ) 
 /\  F : ( M ... N ) --> RR )  ->  ( F `  K )  <_  sup ( ran  F ,  RR ,  <  ) )
 
Theoremnn0ennn 10919 The nonnegative integers are equinumerous to the natural numbers. (Contributed by NM, 19-Jul-2004.)
 |- 
 NN0  ~~  NN
 
Theoremnnenom 10920 The set of natural numbers (as a subset of complex numbers) is equinumerous to omega (the set of finite ordinal numbers). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |- 
 NN  ~~  om
 
Theoremuzindi 10921* Indirect strong induction on the upper integers. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  T  e.  ( ZZ>= `  L ) )   &    |-  ( ( ph  /\  R  e.  ( L
 ... T )  /\  A. y ( S  e.  ( L..^ R )  ->  ch ) )  ->  ps )   &    |-  ( x  =  y  ->  ( ps  <->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( x  =  y  ->  R  =  S )   &    |-  ( x  =  A  ->  R  =  T )   =>    |-  ( ph  ->  th )
 
Theoremaxdc4uzlem 10922* Lemma for axdc4uz 10923. (Contributed by Mario Carneiro, 8-Jan-2014.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  M  e.  ZZ   &    |-  Z  =  ( ZZ>= `  M )   &    |-  A  e.  _V   &    |-  G  =  ( rec ( ( y  e.  _V  |->  ( y  +  1 ) ) ,  M )  |`  om )   &    |-  H  =  ( n  e.  om ,  x  e.  A  |->  ( ( G `  n ) F x ) )   =>    |-  ( ( C  e.  A  /\  F : ( Z  X.  A ) --> ( ~P A  \  { (/) } ) ) 
 ->  E. g ( g : Z --> A  /\  ( g `  M )  =  C  /\  A. k  e.  Z  ( g `  ( k  +  1 ) )  e.  ( k F ( g `  k
 ) ) ) )
 
Theoremaxdc4uz 10923* A version of axdc4 7966 that works on a set of upper integers instead of  om. (Contributed by Mario Carneiro, 8-Jan-2014.)
 |-  M  e.  ZZ   &    |-  Z  =  ( ZZ>= `  M )   =>    |-  (
 ( A  e.  V  /\  C  e.  A  /\  F : ( Z  X.  A ) --> ( ~P A  \  { (/) } )
 )  ->  E. g
 ( g : Z --> A  /\  ( g `  M )  =  C  /\  A. k  e.  Z  ( g `  (
 k  +  1 ) )  e.  ( k F ( g `  k ) ) ) )
 
Syntaxcseq 10924 Extend class notation with recursive sequence builder.
 class  seq  M (  .+  ,  F )
 
Definitiondf-seq 10925* Define a general-purpose operation that builds an recursive sequence (i.e. a function on the natural numbers  NN or some other upper integer set) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by seq1 10937 and seqp1 10939. Typically, those are the main theorems that would be used in practice.

The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation  +, an input sequence  F with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence  seq  1 (  +  ,  F ) with values 1, 3/2, 7/4, 15/8,.., so that  (  seq  1
(  +  ,  F
) `  1 )  =  1,  (  seq  1 (  +  ,  F ) `  2
)  = 3/2, etc. In other words, 
seq  M (  +  ,  F ) transforms a sequence  F into an infinite series.  seq  M (  +  ,  F )  ~~>  2 means "the sum of F(n) from n = M to infinity is 2." Since limits are unique (climuni 11903), by climdm 11905 the "sum of F(n) from n = 1 to infinity" can be expressed as  (  ~~>  `  seq  1
(  +  ,  F
) ) (provided the sequence converges) and evaluates to 2 in this example.

Internally, the  rec function generates as its values a set of ordered pairs starting at 
<. M ,  ( F `
 M ) >., with the first member of each pair incremented by one in each successive value. So, the range of  rec is exactly the sequence we want, and we just extract the range (restricted to omega) and throw away the domain.

