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Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremimsub 10901 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚) โ†’ (โ„‘โ€˜(๐ด โˆ’ ๐ต)) = ((โ„‘โ€˜๐ด) โˆ’ (โ„‘โ€˜๐ต)))
 
Theoremimmul 10902 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚) โ†’ (โ„‘โ€˜(๐ด ยท ๐ต)) = (((โ„œโ€˜๐ด) ยท (โ„‘โ€˜๐ต)) + ((โ„‘โ€˜๐ด) ยท (โ„œโ€˜๐ต))))
 
Theoremimmul2 10903 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
((๐ด โˆˆ โ„ โˆง ๐ต โˆˆ โ„‚) โ†’ (โ„‘โ€˜(๐ด ยท ๐ต)) = (๐ด ยท (โ„‘โ€˜๐ต)))
 
Theoremimdivap 10904 Imaginary part of a division. Related to immul2 10903. (Contributed by Jim Kingdon, 14-Jun-2020.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„ โˆง ๐ต # 0) โ†’ (โ„‘โ€˜(๐ด / ๐ต)) = ((โ„‘โ€˜๐ด) / ๐ต))
 
Theoremcjre 10905 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)
(๐ด โˆˆ โ„ โ†’ (โˆ—โ€˜๐ด) = ๐ด)
 
Theoremcjcj 10906 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.)
(๐ด โˆˆ โ„‚ โ†’ (โˆ—โ€˜(โˆ—โ€˜๐ด)) = ๐ด)
 
Theoremcjadd 10907 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.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚) โ†’ (โˆ—โ€˜(๐ด + ๐ต)) = ((โˆ—โ€˜๐ด) + (โˆ—โ€˜๐ต)))
 
Theoremcjmul 10908 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.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚) โ†’ (โˆ—โ€˜(๐ด ยท ๐ต)) = ((โˆ—โ€˜๐ด) ยท (โˆ—โ€˜๐ต)))
 
Theoremipcnval 10909 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚) โ†’ (โ„œโ€˜(๐ด ยท (โˆ—โ€˜๐ต))) = (((โ„œโ€˜๐ด) ยท (โ„œโ€˜๐ต)) + ((โ„‘โ€˜๐ด) ยท (โ„‘โ€˜๐ต))))
 
Theoremcjmulrcl 10910 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(๐ด โˆˆ โ„‚ โ†’ (๐ด ยท (โˆ—โ€˜๐ด)) โˆˆ โ„)
 
Theoremcjmulval 10911 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
(๐ด โˆˆ โ„‚ โ†’ (๐ด ยท (โˆ—โ€˜๐ด)) = (((โ„œโ€˜๐ด)โ†‘2) + ((โ„‘โ€˜๐ด)โ†‘2)))
 
Theoremcjmulge0 10912 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(๐ด โˆˆ โ„‚ โ†’ 0 โ‰ค (๐ด ยท (โˆ—โ€˜๐ด)))
 
Theoremcjneg 10913 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
(๐ด โˆˆ โ„‚ โ†’ (โˆ—โ€˜-๐ด) = -(โˆ—โ€˜๐ด))
 
Theoremaddcj 10914 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.)
(๐ด โˆˆ โ„‚ โ†’ (๐ด + (โˆ—โ€˜๐ด)) = (2 ยท (โ„œโ€˜๐ด)))
 
Theoremcjsub 10915 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚) โ†’ (โˆ—โ€˜(๐ด โˆ’ ๐ต)) = ((โˆ—โ€˜๐ด) โˆ’ (โˆ—โ€˜๐ต)))
 
Theoremcjexp 10916 Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
((๐ด โˆˆ โ„‚ โˆง ๐‘ โˆˆ โ„•0) โ†’ (โˆ—โ€˜(๐ดโ†‘๐‘)) = ((โˆ—โ€˜๐ด)โ†‘๐‘))
 
Theoremimval2 10917 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)
(๐ด โˆˆ โ„‚ โ†’ (โ„‘โ€˜๐ด) = ((๐ด โˆ’ (โˆ—โ€˜๐ด)) / (2 ยท i)))
 
Theoremre0 10918 The real part of zero. (Contributed by NM, 27-Jul-1999.)
(โ„œโ€˜0) = 0
 
