mmtheorems68 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6701-6800 - Page 68 of 107

Type	Label	Description
Statement
Definition	df-re 6701	Define a function whose value is the real part of a complex number. See revalt 6705 for its value, recl 6716 for its closure, and replimt 6711 for its use in decomposing a complex number.
|- Re = {<.x, y>. | (x e. CC /\ y = U.{z e. RR | E.w e. RR x = (z + (i x. w))})}
Definition	df-im 6702	Define a function whose value is the imaginary part of a complex number. See imvalt 6706 for its value, imcl 6717 for its closure, and replimt 6711 for its use in decomposing a complex number.
|- Im = {<.x, y>. | (x e. CC /\ y = U.{w e. RR | E.z e. RR x = (z + (i x. w))})}
Definition	df-cj 6703	Define the complex conjugate function. See cjcl 6718 for its closure and cjvalt 6714 for its value.
|- * = {<.x, y>. | (x e. CC /\ y = ((Re` x) - (i x. (Im` x))))}
Definition	df-abs 6704	Define the function for the absolute value (modulus) of a complex number. See abscl 6793 for its closure and absvalt 6713 or absval2 6795 for its value.
|- abs = {<.x, y>. | (x e. CC /\ y = (sqr` (x x. (*` x))))}
Theorem	revalt 6705	The value of the real part of a complex number.
|- (A e. CC -> (Re` A) = U.{x e. RR | E.y e. RR A = (x + (i x. y))})
Theorem	imvalt 6706	The value of the imaginary part of a complex number.
|- (A e. CC -> (Im` A) = U.{y e. RR | E.x e. RR A = (x + (i x. y))})
Theorem	reclt 6707	The real part of a complex number is real.
|- (A e. CC -> (Re` A) e. RR)
Theorem	imclt 6708	The imaginary part of a complex number is real.
|- (A e. CC -> (Im` A) e. RR)
Theorem	ref 6709	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- Re:CC-->RR
Theorem	imf 6710	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- Im:CC-->RR
Theorem	replimt 6711	Reconstruct a complex number from its real and imaginary parts.
|- (A e. CC -> A = ((Re` A) + (i x. (Im` A))))
Theorem	replimtOLD 6712	Reconstruct a complex number from its real and imaginary parts.
|- (A e. CC -> A = ((Re` A) + ((Im` A) x. i)))
Theorem	absvalt 6713	The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133.
|- (A e. CC -> (abs` A) = (sqr` (A x. (*` A))))
Theorem	cjvalt 6714	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.
|- (A e. CC -> (*` A) = ((Re` A) - (i x. (Im` A))))
Theorem	cjclt 6715	The conjugate of a complex number is a complex number (closure law).
|- (A e. CC -> (*` A) e. CC)
Theorem	recl 6716	The real part of a complex number is real (closure law).
|- A e. CC   =>   |- (Re` A) e. RR
Theorem	imcl 6717	The imaginary part of a complex number is real (closure law).
|- A e. CC   =>   |- (Im` A) e. RR
Theorem	cjcl 6718	Closure law for complex conjugate.
|- A e. CC   =>   |- (*` A) e. CC
Theorem	replim 6719	Construct a complex number from its real and imaginary parts.
|- A e. CC   =>   |- A = ((Re` A) + (i x. (Im` A)))
Theorem	replimOLD 6720	Construct a complex number from its real and imaginary parts.
|- A e. CC   =>   |- A = ((Re` A) + ((Im` A) x. i))
Theorem	crret 6721	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
|- ((A e. RR /\ B e. RR) -> (Re` (A + (i x. B))) = A)
Theorem	crretOLD 6722	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
|- ((A e. RR /\ B e. RR) -> (Re` (A + (B x. i))) = A)
Theorem	crimt 6723	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
|- ((A e. RR /\ B e. RR) -> (Im` (A + (i x. B))) = B)
Theorem	crimtOLD 6724	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
|- ((A e. RR /\ B e. RR) -> (Im` (A + (B x. i))) = B)
Theorem	crre 6725	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
|- A e. RR   &   |- B e. RR   =>   |- (Re` (A + (i x. B))) = A
Theorem	crreOLD 6726	The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
|- A e. RR   &   |- B e. RR   =>   |- (Re` (A + (B x. i))) = A
Theorem	crim 6727	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
|- A e. RR   &   |- B e. RR   =>   |- (Im` (A + (i x. B))) = B
Theorem	crimOLD 6728	The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132.
|- A e. RR   &   |- B e. RR   =>   |- (Im` (A + (B x. i))) = B
Theorem	imret 6729	The imaginary part of a complex number in terms of the real part function.
|- (A e. CC -> (Im` A) = (Re` (-ui x. A)))
Theorem	reim0t 6730	The imaginary part of a real number is 0.