This definition has its roots in a series of theorems from om2uz0i 10888 through om2uzf1oi 10894, originally proved by Raph Levien for use with df-exp 10983 and later generalized for arbitrary recursive sequences. Definition df-sum 12036 extracts the summation values from partial (finite) and complete (infinite) series. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 4-Sep-2013.)

 |- 
 seq  M (  .+  ,  F )  =  ( rec ( ( x  e. 
 _V ,  y  e. 
 _V  |->  <. ( x  +  1 ) ,  (
 y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )
 " om )
 
Theoremseqex 10926 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |- 
 seq  M (  .+  ,  F )  e.  _V
 
Theoremseqeq1 10927 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  ( M  =  N  ->  seq  M (  .+  ,  F )  =  seq  N (  .+  ,  F ) )
 
Theoremseqeq2 10928 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  (  .+  =  Q  ->  seq  M (  .+  ,  F )  =  seq  M ( Q ,  F ) )
 
Theoremseqeq3 10929 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  ( F  =  G  ->  seq  M (  .+  ,  F )  =  seq  M (  .+  ,  G ) )
 
Theoremseqeq1d 10930 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  seq  A (  .+  ,  F )  =  seq  B ( 
 .+  ,  F )
 )
 
Theoremseqeq2d 10931 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  seq  M ( A ,  F )  =  seq  M ( B ,  F ) )
 
Theoremseqeq3d 10932 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  seq  M (  .+  ,  A )  =  seq  M ( 
 .+  ,  B )
 )
 
Theoremseqeq123d 10933 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
 |-  ( ph  ->  M  =  N )   &    |-  ( ph  ->  .+  =  Q )   &    |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  seq  M (  .+  ,  F )  =  seq  N ( Q ,  G ) )
 
Theoremnfseq 10934 Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x M   &    |-  F/_ x  .+   &    |-  F/_ x F   =>    |-  F/_ x  seq  M (  .+  ,  F )
 
Theoremseqval 10935* Value of the sequence builder function. (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  R  =  ( rec ( ( x  e. 
 _V ,  y  e. 
 _V  |->  <. ( x  +  1 ) ,  ( x ( z  e. 
 _V ,  w  e. 
 _V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )
 >. ) ,  <. M ,  ( F `  M )
 >. )  |`  om )   =>    |-  seq  M (  .+  ,  F )  =  ran  R
 
Theoremseqfn 10936 The sequence builder function is a function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( M  e.  ZZ  ->  seq  M (  .+  ,  F )  Fn  ( ZZ>=
 `  M ) )
 
Theoremseq1 10937 Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( M  e.  ZZ  ->  (  seq  M ( 
 .+  ,  F ) `  M )  =  ( F `  M ) )
 
Theoremseq1i 10938 Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 30-Apr-2014.)
 |-  M  e.  ZZ   &    |-  ( ph  ->  ( F `  M )  =  A )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  M )  =  A )
 
Theoremseqp1 10939 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  (  seq  M (  .+  ,  F ) `  ( N  +  1 )
 )  =  ( ( 
 seq  M (  .+  ,  F ) `  N )  .+  ( F `  ( N  +  1
 ) ) ) )
 
Theoremseqp1i 10940 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 30-Apr-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  N  e.  Z   &    |-  K  =  ( N  +  1 )   &    |-  ( ph  ->  ( 
 seq  M (  .+  ,  F ) `  N )  =  A )   &    |-  ( ph  ->  ( F `  K )  =  B )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  K )  =  ( A  .+  B ) )
 
Theoremseqm1 10941 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>=
 `  ( M  +  1 ) ) ) 
 ->  (  seq  M ( 
 .+  ,  F ) `  N )  =  ( (  seq  M ( 
 .+  ,  F ) `  ( N  -  1
 ) )  .+  ( F `  N ) ) )
 
Theoremseqcl2 10942* Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  ( F `  M )  e.  C )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D ) )  ->  ( x 
 .+  y )  e.  C )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  x  e.  ( ( M  +  1 )
 ... N ) ) 
 ->  ( F `  x )  e.  D )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `
  N )  e.  C )
 
Theoremseqf2 10943* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  ( F `  M )  e.  C )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D ) )  ->  ( x 
 .+  y )  e.  C )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  x )  e.  D )   =>    |-  ( ph  ->  seq  M (  .+  ,  F ) : Z --> C )
 
Theoremseqcl 10944* Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  x  e.  ( M ... N ) )  ->  ( F `
  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x 
 .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  N )  e.  S )
 
Theoremseqf 10945* Range of the recursive sequence builder (special case of seqf2 10943). (Contributed by Mario Carneiro, 24-Jun-2013.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  x  e.  Z )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x 
 .+  y )  e.  S )   =>    |-  ( ph  ->  seq  M (  .+  ,  F ) : Z --> S )
 
Theoremseqfveq2 10946* Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  K )  =  ( G `  K ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  K ) )   &    |-  ( ( ph  /\  k  e.  ( ( K  +  1 ) ... N ) )  ->  ( F `
  k )  =  ( G `  k
 ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  N )  =  (  seq  K (  .+  ,  G ) `  N ) )
 