Theoremim0 10919 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
(โ„‘โ€˜0) = 0
 
Theoremre1 10920 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(โ„œโ€˜1) = 1
 
Theoremim1 10921 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
(โ„‘โ€˜1) = 0
 
Theoremrei 10922 The real part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(โ„œโ€˜i) = 0
 
Theoremimi 10923 The imaginary part of i. (Contributed by Scott Fenton, 9-Jun-2006.)
(โ„‘โ€˜i) = 1
 
Theoremcj0 10924 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
(โˆ—โ€˜0) = 0
 
Theoremcji 10925 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
(โˆ—โ€˜i) = -i
 
Theoremcjreim 10926 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
((๐ด โˆˆ โ„ โˆง ๐ต โˆˆ โ„) โ†’ (โˆ—โ€˜(๐ด + (i ยท ๐ต))) = (๐ด โˆ’ (i ยท ๐ต)))
 
Theoremcjreim2 10927 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.)
((๐ด โˆˆ โ„ โˆง ๐ต โˆˆ โ„) โ†’ (โˆ—โ€˜(๐ด โˆ’ (i ยท ๐ต))) = (๐ด + (i ยท ๐ต)))
 
Theoremcj11 10928 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚) โ†’ ((โˆ—โ€˜๐ด) = (โˆ—โ€˜๐ต) โ†” ๐ด = ๐ต))
 
Theoremcjap 10929 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚) โ†’ ((โˆ—โ€˜๐ด) # (โˆ—โ€˜๐ต) โ†” ๐ด # ๐ต))
 
Theoremcjap0 10930 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
(๐ด โˆˆ โ„‚ โ†’ (๐ด # 0 โ†” (โˆ—โ€˜๐ด) # 0))
 
Theoremcjne0 10931 A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10930 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)
(๐ด โˆˆ โ„‚ โ†’ (๐ด โ‰  0 โ†” (โˆ—โ€˜๐ด) โ‰  0))
 
Theoremcjdivap 10932 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
((๐ด โˆˆ โ„‚ โˆง ๐ต โˆˆ โ„‚ โˆง ๐ต # 0) โ†’ (โˆ—โ€˜(๐ด / ๐ต)) = ((โˆ—โ€˜๐ด) / (โˆ—โ€˜๐ต)))
 
Theoremcnrecnv 10933* The inverse to the canonical bijection from (โ„ ร— โ„) to โ„‚ from cnref1o 9664. (Contributed by Mario Carneiro, 25-Aug-2014.)
๐น = (๐‘ฅ โˆˆ โ„, ๐‘ฆ โˆˆ โ„ โ†ฆ (๐‘ฅ + (i ยท ๐‘ฆ)))    โ‡’   โ—ก๐น = (๐‘ง โˆˆ โ„‚ โ†ฆ โŸจ(โ„œโ€˜๐‘ง), (โ„‘โ€˜๐‘ง)โŸฉ)
 
Theoremrecli 10934 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
๐ด โˆˆ โ„‚    โ‡’   (โ„œโ€˜๐ด) โˆˆ โ„
 
Theoremimcli 10935 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
๐ด โˆˆ โ„‚    โ‡’   (โ„‘โ€˜๐ด) โˆˆ โ„
 
Theoremcjcli 10936 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
๐ด โˆˆ โ„‚    โ‡’   (โˆ—โ€˜๐ด) โˆˆ โ„‚
 
Theoremreplimi 10937 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
๐ด โˆˆ โ„‚    โ‡’   ๐ด = ((โ„œโ€˜๐ด) + (i ยท (โ„‘โ€˜๐ด)))
 
Theoremcjcji 10938 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.)
๐ด โˆˆ โ„‚    โ‡’   (โˆ—โ€˜(โˆ—โ€˜๐ด)) = ๐ด
 
Theoremreim0bi 10939 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
๐ด โˆˆ โ„‚    โ‡’   (๐ด โˆˆ โ„ โ†” (โ„‘โ€˜๐ด) = 0)
 
Theoremrerebi 10940 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
๐ด โˆˆ โ„‚    โ‡’   (๐ด โˆˆ โ„ โ†” (โ„œโ€˜๐ด) = ๐ด)
 
