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Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremseq3val 10401* Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10404, seq3-1 10403 and seq3p1 10405, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  M ) ,  y  e.  _V  |->  <.
 ( x  +  1 ) ,  ( x ( z  e.  ( ZZ>=
 `  M ) ,  w  e.  S  |->  ( w  .+  ( F `
  ( z  +  1 ) ) ) ) y ) >. ) ,  <. M ,  ( F `  M ) >. )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  seq M (  .+  ,  F )  =  ran  R )
 
Theoremseqvalcd 10402* Value of the sequence builder function. Similar to seq3val 10401 but the classes  D (type of each term) and  C (type of the value we are accumulating) do not need to be the same. (Contributed by Jim Kingdon, 9-Jul-2023.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  M ) ,  y  e.  _V  |->  <.
 ( x  +  1 ) ,  ( x ( z  e.  ( ZZ>=
 `  M ) ,  w  e.  C  |->  ( w  .+  ( F `
  ( z  +  1 ) ) ) ) y ) >. ) ,  <. M ,  ( F `  M ) >. )   &    |-  ( ph  ->  ( F `  M )  e.  C )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D )
 )  ->  ( x  .+  y )  e.  C )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  x )  e.  D )   =>    |-  ( ph  ->  seq M (  .+  ,  F )  =  ran  R )
 
Theoremseq3-1 10403* Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 3-Oct-2022.)
 |-  ( 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  ->  ( 
 seq M (  .+  ,  F ) `  M )  =  ( F `  M ) )
 
Theoremseqf 10404* Range of the recursive sequence builder. (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 )
 
Theoremseq3p1 10405* Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  ( N  +  1 )
 )  =  ( ( 
 seq M (  .+  ,  F ) `  N )  .+  ( F `  ( N  +  1
 ) ) ) )
 
Theoremseqovcd 10406* A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10407 and seq1cd 10408 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.)
 |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  x )  e.  D )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D )
 )  ->  ( x  .+  y )  e.  C )   =>    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  C ) )  ->  ( x ( z  e.  ( ZZ>= `  M ) ,  w  e.  C  |->  ( w  .+  ( F `
  ( z  +  1 ) ) ) ) y )  e.  C )
 
Theoremseqf2 10407* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 7-Jul-2023.)
 |-  ( 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 )
 
Theoremseq1cd 10408* Initial value of the recursive sequence builder. A version of seq3-1 10403 which provides two classes 
D and  C for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 19-Jul-2023.)
 |-  ( ph  ->  ( F `  M )  e.  C )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D ) )  ->  ( x 
 .+  y )  e.  C )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  ( M  +  1 ) ) )  ->  ( F `  x )  e.  D )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  M )  =  ( F `  M ) )
 
Theoremseqp1cd 10409* Value of the sequence builder function at a successor. A version of seq3p1 10405 which provides two classes  D and  C for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 20-Jul-2023.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  ( F `  M )  e.  C )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  D ) )  ->  ( x 
 .+  y )  e.  C )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  +  1
 ) ) )  ->  ( F `  x )  e.  D )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  ( N  +  1 )
 )  =  ( ( 
 seq M (  .+  ,  F ) `  N )  .+  ( F `  ( N  +  1
 ) ) ) )
 
Theoremseq3clss 10410* Closure property of the recursive sequence builder. (Contributed by Jim Kingdon, 28-Sep-2022.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  T )   &    |-  ( ( ph  /\  x  e.  ( M ... N ) )  ->  ( F `
  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x 
 .+  y )  e.  S )   &    |-  ( ph  ->  S 
 C_  T )   &    |-  (
 ( ph  /\  ( x  e.  T  /\  y  e.  T ) )  ->  ( x  .+  y )  e.  T )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  N )  e.  S )
 
Theoremseq3m1 10411* Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  ( M  +  1 )
 ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  N )  =  ( (  seq M (  .+  ,  F ) `  ( N  -  1 ) ) 
 .+  ( F `  N ) ) )
 