Theoremseqfeq2 10947* Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  K )  =  ( G `  K ) )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F )  |`  ( ZZ>= `  K ) )  = 
 seq  K (  .+  ,  G ) )
 
Theoremseqfveq 10948* Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  =  ( G `  k
 ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  N )  =  (  seq  M (  .+  ,  G ) `  N ) )
 
Theoremseqfeq 10949* Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  ->  ( F `  k )  =  ( G `  k ) )   =>    |-  ( ph  ->  seq 
 M (  .+  ,  F )  =  seq  M (  .+  ,  G ) )
 
Theoremseqshft2 10950* Shifting the index set of a sequence. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( F `  k
 )  =  ( G `
  ( k  +  K ) ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  N )  =  (  seq  ( M  +  K ) ( 
 .+  ,  G ) `  ( N  +  K ) ) )
 
Theoremseqres 10951 Restricting its characteristic function to  ( ZZ>= `  M ) does not affect the  seq function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( M  e.  ZZ  ->  seq  M (  .+  ,  ( F  |`  ( ZZ>= `  M ) ) )  =  seq  M ( 
 .+  ,  F )
 )
 
Theoremserf 10952* An infinite series of complex terms is a function from  NN to  CC. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
 
Theoremserfre 10953* An infinite series of real numbers is a function from  NN to  RR. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 RR )   =>    |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> RR )
 
Theoremmonoord 10954* Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( N  -  1 ) ) ) 
 ->  ( F `  k
 )  <_  ( F `  ( k  +  1 ) ) )   =>    |-  ( ph  ->  ( F `  M ) 
 <_  ( F `  N ) )
 
Theoremmonoord2 10955* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( N  -  1 ) ) ) 
 ->  ( F `  (
 k  +  1 ) )  <_  ( F `  k ) )   =>    |-  ( ph  ->  ( F `  N ) 
 <_  ( F `  M ) )
 
Theoremsermono 10956* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-Jun-2013.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  K )
 )   &    |-  ( ( ph  /\  x  e.  ( M ... N ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ( ph  /\  x  e.  ( ( K  +  1 )
 ... N ) ) 
 ->  0  <_  ( F `
  x ) )   =>    |-  ( ph  ->  (  seq  M (  +  ,  F ) `  K )  <_  (  seq  M (  +  ,  F ) `  N ) )
 
Theoremseqsplit 10957* Split a sequence into two sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x  .+  y )  .+  z )  =  ( x  .+  ( y  .+  z ) ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  ( M  +  1 ) ) )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  K ) )   &    |-  ( ( ph  /\  x  e.  ( K
 ... N ) ) 
 ->  ( F `  x )  e.  S )   =>    |-  ( ph  ->  (  seq  K (  .+  ,  F ) `
  N )  =  ( (  seq  K (  .+  ,  F ) `
  M )  .+  (  seq  ( M  +  1 ) (  .+  ,  F ) `  N ) ) )
 
Theoremseq1p 10958* Removing the first term from a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x  .+  y )  .+  z )  =  ( x  .+  ( y  .+  z ) ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  ( M  +  1 ) ) )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  x  e.  ( M ... N ) )  ->  ( F `
  x )  e.  S )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  N )  =  ( ( F `  M )  .+  (  seq  ( M  +  1 ) (  .+  ,  F ) `  N ) ) )
 
Theoremseqcaopr3 10959* Lemma for seqcaopr2 10960. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x Q y )  e.  S )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( F `  k
 )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( G `
  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( H `  k
 )  =  ( ( F `  k ) Q ( G `  k ) ) )   &    |-  ( ( ph  /\  n  e.  ( M..^ N ) )  ->  ( (
 (  seq  M (  .+  ,  F ) `  n ) Q ( 
 seq  M (  .+  ,  G ) `  n ) )  .+  ( ( F `  ( n  +  1 ) ) Q ( G `  ( n  +  1
 ) ) ) )  =  ( ( ( 
 seq  M (  .+  ,  F ) `  n )  .+  ( F `  ( n  +  1
 ) ) ) Q ( (  seq  M (  .+  ,  G ) `
  n )  .+  ( G `  ( n  +  1 ) ) ) ) )   =>    |-  ( ph  ->  ( 
 seq  M (  .+  ,  H ) `  N )  =  ( (  seq  M (  .+  ,  F ) `  N ) Q (  seq  M (  .+  ,  G ) `
  N ) ) )
 