Theoremcjrebi 10941 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.)
๐ด โˆˆ โ„‚    โ‡’   (๐ด โˆˆ โ„ โ†” (โˆ—โ€˜๐ด) = ๐ด)
 
Theoremrecji 10942 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
๐ด โˆˆ โ„‚    โ‡’   (โ„œโ€˜(โˆ—โ€˜๐ด)) = (โ„œโ€˜๐ด)
 
Theoremimcji 10943 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
๐ด โˆˆ โ„‚    โ‡’   (โ„‘โ€˜(โˆ—โ€˜๐ด)) = -(โ„‘โ€˜๐ด)
 
Theoremcjmulrcli 10944 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
๐ด โˆˆ โ„‚    โ‡’   (๐ด ยท (โˆ—โ€˜๐ด)) โˆˆ โ„
 
Theoremcjmulvali 10945 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
๐ด โˆˆ โ„‚    โ‡’   (๐ด ยท (โˆ—โ€˜๐ด)) = (((โ„œโ€˜๐ด)โ†‘2) + ((โ„‘โ€˜๐ด)โ†‘2))
 
Theoremcjmulge0i 10946 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
๐ด โˆˆ โ„‚    โ‡’   0 โ‰ค (๐ด ยท (โˆ—โ€˜๐ด))
 
Theoremrenegi 10947 Real part of negative. (Contributed by NM, 2-Aug-1999.)
๐ด โˆˆ โ„‚    โ‡’   (โ„œโ€˜-๐ด) = -(โ„œโ€˜๐ด)
 
Theoremimnegi 10948 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
๐ด โˆˆ โ„‚    โ‡’   (โ„‘โ€˜-๐ด) = -(โ„‘โ€˜๐ด)
 
Theoremcjnegi 10949 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
๐ด โˆˆ โ„‚    โ‡’   (โˆ—โ€˜-๐ด) = -(โˆ—โ€˜๐ด)
 
Theoremaddcji 10950 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.)
๐ด โˆˆ โ„‚    โ‡’   (๐ด + (โˆ—โ€˜๐ด)) = (2 ยท (โ„œโ€˜๐ด))
 
Theoremreaddi 10951 Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
๐ด โˆˆ โ„‚    &   ๐ต โˆˆ โ„‚    โ‡’   (โ„œโ€˜(๐ด + ๐ต)) = ((โ„œโ€˜๐ด) + (โ„œโ€˜๐ต))
 
Theoremimaddi 10952 Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
๐ด โˆˆ โ„‚    &   ๐ต โˆˆ โ„‚    โ‡’   (โ„‘โ€˜(๐ด + ๐ต)) = ((โ„‘โ€˜๐ด) + (โ„‘โ€˜๐ต))
 
Theoremremuli 10953 Real part of a product. (Contributed by NM, 28-Jul-1999.)
๐ด โˆˆ โ„‚    &   ๐ต โˆˆ โ„‚    โ‡’   (โ„œโ€˜(๐ด ยท ๐ต)) = (((โ„œโ€˜๐ด) ยท (โ„œโ€˜๐ต)) โˆ’ ((โ„‘โ€˜๐ด) ยท (โ„‘โ€˜๐ต)))
 
Theoremimmuli 10954 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
๐ด โˆˆ โ„‚    &   ๐ต โˆˆ โ„‚    โ‡’   (โ„‘โ€˜(๐ด ยท ๐ต)) = (((โ„œโ€˜๐ด) ยท (โ„‘โ€˜๐ต)) + ((โ„‘โ€˜๐ด) ยท (โ„œโ€˜๐ต)))
 
Theoremcjaddi 10955 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
๐ด โˆˆ โ„‚    &   ๐ต โˆˆ โ„‚    โ‡’   (โˆ—โ€˜(๐ด + ๐ต)) = ((โˆ—โ€˜๐ด) + (โˆ—โ€˜๐ต))
 
Theoremcjmuli 10956 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
๐ด โˆˆ โ„‚    &   ๐ต โˆˆ โ„‚    โ‡’   (โˆ—โ€˜(๐ด ยท ๐ต)) = ((โˆ—โ€˜๐ด) ยท (โˆ—โ€˜๐ต))
 
