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Theorem List for Metamath Proof Explorer - 11001-11100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmodadd12d 11001 Additive property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  ( A  mod  E )  =  ( B  mod  E ) )   &    |-  ( ph  ->  ( C  mod  E )  =  ( D  mod  E ) )   =>    |-  ( ph  ->  (
 ( A  +  C )  mod  E )  =  ( ( B  +  D )  mod  E ) )
 
Theoremmodsub12d 11002 Subtraction property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  ( A  mod  E )  =  ( B  mod  E ) )   &    |-  ( ph  ->  ( C  mod  E )  =  ( D  mod  E ) )   =>    |-  ( ph  ->  (
 ( A  -  C )  mod  E )  =  ( ( B  -  D )  mod  E ) )
 
Theoremmoddi 11003 Distribute multiplication over a modulo operation. (Contributed by NM, 11-Nov-2008.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( A  x.  ( B  mod  C ) )  =  ( ( A  x.  B )  mod  ( A  x.  C ) ) )
 
Theoremmodsubdir 11004 Distribute the modulo operation over a subtraction. (Contributed by NM, 30-Dec-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( ( B  mod  C )  <_  ( A  mod  C )  <->  ( ( A  -  B )  mod  C )  =  ( ( A  mod  C )  -  ( B  mod  C ) ) ) )
 
Theoremmodirr 11005 A number modulo an irrational multiple of it is nonzero. (Contributed by NM, 11-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( A  /  B )  e.  ( RR  \  QQ ) )  ->  ( A  mod  B )  =/=  0 )
 
5.6.3  The infinite sequence builder "seq"
 
Theoremom2uz0i 11006* The mapping  G is a one-to-one mapping from  om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number  C (normally 0 for the upper integers  NN0 or 1 for the upper integers  NN), 1 maps to  C + 1, etc. This theorem shows the value of  G at ordinal natural number zero. (This series of theorems generalizes an earlier series for  NN0 contributed by Raph Levien, 10-Apr-2004.) (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  ( G `  (/) )  =  C
 
Theoremom2uzsuci 11007* The value of  G (see om2uz0i 11006) at a successor. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  ( A  e.  om  ->  ( G `  suc  A )  =  ( ( G `  A )  +  1 ) )
 
Theoremom2uzuzi 11008* The value  G (see om2uz0i 11006) at an ordinal natural number is in the upper integers. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  ( A  e.  om  ->  ( G `  A )  e.  ( ZZ>= `  C ) )
 
Theoremom2uzlti 11009* Less-than relation for  G (see om2uz0i 11006). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  e.  B  ->  ( G `  A )  <  ( G `
  B ) ) )
 
Theoremom2uzlt2i 11010* The mapping  G (see om2uz0i 11006) preserves order. (Contributed by NM, 4-May-2005.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  e.  B 
 <->  ( G `  A )  <  ( G `  B ) ) )
 
Theoremom2uzrani 11011* Range of  G (see om2uz0i 11006). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |- 
 ran  G  =  ( ZZ>=
 `  C )
 
Theoremom2uzf1oi 11012*  G (see om2uz0i 11006) is a one-to-one onto mapping. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  G : om -1-1-onto-> ( ZZ>= `  C )
 
Theoremom2uzisoi 11013*  G (see om2uz0i 11006) is an isomorphism from natural ordinals to upper integers. (Contributed by NM, 9-Oct-2008.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  G  Isom  _E  ,  <  ( om ,  ( ZZ>= `  C ) )
 
Theoremom2uzoi 11014* An alternative definition of  G in terms of df-oi 7221. (Contributed by Mario Carneiro, 2-Jun-2015.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  G  = OrdIso (  <  ,  ( ZZ>= `  C )
 )
 
Theoremom2uzrdg 11015* A helper lemma for the value of a recursive definition generator on upper integers (typically either  NN or  NN0) with characteristic function  F ( x ,  y ) and initial value  A. Normally  F is a function on the partition, and  A is a member of the partition. See also comment in om2uz0i 11006. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  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 )   =>    |-  ( B  e.  om  ->  ( R `  B )  =  <. ( G `
  B ) ,  ( 2nd `  ( R `  B ) )
 >. )
 