|- (A e. RR -> (Im` A) = 0)
Theorem	reim0bt 6731	A number is real iff its imaginary part is 0.
|- (A e. CC -> (A e. RR <-> (Im` A) = 0))
Theorem	cjcj 6732	The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133.
|- A e. CC   =>   |- (*` (*` A)) = A
Theorem	reim0b 6733	A number is real iff its imaginary part is 0.
|- A e. CC   =>   |- (A e. RR <-> (Im` A) = 0)
Theorem	rereb 6734	A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133.
|- A e. CC   =>   |- (A e. RR <-> (Re` A) = A)
Theorem	cjreb 6735	A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133.
|- A e. CC   =>   |- (A e. RR <-> (*` A) = A)
Theorem	recj 6736	Real part of a complex conjugate.
|- A e. CC   =>   |- (Re` (*` A)) = (Re` A)
Theorem	imcj 6737	Imaginary part of a complex conjugate.
|- A e. CC   =>   |- (Im` (*` A)) = -u(Im` A)
Theorem	readd 6738	Real part distributes over addition.
|- A e. CC   &   |- B e. CC   =>   |- (Re` (A + B)) = ((Re` A) + (Re` B))
Theorem	imadd 6739	Imaginary part distributes over addition.
|- A e. CC   &   |- B e. CC   =>   |- (Im` (A + B)) = ((Im` A) + (Im` B))
Theorem	remul 6740	Real part of a product.
|- A e. CC   &   |- B e. CC   =>   |- (Re` (A x. B)) = (((Re` A) x. (Re` B)) - ((Im` A) x. (Im` B)))
Theorem	immul 6741	Imaginary part of a product.
|- A e. CC   &   |- B e. CC   =>   |- (Im` (A x. B)) = (((Re` A) x. (Im` B)) + ((Im` A) x. (Re` B)))
Theorem	cjadd 6742	Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133.
|- A e. CC   &   |- B e. CC   =>   |- (*` (A + B)) = ((*` A) + (*` B))
Theorem	cjmul 6743	Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133.
|- A e. CC   &   |- B e. CC   =>   |- (*` (A x. B)) = ((*` A) x. (*` B))
Theorem	ipcn 6744	Standard inner product on complex numbers.
|- A e. CC   &   |- B e. CC   =>   |- (Re` (A x. (*` B))) = (((Re` A) x. (Re` B)) + ((Im` A) x. (Im` B)))
Theorem	cjmulrcl 6745	A complex number times its conjugate is real.
|- A e. CC   =>   |- (A x. (*` A)) e. RR
Theorem	cjmulval 6746	A complex number times its conjugate.
|- A e. CC   =>   |- (A x. (*` A)) = (((Re` A)^2) + ((Im` A)^2))
Theorem	cjmulge0 6747	A complex number times its conjugate is nonnegative.
|- A e. CC   =>   |- 0 <_ (A x. (*` A))
Theorem	reneg 6748	Real part of negative.
|- A e. CC   =>   |- (Re` -uA) = -u(Re` A)
Theorem	negreb 6749	The negative of a real is real.
|- A e. CC   =>   |- (-uA e. RR <-> A e. RR)
Theorem	imneg 6750	Imaginary part of negative.
|- A e. CC   =>   |- (Im` -uA) = -u(Im` A)
Theorem	cjneg 6751	Complex conjugate of negative.
|- A e. CC   =>   |- (*` -uA) = -u(*` A)
Theorem	addcj 6752	A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133.
|- A e. CC   =>   |- (A + (*` A)) = (2 x. (Re` A))
Theorem	reret 6753	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	cjrebt 6754	A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133.
|- (A e. CC -> (A e. RR <-> (*` A) = A))
Theorem	cjmulrclt 6755	A complex number times its conjugate is real.
|- (A e. CC -> (A x. (*` A)) e. RR)
Theorem	cjmulvalt 6756	A complex number times its conjugate.
|- (A e. CC -> (A x. (*` A)) = (((Re` A)^2) + ((Im` A)^2)))
Theorem	cjmulge0t 6757	A complex number times its conjugate is nonnegative.
|- (A e. CC -> 0 <_ (A x. (*` A)))
Theorem	renegt 6758	Real part of negative.
|- (A e. CC -> (Re` -uA) = -u(Re` A))
Theorem	readdt 6759	Real part distributes over addition.
|- ((A e. CC /\ B e. CC) -> (Re` (A + B)) = ((Re` A) + (Re` B)))
Theorem	resubt 6760	Real part distributes over subtraction.
|- ((A e. CC /\ B e. CC) -> (Re` (A - B)) = ((Re` A) - (Re` B)))
Theorem	imnegt 6761	The imaginary part of a negative number.
|- (A e. CC -> (Im` -u