Theoremipcni 10957 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)
๐ด โˆˆ โ„‚    &   ๐ต โˆˆ โ„‚    โ‡’   (โ„œโ€˜(๐ด ยท (โˆ—โ€˜๐ต))) = (((โ„œโ€˜๐ด) ยท (โ„œโ€˜๐ต)) + ((โ„‘โ€˜๐ด) ยท (โ„‘โ€˜๐ต)))
 
Theoremcjdivapi 10958 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
๐ด โˆˆ โ„‚    &   ๐ต โˆˆ โ„‚    โ‡’   (๐ต # 0 โ†’ (โˆ—โ€˜(๐ด / ๐ต)) = ((โˆ—โ€˜๐ด) / (โˆ—โ€˜๐ต)))
 
Theoremcrrei 10959 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
๐ด โˆˆ โ„    &   ๐ต โˆˆ โ„    โ‡’   (โ„œโ€˜(๐ด + (i ยท ๐ต))) = ๐ด
 
Theoremcrimi 10960 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
๐ด โˆˆ โ„    &   ๐ต โˆˆ โ„    โ‡’   (โ„‘โ€˜(๐ด + (i ยท ๐ต))) = ๐ต
 
Theoremrecld 10961 The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„œโ€˜๐ด) โˆˆ โ„)
 
Theoremimcld 10962 The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„‘โ€˜๐ด) โˆˆ โ„)
 
Theoremcjcld 10963 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โˆ—โ€˜๐ด) โˆˆ โ„‚)
 
Theoremreplimd 10964 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ ๐ด = ((โ„œโ€˜๐ด) + (i ยท (โ„‘โ€˜๐ด))))
 
Theoremremimd 10965 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.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โˆ—โ€˜๐ด) = ((โ„œโ€˜๐ด) โˆ’ (i ยท (โ„‘โ€˜๐ด))))
 
Theoremcjcjd 10966 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.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โˆ—โ€˜(โˆ—โ€˜๐ด)) = ๐ด)
 
Theoremreim0bd 10967 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ (โ„‘โ€˜๐ด) = 0)    โ‡’   (๐œ‘ โ†’ ๐ด โˆˆ โ„)
 
Theoremrerebd 10968 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ (โ„œโ€˜๐ด) = ๐ด)    โ‡’   (๐œ‘ โ†’ ๐ด โˆˆ โ„)
 
Theoremcjrebd 10969 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.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ (โˆ—โ€˜๐ด) = ๐ด)    โ‡’   (๐œ‘ โ†’ ๐ด โˆˆ โ„)
 
Theoremcjne0d 10970 A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 10971 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ด โ‰  0)    โ‡’   (๐œ‘ โ†’ (โˆ—โ€˜๐ด) โ‰  0)
 
Theoremcjap0d 10971 A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ด # 0)    โ‡’   (๐œ‘ โ†’ (โˆ—โ€˜๐ด) # 0)
 
Theoremrecjd 10972 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„œโ€˜(โˆ—โ€˜๐ด)) = (โ„œโ€˜๐ด))
 
Theoremimcjd 10973 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„‘โ€˜(โˆ—โ€˜๐ด)) = -(โ„‘โ€˜๐ด))
 
Theoremcjmulrcld 10974 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (๐ด ยท (โˆ—โ€˜๐ด)) โˆˆ โ„)
 
Theoremcjmulvald 10975 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (๐ด ยท (โˆ—โ€˜๐ด)) = (((โ„œโ€˜๐ด)โ†‘2) + ((โ„‘โ€˜๐ด)โ†‘2)))
 
Theoremcjmulge0d 10976 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ 0 โ‰ค (๐ด ยท (โˆ—โ€˜๐ด)))
 
Theoremrenegd 10977 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„œโ€˜-๐ด) = -(โ„œโ€˜๐ด))
 
Theoremimnegd 10978 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„‘โ€˜-๐ด) = -(โ„‘โ€˜๐ด))
 
Theoremcjnegd 10979 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โˆ—โ€˜-๐ด) = -(โˆ—โ€˜๐ด))
 
Theoremaddcjd 10980 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.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (๐ด + (โˆ—โ€˜๐ด)) = (2 ยท (โ„œโ€˜๐ด)))
 
Theoremcjexpd 10981 Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•0)    โ‡’   (๐œ‘ โ†’ (โˆ—โ€˜(๐ดโ†‘๐‘)) = ((โˆ—โ€˜๐ด)โ†‘๐‘))
 
