mmtheorems67 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6601-6700 - Page 67 of 105

Type	Label	Description
Statement
Theorem	sqrsq 6601	Square root of square.
|- A e. RR   =>   |- (0 <_ A -> (sqr` (A^2)) = A)
Theorem	sqsqr 6602	Square of square root.
|- A e. RR   =>   |- (0 <_ A -> ((sqr` A)^2) = A)
Theorem	sqrsqt 6603	Square root of square.
|- ((A e. RR /\ 0 <_ A) -> (sqr` (A^2)) = A)
Theorem	sqsqrt 6604	Square of square root.
|- ((A e. RR /\ 0 <_ A) -> ((sqr` A)^2) = A)
Irrationality of square root of 2
Theorem	sqr2irrlem1 6605	Lemma for irrationality of square root of 2. Technical lemma used to simplify the main induction step.
Theorem	sqr2irrlem2 6606	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.
Theorem	sqr2irrlem3 6607	Main theorem for irrationality of square root of 2. There are no natural numbers such that the square of one is twice the square of the other. Uses strong induction.
|- -. E.x e. NN E.y e. NN (x^2) = (2 x. (y^2))
Theorem	sqr2irrlem4 6608	Lemma for irrationality of square root of 2.
Theorem	sqr2irrlem5 6609	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.
Theorem	sqr2irr 6610	The square root of 2 is irrational.
|- (sqr` 2) e/ QQ
Theorem	sqr2re 6611	The square root of 2 exists and is a real number.
|- (sqr` 2) e. RR
Imaginary and complex number properties
Theorem	irec 6612	The reciprocal of i.
|- (1 / i) = -ui
Theorem	i2 6613	i squared.
|- (i^2) = -u1
Theorem	i3 6614	i cubed.
|- (i^3) = -ui
Theorem	i4 6615	i to the fourth power.
|- (i^4) = 1
Theorem	inelr 6616	The imaginary unit i is not a real number.
|- -. i e. RR
Theorem	crulem 6617	Lemma for cru 6618.
Theorem	cru 6618	The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   &   |- D e. RR   =>   |- ((A + (i x. B)) = (C + (i x. D)) <-> (A = C /\ B = D))
Theorem	crut 6619	The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130.
|- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR)) -> ((A + (i x. B)) = (C + (i x. D)) <-> (A = C /\ B = D)))
Theorem	crutOLD 6620	The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130.
|- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR)) -> ((A + (B x. i)) = (C + (D x. i)) <-> (A = C /\ B = D)))
Theorem	crne0 6621	The real representation of complex numbers is nonzero iff one of its terms is nonzero.
|- A e. RR   &   |- B e. RR   =>   |- ((A =/= 0 \/ B =/= 0) <-> (A + (i x. B)) =/= 0)
Theorem	crmul 6622	Multiplication rule for complex number representation. Remark in [Apostol] p. 361. In normal use, the arguments are the real components of two complex numbers, but the theorem works for complex components as well.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   =>   |- ((A + (i x. B)) x. (C + (i x. D))) = (((A x. C) - (B x. D)) + (i x. ((A x. D) + (B x. C))))
Theorem	crrecz 6623	Reciprocal of a complex number in terms of real and imaginary components. Remark in [Apostol] p. 361.
|- A e. RR   &   |- B e. RR   =>   |- ((A =/= 0 \/ B =/= 0) -> (1 / (A + (i x. B))) = ((A - (i x. B)) / ((A^2) + (B^2))))
Theorem	creur 6624	The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130.
|- (A e. CC -> E!x e. RR E.y e. RR A = (x + (i x. y)))
Theorem	creui 6625	The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130.
|- (A e. CC -> E!y e. RR E.x e. RR A = (x + (i x. y)))
Theorem	rimul 6626	A real number times the imaginary unit is real only if the number is 0.
|- ((A e. RR /\ (i x. A) e. RR) -> A = 0)
Theorem	nthruc 6627	The sequence NN, ZZ, QQ, RR, and CC forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to ZZ but not NN, one-half belongs to QQ but not ZZ, the square root of 2 belongs to RR but not QQ, and finally that the imaginary number i belongs to CC but not RR. See nthruz 6628 for a further refinement.
