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Theorem List for Intuitionistic Logic Explorer - 10201-10300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremresunimafz0 10201 The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
 |-  ( ph  ->  Fun  I
 )   &    |-  ( ph  ->  F : ( 0..^ ( `  F ) ) --> dom  I
 )   &    |-  ( ph  ->  N  e.  ( 0..^ ( `  F ) ) )   =>    |-  ( ph  ->  ( I  |`  ( F " ( 0 ... N ) ) )  =  ( ( I  |`  ( F " ( 0..^ N ) ) )  u.  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } ) )
 
Theoremfnfz0hash 10202 The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
 |-  ( ( N  e.  NN0  /\  F  Fn  ( 0
 ... N ) ) 
 ->  ( `  F )  =  ( N  +  1 ) )
 
Theoremffz0hash 10203 The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.)
 |-  ( ( N  e.  NN0  /\  F : ( 0
 ... N ) --> B ) 
 ->  ( `  F )  =  ( N  +  1 ) )
 
Theoremffzo0hash 10204 The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
 |-  ( ( N  e.  NN0  /\  F  Fn  ( 0..^ N ) )  ->  ( `  F )  =  N )
 
Theoremfnfzo0hash 10205 The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.)
 |-  ( ( N  e.  NN0  /\  F : ( 0..^ N ) --> B ) 
 ->  ( `  F )  =  N )
 
Theoremhashfacen 10206* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  { f  |  f : A -1-1-onto-> C }  ~~  { f  |  f : B -1-1-onto-> D } )
 
Theoremleisorel 10207 Version of isorel 5569 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A 
 C_  RR*  /\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C  <_  D  <->  ( F `  C )  <_  ( F `
  D ) ) )
 
Theoremzfz1isolemsplit 10208 Lemma for zfz1iso 10211. Removing one element from an integer range. (Contributed by Jim Kingdon, 8-Sep-2022.)
 |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  M  e.  X )   =>    |-  ( ph  ->  ( 1 ... ( `  X ) )  =  (
 ( 1 ... ( `  ( X  \  { M } ) ) )  u.  { ( `  X ) } ) )
 
Theoremzfz1isolemiso 10209* Lemma for zfz1iso 10211. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)
 |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  X 
 C_  ZZ )   &    |-  ( ph  ->  M  e.  X )   &    |-  ( ph  ->  A. z  e.  X  z  <_  M )   &    |-  ( ph  ->  G  Isom  <  ,  <  ( ( 1
 ... ( `  ( X  \  { M } )
 ) ) ,  ( X  \  { M }
 ) ) )   &    |-  ( ph  ->  A  e.  (
 1 ... ( `  X ) ) )   &    |-  ( ph  ->  B  e.  (
 1 ... ( `  X ) ) )   =>    |-  ( ph  ->  ( A  <  B  <->  ( ( G  u.  { <. ( `  X ) ,  M >. } ) `  A )  <  ( ( G  u.  { <. ( `  X ) ,  M >. } ) `  B ) ) )
 
Theoremzfz1isolem1 10210* Lemma for zfz1iso 10211. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)
 |-  ( ph  ->  K  e.  om )   &    |-  ( ph  ->  A. y ( ( ( y  C_  ZZ  /\  y  e.  Fin )  /\  y  ~~  K )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  y ) ) ,  y ) ) )   &    |-  ( ph  ->  X  C_  ZZ )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  X 
 ~~  suc  K )   &    |-  ( ph  ->  M  e.  X )   &    |-  ( ph  ->  A. z  e.  X  z  <_  M )   =>    |-  ( ph  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  X ) ) ,  X ) )
 
Theoremzfz1iso 10211* A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)
 |-  ( ( A  C_  ZZ  /\  A  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1
 ... ( `  A )
 ) ,  A ) )
 
Theoremiseqcoll 10212* The function  F contains a sparse set of nonzero values to be summed. The function  G is an order isomorphism from the set of nonzero values of  F to a 1-based finite sequence, and  H collects these nonzero values together. Under these conditions, the sum over the values in  H yields the same result as the sum over the original set  F. (Contributed by Mario Carneiro, 2-Apr-2014.)
 |-  ( ( ph  /\  k  e.  S )  ->  ( Z  .+  k )  =  k )   &    |-  ( ( ph  /\  k  e.  S ) 
 ->  ( k  .+  Z )  =  k )   &    |-  (
 ( ph  /\  ( k  e.  S  /\  n  e.  S ) )  ->  ( k  .+  n )  e.  S )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ph  ->  G  Isom  <  ,  <  (
 ( 1 ... ( `  A ) ) ,  A ) )   &    |-  ( ph  ->  N  e.  (
 1 ... ( `  A ) ) )   &    |-  ( ph  ->  A  C_  ( ZZ>=
 `  M ) )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  1 )
 )  ->  ( H `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ( M ... ( G `  ( `  A ) ) )  \  A ) )  ->  ( F `  k )  =  Z )   &    |-  (
 ( ph  /\  n  e.  ( 1 ... ( `  A ) ) ) 
 ->  ( H `  n )  =  ( F `  ( G `  n ) ) )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ,  S ) `
  ( G `  N ) )  =  (  seq 1 ( 
 .+  ,  H ,  S ) `  N ) )
 
