Type | Label | Description |
Statement |
|
Theorem | immul 10901 |
Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised
by Mario Carneiro, 14-Jul-2014.)
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Theorem | immul2 10902 |
Imaginary part of a product. (Contributed by Mario Carneiro,
2-Aug-2014.)
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        |
|
Theorem | imdivap 10903 |
Imaginary part of a division. Related to immul2 10902. (Contributed by Jim
Kingdon, 14-Jun-2020.)
|
  #                |
|
Theorem | cjre 10904 |
A real number equals its complex conjugate. Proposition 10-3.4(f) of
[Gleason] p. 133. (Contributed by NM,
8-Oct-1999.)
|
    
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|
Theorem | cjcj 10905 |
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.)
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           |
|
Theorem | cjadd 10906 |
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.)
|
      
 
    
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|
Theorem | cjmul 10907 |
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.)
|
      
 
            |
|
Theorem | ipcnval 10908 |
Standard inner product on complex numbers. (Contributed by NM,
29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
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        |
|
Theorem | cjmulrcl 10909 |
A complex number times its conjugate is real. (Contributed by NM,
26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|
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|
Theorem | cjmulval 10910 |
A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.)
(Revised by Mario Carneiro, 14-Jul-2014.)
|
                           |
|
Theorem | cjmulge0 10911 |
A complex number times its conjugate is nonnegative. (Contributed by NM,
26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
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|
Theorem | cjneg 10912 |
Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.)
(Revised by Mario Carneiro, 14-Jul-2014.)
|
     
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|
Theorem | addcj 10913 |
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.)
|
               |
|
Theorem | cjsub 10914 |
Complex conjugate distributes over subtraction. (Contributed by NM,
28-Apr-2005.)
|
      
 
            |
|
Theorem | cjexp 10915 |
Complex conjugate of positive integer exponentiation. (Contributed by
NM, 7-Jun-2006.)
|
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|
Theorem | imval2 10916 |
The imaginary part of a number in terms of complex conjugate.
(Contributed by NM, 30-Apr-2005.)
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|
Theorem | re0 10917 |
The real part of zero. (Contributed by NM, 27-Jul-1999.)
|
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|
Theorem | im0 10918 |
The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
|
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|
Theorem | re1 10919 |
The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
|
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|
Theorem | im1 10920 |
The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
|
     |
|
Theorem | rei 10921 |
The real part of .
(Contributed by Scott Fenton, 9-Jun-2006.)
|
   
 |
|
Theorem | imi 10922 |
The imaginary part of . (Contributed by Scott Fenton,
9-Jun-2006.)
|
   
 |
|
Theorem | cj0 10923 |
The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
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|
Theorem | cji 10924 |
The complex conjugate of the imaginary unit. (Contributed by NM,
26-Mar-2005.)
|
   
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|
Theorem | cjreim 10925 |
The conjugate of a representation of a complex number in terms of real and
imaginary parts. (Contributed by NM, 1-Jul-2005.)
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Theorem | cjreim2 10926 |
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.)
|
      
          |
|
Theorem | cj11 10927 |
Complex conjugate is a one-to-one function. (Contributed by NM,
29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
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|
Theorem | cjap 10928 |
Complex conjugate and apartness. (Contributed by Jim Kingdon,
14-Jun-2020.)
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        #     #    |
|
Theorem | cjap0 10929 |
A number is apart from zero iff its complex conjugate is apart from zero.
(Contributed by Jim Kingdon, 14-Jun-2020.)
|
  #     #
   |
|
Theorem | cjne0 10930 |
A number is nonzero iff its complex conjugate is nonzero. Also see
cjap0 10929 which is similar but for apartness.
(Contributed by NM,
29-Apr-2005.)
|
         |
|
Theorem | cjdivap 10931 |
Complex conjugate distributes over division. (Contributed by Jim Kingdon,
14-Jun-2020.)
|
  #                    |
|
Theorem | cnrecnv 10932* |
The inverse to the canonical bijection from 
 to
from cnref1o 9663. (Contributed by Mario Carneiro,
25-Aug-2014.)
|
   
    
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|
Theorem | recli 10933 |
The real part of a complex number is real (closure law). (Contributed
by NM, 11-May-1999.)
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 |
|
Theorem | imcli 10934 |
The imaginary part of a complex number is real (closure law).
(Contributed by NM, 11-May-1999.)
|
   
