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Theorem List for Metamath Proof Explorer - 11001-11100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremseqfveq 11001* 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 11002* 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 11003* 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 11004 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 11005* 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 11006* 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 11007* 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 11008* 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 11009* 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 11010* 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 11011* 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 11012* Lemma for seqcaopr2 11013. (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 11013* 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 11014* 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 11015* Lemma for seqf1o 11018. (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 11016* Lemma for seqf1o 11018. (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 11017* Lemma for seqf1o 11018. (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 11018* 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 11019* 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 11020* 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 11021* 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 11022* 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 11023* 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 11024* 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 11025* 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 11026* 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 11027* 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 11028* 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 11029 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 11030 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 11031* 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 11032* 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 11033 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 11034* 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 11035 Extend class notation to include exponentiation of a complex number to an integer power.
 class  ^
 
Definitiondf-exp 11036* Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 11039 and expp1 11041 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 11037 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 11038 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 11039 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 11040 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 11041 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 11042 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 11043 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 11044 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 11045* 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 11046* 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 11047 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  NN  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  NN )
 
Theoremnn0expcl 11048 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  NN0 )
 
Theoremzexpcl 11049 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  ZZ )
 
Theoremqexpcl 11050 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  QQ  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  QQ )
 
Theoremreexpcl 11051 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  RR )
 
Theoremexpcl 11052 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  CC )
 
Theoremrpexpcl 11053 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+ )
 
Theoremreexpclz 11054 Closure of exponentiation of reals. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
 |-  ( ( A  e.  RR  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  RR )
 
Theoremqexpclz 11055 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 11056 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
 
Theoremm1expcl 11057 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  ZZ )
 
Theoremexpclzlem 11058 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  ( CC  \  { 0 } )
 )
 
Theoremexpclz 11059 Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  CC )
 
Theoremnn0expcli 11060 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( A ^ N )  e.  NN0
 
Theoremexpm1t 11061 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 11062 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 )
 
Theoremexpeq0 11063 Natural number exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  =  0  <->  A  =  0
 ) )
 
Theoremexpne0 11064 Natural number exponentiation is nonzero iff its mantissa is nonzero. (Contributed by NM, 6-May-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  =/=  0  <->  A  =/=  0
 ) )
 
Theoremexpne0i 11065 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  =/=  0 )
 
Theoremexpgt0 11066 Nonnegative integer exponentiation with a positive mantissa 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 ) )
 
Theoremexpnegz 11067 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 11068 Value of zero raised to a natural number power. (Contributed by NM, 19-Aug-2004.)
 |-  ( N  e.  NN  ->  ( 0 ^ N )  =  0 )
 
Theoremexpge0 11069 Nonnegative integer exponentiation with a nonnegative mantissa 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 11070 Nonnegative integer exponentiation with a mantissa greater than or equal to 1 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 ) )
 
Theoremexpgt1 11071 Natural number exponentiation with a mantissa greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A ) 
 ->  1  <  ( A ^ N ) )
 
Theoremmulexp 11072 Natural number exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
 
Theoremmulexpz 11073 Integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  N  e.  ZZ )  ->  (
 ( A  x.  B ) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
 
Theoremexprec 11074 Nonnegative integer exponentiation of a reciprocal. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( ( 1  /  A ) ^ N )  =  ( 1  /  ( A ^ N ) ) )
 
Theoremexpadd 11075 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)
 |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N )
 )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
 
Theoremexpaddzlem 11076 Lemma for expaddz 11077. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
 
Theoremexpaddz 11077 Sum of exponents law for integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
 
Theoremexpmul 11078 Product of exponents law for natural number exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)
 |-  ( ( A  e.  CC  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N ) )
 
Theoremexpmulz 11079 Product of exponents law for integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 7-Jul-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N ) )
 
Theoremexpsub 11080 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  -  N ) )  =  ( ( A ^ M )  /  ( A ^ N ) ) )
 
Theoremexpp1z 11081 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ ( N  +  1 )
 )  =  ( ( A ^ N )  x.  A ) )
 
Theoremexpm1 11082 Value of a complex number raised to an integer power minus one. (Contributed by NM, 25-Dec-2008.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ ( N  -  1 ) )  =  ( ( A ^ N )  /  A ) )
 
Theoremexpdiv 11083 Nonnegative integer exponentiation of a quotient. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0
 )  /\  N  e.  NN0 )  ->  ( ( A  /  B ) ^ N )  =  (
 ( A ^ N )  /  ( B ^ N ) ) )
 
Theoremltexp2a 11084 Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( 1  <  A  /\  M  <  N ) )  ->  ( A ^ M )  <  ( A ^ N ) )
 
Theoremexpcan 11085 Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  (
 ( A ^ M )  =  ( A ^ N )  <->  M  =  N ) )
 
Theoremltexp2 11086 Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( M  <  N  <->  ( A ^ M )  <  ( A ^ N ) ) )
 
Theoremleexp2 11087 Ordering law for exponentiation. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  ( M  <_  N  <->  ( A ^ M )  <_  ( A ^ N ) ) )
 
Theoremleexp2a 11088 Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( ( A  e.  RR  /\  1  <_  A  /\  N  e.  ( ZZ>= `  M ) )  ->  ( A ^ M ) 
 <_  ( A ^ N ) )
 
Theoremltexp2r 11089 The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( ( ( A  e.  RR+  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  A  <  1
 )  ->  ( M  <  N  <->  ( A ^ N )  <  ( A ^ M ) ) )
 
Theoremleexp2r 11090 Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
 |-  ( ( ( A  e.  RR  /\  M  e.  NN0  /\  N  e.  ( ZZ>= `  M )
 )  /\  ( 0  <_  A  /\  A  <_  1 ) )  ->  ( A ^ N )  <_  ( A ^ M ) )
 
Theoremleexp1a 11091 Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  N  e.  NN0 )  /\  ( 0 
 <_  A  /\  A  <_  B ) )  ->  ( A ^ N )  <_  ( B ^ N ) )
 
Theoremexple1 11092 Nonnegative integer exponentiation with a mantissa between 0 and 1 inclusive is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A  /\  A  <_  1 )  /\  N  e.  NN0 )  ->  ( A ^ N )  <_  1
 )
 
Theoremexpubnd 11093 An upper bound on  A ^ N when  2  <_  A. (Contributed by NM, 19-Dec-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  2  <_  A )  ->  ( A ^ N )  <_  ( ( 2 ^ N )  x.  ( ( A  -  1 ) ^ N ) ) )
 
Theoremsqval 11094 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  =  ( A  x.  A ) )
 
Theoremsqneg 11095 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
 |-  ( A  e.  CC  ->  ( -u A ^ 2
 )  =  ( A ^ 2 ) )
 
Theoremsqsubswap 11096 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B ) ^
 2 )  =  ( ( B  -  A ) ^ 2 ) )
 
Theoremsqcl 11097 Closure of square. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  e.  CC )
 
Theoremsqmul 11098 Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B ) ^
 2 )  =  ( ( A ^ 2
 )  x.  ( B ^ 2 ) ) )
 
Theoremsqeq0 11099 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =  0  <->  A  =  0 )
 )
 
Theoremsqdiv 11100 Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Mario Carneiro, 9-Jul-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( ( A  /  B ) ^ 2
 )  =  ( ( A ^ 2 ) 
 /  ( B ^
 2 ) ) )
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