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Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiseqf1olemnab 10501* Lemma for seq3f1o 10517. (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 10502* Lemma for seq3f1o 10517. (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 10503* Lemma for seq3f1o 10517. (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 10504* Lemma for seq3f1o 10517. 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 10505* Lemma for seq3f1o 10517. 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 10506* Lemma for seq3f1o 10517. 
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 10507* Lemma for seq3f1o 10517. 
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 10508* Lemma for seq3f1o 10517. 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 10509* Lemma for seq3f1o 10517. 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 10510* Lemma for seq3f1o 10517. (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 10511* Lemma for seq3f1o 10517. 
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 10512* Lemma for seq3f1o 10517. 
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 10513* Lemma for seq3f1o 10517. 
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 10514* Lemma for seq3f1o 10517. 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 10515* Lemma for seq3f1o 10517. 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 10516* Lemma for seq3f1o 10517. 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 10517* 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 10518* 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 10519* 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 10520* 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 10521* 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 10522* 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 10523* 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 10524* 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 10525* 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 10526* 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 10527 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 10528 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 10529* 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 10530* 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 10531* 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 10532 Extend class notation to include exponentiation of a complex number to an integer power.
 class  ^
 
Definitiondf-exp 10533* Define exponentiation to nonnegative integer powers. For example,  ( 5 ^ 2 )  =  2 5 (see ex-exp 14750).

This definition is not meant to be used directly; instead, exp0 10537 and expp1 10540 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 10538).

4-Jun-2014: The definition was extended to include negative integer exponents. For example,  ( -u 3 ^
-u 2 )  =  ( 1  /  9
) (ex-exp 14750). 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 10534 Lemma for exp3val 10535. 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 10535 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 10536 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 10537 Value of a complex number raised to the 0th power. Note that under our definition,  0 ^ 0  =  1 (0exp0e1 10538) , 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 10538 The zeroth power of zero equals one. See comment of exp0 10537. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 0 ^ 0
 )  =  1
 
Theoremexp1 10539 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 10540 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 10541 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 10542 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 10543 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 10544* 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 10545* 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 10546 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  NN  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  NN )
 
Theoremnn0expcl 10547 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  NN0  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  NN0 )
 
Theoremzexpcl 10548 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  ZZ )
 
Theoremqexpcl 10549 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
 |-  ( ( A  e.  QQ  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  QQ )
 
Theoremreexpcl 10550 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  RR )
 
Theoremexpcl 10551 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A ^ N )  e.  CC )
 
Theoremrpexpcl 10552 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 10553 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 10554 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 10555 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  { -u 1 ,  1 } )
 
Theoremm1expcl 10556 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( N  e.  ZZ  ->  ( -u 1 ^ N )  e.  ZZ )
 
Theoremexpclzaplem 10557* Closure law for integer exponentiation. Lemma for expclzap 10558 and expap0i 10565. (Contributed by Jim Kingdon, 9-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  { z  e.  CC  |  z #  0 } )
 
Theoremexpclzap 10558 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 10559 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  A  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( A ^ N )  e.  NN0
 
Theoremnn0sqcl 10560 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 10561 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 10562 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 10563 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 10564 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 10564 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 10565 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 10566 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 10567 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 10568 Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
 |-  ( N  e.  NN  ->  ( 0 ^ N )  =  0 )
 
Theoremexpge0 10569 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 10570 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 ) )
 
Theoremexpgt1 10571 A real greater than 1 raised to a positive integer 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 10572 Nonnegative integer 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 ) ) )
 
Theoremmulexpzap 10573 Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 )  /\  N  e.  ZZ )  ->  ( ( A  x.  B ) ^ N )  =  ( ( A ^ N )  x.  ( B ^ N ) ) )
 
Theoremexprecap 10574 Integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( ( 1  /  A ) ^ N )  =  ( 1  /  ( A ^ N ) ) )
 
Theoremexpadd 10575 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 ) ) )
 
Theoremexpaddzaplem 10576 Lemma for expaddzap 10577. (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  RR  /\  -u M  e.  NN )  /\  N  e.  NN0 )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
 
Theoremexpaddzap 10577 Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  +  N ) )  =  ( ( A ^ M )  x.  ( A ^ N ) ) )
 
Theoremexpmul 10578 Product of exponents law for nonnegative integer 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 ) )
 
Theoremexpmulzap 10579 Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  x.  N ) )  =  ( ( A ^ M ) ^ N ) )
 
Theoremm1expeven 10580 Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
 |-  ( N  e.  ZZ  ->  ( -u 1 ^ (
 2  x.  N ) )  =  1 )
 
Theoremexpsubap 10581 Exponent subtraction law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( A ^ ( M  -  N ) )  =  ( ( A ^ M )  /  ( A ^ N ) ) )
 
Theoremexpp1zap 10582 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ ( N  +  1 )
 )  =  ( ( A ^ N )  x.  A ) )
 
Theoremexpm1ap 10583 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( A  e.  CC  /\  A #  0  /\  N  e.  ZZ )  ->  ( A ^ ( N  -  1 ) )  =  ( ( A ^ N )  /  A ) )
 
Theoremexpdivap 10584 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B #  0 ) 
 /\  N  e.  NN0 )  ->  ( ( A 
 /  B ) ^ N )  =  (
 ( A ^ N )  /  ( B ^ N ) ) )
 
Theoremltexp2a 10585 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 ) )
 
Theoremleexp2a 10586 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 ) )
 
Theoremleexp2r 10587 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 10588 Weak base 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 10589 A real between 0 and 1 inclusive raised to a nonnegative integer 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 10590 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 10591 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  =  ( A  x.  A ) )
 
Theoremsqneg 10592 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
 |-  ( A  e.  CC  ->  ( -u A ^ 2
 )  =  ( A ^ 2 ) )
 
Theoremsqsubswap 10593 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 10594 Closure of square. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  CC  ->  ( A ^ 2
 )  e.  CC )
 
Theoremsqmul 10595 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 10596 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =  0  <->  A  =  0 )
 )
 
Theoremsqdivap 10597 Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( ( A  /  B ) ^ 2
 )  =  ( ( A ^ 2 ) 
 /  ( B ^
 2 ) ) )
 
Theoremsqdividap 10598 The square of a complex number apart from zero divided by itself equals that number. (Contributed by AV, 19-Jul-2021.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  ( ( A ^
 2 )  /  A )  =  A )
 
Theoremsqne0 10599 A number is nonzero iff its square is nonzero. See also sqap0 10600 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 )  =/=  0  <->  A  =/=  0 ) )
 
Theoremsqap0 10600 A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
 |-  ( A  e.  CC  ->  ( ( A ^
 2 ) #  0  <->  A #  0 )
 )
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