Theoremseqcaopr2 10960* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x Q y )  e.  S )   &    |-  ( ( ph  /\  ( ( x  e.  S  /\  y  e.  S )  /\  (
 z  e.  S  /\  w  e.  S )
 ) )  ->  (
 ( x Q z )  .+  ( y Q w ) )  =  ( ( x 
 .+  y ) Q ( z  .+  w ) ) )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( G `  k
 )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( H `
  k )  =  ( ( F `  k ) Q ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  H ) `  N )  =  ( (  seq  M (  .+  ,  F ) `  N ) Q (  seq  M (  .+  ,  G ) `
  N ) ) )
 
Theoremseqcaopr 10961* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-May-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  =  ( y  .+  x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( F `  k
 )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( G `
  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( H `  k
 )  =  ( ( F `  k ) 
 .+  ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  H ) `  N )  =  ( (  seq  M (  .+  ,  F ) `
  N )  .+  (  seq  M (  .+  ,  G ) `  N ) ) )
 
Theoremseqf1olem2a 10962* Lemma for seqf1o 10965. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  C )
 )  ->  ( x  .+  y )  =  ( y  .+  x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ph  ->  C 
 C_  S )   &    |-  ( ph  ->  G : A --> C )   &    |-  ( ph  ->  K  e.  A )   &    |-  ( ph  ->  ( M ... N )  C_  A )   =>    |-  ( ph  ->  ( ( G `
  K )  .+  (  seq  M (  .+  ,  G ) `  N ) )  =  (
 (  seq  M (  .+  ,  G ) `  N )  .+  ( G `
  K ) ) )
 
Theoremseqf1olem1 10963* Lemma for seqf1o 10965. (Contributed by Mario Carneiro, 26-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  C )
 )  ->  ( x  .+  y )  =  ( y  .+  x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ph  ->  C 
 C_  S )   &    |-  ( ph  ->  F : ( M ... ( N  +  1 ) ) -1-1-onto-> ( M ... ( N  +  1 ) ) )   &    |-  ( ph  ->  G : ( M ... ( N  +  1
 ) ) --> C )   &    |-  J  =  ( k  e.  ( M ... N )  |->  ( F `  if ( k  <  K ,  k ,  ( k  +  1 ) ) ) )   &    |-  K  =  ( `' F `  ( N  +  1 ) )   =>    |-  ( ph  ->  J :
 ( M ... N )
 -1-1-onto-> ( M ... N ) )
 
Theoremseqf1olem2 10964* Lemma for seqf1o 10965. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  C )
 )  ->  ( x  .+  y )  =  ( y  .+  x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ph  ->  C 
 C_  S )   &    |-  ( ph  ->  F : ( M ... ( N  +  1 ) ) -1-1-onto-> ( M ... ( N  +  1 ) ) )   &    |-  ( ph  ->  G : ( M ... ( N  +  1
 ) ) --> C )   &    |-  J  =  ( k  e.  ( M ... N )  |->  ( F `  if ( k  <  K ,  k ,  ( k  +  1 ) ) ) )   &    |-  K  =  ( `' F `  ( N  +  1 ) )   &    |-  ( ph  ->  A. g A. f ( ( f : ( M ... N ) -1-1-onto-> ( M ... N )  /\  g : ( M ... N ) --> C )  ->  (  seq  M (  .+  ,  ( g  o.  f
 ) ) `  N )  =  (  seq  M (  .+  ,  g
 ) `  N )
 ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  ( G  o.  F ) ) `  ( N  +  1 )
 )  =  (  seq  M (  .+  ,  G ) `  ( N  +  1 ) ) )
 
Theoremseqf1o 10965* Rearrange a sum via an arbitrary bijection on  ( M ... N ). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  C )
 )  ->  ( x  .+  y )  =  ( y  .+  x ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ph  ->  C 
 C_  S )   &    |-  ( ph  ->  F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ( ph  /\  x  e.  ( M
 ... N ) ) 
 ->  ( G `  x )  e.  C )   &    |-  (
 ( ph  /\  k  e.  ( M ... N ) )  ->  ( H `
  k )  =  ( G `  ( F `  k ) ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  H ) `  N )  =  (  seq  M (  .+  ,  G ) `  N ) )
 
Theoremseradd 10966* The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( G `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( M ... N ) )  ->  ( H `
  k )  =  ( ( F `  k )  +  ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq  M (  +  ,  H ) `  N )  =  ( (  seq  M (  +  ,  F ) `  N )  +  (  seq  M (  +  ,  G ) `  N ) ) )
 