3.7  Elementary real and complex functions
 
3.7.1  The "shift" operation
 
Syntaxcshi 10213 Extend class notation with function shifter.
 class  shift
 
Definitiondf-shft 10214* Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of  CC) and produces a new function on  CC. See shftval 10224 for its value. (Contributed by NM, 20-Jul-2005.)
 |- 
 shift  =  ( f  e.  _V ,  x  e. 
 CC  |->  { <. y ,  z >.  |  ( y  e. 
 CC  /\  ( y  -  x ) f z ) } )
 
Theoremshftlem 10215* Two ways to write a shifted set  ( B  +  A
). (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  CC  /\  B  C_  CC )  ->  { x  e. 
 CC  |  ( x  -  A )  e.  B }  =  { x  |  E. y  e.  B  x  =  ( y  +  A ) } )
 
Theoremshftuz 10216* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  { x  e. 
 CC  |  ( x  -  A )  e.  ( ZZ>= `  B ) }  =  ( ZZ>= `  ( B  +  A ) ) )
 
Theoremshftfvalg 10217* The value of the sequence shifter operation is a function on  CC.  A is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  CC  /\  F  e.  V )  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }
 )
 
Theoremovshftex 10218 Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)
 |-  ( ( F  e.  V  /\  A  e.  CC )  ->  ( F  shift  A )  e.  _V )
 
Theoremshftfibg 10219 Value of a fiber of the relation  F. (Contributed by Jim Kingdon, 15-Aug-2021.)
 |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) " { B } )  =  ( F " { ( B  -  A ) }
 ) )
 
Theoremshftfval 10220* The value of the sequence shifter operation is a function on  CC.  A is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }
 )
 
Theoremshftdm 10221* Domain of a relation shifted by  A. The set on the right is more commonly notated as  ( dom  F  +  A ) (meaning add  A to every element of  dom  F). (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( A  e.  CC  ->  dom  ( F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e.  dom  F }
 )
 
Theoremshftfib 10222 Value of a fiber of the relation  F. (Contributed by Mario Carneiro, 4-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A )
 " { B }
 )  =  ( F
 " { ( B  -  A ) }
 ) )
 
Theoremshftfn 10223* Functionality and domain of a sequence shifted by  A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  Fn 
 { x  e.  CC  |  ( x  -  A )  e.  B }
 )
 
Theoremshftval 10224 Value of a sequence shifted by  A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A ) `
  B )  =  ( F `  ( B  -  A ) ) )
 
Theoremshftval2 10225 Value of a sequence shifted by  A  -  B. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( F  shift  ( A  -  B ) ) `  ( A  +  C ) )  =  ( F `  ( B  +  C ) ) )
 
Theoremshftval3 10226 Value of a sequence shifted by  A  -  B. (Contributed by NM, 20-Jul-2005.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  ( A  -  B ) ) `
  A )  =  ( F `  B ) )
 
Theoremshftval4 10227 Value of a sequence shifted by  -u A. (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  -u A ) `  B )  =  ( F `  ( A  +  B )
 ) )
 
Theoremshftval5 10228 Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A ) `
  ( B  +  A ) )  =  ( F `  B ) )
 
Theoremshftf 10229* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( F : B --> C  /\  A  e.  CC )  ->  ( F  shift  A ) : { x  e. 
 CC  |  ( x  -  A )  e.  B } --> C )
 
Theorem2shfti 10230 Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A ) 
 shift  B )  =  ( F  shift  ( A  +  B ) ) )
 
Theoremshftidt2 10231 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( F  shift  0 )  =  ( F  |`  CC )
 
Theoremshftidt 10232 Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( A  e.  CC  ->  ( ( F 
 shift  0 ) `  A )  =  ( F `  A ) )
 
Theoremshftcan1 10233 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( ( F  shift  A )  shift  -u A ) `  B )  =  ( F `  B ) )
 
Theoremshftcan2 10234 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( ( F  shift  -u A )  shift  A ) `
  B )  =  ( F `  B ) )
 