Theoremuzrdglem 11016* A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  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 )   =>    |-  ( B  e.  ( ZZ>=
 `  C )  ->  <. B ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >.  e.  ran  R )
 
Theoremuzrdgfni 11017* The recursive definition generator on upper integers is a function. See comment in om2uzrdg 11015. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  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   =>    |-  S  Fn  ( ZZ>= `  C )
 
Theoremuzrdg0i 11018* Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 11015. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  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   =>    |-  ( S `  C )  =  A
 
Theoremuzrdgsuci 11019* Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 11015. (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 11020  < 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 11021 Less than well orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
 |- 
 <  We  NN
 
Theoremltwefz 11022 Less than well orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |- 
 <  We  ( M ... N )
 
Theoremuzenom 11023 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  Z  ~~  om )
 
Theoremuzinf 11024 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 11025* 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 11026 The cardinality of a finite set of sequential integers. (See om2uz0i 11006 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 11027 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 11028 The cardinality of a finite set of sequential integers. (See om2uz0i 11006 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 11029  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 11030 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 11031 Commonly used special case of fzfi 11030. (Contributed by Mario Carneiro, 25-May-2014.)
 |-  ( ph  ->  ( M ... N )  e. 
 Fin )
 
Theoremfzofi 11032 Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( M..^ N )  e.  Fin
 
Theoremfsequb 11033* 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 11034* 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 11035 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 11036 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 11037 The nonnegative integers are equinumerous to the natural numbers. (Contributed by NM, 19-Jul-2004.)
 |- 
 NN0  ~~  NN
 
Theoremnnenom 11038 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 11039* 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 11040* Lemma for axdc4uz 11041. (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 11041* A version of axdc4 8078 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 11042 Extend class notation with recursive sequence builder.
 class  seq  M (  .+  ,  F )
 
Definitiondf-seq 11043* 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 11055 and seqp1 11057. 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 12022), by climdm 12024 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 11006 through om2uzf1oi 11012, originally proved by Raph Levien for use with df-exp 11101 and later generalized for arbitrary recursive sequences. Definition df-sum 12155 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 11044 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |- 
 seq  M (  .+  ,  F )  e.  _V
 
Theoremseqeq1 11045 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  ( M  =  N  ->  seq  M (  .+  ,  F )  =  seq  N (  .+  ,  F ) )
 
Theoremseqeq2 11046 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  (  .+  =  Q  ->  seq  M (  .+  ,  F )  =  seq  M ( Q ,  F ) )
 
Theoremseqeq3 11047 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
 |-  ( F  =  G  ->  seq  M (  .+  ,  F )  =  seq  M (  .+  ,  G ) )
 
Theoremseqeq1d 11048 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 11049 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 11050 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 11051 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 11052 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 11053* 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 11054 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 11055 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 11056 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 11057 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 11058 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 11059 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 11060* 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 11061* 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 11062* 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 11063* Range of the recursive sequence builder (special case of seqf2 11061). (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 11064* 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 11065* 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 11066* 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 11067* 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 11068* 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 11069 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 11070* 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 11071* 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 11072* 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 11073* 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 11074* 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 11075* 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 11076* 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 11077* Lemma for seqcaopr2 11078. (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 11078* 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 11079* 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 11080* Lemma for seqf1o 11083. (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 11081* Lemma for seqf1o 11083. (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 11082* Lemma for seqf1o 11083. (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 11083* 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 11084* 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 11085* 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 11086* 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 11087* 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 11088* 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 11089* 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 11090* 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 11091* 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 11092* 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 11093* 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 11094 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 11095 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 11096* 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 11097* 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 11098 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 11099* 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 11100 Extend class notation to include exponentiation of a complex number to an integer power.
 class  ^
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