P. 107 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 10601-10700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	shftvalg 10601	Value of a sequence shifted by . (Contributed by Scott Fenton, 16-Dec-2017.)
$/\$ $/\$

Theorem	shftval4g 10602	Value of a sequence shifted by . (Contributed by Jim Kingdon, 19-Aug-2021.)
$/\$ $/\$

Theorem	seq3shft 10603*	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 10604	Extend class notation to include complex conjugate function.


Syntax	cre 10605	Extend class notation to include real part of a complex number.


Syntax	cim 10606	Extend class notation to include imaginary part of a complex number.


Definition	df-cj 10607*	Define the complex conjugate function. See cjcli 10678 for its closure and cjval 10610 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
$/\$

Definition	df-re 10608	Define a function whose value is the real part of a complex number. See reval 10614 for its value, recli 10676 for its closure, and replim 10624 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)


Definition	df-im 10609	Define a function whose value is the imaginary part of a complex number. See imval 10615 for its value, imcli 10677 for its closure, and replim 10624 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)


Theorem	cjval 10610*	The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjth 10611	The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjf 10612	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)


Theorem	cjcl 10613	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 10614	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 10615	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 10616	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 10617	The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	recl 10618	The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imcl 10619	The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	ref 10620	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imf 10621	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	crre 10622	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 10623	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 10624	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 10625	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 10626	The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)


Theorem	reim0b 10627	A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)


Theorem	rereb 10628	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 10629	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 10630	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 10631	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 10632	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	reneg 10633	Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	readd 10634	Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	resub 10635	Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
$/\$

Theorem	remullem 10636	Lemma for remul 10637, immul 10644, and cjmul 10650. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$ $/\$ $/\$

Theorem	remul 10637	Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	remul2 10638	Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	redivap 10639	Real part of a division. Related to remul2 10638. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	imcj 10640	Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imneg 10641	The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imadd 10642	Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	imsub 10643	Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
$/\$

Theorem	immul 10644	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	immul2 10645	Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	imdivap 10646	Imaginary part of a division. Related to immul2 10645. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	cjre 10647	A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)


Theorem	cjcj 10648	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 10649	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 10650	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 10651	Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	cjmulrcl 10652	A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulval 10653	A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulge0 10654	A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjneg 10655	Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	addcj 10656	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 10657	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
$/\$

Theorem	cjexp 10658	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
$/\$

Theorem	imval2 10659	The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)


Theorem	re0 10660	The real part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	im0 10661	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	re1 10662	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	im1 10663	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	rei 10664	The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	imi 10665	The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	cj0 10666	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	cji 10667	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)


Theorem	cjreim 10668	The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
$/\$

Theorem	cjreim2 10669	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 10670	Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
$/\$

Theorem	cjap 10671	Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ # #

Theorem	cjap0 10672	A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
# #

Theorem	cjne0 10673	A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10672 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)


Theorem	cjdivap 10674	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	cnrecnv 10675*	The inverse to the canonical bijection from to from cnref1o 9433. (Contributed by Mario Carneiro, 25-Aug-2014.)


Theorem	recli 10676	The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	imcli 10677	The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	cjcli 10678	Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)


Theorem	replimi 10679	Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)


Theorem	cjcji 10680	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 10681	A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)


Theorem	rerebi 10682	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)


Theorem	cjrebi 10683	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 10684	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	imcji 10685	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulrcli 10686	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)


Theorem	cjmulvali 10687	A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulge0i 10688	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)


Theorem	renegi 10689	Real part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	imnegi 10690	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	cjnegi 10691	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	addcji 10692	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 10693	Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	imaddi 10694	Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	remuli 10695	Real part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	immuli 10696	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	cjaddi 10697	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)


Theorem	cjmuli 10698	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)


Theorem	ipcni 10699	Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)


Theorem	cjdivapi 10700	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

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