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Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcrre 10901 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 10902 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 10903 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 10904 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 10905 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 10906 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 10907 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 10908 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 10909 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 10910 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 10911 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( Re `  ( * `  A ) )  =  ( Re `  A ) )
 
Theoremreneg 10912 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 10913 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 10914 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 10915 Lemma for remul 10916, immul 10923, and cjmul 10929. (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 10916 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 10917 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 10918 Real part of a division. Related to remul2 10917. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  ( Re `  ( A 
 /  B ) )  =  ( ( Re
 `  A )  /  B ) )
 
Theoremimcj 10919 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 10920 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 10921 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 10922 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 10923 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 10924 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 10925 Imaginary part of a division. Related to immul2 10924. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  RR  /\  B #  0 )  ->  ( Im `  ( A 
 /  B ) )  =  ( ( Im
 `  A )  /  B ) )
 
Theoremcjre 10926 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 10927 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 10928 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 10929 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 10930 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 10931 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 10932 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 10933 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 10934 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 10935 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 10936 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  -  B ) )  =  (
 ( * `  A )  -  ( * `  B ) ) )
 
Theoremcjexp 10937 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
 |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  ( * `  ( A ^ N ) )  =  ( ( * `  A ) ^ N ) )
 
Theoremimval2 10938 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 10939 The real part of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( Re `  0
 )  =  0
 
Theoremim0 10940 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( Im `  0
 )  =  0
 
Theoremre1 10941 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Re `  1
 )  =  1
 
Theoremim1 10942 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Im `  1
 )  =  0
 
Theoremrei 10943 The real part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Re `  _i )  =  0
 
Theoremimi 10944 The imaginary part of  _i. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( Im `  _i )  =  1
 
Theoremcj0 10945 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
 |-  ( * `  0
 )  =  0
 
Theoremcji 10946 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
 |-  ( * `  _i )  =  -u _i
 
Theoremcjreim 10947 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 10948 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 10949 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 10950 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `
  A ) #  ( * `  B )  <->  A #  B ) )
 
Theoremcjap0 10951 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 10952 A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10951 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)
 |-  ( A  e.  CC  ->  ( A  =/=  0  <->  ( * `  A )  =/=  0 ) )
 
Theoremcjdivap 10953 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 10954* The inverse to the canonical bijection from  ( RR  X.  RR ) to  CC from cnref1o 9682. (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 10955 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 10956 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 10957 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( * `  A )  e.  CC
 
Theoremreplimi 10958 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 10959 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 10960 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 10961 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 10962 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 10963 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  ( * `  A ) )  =  ( Re
 `  A )
 
Theoremimcji 10964 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  ( * `  A ) )  =  -u ( Im `  A )
 
Theoremcjmulrcli 10965 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  x.  ( * `  A ) )  e.  RR
 
Theoremcjmulvali 10966 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 10967 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
 |-  A  e.  CC   =>    |-  0  <_  ( A  x.  ( * `  A ) )
 
Theoremrenegi 10968 Real part of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( Re `  -u A )  =  -u ( Re `  A )
 
Theoremimnegi 10969 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( Im `  -u A )  =  -u ( Im `  A )
 
Theoremcjnegi 10970 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( * `  -u A )  =  -u ( * `  A )
 
Theoremaddcji 10971 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 10972 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 10973 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 10974 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 10975 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 10976 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 10977 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 10978 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 10979 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 10980 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 10981 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 10982 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 10983 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 10984 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  A )  e. 
 CC )
 
Theoremreplimd 10985 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 10986 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 10987 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 10988 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 10989 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 10990 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 10991 A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 10992 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  ( * `  A )  =/=  0 )
 
Theoremcjap0d 10992 A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  ( * `  A ) #  0 )
 
Theoremrecjd 10993 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  ( * `  A ) )  =  ( Re `  A ) )
 
Theoremimcjd 10994 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  ( * `  A ) )  =  -u ( Im `  A ) )
 
Theoremcjmulrcld 10995 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 10996 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 10997 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 10998 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Re `  -u A )  =  -u ( Re `  A ) )
 
Theoremimnegd 10999 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( Im `  -u A )  =  -u ( Im `  A ) )
 
Theoremcjnegd 11000 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( * `  -u A )  =  -u ( * `  A ) )
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