Theoremseq3fveq2 10412* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  K )  =  ( G `  K ) )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  K )
 )  ->  ( G `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  K )
 )   &    |-  ( ( ph  /\  k  e.  ( ( K  +  1 ) ... N ) )  ->  ( F `
  k )  =  ( G `  k
 ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  N )  =  (  seq K (  .+  ,  G ) `  N ) )
 
Theoremseq3feq2 10413* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  K )  =  ( G `  K ) )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  K )
 )  ->  ( G `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  ( K  +  1 ) ) )  ->  ( F `  k )  =  ( G `  k ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F )  |`  ( ZZ>= `  K ) )  = 
 seq K (  .+  ,  G ) )
 
Theoremseq3fveq 10414* Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  =  ( G `  k
 ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( G `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  N )  =  (  seq M (  .+  ,  G ) `  N ) )
 
Theoremseq3feq 10415* Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  =  ( G `  k ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x 
 .+  y )  e.  S )   =>    |-  ( ph  ->  seq M (  .+  ,  F )  =  seq M ( 
 .+  ,  G )
 )
 
Theoremseq3shft2 10416* Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  K  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( F `  k
 )  =  ( G `
  ( k  +  K ) ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  +  K ) ) ) 
 ->  ( G `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  N )  =  (  seq ( M  +  K ) (  .+  ,  G ) `  ( N  +  K ) ) )
 
Theoremserf 10417* 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 10418* 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 10419* 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 10420* 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 ) )
 
Theoremser3mono 10421* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
 |-  ( ph  ->  K  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  K )
 )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  RR )   &    |-  ( ( ph  /\  x  e.  ( ( K  +  1 ) ... N ) )  ->  0  <_  ( F `  x ) )   =>    |-  ( ph  ->  (  seq M (  +  ,  F ) `  K )  <_  (  seq M (  +  ,  F ) `  N ) )
 
Theoremseq3split 10422* Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.)
 |-  ( ( 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.  ( ZZ>= `  K ) )  ->  ( F `  x )  e.  S )   =>    |-  ( ph  ->  ( 
 seq K (  .+  ,  F ) `  N )  =  ( (  seq K (  .+  ,  F ) `  M )  .+  (  seq ( M  +  1 )
 (  .+  ,  F ) `  N ) ) )
 
Theoremseq3-1p 10423* Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.)
 |-  ( ( 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.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  N )  =  ( ( F `  M )  .+  (  seq ( M  +  1 ) (  .+  ,  F ) `  N ) ) )
 
Theoremseq3caopr3 10424* Lemma for seq3caopr2 10425. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
 |-  ( ( 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.  ( ZZ>= `  M ) )  ->  ( F `  k )  e.  S )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( G `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( 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 ) ) )
 
Theoremseq3caopr2 10425* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
 |-  ( ( 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.  ( ZZ>= `  M )
 )  ->  ( F `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( G `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( H `  k )  =  ( ( F `  k
 ) Q ( G `
  k ) ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  H ) `  N )  =  ( (  seq M (  .+  ,  F ) `  N ) Q (  seq M (  .+  ,  G ) `
  N ) ) )
 
Theoremseq3caopr 10426* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
 |-  ( ( 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.  ( ZZ>= `  M ) )  ->  ( F `  k )  e.  S )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( G `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( H `  k )  =  ( ( F `  k
 )  .+  ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  H ) `  N )  =  ( (  seq M (  .+  ,  F ) `
  N )  .+  (  seq M (  .+  ,  G ) `  N ) ) )
 
Theoremiseqf1olemkle 10427* Lemma for seq3f1o 10447. (Contributed by Jim Kingdon, 21-Aug-2022.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `
  x )  =  x )   =>    |-  ( ph  ->  K  <_  ( `' J `  K ) )
 
Theoremiseqf1olemklt 10428* Lemma for seq3f1o 10447. (Contributed by Jim Kingdon, 21-Aug-2022.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `
  x )  =  x )   &    |-  ( ph  ->  K  =/=  ( `' J `  K ) )   =>    |-  ( ph  ->  K  <  ( `' J `  K ) )
 
Theoremiseqf1olemqcl 10429 Lemma for seq3f1o 10447. (Contributed by Jim Kingdon, 27-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A  e.  ( M ... N ) )   =>    |-  ( ph  ->  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K ,  ( J `  ( A  -  1 ) ) ) ,  ( J `
  A ) )  e.  ( M ... N ) )
 
