P. 117 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 11601-11700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	shftcan1 11601	Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
$/\$

Theorem	shftcan2 11602	Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
$/\$

Theorem	shftvalg 11603	Value of a sequence shifted by . (Contributed by Scott Fenton, 16-Dec-2017.)
$/\$ $/\$

Theorem	shftval4g 11604	Value of a sequence shifted by . (Contributed by Jim Kingdon, 19-Aug-2021.)
$/\$ $/\$

Theorem	seq3shft 11605*	Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 17-Oct-2022.)
$/\$ $/\$ $/\$

4.8.2 Real and imaginary parts; conjugate

Syntax	ccj 11606	Extend class notation to include complex conjugate function.


Syntax	cre 11607	Extend class notation to include real part of a complex number.


Syntax	cim 11608	Extend class notation to include imaginary part of a complex number.


Definition	df-cj 11609*	Define the complex conjugate function. See cjcli 11681 for its closure and cjval 11612 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
$/\$

Definition	df-re 11610	Define a function whose value is the real part of a complex number. See reval 11616 for its value, recli 11679 for its closure, and replim 11626 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)


Definition	df-im 11611	Define a function whose value is the imaginary part of a complex number. See imval 11617 for its value, imcli 11680 for its closure, and replim 11626 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)


Theorem	cjval 11612*	The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjth 11613	The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjf 11614	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)


Theorem	cjcl 11615	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 11616	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 11617	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 11618	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 11619	The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	recl 11620	The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imcl 11621	The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	ref 11622	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imf 11623	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	crre 11624	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 11625	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 11626	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 11627	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 11628	The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)


Theorem	reim0b 11629	A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)


Theorem	rereb 11630	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 11631	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 11632	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 11633	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 11634	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	reneg 11635	Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	readd 11636	Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	resub 11637	Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
$/\$

Theorem	remullem 11638	Lemma for remul 11639, immul 11646, and cjmul 11652. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$ $/\$ $/\$

Theorem	remul 11639	Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	remul2 11640	Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	redivap 11641	Real part of a division. Related to remul2 11640. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	imcj 11642	Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imneg 11643	The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imadd 11644	Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	imsub 11645	Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
$/\$

Theorem	immul 11646	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	immul2 11647	Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	imdivap 11648	Imaginary part of a division. Related to immul2 11647. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	cjre 11649	A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)


Theorem	cjcj 11650	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 11651	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 11652	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 11653	Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	cjmulrcl 11654	A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulval 11655	A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulge0 11656	A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjneg 11657	Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	addcj 11658	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 11659	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
$/\$

Theorem	cjexp 11660	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
$/\$

Theorem	imval2 11661	The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)


Theorem	sq01 11662	If a complex number equals its square, it must be 0 or 1. (Contributed by NM, 6-Jun-2006.)
$\/$

Theorem	re0 11663	The real part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	im0 11664	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	re1 11665	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	im1 11666	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	rei 11667	The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	imi 11668	The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	cj0 11669	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	cji 11670	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)


Theorem	cjreim 11671	The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
$/\$

Theorem	cjreim2 11672	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 11673	Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
$/\$

Theorem	cjap 11674	Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ # #

Theorem	cjap0 11675	A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
# #

Theorem	cjne0 11676	A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 11675 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)


Theorem	cjdivap 11677	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	cnrecnv 11678*	The inverse to the canonical bijection from to from cnref1o 10053. (Contributed by Mario Carneiro, 25-Aug-2014.)


Theorem	recli 11679	The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	imcli 11680	The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	cjcli 11681	Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)


Theorem	replimi 11682	Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)


Theorem	cjcji 11683	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 11684	A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)


Theorem	rerebi 11685	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)


Theorem	cjrebi 11686	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 11687	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	imcji 11688	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulrcli 11689	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)


Theorem	cjmulvali 11690	A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulge0i 11691	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)


Theorem	renegi 11692	Real part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	imnegi 11693	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	cjnegi 11694	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	addcji 11695	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 11696	Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	imaddi 11697	Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	remuli 11698	Real part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	immuli 11699	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	cjaddi 11700	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)

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