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Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorems2fv1 11501 Extract the second symbol from a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( B  e.  V  ->  ( <" A B "> `  1 )  =  B )
 
Theorems2len 11502 The length of a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B "> )  =  2
 
Theorems3fv0 11503 Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  ( A  e.  V  ->  ( <" A B C "> `  0
 )  =  A )
 
Theorems3fv1 11504 Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  ( B  e.  V  ->  ( <" A B C "> `  1
 )  =  B )
 
Theorems3fv2 11505 Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  ( C  e.  V  ->  ( <" A B C "> `  2
 )  =  C )
 
Theorems3len 11506 The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C "> )  =  3
 
Theorems4len 11507 The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D "> )  =  4
 
Theorems5len 11508 The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E "> )  =  5
 
Theorems6len 11509 The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F "> )  =  6
 
Theorems7len 11510 The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F G "> )  =  7
 
Theorems8len 11511 The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F G H "> )  =  8
 
Theorems2co 11512 Mapping a doubleton by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  F : X --> Y )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   =>    |-  ( ph  ->  ( F  o.  <" A B "> )  =  <" ( F `  A ) ( F `  B ) "> )
 
Theorems3co 11513 Mapping a length 3 string by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  F : X --> Y )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   =>    |-  ( ph  ->  ( F  o.  <" A B C "> )  =  <" ( F `
  A ) ( F `  B ) ( F `  C ) "> )
 
Theorems0s1 11514 Concatenation of fixed length strings. (This special case of ccatlid 11399 is provided to complete the pattern s0s1 11514, df-s2 11463, df-s3 11464, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
 |- 
 <" A ">  =  ( (/) concat  <" A "> )
 
Theorems1s2 11515 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C ">  =  ( <" A "> concat  <" B C "> )
 
Theorems1s3 11516 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  =  ( <" A "> concat  <" B C D "> )
 
Theorems1s4 11517 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E ">  =  ( <" A "> concat 
 <" B C D E "> )
 
Theorems1s5 11518 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F ">  =  ( <" A "> concat 
 <" B C D E F "> )
 
Theorems1s6 11519 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G ">  =  ( <" A "> concat  <" B C D E F G "> )
 
Theorems1s7 11520 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G H ">  =  ( <" A "> concat  <" B C D E F G H "> )
 
Theorems2s2 11521 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  =  ( <" A B "> concat 
 <" C D "> )
 
Theorems4s2 11522 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F ">  =  ( <" A B C D "> concat  <" E F "> )
 
Theorems4s3 11523 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G ">  =  ( <" A B C D "> concat  <" E F G "> )
 
Theorems4s4 11524 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G H ">  =  ( <" A B C D "> concat  <" E F G H "> )
 
Theoremswrds2 11525 Extract two adjacent symbols from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( ( W  e. Word  A 
 /\  I  e.  NN0  /\  ( I  +  1 )  e.  ( 0..^ ( # `  W ) ) )  ->  ( W substr  <. I ,  ( I  +  2
 ) >. )  =  <" ( W `  I
 ) ( W `  ( I  +  1
 ) ) "> )
 
5.7  Elementary real and complex functions
 
5.7.1  The "shift" operation
 
Syntaxcshi 11526 Extend class notation with function shifter.
 class  shift
 
Definitiondf-shft 11527* 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 11534 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 11528* 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 11529* 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 ) ) )
 
Theoremshftfval 11530* 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 11531* 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 11532 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 11533* 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 11534 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 11535 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 11536 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 11537 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 11538 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 11539* 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 11540 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 11541 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( F  shift  0 )  =  ( F  |`  CC )
 
Theoremshftidt 11542 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 11543 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 11544 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 ) )
 
Theoremseqshft 11545 Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-Feb-2014.)
 |-  F  e.  _V   =>    |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  seq  M (  .+  ,  ( F 
 shift  N ) )  =  (  seq  ( M  -  N ) ( 
 .+  ,  F )  shift  N ) )
 
5.7.2  Real and imaginary parts; conjugate
 
Syntaxccj 11546 Extend class notation to include complex conjugate function.
 class  *
 
Syntaxcre 11547 Extend class notation to include real part of a complex number.
 class  Re
 
Syntaxcim 11548 Extend class notation to include imaginary part of a complex number.
 class  Im
 
Definitiondf-cj 11549* Define the complex conjugate function. See cjcli 11619 for its closure and cjval 11552 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 11550 Define a function whose value is the real part of a complex number. See reval 11556 for its value, recli 11617 for its closure, and replim 11566 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Re  =  ( x  e.  CC  |->  ( ( x  +  ( * `
  x ) ) 
 /  2 ) )
 
Definitiondf-im 11551 Define a function whose value is the imaginary part of a complex number. See imval 11557 for its value, imcli 11618 for its closure, and replim 11566 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Im  =  ( x  e.  CC  |->  ( Re
 `  ( x  /  _i ) ) )
 
Theoremcjval 11552* 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 11553 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 11554 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  * : CC --> CC
 
Theoremcjcl 11555 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 11556 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 11557 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 11558 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 11559 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 11560 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 11561 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 11562 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 11563 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 11564 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 11565 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 11566 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 11567 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 11568 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 11569 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 11570 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 ) )
 
Theoremmulre 11571 A product with a nonzero real multiplier is real iff the multiplicand is real. (Contributed by NM, 21-Aug-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 ) 
 ->  ( A  e.  RR  <->  ( B  x.  A )  e. 
 RR ) )
 
Theoremrere 11572 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 11573 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 11574 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( Re `  ( * `  A ) )  =  ( Re `  A ) )
 
Theoremreneg 11575 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 11576 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 11577 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 11578 Lemma for remul 11579, immul 11586, and cjmul 11592. (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 11579 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 11580 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 ) ) )
 
Theoremrediv 11581 Real part of a division. Related to remul2 11580. (Contributed by David A. Wheeler, 10-Jun-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 ) 
 ->  ( Re `  ( A  /  B ) )  =  ( ( Re
 `  A )  /  B ) )
 
Theoremimcj 11582 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 11583 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 11584 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 11585 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 11586 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 11587 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 ) ) )
 
Theoremimdiv 11588 Imaginary part of a division. Related to immul2 11587. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B  =/=  0 ) 
 ->  ( Im `  ( A  /  B ) )  =  ( ( Im
 `  A )  /  B ) )
 
Theoremcjre 11589 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 11590 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 11591 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 11592 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 11593 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 11594 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 11595 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 11596 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 11597 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 11598 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 11599 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  -  B ) )  =  (
 ( * `  A )  -  ( * `  B ) ) )
 
Theoremcjexp 11600 Complex conjugate of natural number exponentiation. (Contributed by NM, 7-Jun-2006.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( * `  ( A ^ N ) )  =  ( ( * `  A ) ^ N ) )
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