P. 115 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 11401-11500 *Has distinct variable group(s)
Type	Label	Description
Statement

Syntax	cim 11401	Extend class notation to include imaginary part of a complex number.


Definition	df-cj 11402*	Define the complex conjugate function. See cjcli 11473 for its closure and cjval 11405 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
$/\$

Definition	df-re 11403	Define a function whose value is the real part of a complex number. See reval 11409 for its value, recli 11471 for its closure, and replim 11419 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)


Definition	df-im 11404	Define a function whose value is the imaginary part of a complex number. See imval 11410 for its value, imcli 11472 for its closure, and replim 11419 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)


Theorem	cjval 11405*	The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjth 11406	The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjf 11407	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)


Theorem	cjcl 11408	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 11409	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 11410	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 11411	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 11412	The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	recl 11413	The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imcl 11414	The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	ref 11415	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imf 11416	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	crre 11417	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 11418	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 11419	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 11420	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 11421	The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)


Theorem	reim0b 11422	A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)


Theorem	rereb 11423	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 11424	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 11425	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 11426	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 11427	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	reneg 11428	Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	readd 11429	Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	resub 11430	Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
$/\$

Theorem	remullem 11431	Lemma for remul 11432, immul 11439, and cjmul 11445. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$ $/\$ $/\$

Theorem	remul 11432	Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	remul2 11433	Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	redivap 11434	Real part of a division. Related to remul2 11433. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	imcj 11435	Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imneg 11436	The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imadd 11437	Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	imsub 11438	Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
$/\$

Theorem	immul 11439	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	immul2 11440	Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	imdivap 11441	Imaginary part of a division. Related to immul2 11440. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	cjre 11442	A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)


Theorem	cjcj 11443	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 11444	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 11445	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 11446	Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	cjmulrcl 11447	A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulval 11448	A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulge0 11449	A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjneg 11450	Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	addcj 11451	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 11452	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
$/\$

Theorem	cjexp 11453	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
$/\$

Theorem	imval2 11454	The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)


Theorem	re0 11455	The real part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	im0 11456	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	re1 11457	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	im1 11458	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	rei 11459	The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	imi 11460	The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	cj0 11461	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	cji 11462	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)


Theorem	cjreim 11463	The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
$/\$

Theorem	cjreim2 11464	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 11465	Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
$/\$

Theorem	cjap 11466	Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ # #

Theorem	cjap0 11467	A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
# #

Theorem	cjne0 11468	A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 11467 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)


Theorem	cjdivap 11469	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	cnrecnv 11470*	The inverse to the canonical bijection from to from cnref1o 9884. (Contributed by Mario Carneiro, 25-Aug-2014.)


Theorem	recli 11471	The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	imcli 11472	The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	cjcli 11473	Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)


Theorem	replimi 11474	Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)


Theorem	cjcji 11475	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 11476	A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)


Theorem	rerebi 11477	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)


Theorem	cjrebi 11478	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 11479	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	imcji 11480	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulrcli 11481	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)


Theorem	cjmulvali 11482	A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulge0i 11483	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)


Theorem	renegi 11484	Real part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	imnegi 11485	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	cjnegi 11486	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	addcji 11487	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 11488	Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	imaddi 11489	Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	remuli 11490	Real part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	immuli 11491	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	cjaddi 11492	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)


Theorem	cjmuli 11493	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)


Theorem	ipcni 11494	Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)


Theorem	cjdivapi 11495	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theorem	crrei 11496	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)


Theorem	crimi 11497	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)


Theorem	recld 11498	The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)


Theorem	imcld 11499	The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)


Theorem	cjcld 11500	Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

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