Theoremiseqf1olemqval 10430* Lemma for seq3f1o 10447. Value of the function  Q. (Contributed by Jim Kingdon, 28-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A  e.  ( M ... N ) )   &    |-  Q  =  ( u  e.  ( M
 ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   =>    |-  ( ph  ->  ( Q `  A )  =  if ( A  e.  ( K ... ( `' J `  K ) ) ,  if ( A  =  K ,  K ,  ( J `  ( A  -  1
 ) ) ) ,  ( J `  A ) ) )
 
Theoremiseqf1olemnab 10431* Lemma for seq3f1o 10447. (Contributed by Jim Kingdon, 27-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A  e.  ( M ... N ) )   &    |-  ( ph  ->  B  e.  ( M ... N ) )   &    |-  ( ph  ->  ( Q `  A )  =  ( Q `  B ) )   &    |-  Q  =  ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   =>    |-  ( ph  ->  -.  ( A  e.  ( K ... ( `' J `  K ) )  /\  -.  B  e.  ( K
 ... ( `' J `  K ) ) ) )
 
Theoremiseqf1olemab 10432* Lemma for seq3f1o 10447. (Contributed by Jim Kingdon, 27-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A  e.  ( M ... N ) )   &    |-  ( ph  ->  B  e.  ( M ... N ) )   &    |-  ( ph  ->  ( Q `  A )  =  ( Q `  B ) )   &    |-  Q  =  ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   &    |-  ( ph  ->  A  e.  ( K ... ( `' J `  K ) ) )   &    |-  ( ph  ->  B  e.  ( K ... ( `' J `  K ) ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremiseqf1olemnanb 10433* Lemma for seq3f1o 10447. (Contributed by Jim Kingdon, 27-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A  e.  ( M ... N ) )   &    |-  ( ph  ->  B  e.  ( M ... N ) )   &    |-  ( ph  ->  ( Q `  A )  =  ( Q `  B ) )   &    |-  Q  =  ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   &    |-  ( ph  ->  -.  A  e.  ( K
 ... ( `' J `  K ) ) )   &    |-  ( ph  ->  -.  B  e.  ( K ... ( `' J `  K ) ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremiseqf1olemqf 10434* Lemma for seq3f1o 10447. Domain and codomain of  Q. (Contributed by Jim Kingdon, 26-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  Q  =  ( u  e.  ( M
 ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   =>    |-  ( ph  ->  Q : ( M ... N ) --> ( M ... N ) )
 
Theoremiseqf1olemmo 10435* Lemma for seq3f1o 10447. Showing that  Q is one-to-one. (Contributed by Jim Kingdon, 27-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  Q  =  ( u  e.  ( M
 ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   &    |-  ( ph  ->  A  e.  ( M ... N ) )   &    |-  ( ph  ->  B  e.  ( M ... N ) )   &    |-  ( ph  ->  ( Q `  A )  =  ( Q `  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremiseqf1olemqf1o 10436* Lemma for seq3f1o 10447. 
Q is a permutation of  ( M ... N
).  Q is formed from the constant portion of  J, followed by the single element  K (at position  K), followed by the rest of J (with the  K deleted and the elements before  K moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  Q  =  ( u  e.  ( M
 ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   =>    |-  ( ph  ->  Q : ( M ... N ) -1-1-onto-> ( M ... N ) )
 
Theoremiseqf1olemqk 10437* Lemma for seq3f1o 10447. 
Q is constant for one more position than  J is. (Contributed by Jim Kingdon, 21-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  Q  =  ( u  e.  ( M
 ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   &    |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `
  x )  =  x )   =>    |-  ( ph  ->  A. x  e.  ( M ... K ) ( Q `  x )  =  x )
 
Theoremiseqf1olemjpcl 10438* Lemma for seq3f1o 10447. A closure lemma involving  J and  P. (Contributed by Jim Kingdon, 29-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  Q  =  ( u  e.  ( M
 ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( G `  x )  e.  S )   &    |-  P  =  ( x  e.  ( ZZ>=
 `  M )  |->  if ( x  <_  N ,  ( G `  (
 f `  x )
 ) ,  ( G `
  M ) ) )   =>    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( [_ J  /  f ]_ P `  x )  e.  S )
 
