P. 110 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	crre 10901	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.)
|- ( ( A e. RR /\ B e. RR ) -> ( Re ` ( A + ( _i x. B ) ) ) = A )
Theorem	crim 10902	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.)
|- ( ( A e. RR /\ B e. RR ) -> ( Im ` ( A + ( _i x. B ) ) ) = B )
Theorem	replim 10903	Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
|- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
Theorem	remim 10904	Value of the conjugate of a complex number. The value is the real part minus _i 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.)
|- ( A e. CC -> ( * ` A ) = ( ( Re ` A ) - ( _i x. ( Im ` A ) ) ) )
Theorem	reim0 10905	The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
|- ( A e. RR -> ( Im ` A ) = 0 )
Theorem	reim0b 10906	A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)
|- ( A e. CC -> ( A e. RR <-> ( Im ` A ) = 0 ) )
Theorem	rereb 10907	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.)
|- ( A e. CC -> ( A e. RR <-> ( Re ` A ) = A ) )
Theorem	mulreap 10908	A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( A e. RR <-> ( B x. A ) e. RR ) )
Theorem	rere 10909	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.)
|- ( A e. RR -> ( Re ` A ) = A )
Theorem	cjreb 10910	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.)
|- ( A e. CC -> ( A e. RR <-> ( * ` A ) = A ) )
Theorem	recj 10911	Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( Re ` ( * ` A ) ) = ( Re ` A ) )
Theorem	reneg 10912	Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( Re ` -u A ) = -u ( Re ` A ) )
Theorem	readd 10913	Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A + B ) ) = ( ( Re ` A ) + ( Re ` B ) ) )
Theorem	resub 10914	Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )
Theorem	remullem 10915	Lemma for remul 10916, immul 10923, and cjmul 10929. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( Re ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) ) /\ ( Im ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) ) /\ ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) ) )
Theorem	remul 10916	Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) ) )
Theorem	remul2 10917	Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR /\ B e. CC ) -> ( Re ` ( A x. B ) ) = ( A x. ( Re ` B ) ) )
Theorem	redivap 10918	Real part of a division. Related to remul2 10917. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Re ` ( A / B ) ) = ( ( Re ` A ) / B ) )
Theorem	imcj 10919	Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( Im ` ( * ` A ) ) = -u ( Im ` A ) )
Theorem	imneg 10920	The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( Im ` -u A ) = -u ( Im ` A ) )
Theorem	imadd 10921	Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( Im ` ( A + B ) ) = ( ( Im ` A ) + ( Im ` B ) ) )
Theorem	imsub 10922	Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( Im ` ( A - B ) ) = ( ( Im ` A ) - ( Im ` B ) ) )
Theorem	immul 10923	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( Im ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) ) )
Theorem	immul2 10924	Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR /\ B e. CC ) -> ( Im ` ( A x. B ) ) = ( A x. ( Im ` B ) ) )
Theorem	imdivap 10925	Imaginary part of a division. Related to immul2 10924. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Im ` ( A / B ) ) = ( ( Im ` A ) / B ) )
Theorem	cjre 10926	A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)
|- ( A e. RR -> ( * ` A ) = A )
Theorem	cjcj 10927	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.)
|- ( A e. CC -> ( * ` ( * ` A ) ) = A )
Theorem	cjadd 10928	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.)
|- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) ) )
Theorem	cjmul 10929	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.)
|- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) )
Theorem	ipcnval 10930	Standard inner product on complex numbers. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) ) )
Theorem	cjmulrcl 10931	A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( A x. ( * ` A ) ) e. RR )
Theorem	cjmulval 10932	A complex number times its conjugate. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )
Theorem	cjmulge0 10933	A complex number times its conjugate is nonnegative. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> 0 <_ ( A x. ( * ` A ) ) )
Theorem	cjneg 10934	Complex conjugate of negative. (Contributed by NM, 27-Feb-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( * ` -u A ) = -u ( * ` A ) )
Theorem	addcj 10935	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.)
|- ( A e. CC -> ( A + ( * ` A ) ) = ( 2 x. ( Re ` A ) ) )
Theorem	cjsub 10936	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A - B ) ) = ( ( * ` A ) - ( * ` B ) ) )
Theorem	cjexp 10937	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) )
Theorem	imval2 10938	The imaginary part of a number in terms of complex conjugate. (Contributed by NM, 30-Apr-2005.)
|- ( A e. CC -> ( Im ` A ) = ( ( A - ( * ` A ) ) / ( 2 x. _i ) ) )
Theorem	re0 10939	The real part of zero. (Contributed by NM, 27-Jul-1999.)
|- ( Re ` 0 ) = 0
Theorem	im0 10940	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
|- ( Im ` 0 ) = 0
Theorem	re1 10941	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Re ` 1 ) = 1
Theorem	im1 10942	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Im ` 1 ) = 0
Theorem	rei 10943	The real part of _i. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Re ` _i ) = 0
Theorem	imi 10944	The imaginary part of _i. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Im ` _i ) = 1
Theorem	cj0 10945	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
|- ( * ` 0 ) = 0
Theorem	cji 10946	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
|- ( * ` _i ) = -u _i
Theorem	cjreim 10947	The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A + ( _i x. B ) ) ) = ( A - ( _i x. B ) ) )
Theorem	cjreim2 10948	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.)
|- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A - ( _i x. B ) ) ) = ( A + ( _i x. B ) ) )
Theorem	cj11 10949	Complex conjugate is a one-to-one function. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Eric Schmidt, 2-Jul-2009.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) = ( * ` B ) <-> A = B ) )
Theorem	cjap 10950	Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) # ( * ` B ) <-> A # B ) )
Theorem	cjap0 10951	A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( A e. CC -> ( A # 0 <-> ( * ` A ) # 0 ) )
Theorem	cjne0 10952	A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 10951 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)
|- ( A e. CC -> ( A =/= 0 <-> ( * ` A ) =/= 0 ) )
Theorem	cjdivap 10953	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( * ` ( A / B ) ) = ( ( * ` A ) / ( * ` B ) ) )
Theorem	cnrecnv 10954*	The inverse to the canonical bijection from ( RR X. RR ) to CC from cnref1o 9682. (Contributed by Mario Carneiro, 25-Aug-2014.)
|- F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) )   =>    |- `' F = ( z e. CC |-> <. ( Re ` z ) , ( Im ` z ) >. )
Theorem	recli 10955	The real part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
|- A e. CC   =>    |- ( Re ` A ) e. RR
Theorem	imcli 10956	The imaginary part of a complex number is real (closure law). (Contributed by NM, 11-May-1999.)
|- A e. CC   =>    |- ( Im ` A ) e. RR
Theorem	cjcli 10957	Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
|- A e. CC   =>    |- ( * ` A ) e. CC
Theorem	replimi 10958	Construct a complex number from its real and imaginary parts. (Contributed by NM, 1-Oct-1999.)
|- A e. CC   =>    |- A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) )
Theorem	cjcji 10959	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.)
|- A e. CC   =>    |- ( * ` ( * ` A ) ) = A
Theorem	reim0bi 10960	A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999.)
|- A e. CC   =>    |- ( A e. RR <-> ( Im ` A ) = 0 )
Theorem	rerebi 10961	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 27-Oct-1999.)
|- A e. CC   =>    |- ( A e. RR <-> ( Re ` A ) = A )
Theorem	cjrebi 10962	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.)
|- A e. CC   =>    |- ( A e. RR <-> ( * ` A ) = A )
Theorem	recji 10963	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   =>    |- ( Re ` ( * ` A ) ) = ( Re ` A )
Theorem	imcji 10964	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   =>    |- ( Im ` ( * ` A ) ) = -u ( Im ` A )
Theorem	cjmulrcli 10965	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
|- A e. CC   =>    |- ( A x. ( * ` A ) ) e. RR
Theorem	cjmulvali 10966	A complex number times its conjugate. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   =>    |- ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) )
Theorem	cjmulge0i 10967	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
|- A e. CC   =>    |- 0 <_ ( A x. ( * ` A ) )
Theorem	renegi 10968	Real part of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( Re ` -u A ) = -u ( Re ` A )
Theorem	imnegi 10969	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( Im ` -u A ) = -u ( Im ` A )
Theorem	cjnegi 10970	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( * ` -u A ) = -u ( * ` A )
Theorem	addcji 10971	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.)
|- A e. CC   =>    |- ( A + ( * ` A ) ) = ( 2 x. ( Re ` A ) )
Theorem	readdi 10972	Real part distributes over addition. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( Re ` ( A + B ) ) = ( ( Re ` A ) + ( Re ` B ) )
Theorem	imaddi 10973	Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( Im ` ( A + B ) ) = ( ( Im ` A ) + ( Im ` B ) )
Theorem	remuli 10974	Real part of a product. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( Re ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) )
Theorem	immuli 10975	Imaginary part of a product. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( Im ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) )
Theorem	cjaddi 10976	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) )
Theorem	cjmuli 10977	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by NM, 28-Jul-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) )
Theorem	ipcni 10978	Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) )
Theorem	cjdivapi 10979	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- A e. CC   &    |- B e. CC   =>    |- ( B # 0 -> ( * ` ( A / B ) ) = ( ( * ` A ) / ( * ` B ) ) )
Theorem	crrei 10980	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( Re ` ( A + ( _i x. B ) ) ) = A
Theorem	crimi 10981	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( Im ` ( A + ( _i x. B ) ) ) = B
Theorem	recld 10982	The real part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` A ) e. RR )
Theorem	imcld 10983	The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Im ` A ) e. RR )
Theorem	cjcld 10984	Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` A ) e. CC )
Theorem	replimd 10985	Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
Theorem	remimd 10986	Value of the conjugate of a complex number. The value is the real part minus _i times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` A ) = ( ( Re ` A ) - ( _i x. ( Im ` A ) ) ) )
Theorem	cjcjd 10987	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.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` ( * ` A ) ) = A )
Theorem	reim0bd 10988	A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( Im ` A ) = 0 )   =>    |- ( ph -> A e. RR )
Theorem	rerebd 10989	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( Re ` A ) = A )   =>    |- ( ph -> A e. RR )
Theorem	cjrebd 10990	A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( * ` A ) = A )   =>    |- ( ph -> A e. RR )
Theorem	cjne0d 10991	A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 10992 which is similar but for apartness. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> A =/= 0 )   =>    |- ( ph -> ( * ` A ) =/= 0 )
Theorem	cjap0d 10992	A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( * ` A ) # 0 )
Theorem	recjd 10993	Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` ( * ` A ) ) = ( Re ` A ) )
Theorem	imcjd 10994	Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Im ` ( * ` A ) ) = -u ( Im ` A ) )
Theorem	cjmulrcld 10995	A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. ( * ` A ) ) e. RR )
Theorem	cjmulvald 10996	A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )
Theorem	cjmulge0d 10997	A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> 0 <_ ( A x. ( * ` A ) ) )
Theorem	renegd 10998	Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` -u A ) = -u ( Re ` A ) )
Theorem	imnegd 10999	Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Im ` -u A ) = -u ( Im ` A ) )
Theorem	cjnegd 11000	Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` -u A ) = -u ( * ` A ) )