Theoremsersub 10967* The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( G `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( M ... N ) )  ->  ( H `
  k )  =  ( ( F `  k )  -  ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq  M (  +  ,  H ) `  N )  =  ( (  seq  M (  +  ,  F ) `  N )  -  (  seq  M (  +  ,  G ) `  N ) ) )
 
Theoremseqid3 10968* A sequence that consists entirely of zeroes (or whatever the identity  Z is for operation  .+) sums to zero. (Contributed by Mario Carneiro, 15-Dec-2014.)
 |-  ( ph  ->  ( Z  .+  Z )  =  Z )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  x  e.  ( M
 ... N ) ) 
 ->  ( F `  x )  =  Z )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `
  N )  =  Z )
 
Theoremseqid 10969* Discard the first few terms of a sequence that starts with all zeroes (or whatever the identity  Z is for operation  .+). (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ( ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  x )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  M ) )   &    |-  ( ph  ->  ( F `  N )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  x )  =  Z )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F )  |`  ( ZZ>= `  N ) )  = 
 seq  N (  .+  ,  F ) )
 
Theoremseqid2 10970* The last few terms of a sequence that ends with all zeroes (or whatever the identity  Z is for operation  .+) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ( ph  /\  x  e.  S )  ->  ( x  .+  Z )  =  x )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )   &    |-  ( ph  ->  ( 
 seq  M (  .+  ,  F ) `  K )  e.  S )   &    |-  (
 ( ph  /\  x  e.  ( ( K  +  1 ) ... N ) )  ->  ( F `
  x )  =  Z )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  N ) )
 
Theoremseqhomo 10971* Apply a homomorphism to a sequence. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( M ... N ) )  ->  ( F `
  x )  e.  S )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( H `
  ( x  .+  y ) )  =  ( ( H `  x ) Q ( H `  y ) ) )   &    |-  ( ( ph  /\  x  e.  ( M
 ... N ) ) 
 ->  ( H `  ( F `  x ) )  =  ( G `  x ) )   =>    |-  ( ph  ->  ( H `  (  seq  M (  .+  ,  F ) `  N ) )  =  (  seq  M ( Q ,  G ) `
  N ) )
 
Theoremseqz 10972* If the operation  .+ has an absorbing element  Z (a.k.a. zero element), then any sequence containing a  Z evaluates to  Z. (Contributed by Mario Carneiro, 27-May-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( M ... N ) )  ->  ( F `
  x )  e.  S )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( Z  .+  x )  =  Z )   &    |-  (
 ( ph  /\  x  e.  S )  ->  ( x  .+  Z )  =  Z )   &    |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  N  e.  V )   &    |-  ( ph  ->  ( F `  K )  =  Z )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  N )  =  Z )
 
Theoremseqfeq4 10973* Equality of series under different addition operations which agree on an an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  x  e.  ( M ... N ) )  ->  ( F `
  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x 
 .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x 
 .+  y )  =  ( x Q y ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  N )  =  (  seq  M ( Q ,  F ) `  N ) )
 
Theoremseqfeq3 10974* Equality of series under different addition operations which agree on an an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  =  ( x Q y ) )   =>    |-  ( ph  ->  seq 
 M (  .+  ,  F )  =  seq  M ( Q ,  F ) )
 
Theoremseqdistr 10975* The distributive property for series. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( C T ( x  .+  y ) )  =  ( ( C T x )  .+  ( C T y ) ) )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  x  e.  ( M
 ... N ) ) 
 ->  ( G `  x )  e.  S )   &    |-  (
 ( ph  /\  x  e.  ( M ... N ) )  ->  ( F `
  x )  =  ( C T ( G `  x ) ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  N )  =  ( C T (  seq  M ( 
 .+  ,  G ) `  N ) ) )
 
Theoremser0 10976 The value of the partial sums in a zero-valued infinite series. (Contributed by Mario Carneiro, 31-Aug-2013.) (Revised by Mario Carneiro, 15-Dec-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( N  e.  Z  ->  (  seq  M (  +  ,  ( Z  X.  { 0 } ) ) `  N )  =  0 )
 
Theoremser0f 10977 A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  seq  M (  +  ,  ( Z  X.  {
 0 } ) )  =  ( Z  X.  { 0 } ) )
 
Theoremserge0 10978* A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  0  <_  ( F `
  k ) )   =>    |-  ( ph  ->  0  <_  ( 
 seq  M (  +  ,  F ) `  N ) )
 