Theoremiseqf1olemqpcl 10439* Lemma for seq3f1o 10447. A closure lemma involving  Q and  P. (Contributed by Jim Kingdon, 29-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  Q  =  ( u  e.  ( M
 ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( G `  x )  e.  S )   &    |-  P  =  ( x  e.  ( ZZ>=
 `  M )  |->  if ( x  <_  N ,  ( G `  (
 f `  x )
 ) ,  ( G `
  M ) ) )   =>    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( [_ Q  /  f ]_ P `  x )  e.  S )
 
Theoremiseqf1olemfvp 10440* Lemma for seq3f1o 10447. (Contributed by Jim Kingdon, 30-Aug-2022.)
 |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  T : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A  e.  ( M ... N ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( G `  x )  e.  S )   &    |-  P  =  ( x  e.  ( ZZ>=
 `  M )  |->  if ( x  <_  N ,  ( G `  (
 f `  x )
 ) ,  ( G `
  M ) ) )   =>    |-  ( ph  ->  ( [_ T  /  f ]_ P `  A )  =  ( G `  ( T `  A ) ) )
 
Theoremseq3f1olemqsumkj 10441* Lemma for seq3f1o 10447. 
Q gives the same sum as 
J in the range  ( K ... ( `' J `  K ) ). (Contributed by Jim Kingdon, 29-Aug-2022.)
 |-  ( ( 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  ->  F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( G `  x )  e.  S )   &    |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J :
 ( M ... N )
 -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `
  x )  =  x )   &    |-  ( ph  ->  K  =/=  ( `' J `  K ) )   &    |-  Q  =  ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   &    |-  P  =  ( x  e.  ( ZZ>= `  M )  |->  if ( x  <_  N ,  ( G `  ( f `  x ) ) ,  ( G `  M ) ) )   =>    |-  ( ph  ->  ( 
 seq K (  .+  , 
 [_ J  /  f ]_ P ) `  ( `' J `  K ) )  =  (  seq K (  .+  ,  [_ Q  /  f ]_ P ) `  ( `' J `  K ) ) )
 
Theoremseq3f1olemqsumk 10442* Lemma for seq3f1o 10447. 
Q gives the same sum as 
J in the range  ( K ... N ). (Contributed by Jim Kingdon, 22-Aug-2022.)
 |-  ( ( 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  ->  F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( G `  x )  e.  S )   &    |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J :
 ( M ... N )
 -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `
  x )  =  x )   &    |-  ( ph  ->  K  =/=  ( `' J `  K ) )   &    |-  Q  =  ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   &    |-  P  =  ( x  e.  ( ZZ>= `  M )  |->  if ( x  <_  N ,  ( G `  ( f `  x ) ) ,  ( G `  M ) ) )   =>    |-  ( ph  ->  ( 
 seq K (  .+  , 
 [_ J  /  f ]_ P ) `  N )  =  (  seq K (  .+  ,  [_ Q  /  f ]_ P ) `  N ) )
 
Theoremseq3f1olemqsum 10443* Lemma for seq3f1o 10447. 
Q gives the same sum as 
J. (Contributed by Jim Kingdon, 21-Aug-2022.)
 |-  ( ( 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  ->  F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( G `  x )  e.  S )   &    |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J :
 ( M ... N )
 -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `
  x )  =  x )   &    |-  ( ph  ->  K  =/=  ( `' J `  K ) )   &    |-  Q  =  ( u  e.  ( M ... N )  |->  if ( u  e.  ( K ... ( `' J `  K ) ) ,  if ( u  =  K ,  K ,  ( J `  ( u  -  1 ) ) ) ,  ( J `
  u ) ) )   &    |-  P  =  ( x  e.  ( ZZ>= `  M )  |->  if ( x  <_  N ,  ( G `  ( f `  x ) ) ,  ( G `  M ) ) )   =>    |-  ( ph  ->  ( 
 seq M (  .+  , 
 [_ J  /  f ]_ P ) `  N )  =  (  seq M (  .+  ,  [_ Q  /  f ]_ P ) `  N ) )
 