Theoremshftvalg 10235 Value of a sequence shifted by  A. (Contributed by Scott Fenton, 16-Dec-2017.)
 |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) `  B )  =  ( F `  ( B  -  A ) ) )
 
Theoremshftval4g 10236 Value of a sequence shifted by  -u A. (Contributed by Jim Kingdon, 19-Aug-2021.)
 |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  -u A ) `  B )  =  ( F `  ( A  +  B ) ) )
 
Theoremseq3shft 10237* Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 17-Oct-2022.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  -  N ) ) ) 
 ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  seq
 M (  .+  ,  ( F  shift  N ) )  =  (  seq ( M  -  N ) (  .+  ,  F )  shift  N ) )
 
3.7.2  Real and imaginary parts; conjugate
 
Syntaxccj 10238 Extend class notation to include complex conjugate function.
 class  *
 
Syntaxcre 10239 Extend class notation to include real part of a complex number.
 class  Re
 
Syntaxcim 10240 Extend class notation to include imaginary part of a complex number.
 class  Im
 
Definitiondf-cj 10241* Define the complex conjugate function. See cjcli 10312 for its closure and cjval 10244 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  *  =  ( x  e.  CC  |->  ( iota_ y  e.  CC  ( ( x  +  y )  e.  RR  /\  ( _i  x.  ( x  -  y ) )  e. 
 RR ) ) )
 
Definitiondf-re 10242 Define a function whose value is the real part of a complex number. See reval 10248 for its value, recli 10310 for its closure, and replim 10258 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Re  =  ( x  e.  CC  |->  ( ( x  +  ( * `
  x ) ) 
 /  2 ) )
 
Definitiondf-im 10243 Define a function whose value is the imaginary part of a complex number. See imval 10249 for its value, imcli 10311 for its closure, and replim 10258 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Im  =  ( x  e.  CC  |->  ( Re
 `  ( x  /  _i ) ) )
 
Theoremcjval 10244* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( * `  A )  =  ( iota_ x  e. 
 CC  ( ( A  +  x )  e. 
 RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) ) )
 
Theoremcjth 10245 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( ( A  +  ( * `  A ) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) )
 
Theoremcjf 10246 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  * : CC --> CC
 
Theoremcjcl 10247 The conjugate of a complex number is a complex number (closure law). (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( * `  A )  e.  CC )
 
Theoremreval 10248 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Re `  A )  =  ( ( A  +  ( * `  A ) )  / 
 2 ) )
 
Theoremimval 10249 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Im `  A )  =  ( Re `  ( A  /  _i ) ) )
 
Theoremimre 10250 The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Im `  A )  =  ( Re `  ( -u _i  x.  A ) ) )
 
Theoremreim 10251 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  ( Re `  A )  =  ( Im `  ( _i  x.  A ) ) )
 
Theoremrecl 10252 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Re `  A )  e.  RR )
 
Theoremimcl 10253 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Im `  A )  e.  RR )
 
Theoremref 10254 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  Re : CC --> RR
 
Theoremimf 10255 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  Im : CC --> RR
 
Theoremcrre 10256 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A )
 
Theoremcrim 10257 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B )
 
Theoremreplim 10258 Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( A  e.  CC  ->  A  =  ( ( Re `  A )  +  ( _i  x.  ( Im `  A ) ) ) )
 
Theoremremim 10259 Value of the conjugate of a complex number. The value is the real part minus  _i times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( A  e.  CC  ->  ( * `  A )  =  ( ( Re `  A )  -  ( _i  x.  ( Im `  A ) ) ) )
 
Theoremreim0 10260 The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( A  e.  RR  ->  ( Im `  A )  =  0 )
 
Theoremreim0b 10261 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)
 |-  ( A  e.  CC  ->  ( A  e.  RR  <->  ( Im `  A )  =  0 ) )
 
Theoremrereb 10262 A number is real iff it equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 20-Aug-2008.)
 |-  ( A  e.  CC  ->  ( A  e.  RR  <->  ( Re `  A )  =  A ) )
 
Theoremmulreap 10263 A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  ( A  e.  RR  <->  ( B  x.  A )  e. 
 RR ) )
 
Theoremrere 10264 A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  ( A  e.  RR  ->  ( Re `  A )  =  A )
 
Theoremcjreb 10265 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( A  e.  RR  <->  ( * `  A )  =  A ) )
 
Theoremrecj 10266 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( Re `  ( * `  A ) )  =  ( Re `  A ) )
 
Theoremreneg 10267 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( Re `  -u A )  =  -u ( Re
 `  A ) )
 
Theoremreadd 10268 Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  +  B ) )  =  (
 ( Re `  A )  +  ( Re `  B ) ) )
 