Theoremserle 10979* Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 27-May-2014.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 RR )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( G `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  <_  ( G `  k ) )   =>    |-  ( ph  ->  (  seq  M (  +  ,  F ) `  N )  <_  (  seq  M (  +  ,  G ) `  N ) )
 
Theoremser1const 10980 Value of the partial series sum of a constant function. (Contributed by NM, 8-Aug-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq  1
 (  +  ,  ( NN  X.  { A }
 ) ) `  N )  =  ( N  x.  A ) )
 
Theoremseqof 10981* Distribute function operation through a sequence. Note that  G
( z ) is an implicit function on  z. (Contributed by Mario Carneiro, 3-Mar-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  x  e.  ( M
 ... N ) ) 
 ->  ( F `  x )  =  ( z  e.  A  |->  ( G `  x ) ) )   =>    |-  ( ph  ->  (  seq  M (  o F  .+  ,  F ) `  N )  =  ( z  e.  A  |->  (  seq  M (  .+  ,  G ) `
  N ) ) )
 
5.6.4  Integer powers
 
Syntaxcexp 10982 Extend class notation to include exponentiation of a complex number to an integer power.
 class  ^
 
Definitiondf-exp 10983* Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 10986 and expp1 10988 provide a the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that  0 ^ 0  =  1 per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014: The definition was extended to include negative integer exponents. The case  x  =  0 ,  y  <  0 gives the value  ( 1  /  0 ), so we will avoid this case in our theorems. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.)
 |- 
 ^  =  ( x  e.  CC ,  y  e.  ZZ  |->  if ( y  =  0 ,  1 ,  if ( 0  < 
 y ,  (  seq  1 (  x.  ,  ( NN  X.  { x }
 ) ) `  y
 ) ,  ( 1 
 /  (  seq  1
 (  x.  ,  ( NN  X.  { x }
 ) ) `  -u y
 ) ) ) ) )
 
Theoremexpval 10984 Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( A ^ N )  =  if ( N  =  0 ,  1 ,  if ( 0  <  N ,  (  seq  1 (  x.  ,  ( NN 
 X.  { A } )
 ) `  N ) ,  ( 1  /  (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `  -u N ) ) ) ) )
 
Theoremexpnnval 10985 Value of exponentiation to positive integer powers. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N )  =  (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `  N ) )
 
Theoremexp0 10986 Value of a complex number raised to the 0th power. Note that under our definition,  0 ^ 0  =  1, following the convention used by Gleason. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( A  e.  CC  ->  ( A ^ 0
 )  =  1 )
 
Theoremexp1 10987 Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)
 |-  ( A  e.  CC  ->  ( A ^ 1
 )  =  A )
 
Theoremexpp1 10988 Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A ^
 ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
 
Theoremexpneg 10989 Value of a complex number raised to a negative integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
 
Theoremexpneg2 10990 Value of a complex number raised to a negative integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  N  e.  CC  /\  -u N  e.  NN0 )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
 
Theoremexpn1 10991 A number to the negative one power is the reciprocal. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( A  e.  CC  ->  ( A ^ -u 1
 )  =  ( 1 
 /  A ) )
 
Theoremexpcllem 10992* Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
 |-  F  C_  CC   &    |-  ( ( x  e.  F  /\  y  e.  F )  ->  ( x  x.  y )  e.  F )   &    |-  1  e.  F   =>    |-  (
 ( A  e.  F  /\  B  e.  NN0 )  ->  ( A ^ B )  e.  F )
 
Theoremexpcl2lem 10993* Lemma for proving integer exponentiation closure laws. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
 |-  F  C_  CC   &    |-  ( ( x  e.  F  /\  y  e.  F )  ->  ( x  x.  y )  e.  F )   &    |-  1  e.  F   &    |-  (
 ( x  e.  F  /\  x  =/=  0
 )  ->  ( 1  /  x )  e.  F )   =>    |-  ( ( A  e.  F  /\  A  =/=  0  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  F )
 
Theoremnnexpcl 10994 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  NN  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  NN )
 
Theoremnn0expcl 10995 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  NN0 )
 
Theoremzexpcl 10996 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  ZZ )
 
Theoremqexpcl 10997 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  QQ  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  QQ )
 
Theoremreexpcl 10998 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  RR )
 
Theoremexpcl 10999 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  CC )
 
Theoremrpexpcl 11000 Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
 |-  ( ( A  e.  RR+  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  RR+ )
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