P. 109 of Theorem List - Intuitionistic Logic Explorer

Home	Intuitionistic Logic Explorer Theorem List (p. 109 of 141)	< Previous Next >
		Browser slow? Try the Unicode version.
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List

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

Definition	df-im 10801	Define a function whose value is the imaginary part of a complex number. See imval 10807 for its value, imcli 10869 for its closure, and replim 10816 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)


Theorem	cjval 10802*	The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjth 10803	The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
$/\$

Theorem	cjf 10804	Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)


Theorem	cjcl 10805	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 10806	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 10807	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 10808	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 10809	The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	recl 10810	The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imcl 10811	The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	ref 10812	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	imf 10813	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)


Theorem	crre 10814	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 10815	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 10816	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 10817	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 10818	The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)


Theorem	reim0b 10819	A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)


Theorem	rereb 10820	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 10821	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 10822	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 10823	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 10824	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	reneg 10825	Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	readd 10826	Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	resub 10827	Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
$/\$

Theorem	remullem 10828	Lemma for remul 10829, immul 10836, and cjmul 10842. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$ $/\$ $/\$

Theorem	remul 10829	Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	remul2 10830	Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	redivap 10831	Real part of a division. Related to remul2 10830. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	imcj 10832	Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imneg 10833	The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	imadd 10834	Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	imsub 10835	Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
$/\$

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

Theorem	immul2 10837	Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
$/\$

Theorem	imdivap 10838	Imaginary part of a division. Related to immul2 10837. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	cjre 10839	A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)


Theorem	cjcj 10840	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 10841	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 10842	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 10843	Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	cjmulrcl 10844	A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulval 10845	A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjmulge0 10846	A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	cjneg 10847	Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	addcj 10848	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 10849	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
$/\$

Theorem	cjexp 10850	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
$/\$

Theorem	imval2 10851	The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)


Theorem	re0 10852	The real part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	im0 10853	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	re1 10854	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	im1 10855	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	rei 10856	The real part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	imi 10857	The imaginary part of . (Contributed by Scott Fenton, 9-Jun-2006.)


Theorem	cj0 10858	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)


Theorem	cji 10859	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)


Theorem	cjreim 10860	The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
$/\$

Theorem	cjreim2 10861	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 10862	Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
$/\$

Theorem	cjap 10863	Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ # #

Theorem	cjap0 10864	A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
# #

Theorem	cjne0 10865	A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10864 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)


Theorem	cjdivap 10866	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
$/\$ $/\$ #

Theorem	cnrecnv 10867*	The inverse to the canonical bijection from to from cnref1o 9602. (Contributed by Mario Carneiro, 25-Aug-2014.)


Theorem	recli 10868	The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	imcli 10869	The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)


Theorem	cjcli 10870	Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)


Theorem	replimi 10871	Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)


Theorem	cjcji 10872	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 10873	A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)


Theorem	rerebi 10874	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)


Theorem	cjrebi 10875	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 10876	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	imcji 10877	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulrcli 10878	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)


Theorem	cjmulvali 10879	A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)


Theorem	cjmulge0i 10880	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)


Theorem	renegi 10881	Real part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	imnegi 10882	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	cjnegi 10883	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)


Theorem	addcji 10884	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 10885	Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	imaddi 10886	Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)


Theorem	remuli 10887	Real part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	immuli 10888	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)


Theorem	cjaddi 10889	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)


Theorem	cjmuli 10890	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)


Theorem	ipcni 10891	Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)


Theorem	cjdivapi 10892	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
#

Theorem	crrei 10893	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)


Theorem	crimi 10894	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)


Theorem	recld 10895	The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)


Theorem	imcld 10896	The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)


Theorem	cjcld 10897	Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)


Theorem	replimd 10898	Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)


Theorem	remimd 10899	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 Mario Carneiro, 29-May-2016.)


Theorem	cjcjd 10900	The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

< Previous Next >

Page List Jump to page: Contents 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14078

< Previous Next >