|- ((NN (. ZZ /\ ZZ (. QQ) /\ (QQ (. RR /\ RR (. CC))
Theorem	nthruz 6628	The sequence NN, NN0, and ZZ forms a chain of proper subsets. In each case the proper subset relationship is shown by demonstrating a number that belongs to one set but not the other. We show that zero belongs to NN0 but not NN and minus one belongs to ZZ but not NN0. This theorem refines the chain of proper subsets nthruc 6627.
|- (NN (. NN0 /\ NN0 (. ZZ)
Real and imaginary parts; conjugate; absolute value
Syntax	cre 6629	Extend class notation to include real part of a complex number.
class Re
Syntax	cim 6630	Extend class notation to include imaginary part of a complex number.
class Im
Syntax	ccj 6631	Extend class notation to include complex conjugate function.
class *
Syntax	cabs 6632	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs
Definition	df-re 6633	Define a function whose value is the real part of a complex number. See revalt 6637 for its value, recl 6648 for its closure, and replimt 6643 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 6634	Define a function whose value is the imaginary part of a complex number. See imvalt 6638 for its value, imcl 6649 for its closure, and replimt 6643 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 6635	Define the complex conjugate function. See cjcl 6650 for its closure and cjvalt 6646 for its value.
|- * = {<.x, y>. | (x e. CC /\ y = ((Re` x) - (i x. (Im` x))))}
Definition	df-abs 6636	Define the function for the absolute value (modulus) of a complex number. See abscl 6725 for its closure and absvalt 6645 or absval2 6727 for its value.
|- abs = {<.x, y>. | (x e. CC /\ y = (sqr` (x x. (*` x))))}
Theorem	revalt 6637	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 6638	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 6639	The real part of a complex number is real.
|- (A e. CC -> (Re` A) e. RR)
Theorem	imclt 6640	The imaginary part of a complex number is real.
|- (A e. CC -> (Im` A) e. RR)
Theorem	ref 6641	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- Re:CC-->RR
Theorem	imf 6642	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- Im:CC-->RR
Theorem	replimt 6643	Reconstruct a complex number from its real and imaginary parts.
|- (A e. CC -> A = ((Re` A) + (i x. (Im` A))))
Theorem	replimtOLD 6644	Reconstruct a complex number from its real and imaginary parts.
|- (A e. CC -> A = ((Re` A) + ((Im` A) x. i)))
Theorem	absvalt 6645	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 6646	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 6647	The conjugate of a complex number is a complex number (closure law).
|- (A e. CC -> (*` A) e. CC)
Theorem	recl 6648	The real part of a complex number is real (closure law).
|- A e. CC   =>   |- (Re` A) e. RR
Theorem	imcl 6649	The imaginary part of a complex number is real (closure law).
|- A e. CC   =>   |- (Im` A) e. RR
Theorem	cjcl 6650	Closure law for complex conjugate.
|- A e. CC   =>   |- (*` A) e. CC
Theorem	replim 6651	Construct a complex number from its real and imaginary parts.
|- A e. CC   =>   |- A = ((Re` A) + (i x. (Im` A)))
Theorem	replimOLD 6652	Construct a complex number from its real and imaginary parts.
|- A e. CC   =>   |- A = ((Re` A) + ((Im` A) x. i))
Theorem	crret 6653	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 6654	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 6655	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 6656	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 6657	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 6658	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 6659	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 6660	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 6661	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 6662	The imaginary part of a real number is 0.
|- (A e. RR -> (Im` A) = 0)
Theorem	reim0bt 6663	A number is real iff its imaginary part is 0.
|- (A e. CC -> (A e. RR <-> (Im` A) = 0))
Theorem	cjcj 6664	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 6665	A number is real iff its imaginary part is 0.
|- A e. CC   =>   |- (A e. RR <-> (Im` A) = 0)
Theorem	rereb 6666	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 6667	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