Theoremreaddd 10982 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„œโ€˜(๐ด + ๐ต)) = ((โ„œโ€˜๐ด) + (โ„œโ€˜๐ต)))
 
Theoremimaddd 10983 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„‘โ€˜(๐ด + ๐ต)) = ((โ„‘โ€˜๐ด) + (โ„‘โ€˜๐ต)))
 
Theoremresubd 10984 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„œโ€˜(๐ด โˆ’ ๐ต)) = ((โ„œโ€˜๐ด) โˆ’ (โ„œโ€˜๐ต)))
 
Theoremimsubd 10985 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„‘โ€˜(๐ด โˆ’ ๐ต)) = ((โ„‘โ€˜๐ด) โˆ’ (โ„‘โ€˜๐ต)))
 
Theoremremuld 10986 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„œโ€˜(๐ด ยท ๐ต)) = (((โ„œโ€˜๐ด) ยท (โ„œโ€˜๐ต)) โˆ’ ((โ„‘โ€˜๐ด) ยท (โ„‘โ€˜๐ต))))
 
Theoremimmuld 10987 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„‘โ€˜(๐ด ยท ๐ต)) = (((โ„œโ€˜๐ด) ยท (โ„‘โ€˜๐ต)) + ((โ„‘โ€˜๐ด) ยท (โ„œโ€˜๐ต))))
 
Theoremcjaddd 10988 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โˆ—โ€˜(๐ด + ๐ต)) = ((โˆ—โ€˜๐ด) + (โˆ—โ€˜๐ต)))
 
Theoremcjmuld 10989 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โˆ—โ€˜(๐ด ยท ๐ต)) = ((โˆ—โ€˜๐ด) ยท (โˆ—โ€˜๐ต)))
 
Theoremipcnd 10990 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„œโ€˜(๐ด ยท (โˆ—โ€˜๐ต))) = (((โ„œโ€˜๐ด) ยท (โ„œโ€˜๐ต)) + ((โ„‘โ€˜๐ด) ยท (โ„‘โ€˜๐ต))))
 
Theoremcjdivapd 10991 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ต # 0)    โ‡’   (๐œ‘ โ†’ (โˆ—โ€˜(๐ด / ๐ต)) = ((โˆ—โ€˜๐ด) / (โˆ—โ€˜๐ต)))
 
Theoremrered 10992 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.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    โ‡’   (๐œ‘ โ†’ (โ„œโ€˜๐ด) = ๐ด)
 
Theoremreim0d 10993 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    โ‡’   (๐œ‘ โ†’ (โ„‘โ€˜๐ด) = 0)
 
Theoremcjred 10994 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    โ‡’   (๐œ‘ โ†’ (โˆ—โ€˜๐ด) = ๐ด)
 
Theoremremul2d 10995 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„œโ€˜(๐ด ยท ๐ต)) = (๐ด ยท (โ„œโ€˜๐ต)))
 
Theoremimmul2d 10996 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    โ‡’   (๐œ‘ โ†’ (โ„‘โ€˜(๐ด ยท ๐ต)) = (๐ด ยท (โ„‘โ€˜๐ต)))
 
Theoremredivapd 10997 Real part of a division. Related to remul2 10896. (Contributed by Jim Kingdon, 15-Jun-2020.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ด # 0)    โ‡’   (๐œ‘ โ†’ (โ„œโ€˜(๐ต / ๐ด)) = ((โ„œโ€˜๐ต) / ๐ด))
 
Theoremimdivapd 10998 Imaginary part of a division. Related to remul2 10896. (Contributed by Jim Kingdon, 15-Jun-2020.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„‚)    &   (๐œ‘ โ†’ ๐ด # 0)    โ‡’   (๐œ‘ โ†’ (โ„‘โ€˜(๐ต / ๐ด)) = ((โ„‘โ€˜๐ต) / ๐ด))
 
Theoremcrred 10999 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„)    โ‡’   (๐œ‘ โ†’ (โ„œโ€˜(๐ด + (i ยท ๐ต))) = ๐ด)
 
Theoremcrimd 11000 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„)    โ‡’   (๐œ‘ โ†’ (โ„‘โ€˜(๐ด + (i ยท ๐ต))) = ๐ต)
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