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|
Theorem | cjcli 10935 |
Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
|
   
 |
|
Theorem | replimi 10936 |
Construct a complex number from its real and imaginary parts.
(Contributed by NM, 1-Oct-1999.)
|
    
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|
Theorem | cjcji 10937 |
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.)
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         |
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Theorem | reim0bi 10938 |
A number is real iff its imaginary part is 0. (Contributed by NM,
29-May-1999.)
|
    
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|
Theorem | rerebi 10939 |
A real number equals its real part. Proposition 10-3.4(f) of [Gleason]
p. 133. (Contributed by NM, 27-Oct-1999.)
|
    
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|
Theorem | cjrebi 10940 |
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.)
|
    
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|
Theorem | recji 10941 |
Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
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Theorem | imcji 10942 |
Imaginary part of a complex conjugate. (Contributed by NM,
2-Oct-1999.)
|
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|
Theorem | cjmulrcli 10943 |
A complex number times its conjugate is real. (Contributed by NM,
11-May-1999.)
|
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Theorem | cjmulvali 10944 |
A complex number times its conjugate. (Contributed by NM,
2-Oct-1999.)
|
                         |
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Theorem | cjmulge0i 10945 |
A complex number times its conjugate is nonnegative. (Contributed by
NM, 28-May-1999.)
|
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Theorem | renegi 10946 |
Real part of negative. (Contributed by NM, 2-Aug-1999.)
|
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Theorem | imnegi 10947 |
Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
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Theorem | cjnegi 10948 |
Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
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Theorem | addcji 10949 |
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.)
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Theorem | readdi 10950 |
Real part distributes over addition. (Contributed by NM,
28-Jul-1999.)
|
   
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Theorem | imaddi 10951 |
Imaginary part distributes over addition. (Contributed by NM,
28-Jul-1999.)
|
   
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|
Theorem | remuli 10952 |
Real part of a product. (Contributed by NM, 28-Jul-1999.)
|
                     
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Theorem | immuli 10953 |
Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
|
                     
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|
Theorem | cjaddi 10954 |
Complex conjugate distributes over addition. Proposition 10-3.4(a) of
[Gleason] p. 133. (Contributed by NM,
28-Jul-1999.)
|
   
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|
Theorem | cjmuli 10955 |
Complex conjugate distributes over multiplication. Proposition
10-3.4(c) of [Gleason] p. 133.
(Contributed by NM, 28-Jul-1999.)
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|
Theorem | ipcni 10956 |
Standard inner product on complex numbers. (Contributed by NM,
2-Oct-1999.)
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Theorem | cjdivapi 10957 |
Complex conjugate distributes over division. (Contributed by Jim
Kingdon, 14-Jun-2020.)
|
 #                   |
|
Theorem | crrei 10958 |
The real part of a complex number representation. Definition 10-3.1 of
[Gleason] p. 132. (Contributed by NM,
10-May-1999.)
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Theorem | crimi 10959 |
The imaginary part of a complex number representation. Definition
10-3.1 of [Gleason] p. 132.
(Contributed by NM, 10-May-1999.)
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Theorem | recld 10960 |
The real part of a complex number is real (closure law). (Contributed
by Mario Carneiro, 29-May-2016.)
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Theorem | imcld 10961 |
The imaginary part of a complex number is real (closure law).
(Contributed by Mario Carneiro, 29-May-2016.)
|
         |
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Theorem | cjcld 10962 |
Closure law for complex conjugate. (Contributed by Mario Carneiro,
29-May-2016.)
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         |
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Theorem | replimd 10963 |
Construct a complex number from its real and imaginary parts.
(Contributed by Mario Carneiro, 29-May-2016.)
|
       
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Theorem | remimd 10964 |
Value of the conjugate of a complex number. The value is the real part
minus times
the imaginary part. Definition 10-3.2 of [Gleason]
p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
|
           
