P. 109 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 10801-10900 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	shftval4g 10801	Value of a sequence shifted by . (Contributed by Jim Kingdon, 19-Aug-2021.)
$/\$ $/\$

Theorem	seq3shft 10802*	Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 17-Oct-2022.)
$/\$ $/\$ $/\$

4.7.2 Real and imaginary parts; conjugate

Syntax	ccj 10803	Extend class notation to include complex conjugate function.


Syntax	cre 10804	Extend class notation to include real part of a complex number.


Syntax	cim 10805	Extend class notation to include imaginary part of a complex number.


Definition	df-cj 10806*	Define the complex conjugate function. See cjcli 10877 for its closure and cjval 10809 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
$/\$

Definition	df-re 10807	Define a function whose value is the real part of a complex number. See reval 10813 for its value, recli 10875 for its closure, and replim 10823 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)


Definition	df-im 10808	Define a function whose value is the imaginary part of a complex number. See imval 10814 for its value, imcli 10876 for its closure, and replim 10823 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)


Theorem	cjval 10809*	The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjth 10810	The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjf 10811	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)


Theorem	cjcl 10812	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 10813	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 10814	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 10815	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 10816	The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	recl 10817	The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imcl 10818	The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	ref 10819	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imf 10820	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	crre 10821	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 10822	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 10823	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 10824	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 10825	The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)


Theorem	reim0b 10826	A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)


Theorem	rereb 10827	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	mulreap 10828	A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	rere 10829	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 10830	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 10831	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	reneg 10832	Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	readd 10833	Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	resub 10834	Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
$/\$

Theorem	remullem 10835	Lemma for remul 10836, immul 10843, and cjmul 10849. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$ $/\$ $/\$

Theorem	remul 10836	Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	remul2 10837	Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	redivap 10838	Real part of a division. Related to remul2 10837. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	imcj 10839	Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imneg 10840	The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imadd 10841	Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	imsub 10842	Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
$/\$

Theorem	immul 10843	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	immul2 10844	Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	imdivap 10845	Imaginary part of a division. Related to immul2 10844. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	cjre 10846	A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)


Theorem	cjcj 10847	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 10848	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 10849	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 10850	Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	cjmulrcl 10851	A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulval 10852	A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulge0 10853	A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjneg 10854	Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	addcj 10855	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 10856	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
$/\$

Theorem	cjexp 10857	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
$/\$

Theorem	imval2 10858	The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)


Theorem	re0 10859	The real part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	im0 10860	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	re1 10861	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	im1 10862	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	rei 10863	The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	imi 10864	The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	cj0 10865	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	cji 10866	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)


Theorem	cjreim 10867	The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
$/\$

Theorem	cjreim2 10868	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 10869	Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
$/\$

Theorem	cjap 10870	Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ # #

Theorem	cjap0 10871	A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
# #

Theorem	cjne0 10872	A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10871 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)


Theorem	cjdivap 10873	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	cnrecnv 10874*	The inverse to the canonical bijection from to from cnref1o 9609. (Contributed by Mario Carneiro, 25-Aug-2014.)


Theorem	recli 10875	The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	imcli 10876	The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	cjcli 10877	Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)


Theorem	replimi 10878	Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)


Theorem	cjcji 10879	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 10880	A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)


Theorem	rerebi 10881	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)


Theorem	cjrebi 10882	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 10883	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	imcji 10884	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulrcli 10885	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)


Theorem	cjmulvali 10886	A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulge0i 10887	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)


Theorem	renegi 10888	Real part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	imnegi 10889	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	cjnegi 10890	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	addcji 10891	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 10892	Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	imaddi 10893	Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	remuli 10894	Real part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	immuli 10895	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	cjaddi 10896	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)


Theorem	cjmuli 10897	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)


Theorem	ipcni 10898	Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)


Theorem	cjdivapi 10899	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theorem	crrei 10900	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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