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Theorem List for Metamath Proof Explorer - 11901-12000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremshftcan2 11901 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremseqshft 11902 Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-Feb-2014.)

5.7.2  Real and imaginary parts; conjugate

Syntaxccj 11903 Extend class notation to include complex conjugate function.

Syntaxcre 11904 Extend class notation to include real part of a complex number.

Syntaxcim 11905 Extend class notation to include imaginary part of a complex number.

Definitiondf-cj 11906* Define the complex conjugate function. See cjcli 11976 for its closure and cjval 11909 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Definitiondf-re 11907 Define a function whose value is the real part of a complex number. See reval 11913 for its value, recli 11974 for its closure, and replim 11923 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)

Definitiondf-im 11908 Define a function whose value is the imaginary part of a complex number. See imval 11914 for its value, imcli 11975 for its closure, and replim 11923 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)

Theoremcjval 11909* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjth 11910 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjf 11911 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjcl 11912 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.)

Theoremreval 11913 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimval 11914 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimre 11915 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.)

Theoremreim 11916 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremrecl 11917 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimcl 11918 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremref 11919 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimf 11920 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremcrre 11921 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.)

Theoremcrim 11922 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.)

Theoremreplim 11923 Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremremim 11924 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 NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremreim0 11925 The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremreim0b 11926 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)

Theoremrereb 11927 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.)

Theoremmulre 11928 A product with a nonzero real multiplier is real iff the multiplicand is real. (Contributed by NM, 21-Aug-2008.)

Theoremrere 11929 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.)

Theoremcjreb 11930 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.)

Theoremrecj 11931 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremreneg 11932 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremreadd 11933 Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremresub 11934 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)

Theoremremullem 11935 Lemma for remul 11936, immul 11943, and cjmul 11949. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremremul 11936 Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremremul2 11937 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremrediv 11938 Real part of a division. Related to remul2 11937. (Contributed by David A. Wheeler, 10-Jun-2015.)

Theoremimcj 11939 Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimneg 11940 The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimadd 11941 Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimsub 11942 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)

Theoremimmul 11943 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimmul2 11944 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremimdiv 11945 Imaginary part of a division. Related to immul2 11944. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremcjre 11946 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)

Theoremcjcj 11947 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.)

Theoremcjadd 11948 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.)

Theoremcjmul 11949 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.)

Theoremipcnval 11950 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulrcl 11951 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulval 11952 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulge0 11953 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjneg 11954 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremaddcj 11955 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.)

Theoremcjsub 11956 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)

Theoremcjexp 11957 Complex conjugate of natural number exponentiation. (Contributed by NM, 7-Jun-2006.)

Theoremimval2 11958 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)

Theoremre0 11959 The real part of zero. (Contributed by NM, 27-Jul-1999.)

Theoremim0 11960 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)

Theoremre1 11961 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremim1 11962 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremrei 11963 The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremimi 11964 The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremcj0 11965 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)

Theoremcji 11966 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremcjreim 11967 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)

Theoremcjreim2 11968 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.)

Theoremcj11 11969 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)

Theoremcjne0 11970 A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)

Theoremcjdiv 11971 Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremcnrecnv 11972* The inverse to the canonical bijection from to from cnref1o 10609. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremsqeqd 11973 A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.)

Theoremrecli 11974 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremimcli 11975 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremcjcli 11976 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)

Theoremreplimi 11977 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)

Theoremcjcji 11978 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.)

Theoremreim0bi 11979 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)

Theoremrerebi 11980 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)

Theoremcjrebi 11981 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.)

Theoremrecji 11982 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremimcji 11983 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremcjmulrcli 11984 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)

Theoremcjmulvali 11985 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremcjmulge0i 11986 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)

Theoremrenegi 11987 Real part of negative. (Contributed by NM, 2-Aug-1999.)

Theoremimnegi 11988 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)

Theoremcjnegi 11989 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)

Theoremaddcji 11990 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.)

Theoremreaddi 11991 Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)

Theoremimaddi 11992 Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)

Theoremremuli 11993 Real part of a product. (Contributed by NM, 28-Jul-1999.)

Theoremimmuli 11994 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)

Theoremcjaddi 11995 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

Theoremcjmuli 11996 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

Theoremipcni 11997 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)

Theoremcjdivi 11998 Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremcrrei 11999 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)

Theoremcrimi 12000 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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