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Theorem List for Intuitionistic Logic Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-im 10801 Define a function whose value is the imaginary part of a complex number. See imval 10807 for its value, imcli 10869 for its closure, and replim 10816 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Im  =  ( x  e.  CC  |->  ( Re
 `  ( x  /  _i ) ) )
 
Theoremcjval 10802* 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 10803 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 10804 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  * : CC --> CC
 
Theoremcjcl 10805 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 10806 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 10807 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 10808 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 10809 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 10810 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 10811 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 10812 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 10813 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 10814 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 10815 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 10816 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 10817 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 10818 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 10819 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 10820 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 10821 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 10822 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 10823 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 10824 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( Re `  ( * `  A ) )  =  ( Re `  A ) )
 
Theoremreneg 10825 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 10826 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 10827 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 10828 Lemma for remul 10829, immul 10836, and cjmul 10842. (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 10829 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 10830 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 10831 Real part of a division. Related to remul2 10830. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  ( Re `  ( A 
 /  B ) )  =  ( ( Re
 `  A )  /  B ) )
 
Theoremimcj 10832 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 10833 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 10834 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 10835 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 10836 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 10837 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 10838 Imaginary part of a division. Related to immul2 10837. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  ( Im `  ( A 
 /  B ) )  =  ( ( Im
 `  A )  /  B ) )
 
Theoremcjre 10839 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 10840 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 10841 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 10842 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 10843 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 10844 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 10845 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 10846 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 10847 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 10848 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 10849 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  -  B ) )  =  (
 ( * `  A )  -  ( * `  B ) ) )
 
Theoremcjexp 10850 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( * `  ( A ^ N ) )  =  ( ( * `  A ) ^ N ) )
 
Theoremimval2 10851 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 10852 The real part of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( Re `  0
 )  =  0
 
Theoremim0 10853 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( Im `  0
 )  =  0
 
Theoremre1 10854 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Re `  1
 )  =  1
 
Theoremim1 10855 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Im `  1
 )  =  0
 
Theoremrei 10856 The real part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Re `  _i )  =  0
 
Theoremimi 10857 The imaginary part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Im `  _i )  =  1
 
Theoremcj0 10858 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( * `  0
 )  =  0
 
Theoremcji 10859 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
 |-  ( * `  _i )  =  -u _i
 
Theoremcjreim 10860 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 10861 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 10862 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 ) )
 
Theoremcjap 10863 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `
  A ) #  ( * `  B )  <->  A #  B ) )
 
Theoremcjap0 10864 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( A  e.  CC  ->  ( A #  0  <->  ( * `  A ) #  0 )
 )
 
Theoremcjne0 10865 A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10864 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)
 |-  ( A  e.  CC  ->  ( A  =/=  0  <->  ( * `  A )  =/=  0 ) )
 
Theoremcjdivap 10866 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( * `  ( A 
 /  B ) )  =  ( ( * `
  A )  /  ( * `  B ) ) )
 
Theoremcnrecnv 10867* The inverse to the canonical bijection from  ( RR  X.  RR ) to  CC from cnref1o 9602. (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 ) >. )
 
Theoremrecli 10868 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 10869 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 10870 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( * `  A )  e.  CC
 
Theoremreplimi 10871 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 10872 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 10873 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 10874 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 10875 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 10876 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  ( * `  A ) )  =  ( Re
 `  A )
 
Theoremimcji 10877 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  ( * `  A ) )  =  -u ( Im `  A )
 
Theoremcjmulrcli 10878 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  x.  ( * `  A ) )  e.  RR
 
Theoremcjmulvali 10879 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 10880 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
 |-  A  e.  CC   =>    |-  0  <_  ( A  x.  ( * `  A ) )
 
Theoremrenegi 10881 Real part of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  -u A )  =  -u ( Re `  A )
 
Theoremimnegi 10882 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  -u A )  =  -u ( Im `  A )
 
Theoremcjnegi 10883 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( * `  -u A )  =  -u ( * `  A )
 
Theoremaddcji 10884 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 10885 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 10886 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 10887 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 10888 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 10889 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 10890 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 10891 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 ) ) )
 
Theoremcjdivapi 10892 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( B #  0  ->  ( * `  ( A 
 /  B ) )  =  ( ( * `
  A )  /  ( * `  B ) ) )
 
Theoremcrrei 10893 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 10894 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 10895 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 10896 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 10897 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  A )  e. 
 CC )
 
Theoremreplimd 10898 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 10899 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 10900 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 )
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