Theoremseq3f1olemstep 10444* Lemma for seq3f1o 10447. Given a permutation which is constant up to a point, supply a new one which is constant for one more position. (Contributed by Jim Kingdon, 19-Aug-2022.)
 |-  ( ( 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  ->  F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( G `  x )  e.  S )   &    |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  J :
 ( M ... N )
 -1-1-onto-> ( M ... N ) )   &    |-  ( ph  ->  A. x  e.  ( M..^ K ) ( J `
  x )  =  x )   &    |-  ( ph  ->  ( 
 seq M (  .+  , 
 [_ J  /  f ]_ P ) `  N )  =  (  seq M (  .+  ,  L ) `  N ) )   &    |-  P  =  ( x  e.  ( ZZ>= `  M )  |->  if ( x  <_  N ,  ( G `  ( f `  x ) ) ,  ( G `  M ) ) )   =>    |-  ( ph  ->  E. f
 ( f : ( M ... N ) -1-1-onto-> ( M ... N ) 
 /\  A. x  e.  ( M ... K ) ( f `  x )  =  x  /\  (  seq M (  .+  ,  P ) `  N )  =  (  seq M (  .+  ,  L ) `  N ) ) )
 
Theoremseq3f1olemp 10445* Lemma for seq3f1o 10447. Existence of a constant permutation of  ( M ... N ) which leads to the same sum as the permutation  F itself. (Contributed by Jim Kingdon, 18-Aug-2022.)
 |-  ( ( 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  ->  F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( G `  x )  e.  S )   &    |-  L  =  ( x  e.  ( ZZ>=
 `  M )  |->  if ( x  <_  N ,  ( G `  ( F `  x ) ) ,  ( G `  M ) ) )   &    |-  P  =  ( x  e.  ( ZZ>= `  M )  |->  if ( x  <_  N ,  ( G `  ( f `  x ) ) ,  ( G `  M ) ) )   =>    |-  ( ph  ->  E. f
 ( f : ( M ... N ) -1-1-onto-> ( M ... N ) 
 /\  A. x  e.  ( M ... N ) ( f `  x )  =  x  /\  (  seq M (  .+  ,  P ) `  N )  =  (  seq M (  .+  ,  L ) `  N ) ) )
 
Theoremseq3f1oleml 10446* Lemma for seq3f1o 10447. This is more or less the result, but stated in terms of  F and  G without  H.  L and  H may differ in terms of what happens to terms after  N. The terms after  N don't matter for the value at  N but we need some definition given the way our theorems concerning  seq work. (Contributed by Jim Kingdon, 17-Aug-2022.)
 |-  ( ( 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  ->  F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( G `  x )  e.  S )   &    |-  L  =  ( x  e.  ( ZZ>=
 `  M )  |->  if ( x  <_  N ,  ( G `  ( F `  x ) ) ,  ( G `  M ) ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  L ) `  N )  =  (  seq M ( 
 .+  ,  G ) `  N ) )
 
Theoremseq3f1o 10447* Rearrange a sum via an arbitrary bijection on  ( M ... N ). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 3-Nov-2022.)
 |-  ( ( 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  ->  F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M ) )  ->  ( G `  x )  e.  S )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( H `  x )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( H `
  k )  =  ( G `  ( F `  k ) ) )   =>    |-  ( ph  ->  (  seq M (  .+  ,  H ) `  N )  =  (  seq M (  .+  ,  G ) `  N ) )
 
Theoremser3add 10448* The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 4-Oct-2022.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( G `  k )  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( H `  k )  =  ( ( F `  k
 )  +  ( G `
  k ) ) )   =>    |-  ( ph  ->  (  seq M (  +  ,  H ) `  N )  =  ( (  seq M (  +  ,  F ) `  N )  +  (  seq M (  +  ,  G ) `  N ) ) )
 
