P. 98 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9701-9800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	immul2 9701	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 9702	Imaginary part of a division. Related to immul2 9701. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Im ` ( A / B ) ) = ( ( Im ` A ) / B ) )
Theorem	cjre 9703	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 9704	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 9705	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 9706	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 9707	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 9708	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 9709	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 9710	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 9711	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 9712	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 9713	Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A - B ) ) = ( ( * ` A ) - ( * ` B ) ) )
Theorem	cjexp 9714	Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) )
Theorem	imval2 9715	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 9716	The real part of zero. (Contributed by NM, 27-Jul-1999.)
|- ( Re ` 0 ) = 0
Theorem	im0 9717	The imaginary part of zero. (Contributed by NM, 27-Jul-1999.)
|- ( Im ` 0 ) = 0
Theorem	re1 9718	The real part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Re ` 1 ) = 1
Theorem	im1 9719	The imaginary part of one. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Im ` 1 ) = 0
Theorem	rei 9720	The real part of _i. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Re ` _i ) = 0
Theorem	imi 9721	The imaginary part of _i. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( Im ` _i ) = 1
Theorem	cj0 9722	The conjugate of zero. (Contributed by NM, 27-Jul-1999.)
|- ( * ` 0 ) = 0
Theorem	cji 9723	The complex conjugate of the imaginary unit. (Contributed by NM, 26-Mar-2005.)
|- ( * ` _i ) = -u _i
Theorem	cjreim 9724	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 9725	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 9726	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 9727	Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) # ( * ` B ) <-> A # B ) )
Theorem	cjap0 9728	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 9729	A number is nonzero iff its complex conjugate is nonzero. Also see cjap0 9728 which is similar but for apartness. (Contributed by NM, 29-Apr-2005.)
|- ( A e. CC -> ( A =/= 0 <-> ( * ` A ) =/= 0 ) )
Theorem	cjdivap 9730	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 9731*	The inverse to the canonical bijection from ( RR X. RR ) to CC from cnref1o 8679. (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 9732	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 9733	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 9734	Closure law for complex conjugate. (Contributed by NM, 11-May-1999.)
|- A e. CC   =>    |- ( * ` A ) e. CC
Theorem	replimi 9735	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 9736	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 9737	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 9738	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 9739	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 9740	Real part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   =>    |- ( Re ` ( * ` A ) ) = ( Re ` A )
Theorem	imcji 9741	Imaginary part of a complex conjugate. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   =>    |- ( Im ` ( * ` A ) ) = -u ( Im ` A )
Theorem	cjmulrcli 9742	A complex number times its conjugate is real. (Contributed by NM, 11-May-1999.)
|- A e. CC   =>    |- ( A x. ( * ` A ) ) e. RR
Theorem	cjmulvali 9743	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 9744	A complex number times its conjugate is nonnegative. (Contributed by NM, 28-May-1999.)
|- A e. CC   =>    |- 0 <_ ( A x. ( * ` A ) )
Theorem	renegi 9745	Real part of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( Re ` -u A ) = -u ( Re ` A )
Theorem	imnegi 9746	Imaginary part of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( Im ` -u A ) = -u ( Im ` A )
Theorem	cjnegi 9747	Complex conjugate of negative. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( * ` -u A ) = -u ( * ` A )
Theorem	addcji 9748	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 9749	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 9750	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 9751	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 9752	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 9753	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 9754	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 9755	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 9756	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 9757	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 9758	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 9759	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 9760	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 9761	Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` A ) e. CC )
Theorem	replimd 9762	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 9763	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 9764	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 9765	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 9766	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 9767	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 9768	A number which is nonzero has a complex conjugate which is nonzero. Also see cjap0d 9769 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 9769	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 9770	Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` ( * ` A ) ) = ( Re ` A ) )
Theorem	imcjd 9771	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 9772	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 9773	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 9774	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 9775	Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Re ` -u A ) = -u ( Re ` A ) )
Theorem	imnegd 9776	Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( Im ` -u A ) = -u ( Im ` A ) )
Theorem	cjnegd 9777	Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( * ` -u A ) = -u ( * ` A ) )
Theorem	addcjd 9778	A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A + ( * ` A ) ) = ( 2 x. ( Re ` A ) ) )
Theorem	cjexpd 9779	Complex conjugate of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> N e. NN0 )   =>    |- ( ph -> ( * ` ( A ^ N ) ) = ( ( * ` A ) ^ N ) )
Theorem	readdd 9780	Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A + B ) ) = ( ( Re ` A ) + ( Re ` B ) ) )
Theorem	imaddd 9781	Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A + B ) ) = ( ( Im ` A ) + ( Im ` B ) ) )
Theorem	resubd 9782	Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A - B ) ) = ( ( Re ` A ) - ( Re ` B ) ) )
Theorem	imsubd 9783	Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A - B ) ) = ( ( Im ` A ) - ( Im ` B ) ) )
Theorem	remuld 9784	Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) ) )
Theorem	immuld 9785	Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Im ` B ) ) + ( ( Im ` A ) x. ( Re ` B ) ) ) )
Theorem	cjaddd 9786	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) ) )
Theorem	cjmuld 9787	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) )
Theorem	ipcnd 9788	Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) ) )
Theorem	cjdivapd 9789	Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( * ` ( A / B ) ) = ( ( * ` A ) / ( * ` B ) ) )
Theorem	rered 9790	A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( Re ` A ) = A )
Theorem	reim0d 9791	The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( Im ` A ) = 0 )
Theorem	cjred 9792	A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( * ` A ) = A )
Theorem	remul2d 9793	Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Re ` ( A x. B ) ) = ( A x. ( Re ` B ) ) )
Theorem	immul2d 9794	Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( Im ` ( A x. B ) ) = ( A x. ( Im ` B ) ) )
Theorem	redivapd 9795	Real part of a division. Related to remul2 9694. (Contributed by Jim Kingdon, 15-Jun-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( Re ` ( B / A ) ) = ( ( Re ` B ) / A ) )
Theorem	imdivapd 9796	Imaginary part of a division. Related to remul2 9694. (Contributed by Jim Kingdon, 15-Jun-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( Im ` ( B / A ) ) = ( ( Im ` B ) / A ) )
Theorem	crred 9797	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( Re ` ( A + ( _i x. B ) ) ) = A )
Theorem	crimd 9798	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( Im ` ( A + ( _i x. B ) ) ) = B )
3.7.3  Sequence convergence
Theorem	caucvgrelemrec 9799*	Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
|- ( ( A e. RR /\ A # 0 ) -> ( iota_ r e. RR ( A x. r ) = 1 ) = ( 1 / A ) )
Theorem	caucvgrelemcau 9800*	Lemma for caucvgre 9801. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( 1 / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( 1 / n ) ) ) )   =>    |- ( ph -> A. n e. NN A. k e. NN ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )