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Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcjre 11501 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 11502 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 11503 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 11504 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 11505 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 11506 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 11507 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 11508 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 11509 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 11510 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 11511 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  -  B ) )  =  (
 ( * `  A )  -  ( * `  B ) ) )
 
Theoremcjexp 11512 Complex conjugate of natural number exponentiation. (Contributed by NM, 7-Jun-2006.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( * `  ( A ^ N ) )  =  ( ( * `  A ) ^ N ) )
 
Theoremimval2 11513 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 11514 The real part of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( Re `  0
 )  =  0
 
Theoremim0 11515 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( Im `  0
 )  =  0
 
Theoremre1 11516 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Re `  1
 )  =  1
 
Theoremim1 11517 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Im `  1
 )  =  0
 
Theoremrei 11518 The real part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Re `  _i )  =  0
 
Theoremimi 11519 The imaginary part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Im `  _i )  =  1
 
Theoremcj0 11520 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( * `  0
 )  =  0
 
Theoremcji 11521 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
 |-  ( * `  _i )  =  -u _i
 
Theoremcjreim 11522 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  +  ( _i  x.  B ) ) )  =  ( A  -  ( _i  x.  B ) ) )
 
Theoremcjreim2 11523 The conjugate of the representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( * `  ( A  -  ( _i  x.  B ) ) )  =  ( A  +  ( _i  x.  B ) ) )
 
Theoremcj11 11524 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `
  A )  =  ( * `  B ) 
 <->  A  =  B ) )
 
Theoremcjne0 11525 A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)
 |-  ( A  e.  CC  ->  ( A  =/=  0  <->  ( * `  A )  =/=  0 ) )
 
Theoremcjdiv 11526 Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) 
 ->  ( * `  ( A  /  B ) )  =  ( ( * `
  A )  /  ( * `  B ) ) )
 
Theoremcnrecnv 11527* The inverse to the canonical bijection from  ( RR  X.  RR ) to  CC from cnref1o 10228. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y
 ) ) )   =>    |-  `' F  =  ( z  e.  CC  |->  <.
 ( Re `  z
 ) ,  ( Im
 `  z ) >. )
 
Theoremsqeqd 11528 A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A ^
 2 )  =  ( B ^ 2 ) )   &    |-  ( ph  ->  0 
 <_  ( Re `  A ) )   &    |-  ( ph  ->  0 
 <_  ( Re `  B ) )   &    |-  ( ( ph  /\  ( Re `  A )  =  0  /\  ( Re `  B )  =  0 )  ->  A  =  B )   =>    |-  ( ph  ->  A  =  B )
 
Theoremrecli 11529 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  A )  e.  RR
 
Theoremimcli 11530 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  A )  e.  RR
 
Theoremcjcli 11531 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( * `  A )  e.  CC
 
Theoremreplimi 11532 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   =>    |-  A  =  ( ( Re `  A )  +  ( _i  x.  ( Im `  A ) ) )
 
Theoremcjcji 11533 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( * `  ( * `  A ) )  =  A
 
Theoremreim0bi 11534 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  e.  RR 
 <->  ( Im `  A )  =  0 )
 
Theoremrerebi 11535 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( A  e.  RR 
 <->  ( Re `  A )  =  A )
 
Theoremcjrebi 11536 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( A  e.  RR 
 <->  ( * `  A )  =  A )
 
Theoremrecji 11537 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  ( * `  A ) )  =  ( Re
 `  A )
 
Theoremimcji 11538 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  ( * `  A ) )  =  -u ( Im `  A )
 
Theoremcjmulrcli 11539 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  x.  ( * `  A ) )  e.  RR
 
Theoremcjmulvali 11540 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( A  x.  ( * `  A ) )  =  ( ( ( Re `  A ) ^ 2 )  +  ( ( Im `  A ) ^ 2
 ) )
 
Theoremcjmulge0i 11541 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
 |-  A  e.  CC   =>    |-  0  <_  ( A  x.  ( * `  A ) )
 
Theoremrenegi 11542 Real part of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  -u A )  =  -u ( Re `  A )
 
Theoremimnegi 11543 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  -u A )  =  -u ( Im `  A )
 
Theoremcjnegi 11544 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( * `  -u A )  =  -u ( * `  A )
 
Theoremaddcji 11545 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( A  +  ( * `  A ) )  =  ( 2  x.  ( Re `  A ) )
 
Theoremreaddi 11546 Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Re `  ( A  +  B )
 )  =  ( ( Re `  A )  +  ( Re `  B ) )
 
Theoremimaddi 11547 Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Im `  ( A  +  B )
 )  =  ( ( Im `  A )  +  ( Im `  B ) )
 
Theoremremuli 11548 Real part of a product. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Re `  ( A  x.  B ) )  =  ( ( ( Re `  A )  x.  ( Re `  B ) )  -  ( ( Im `  A )  x.  ( Im `  B ) ) )
 
Theoremimmuli 11549 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Im `  ( A  x.  B ) )  =  ( ( ( Re `  A )  x.  ( Im `  B ) )  +  ( ( Im `  A )  x.  ( Re `  B ) ) )
 
Theoremcjaddi 11550 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( * `  ( A  +  B )
 )  =  ( ( * `  A )  +  ( * `  B ) )
 
Theoremcjmuli 11551 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( * `  ( A  x.  B ) )  =  ( ( * `
  A )  x.  ( * `  B ) )
 
Theoremipcni 11552 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( Re `  ( A  x.  ( * `  B ) ) )  =  ( ( ( Re `  A )  x.  ( Re `  B ) )  +  ( ( Im `  A )  x.  ( Im `  B ) ) )
 
Theoremcjdivi 11553 Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B  =/=  0  ->  ( * `  ( A  /  B ) )  =  ( ( * `
  A )  /  ( * `  B ) ) )
 
Theoremcrrei 11554 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A
 
Theoremcrimi 11555 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B
 
Theoremrecld 11556 The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  A )  e. 
 RR )
 
Theoremimcld 11557 The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  A )  e. 
 RR )
 
Theoremcjcld 11558 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  A )  e. 
 CC )
 
Theoremreplimd 11559 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  A  =  ( ( Re `  A )  +  ( _i  x.  ( Im `  A ) ) ) )
 
Theoremremimd 11560 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 Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  A )  =  ( ( Re `  A )  -  ( _i  x.  ( Im `  A ) ) ) )
 
Theoremcjcjd 11561 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  ( * `  A ) )  =  A )
 
Theoremreim0bd 11562 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( Im `  A )  =  0 )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremrerebd 11563 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( Re `  A )  =  A )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremcjrebd 11564 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  ( * `  A )  =  A )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremcjne0d 11565 A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( * `  A )  =/=  0 )
 
Theoremrecjd 11566 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( * `  A ) )  =  ( Re `  A ) )
 
Theoremimcjd 11567 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( * `  A ) )  =  -u ( Im `  A ) )
 
Theoremcjmulrcld 11568 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( * `  A ) )  e. 
 RR )
 
Theoremcjmulvald 11569 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( * `  A ) )  =  ( ( ( Re
 `  A ) ^
 2 )  +  (
 ( Im `  A ) ^ 2 ) ) )
 
Theoremcjmulge0d 11570 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  0  <_  ( A  x.  ( * `  A ) ) )
 
Theoremrenegd 11571 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  -u A )  =  -u ( Re `  A ) )
 
Theoremimnegd 11572 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  -u A )  =  -u ( Im `  A ) )
 
Theoremcjnegd 11573 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  -u A )  =  -u ( * `  A ) )
 
Theoremaddcjd 11574 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  ( * `  A ) )  =  ( 2  x.  ( Re `  A ) ) )
 
Theoremcjexpd 11575 Complex conjugate of natural number exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( * `  ( A ^ N ) )  =  ( ( * `  A ) ^ N ) )
 
Theoremreaddd 11576 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  +  B ) )  =  ( ( Re
 `  A )  +  ( Re `  B ) ) )
 
Theoremimaddd 11577 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  +  B ) )  =  ( ( Im
 `  A )  +  ( Im `  B ) ) )
 
Theoremresubd 11578 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  -  B ) )  =  ( ( Re
 `  A )  -  ( Re `  B ) ) )
 
Theoremimsubd 11579 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  -  B ) )  =  ( ( Im
 `  A )  -  ( Im `  B ) ) )
 
Theoremremuld 11580 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  x.  B ) )  =  ( ( ( Re `  A )  x.  ( Re `  B ) )  -  ( ( Im `  A )  x.  ( Im `  B ) ) ) )
 
Theoremimmuld 11581 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  x.  B ) )  =  ( ( ( Re `  A )  x.  ( Im `  B ) )  +  ( ( Im `  A )  x.  ( Re `  B ) ) ) )
 
Theoremcjaddd 11582 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( * `  ( A  +  B ) )  =  ( ( * `
  A )  +  ( * `  B ) ) )
 
Theoremcjmuld 11583 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( * `  ( A  x.  B ) )  =  ( ( * `
  A )  x.  ( * `  B ) ) )
 
Theoremipcnd 11584 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  x.  ( * `  B ) ) )  =  ( ( ( Re `  A )  x.  ( Re `  B ) )  +  ( ( Im `  A )  x.  ( Im `  B ) ) ) )
 
Theoremcjdivd 11585 Complex conjugate distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  =/=  0
 )   =>    |-  ( ph  ->  ( * `  ( A  /  B ) )  =  ( ( * `  A )  /  ( * `  B ) ) )
 
Theoremrered 11586 A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( Re `  A )  =  A )
 
Theoremreim0d 11587 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( Im `  A )  =  0 )
 
Theoremcjred 11588 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( * `  A )  =  A )
 
Theoremremul2d 11589 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( A  x.  B ) )  =  ( A  x.  ( Re `  B ) ) )
 
Theoremimmul2d 11590 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( A  x.  B ) )  =  ( A  x.  ( Im `  B ) ) )
 
Theoremredivd 11591 Real part of a division. Related to remul2 11492. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   =>    |-  ( ph  ->  ( Re `  ( B  /  A ) )  =  ( ( Re `  B )  /  A ) )
 
Theoremimdivd 11592 Imaginary part of a division. Related to remul2 11492. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  0
 )   =>    |-  ( ph  ->  ( Im `  ( B  /  A ) )  =  ( ( Im `  B )  /  A ) )
 
Theoremcrred 11593 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A )
 
Theoremcrimd 11594 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B )
 
5.7.3  Square root; absolute value
 
Syntaxcsqr 11595 Extend class notation to include square root of a complex number.
 class  sqr
 
Syntaxcabs 11596 Extend class notation to include a function for the absolute value (modulus) of a complex number.
 class  abs
 
Definitiondf-sqr 11597* Define a function whose value is the square root of a complex number. Since  ( y ^
2 )  =  x iff  ( -u y ^
2 )  =  x, we ensure uniqueness by restricting the range to numbers with positive real part, or numbers with 0 real part and nonnegative imaginary part. A description can be found under "Principal square root of a complex number" at http://en.wikipedia.org/wiki/Square_root. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrcl 11722 for its closure, sqrval 11599 for its value, sqrth 11725 and sqsqri 11736 for its relationship to squares, and sqr11i 11745 for uniqueness. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 8-Jul-2013.)

 |- 
 sqr  =  ( x  e.  CC  |->  ( iota_ y  e. 
 CC ( ( y ^ 2 )  =  x  /\  0  <_  ( Re `  y ) 
 /\  ( _i  x.  y )  e/  RR+ )
 ) )
 
Definitiondf-abs 11598 Define the function for the absolute value (modulus) of a complex number. See abscli 11755 for its closure and absval 11600 or absval2i 11757 for its value. (Contributed by NM, 27-Jul-1999.)
 |- 
 abs  =  ( x  e.  CC  |->  ( sqr `  ( x  x.  ( * `  x ) ) ) )
 
Theoremsqrval 11599* Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.)
 |-  ( A  e.  CC  ->  ( sqr `  A )  =  ( iota_ x  e. 
 CC ( ( x ^ 2 )  =  A  /\  0  <_  ( Re `  x ) 
 /\  ( _i  x.  x )  e/  RR+ )
 ) )
 
Theoremabsval 11600 The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( A  e.  CC  ->  ( abs `  A )  =  ( sqr `  ( A  x.  ( * `  A ) ) ) )
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