Theoremser3sub 10449* The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( G `  k )  e.  CC )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( H `  k )  =  ( ( F `  k
 )  -  ( G `
  k ) ) )   =>    |-  ( ph  ->  (  seq M (  +  ,  H ) `  N )  =  ( (  seq M (  +  ,  F ) `  N )  -  (  seq M (  +  ,  G ) `  N ) ) )
 
Theoremseq3id3 10450* A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a  .+ -idempotent sums (or " .+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Jim Kingdon, 8-Apr-2023.)
 |-  ( ph  ->  ( Z  .+  Z )  =  Z )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  x  e.  ( M
 ... N ) ) 
 ->  ( F `  x )  =  Z )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  N )  =  Z )
 
Theoremseq3id 10451* Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for  .+) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
 |-  ( ( 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  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F )  |`  ( ZZ>= `  N ) )  = 
 seq N (  .+  ,  F ) )
 
Theoremseq3id2 10452* The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for  .+) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 12-Nov-2022.)
 |-  ( ( 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  /\  x  e.  ( ZZ>= `  M ) )  ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  K )  =  (  seq M (  .+  ,  F ) `  N ) )
 
Theoremseq3homo 10453* Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 10-Oct-2022.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( 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.  ( ZZ>= `  M )
 )  ->  ( H `  ( F `  x ) )  =  ( G `  x ) )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( G `  x )  e.  S )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x Q y )  e.  S )   =>    |-  ( ph  ->  ( H `  (  seq M (  .+  ,  F ) `
  N ) )  =  (  seq M ( Q ,  G ) `
  N ) )
 
Theoremseq3z 10454* 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.) (Revised by Jim Kingdon, 23-Apr-2023.)
 |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S )
 )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  e.  S )   &    |-  ( ( ph  /\  x  e.  S )  ->  ( Z  .+  x )  =  Z )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( x  .+  Z )  =  Z )   &    |-  ( ph  ->  K  e.  ( M ... N ) )   &    |-  ( ph  ->  ( F `  K )  =  Z )   =>    |-  ( ph  ->  (  seq M (  .+  ,  F ) `  N )  =  Z )
 
Theoremseqfeq3 10455* Equality of series under different addition operations which agree on 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 ) )
 
Theoremseq3distr 10456* The distributive property for series. (Contributed by Jim Kingdon, 10-Oct-2022.)
 |-  ( ( 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.  ( ZZ>= `  M ) )  ->  ( G `  x )  e.  S )   &    |-  (
 ( ph  /\  x  e.  ( ZZ>= `  M )
 )  ->  ( F `  x )  =  ( C T ( G `
  x ) ) )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x T y )  e.  S )   &    |-  ( ph  ->  C  e.  S )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  N )  =  ( C T (  seq M ( 
 .+  ,  G ) `  N ) ) )
 
Theoremser0 10457 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 10458 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 } ) )
 
Theoremfser0const 10459* Simplifying an expression which turns out just to be a constant zero sequence. (Contributed by Jim Kingdon, 16-Sep-2022.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( N  e.  Z  ->  ( n  e.  Z  |->  if ( n  <_  N ,  ( ( Z  X.  { 0 } ) `  n ) ,  0 ) )  =  ( Z  X.  { 0 } ) )
 
Theoremser3ge0 10460* 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.  ( ZZ>= `  M )
 )  ->  ( F `  k )  e.  RR )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  0  <_  ( F `  k ) )   =>    |-  ( ph  ->  0  <_  (  seq M (  +  ,  F ) `
  N ) )
 
Theoremser3le 10461* Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  e.  RR )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( G `  k )  e.  RR )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  <_  ( G `  k ) )   =>    |-  ( ph  ->  (  seq M (  +  ,  F ) `  N )  <_  (  seq M (  +  ,  G ) `  N ) )
 
4.6.6  Integer powers
 
Syntaxcexp 10462 Extend class notation to include exponentiation of a complex number to an integer power.
 class  ^
 
Definitiondf-exp 10463* Define exponentiation to nonnegative integer powers. For example,  ( 5 ^ 2 )  =  2 5 (see ex-exp 13721).

This definition is not meant to be used directly; instead, exp0 10467 and expp1 10470 provide 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 (see 0exp0e1 10468).