Theoremresub 10269 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  -  B ) )  =  (
 ( Re `  A )  -  ( Re `  B ) ) )
 
Theoremremullem 10270 Lemma for remul 10271, immul 10278, and cjmul 10284. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( Re
 `  ( A  x.  B ) )  =  ( ( ( Re
 `  A )  x.  ( Re `  B ) )  -  (
 ( Im `  A )  x.  ( Im `  B ) ) ) 
 /\  ( Im `  ( A  x.  B ) )  =  (
 ( ( Re `  A )  x.  ( Im `  B ) )  +  ( ( Im
 `  A )  x.  ( Re `  B ) ) )  /\  ( * `  ( A  x.  B ) )  =  ( ( * `
  A )  x.  ( * `  B ) ) ) )
 
Theoremremul 10271 Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  x.  B ) )  =  (
 ( ( Re `  A )  x.  ( Re `  B ) )  -  ( ( Im
 `  A )  x.  ( Im `  B ) ) ) )
 
Theoremremul2 10272 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  CC )  ->  ( Re `  ( A  x.  B ) )  =  ( A  x.  ( Re `  B ) ) )
 
Theoremredivap 10273 Real part of a division. Related to remul2 10272. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  ( Re `  ( A 
 /  B ) )  =  ( ( Re
 `  A )  /  B ) )
 
Theoremimcj 10274 Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( Im `  ( * `  A ) )  =  -u ( Im `  A ) )
 
Theoremimneg 10275 The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( Im `  -u A )  =  -u ( Im
 `  A ) )
 
Theoremimadd 10276 Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Im `  ( A  +  B ) )  =  (
 ( Im `  A )  +  ( Im `  B ) ) )
 
Theoremimsub 10277 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Im `  ( A  -  B ) )  =  (
 ( Im `  A )  -  ( Im `  B ) ) )
 
Theoremimmul 10278 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Im `  ( A  x.  B ) )  =  (
 ( ( Re `  A )  x.  ( Im `  B ) )  +  ( ( Im
 `  A )  x.  ( Re `  B ) ) ) )
 
Theoremimmul2 10279 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  CC )  ->  ( Im `  ( A  x.  B ) )  =  ( A  x.  ( Im `  B ) ) )
 
Theoremimdivap 10280 Imaginary part of a division. Related to immul2 10279. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  ( Im `  ( A 
 /  B ) )  =  ( ( Im
 `  A )  /  B ) )
 
Theoremcjre 10281 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)
 |-  ( A  e.  RR  ->  ( * `  A )  =  A )
 
Theoremcjcj 10282 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( * `  ( * `  A ) )  =  A )
 
Theoremcjadd 10283 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  +  B ) )  =  (
 ( * `  A )  +  ( * `  B ) ) )
 
Theoremcjmul 10284 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  B ) )  =  (
 ( * `  A )  x.  ( * `  B ) ) )
 
Theoremipcnval 10285 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  x.  ( * `  B ) ) )  =  ( ( ( Re `  A )  x.  ( Re `  B ) )  +  ( ( Im `  A )  x.  ( Im `  B ) ) ) )
 
Theoremcjmulrcl 10286 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( A  x.  ( * `  A ) )  e.  RR )
 
Theoremcjmulval 10287 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( A  x.  ( * `  A ) )  =  ( ( ( Re `  A ) ^ 2 )  +  ( ( Im `  A ) ^ 2
 ) ) )
 
Theoremcjmulge0 10288 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  0  <_  ( A  x.  ( * `  A ) ) )
 
Theoremcjneg 10289 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( * `  -u A )  =  -u ( * `
  A ) )
 
Theoremaddcj 10290 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( A  +  ( * `  A ) )  =  ( 2  x.  ( Re `  A ) ) )
 
Theoremcjsub 10291 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  -  B ) )  =  (
 ( * `  A )  -  ( * `  B ) ) )
 
Theoremcjexp 10292 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( * `  ( A ^ N ) )  =  ( ( * `  A ) ^ N ) )
 
Theoremimval2 10293 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)
 |-  ( A  e.  CC  ->  ( Im `  A )  =  ( ( A  -  ( * `  A ) )  /  ( 2  x.  _i ) ) )
 
Theoremre0 10294 The real part of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( Re `  0
 )  =  0
 
Theoremim0 10295 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( Im `  0
 )  =  0
 
Theoremre1 10296 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Re `  1
 )  =  1
 
Theoremim1 10297 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Im `  1
 )  =  0
 
Theoremrei 10298 The real part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Re `  _i )  =  0
 
Theoremimi 10299 The imaginary part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Im `  _i )  =  1
 
Theoremcj0 10300 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( * `  0
 )  =  0
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