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Theorem | cjcjd 10965 |
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.)
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Theorem | reim0bd 10966 |
A number is real iff its imaginary part is 0. (Contributed by Mario
Carneiro, 29-May-2016.)
|
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Theorem | rerebd 10967 |
A real number equals its real part. Proposition 10-3.4(f) of
[Gleason] p. 133. (Contributed by
Mario Carneiro, 29-May-2016.)
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Theorem | cjrebd 10968 |
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.)
|
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Theorem | cjne0d 10969 |
A number which is nonzero has a complex conjugate which is nonzero.
Also see cjap0d 10970 which is similar but for apartness.
(Contributed by
Mario Carneiro, 29-May-2016.)
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Theorem | cjap0d 10970 |
A number which is apart from zero has a complex conjugate which is
apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
|
   #       #   |
|
Theorem | recjd 10971 |
Real part of a complex conjugate. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | imcjd 10972 |
Imaginary part of a complex conjugate. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | cjmulrcld 10973 |
A complex number times its conjugate is real. (Contributed by Mario
Carneiro, 29-May-2016.)
|
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Theorem | cjmulvald 10974 |
A complex number times its conjugate. (Contributed by Mario Carneiro,
29-May-2016.)
|
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Theorem | cjmulge0d 10975 |
A complex number times its conjugate is nonnegative. (Contributed by
Mario Carneiro, 29-May-2016.)
|
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Theorem | renegd 10976 |
Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
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Theorem | imnegd 10977 |
Imaginary part of negative. (Contributed by Mario Carneiro,
29-May-2016.)
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Theorem | cjnegd 10978 |
Complex conjugate of negative. (Contributed by Mario Carneiro,
29-May-2016.)
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|
Theorem | addcjd 10979 |
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.)
|
   
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Theorem | cjexpd 10980 |
Complex conjugate of positive integer exponentiation. (Contributed by
Mario Carneiro, 29-May-2016.)
|
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|
Theorem | readdd 10981 |
Real part distributes over addition. (Contributed by Mario Carneiro,
29-May-2016.)
|
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Theorem | imaddd 10982 |
Imaginary part distributes over addition. (Contributed by Mario
Carneiro, 29-May-2016.)
|
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|
Theorem | resubd 10983 |
Real part distributes over subtraction. (Contributed by Mario Carneiro,
29-May-2016.)
|
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Theorem | imsubd 10984 |
Imaginary part distributes over subtraction. (Contributed by Mario
Carneiro, 29-May-2016.)
|
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|
Theorem | remuld 10985 |
Real part of a product. (Contributed by Mario Carneiro,
29-May-2016.)
|
                          
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|
Theorem | immuld 10986 |
Imaginary part of a product. (Contributed by Mario Carneiro,
29-May-2016.)
|
                          
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|
Theorem | cjaddd 10987 |
Complex conjugate distributes over addition. Proposition 10-3.4(a) of
[Gleason] p. 133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
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Theorem | cjmuld 10988 |
Complex conjugate distributes over multiplication. Proposition
10-3.4(c) of [Gleason] p. 133.
(Contributed by Mario Carneiro,
29-May-2016.)
|
                       |
|
Theorem | ipcnd 10989 |
Standard inner product on complex numbers. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                              
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|
Theorem | cjdivapd 10990 |
Complex conjugate distributes over division. (Contributed by Jim
Kingdon, 15-Jun-2020.)
|
     #
     
              |
|
Theorem | rered 10991 |
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.)
|
         |
|
Theorem | reim0d 10992 |
The imaginary part of a real number is 0. (Contributed by Mario
Carneiro, 29-May-2016.)
|
         |
|
Theorem | cjred 10993 |
A real number equals its complex conjugate. Proposition 10-3.4(f) of
[Gleason] p. 133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
         |
|
Theorem | remul2d 10994 |
Real part of a product. (Contributed by Mario Carneiro,
29-May-2016.)
|
                   |
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Theorem | immul2d 10995 |
Imaginary part of a product. (Contributed by Mario Carneiro,
29-May-2016.)
|
                   |
|
Theorem | redivapd 10996 |
Real part of a division. Related to remul2 10895. (Contributed by Jim
Kingdon, 15-Jun-2020.)
|
     #
     
          |
|
Theorem | imdivapd 10997 |
Imaginary part of a division. Related to remul2 10895. (Contributed by
Jim Kingdon, 15-Jun-2020.)
|
     #
     
          |
|
Theorem | crred 10998 |
The real part of a complex number representation. Definition 10-3.1 of
[Gleason] p. 132. (Contributed by Mario
Carneiro, 29-May-2016.)
|
               |
|
Theorem | crimd 10999 |
The imaginary part of a complex number representation. Definition
10-3.1 of [Gleason] p. 132.
(Contributed by Mario Carneiro,
29-May-2016.)
|
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Theorem | cnreim 11000 |
Complex apartness in terms of real and imaginary parts. See also
apreim 8573 which is similar but with different notation.
(Contributed by
Jim Kingdon, 16-Dec-2023.)
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    #      #         #
        |