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

Theoremcjadd 11901 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 11902 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 11903 Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulrcl 11904 A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulval 11905 A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjmulge0 11906 A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremcjneg 11907 Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremaddcj 11908 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 11909 Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)

Theoremcjexp 11910 Complex conjugate of natural number exponentiation. (Contributed by NM, 7-Jun-2006.)

Theoremimval2 11911 The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)

Theoremre0 11912 The real part of zero. (Contributed by NM, 27-Jul-1999.)

Theoremim0 11913 The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)

Theoremre1 11914 The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremim1 11915 The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremrei 11916 The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremimi 11917 The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)

Theoremcj0 11918 The conjugate of zero. (Contributed by NM, 27-Jul-1999.)

Theoremcji 11919 The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremcjreim 11920 The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)

Theoremcjreim2 11921 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 11922 Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)

Theoremcjne0 11923 A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)

Theoremcjdiv 11924 Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremcnrecnv 11925* The inverse to the canonical bijection from to from cnref1o 10563. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremsqeqd 11926 A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.)

Theoremrecli 11927 The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremimcli 11928 The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)

Theoremcjcli 11929 Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)

Theoremreplimi 11930 Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)

Theoremcjcji 11931 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 11932 A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)

Theoremrerebi 11933 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)

Theoremcjrebi 11934 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 11935 Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremimcji 11936 Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremcjmulrcli 11937 A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)

Theoremcjmulvali 11938 A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)

Theoremcjmulge0i 11939 A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)

Theoremrenegi 11940 Real part of negative. (Contributed by NM, 2-Aug-1999.)

Theoremimnegi 11941 Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)

Theoremcjnegi 11942 Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)

Theoremaddcji 11943 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 11944 Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)

Theoremimaddi 11945 Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)

Theoremremuli 11946 Real part of a product. (Contributed by NM, 28-Jul-1999.)

Theoremimmuli 11947 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)

Theoremcjaddi 11948 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

Theoremcjmuli 11949 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

Theoremipcni 11950 Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)

Theoremcjdivi 11951 Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremcrrei 11952 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)

Theoremcrimi 11953 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)

Theoremrecld 11954 The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimcld 11955 The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjcld 11956 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreplimd 11957 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremimd 11958 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.)

Theoremcjcjd 11959 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.)

Theoremreim0bd 11960 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrerebd 11961 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjrebd 11962 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.)

Theoremcjne0d 11963 A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrecjd 11964 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimcjd 11965 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulrcld 11966 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulvald 11967 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulge0d 11968 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrenegd 11969 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimnegd 11970 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjnegd 11971 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremaddcjd 11972 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.)

Theoremcjexpd 11973 Complex conjugate of natural number exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreaddd 11974 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimaddd 11975 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremresubd 11976 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimsubd 11977 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremuld 11978 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimmuld 11979 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjaddd 11980 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmuld 11981 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremipcnd 11982 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjdivd 11983 Complex conjugate distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrered 11984 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.)

Theoremreim0d 11985 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjred 11986 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremul2d 11987 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimmul2d 11988 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremredivd 11989 Real part of a division. Related to remul2 11890. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimdivd 11990 Imaginary part of a division. Related to remul2 11890. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcrred 11991 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcrimd 11992 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)

5.7.3  Square root; absolute value

Syntaxcsqr 11993 Extend class notation to include square root of a complex number.

Syntaxcabs 11994 Extend class notation to include a function for the absolute value (modulus) of a complex number.

Definitiondf-sqr 11995* Define a function whose value is the square root of a complex number. Since iff , we ensure uniqueness by restricting the range to numbers with positive real part, or numbers with 0 real part and nonnegative imaginary part. A description can be found under "Principal square root of a complex number" at http://en.wikipedia.org/wiki/Square_root. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrcl 12120 for its closure, sqrval 11997 for its value, sqrth 12123 and sqsqri 12134 for its relationship to squares, and sqr11i 12143 for uniqueness. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 8-Jul-2013.)

Definitiondf-abs 11996 Define the function for the absolute value (modulus) of a complex number. See abscli 12153 for its closure and absval 11998 or absval2i 12155 for its value. (Contributed by NM, 27-Jul-1999.)

Theoremsqrval 11997* Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.)

Theoremabsval 11998 The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremrennim 11999 A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)

Theoremcnpart 12000 The specification of restriction to the right half-plane partitions the complex plane without 0 into two disjoint pieces, which are related by a reflection about the origin (under the map ). (Contributed by Mario Carneiro, 8-Jul-2013.)

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