P. 133 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 13201-13300 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	sgnp 13201	Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
$/\$ sgn

Theorem	sgnrrp 13202	Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
sgn

Theorem	sgn1 13203	Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
sgn

Theorem	sgnpnf 13204	Proof that the signum of is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
sgn

Theorem	sgnn 13205	Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
$/\$ sgn

Theorem	sgnmnf 13206	Proof that the signum of is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
sgn

5.9.3 Real and imaginary parts; conjugate

Syntax	ccj 13207	Extend class notation to include complex conjugate function.


Syntax	cre 13208	Extend class notation to include real part of a complex number.


Syntax	cim 13209	Extend class notation to include imaginary part of a complex number.


Definition	df-cj 13210*	Define the complex conjugate function. See cjcli 13280 for its closure and cjval 13213 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
$/\$

Definition	df-re 13211	Define a function whose value is the real part of a complex number. See reval 13217 for its value, recli 13278 for its closure, and replim 13227 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)


Definition	df-im 13212	Define a function whose value is the imaginary part of a complex number. See imval 13218 for its value, imcli 13279 for its closure, and replim 13227 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)


Theorem	cjval 13213*	The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjth 13214	The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjf 13215	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)


Theorem	cjcl 13216	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.)


Theorem	reval 13217	The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imval 13218	The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imre 13219	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.)


Theorem	reim 13220	The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	recl 13221	The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imcl 13222	The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	ref 13223	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imf 13224	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	crre 13225	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.)
$/\$

Theorem	crim 13226	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.)
$/\$

Theorem	replim 13227	Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)


Theorem	remim 13228	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.)


Theorem	reim0 13229	The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)


Theorem	reim0b 13230	A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)


Theorem	rereb 13231	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.)


Theorem	mulre 13232	A product with a nonzero real multiplier is real iff the multiplicand is real. (Contributed by NM, 21-Aug-2008.)
$/\$ $/\$

Theorem	rere 13233	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.)


Theorem	cjreb 13234	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.)


Theorem	recj 13235	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	reneg 13236	Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	readd 13237	Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	resub 13238	Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
$/\$

Theorem	remullem 13239	Lemma for remul 13240, immul 13247, and cjmul 13253. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$ $/\$ $/\$

Theorem	remul 13240	Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	remul2 13241	Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	rediv 13242	Real part of a division. Related to remul2 13241. (Contributed by David A. Wheeler, 10-Jun-2015.)
$/\$ $/\$

Theorem	imcj 13243	Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imneg 13244	The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imadd 13245	Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	imsub 13246	Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
$/\$

Theorem	immul 13247	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	immul2 13248	Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	imdiv 13249	Imaginary part of a division. Related to immul2 13248. (Contributed by Mario Carneiro, 20-Jun-2015.)
$/\$ $/\$

Theorem	cjre 13250	A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)


Theorem	cjcj 13251	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.)


Theorem	cjadd 13252	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.)
$/\$

Theorem	cjmul 13253	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 13254	Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	cjmulrcl 13255	A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulval 13256	A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulge0 13257	A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjneg 13258	Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	addcj 13259	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 13260	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
$/\$

Theorem	cjexp 13261	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
$/\$

Theorem	imval2 13262	The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)


Theorem	re0 13263	The real part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	im0 13264	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	re1 13265	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	im1 13266	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	rei 13267	The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	imi 13268	The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	cj0 13269	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	cji 13270	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)


Theorem	cjreim 13271	The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
$/\$

Theorem	cjreim2 13272	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 13273	Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
$/\$

Theorem	cjne0 13274	A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.)


Theorem	cjdiv 13275	Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)
$/\$ $/\$

Theorem	cnrecnv 13276*	The inverse to the canonical bijection from to from cnref1o 11325. (Contributed by Mario Carneiro, 25-Aug-2014.)


Theorem	sqeqd 13277	A deduction for showing two numbers whose squares are equal are themselves equal. (Contributed by Mario Carneiro, 3-Apr-2015.)
$/\$ $/\$

Theorem	recli 13278	The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	imcli 13279	The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	cjcli 13280	Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)


Theorem	replimi 13281	Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)


Theorem	cjcji 13282	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.)


Theorem	reim0bi 13283	A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)


Theorem	rerebi 13284	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)


Theorem	cjrebi 13285	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.)


Theorem	recji 13286	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	imcji 13287	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulrcli 13288	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)


Theorem	cjmulvali 13289	A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulge0i 13290	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)


Theorem	renegi 13291	Real part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	imnegi 13292	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	cjnegi 13293	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	addcji 13294	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.)


Theorem	readdi 13295	Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	imaddi 13296	Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	remuli 13297	Real part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	immuli 13298	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	cjaddi 13299	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)


Theorem	cjmuli 13300	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

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