4-Jun-2014: The definition was extended to include negative integer exponents. For example,  ( -u 3 ^
-u 2 )  =  ( 1  /  9
) (ex-exp 13721). 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
 ) ) ) ) )
 
Theoremexp3vallem 10464 Lemma for exp3val 10465. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingdon, 7-Jun-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  ( 
 seq 1 (  x. 
 ,  ( NN  X.  { A } ) ) `
  N ) #  0 )
 
Theoremexp3val 10465 Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
 |-  ( ( A  e.  CC  /\  N  e.  ZZ  /\  ( A #  0  \/  0  <_  N )
 )  ->  ( 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 10466 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 10467 Value of a complex number raised to the 0th power. Note that under our definition,  0 ^ 0  =  1 (0exp0e1 10468) , 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 )
 
Theorem0exp0e1 10468 The zeroth power of zero equals one. See comment of exp0 10467. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 0 ^ 0
 )  =  1
 
Theoremexp1 10469 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 10470 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 ) )
 
Theoremexpnegap0 10471 Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  NN0 )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
 
Theoremexpineg2 10472 Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( N  e.  CC  /\  -u N  e.  NN0 ) )  ->  ( A ^ N )  =  ( 1  /  ( A ^ -u N ) ) )
 
Theoremexpn1ap0 10473 A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( A ^ -u 1
 )  =  ( 1 
 /  A ) )
 
Theoremexpcllem 10474* 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 )
 
Theoremexpcl2lemap 10475* Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
 |-  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 10476 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  NN  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  NN )
 
Theoremnn0expcl 10477 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  NN0 )
 
Theoremzexpcl 10478 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  ZZ )
 
Theoremqexpcl 10479 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  QQ  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  QQ )
 
Theoremreexpcl 10480 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  RR )
 
Theoremexpcl 10481 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  CC )
 
Theoremrpexpcl 10482 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+ )
 
Theoremreexpclzap 10483 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
 |-  ( ( A  e.  RR  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  RR )
 
Theoremqexpclz 10484 Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  ( ( A  e.  QQ  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  QQ )
 
Theoremm1expcl2 10485 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
 
Theoremm1expcl 10486 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  ZZ )
 
Theoremexpclzaplem 10487* Closure law for integer exponentiation. Lemma for expclzap 10488 and expap0i 10495. (Contributed by Jim Kingdon, 9-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  { z  e.  CC  |  z #  0 } )
 
Theoremexpclzap 10488 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
 
Theoremnn0expcli 10489 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( A ^ N )  e.  NN0
 
Theoremnn0sqcl 10490 The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  ( A  e.  NN0  ->  ( A ^ 2 )  e.  NN0 )
 
Theoremexpm1t 10491 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N )  =  (
 ( A ^ ( N  -  1 ) )  x.  A ) )
 
Theorem1exp 10492 Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( N  e.  ZZ  ->  ( 1 ^ N )  =  1 )
 
Theoremexpap0 10493 Positive integer exponentiation is apart from zero iff its base is apart from zero. That it is easier to prove this first, and then prove expeq0 10494 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." (Remark of [Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N ) #  0  <->  A #  0 ) )
 
Theoremexpeq0 10494 Positive integer exponentiation is 0 iff its base is 0. (Contributed by NM, 23-Feb-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  =  0  <->  A  =  0
 ) )
 
Theoremexpap0i 10495 Integer exponentiation is apart from zero if its base is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N ) #  0 )
 
Theoremexpgt0 10496 A positive real raised to an integer power is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  RR  /\  N  e.  ZZ  /\  0  <  A ) 
 ->  0  <  ( A ^ N ) )
 
Theoremexpnegzap 10497 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ -u N )  =  ( 1  /  ( A ^ N ) ) )
 
Theorem0exp 10498 Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
 |-  ( N  e.  NN  ->  ( 0 ^ N )  =  0 )
 
Theoremexpge0 10499 A nonnegative real raised to a nonnegative integer is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  0  <_  A )  ->  0  <_  ( A ^ N ) )
 
Theoremexpge1 10500 A real greater than or equal to 1 raised to a nonnegative integer is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  1  <_  A )  ->  1  <_  ( A ^ N ) )
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