mmtheorems70 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6901-7000 - Page 70 of 123

Type	Label	Description
Statement
Theorem	sqrcli 6901	The square root of a nonnegative real is a real.
|- A e. RR   =>   |- (0 <_ A -> (sqr` A) e. RR)
Theorem	sqrgt0i 6902	The square root of a positive real is positive.
|- A e. RR   =>   |- (0 < A -> 0 < (sqr` A))
Theorem	sqrge0i 6903	The square root of a nonnegative real is nonnegative.
|- A e. RR   =>   |- (0 <_ A -> 0 <_ (sqr` A))
Theorem	sqr11i 6904	The square root function is one-to-one.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> ((sqr` A) = (sqr` B) <-> A = B))
Theorem	sqrmulii 6905	Square root distributes over multiplication.
|- A e. RR   &   |- B e. RR   &   |- 0 <_ A   &   |- 0 <_ B   =>   |- (sqr` (A x. B)) = ((sqr` A) x. (sqr` B))
Theorem	sqrmuli 6906	Square root distributes over multiplication.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> (sqr` (A x. B)) = ((sqr` A) x. (sqr` B)))
Theorem	sqrmsq2i 6907	Relationship between square root and squares.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> ((sqr` A) = B <-> A = (B x. B)))
Theorem	sqrlei 6908	Square root is monotonic.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> (A <_ B <-> (sqr` A) <_ (sqr` B)))
Theorem	sqrlti 6909	Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.)
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> (A < B <-> (sqr` A) < (sqr` B)))
Theorem	sqrmsqi 6910	Square root of square.
|- A e. RR   =>   |- (0 <_ A -> (sqr` (A x. A)) = A)
Theorem	sqrcl 6911	The square root of a nonnegative real is a real.
|- ((A e. RR /\ 0 <_ A) -> (sqr` A) e. RR)
Theorem	sqrgt0 6912	The square root of a positive real is positive.
|- ((A e. RR /\ 0 < A) -> 0 < (sqr` A))
Theorem	sqrge0 6913	The square root of a nonnegative real is nonnegative.
|- ((A e. RR /\ 0 <_ A) -> 0 <_ (sqr` A))
Theorem	sqrle 6914	Square root is monotonic.
|- (((A e. RR /\ B e. RR) /\ (0 <_ A /\ 0 <_ B)) -> (A <_ B <-> (sqr` A) <_ (sqr` B)))
Theorem	sqr00 6915	A square root is zero iff its argument is 0.
|- ((A e. RR /\ 0 <_ A) -> ((sqr` A) = 0 <-> A = 0))
Theorem	rpsqrcl 6916	The square root of a positive real is a postive real.
|- (A e. RR+ -> (sqr` A) e. RR+)
Theorem	sqr1 6917	The square root of 1 is 1.
|- (sqr` 1) = 1
Theorem	sqr4 6918	The square root of 4 is 2.
|- (sqr` 4) = 2
Theorem	sqr9 6919	The square root of 9 is 3.
|- (sqr` 9) = 3
Theorem	sqr2gt1lt2 6920	The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.)
|- (1 < (sqr` 2) /\ (sqr` 2) < 2)
Theorem	sqrsqi 6921	Square root of square.
|- A e. RR   =>   |- (0 <_ A -> (sqr` (A^2)) = A)
Theorem	sqsqri 6922	Square of square root.
|- A e. RR   =>   |- (0 <_ A -> ((sqr` A)^2) = A)
Theorem	sqrsq 6923	Square root of square.
|- ((A e. RR /\ 0 <_ A) -> (sqr` (A^2)) = A)
Theorem	sqsqr 6924	Square of square root.
|- ((A e. RR /\ 0 <_ A) -> ((sqr` A)^2) = A)
Irrationality of square root of 2
Theorem	sqr2irrlem1 6925	Lemma for irrationality of square root of 2. Technical lemma used to simplify the main induction step.
Theorem	sqr2irrlem2 6926	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.
Theorem	sqr2irrlem3 6927	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 6928	Lemma for irrationality of square root of 2.
Theorem	sqr2irrlem5 6929	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.
Theorem	sqr2irr 6930	The square root of 2 is irrational.
|- (sqr` 2) e/ QQ
Theorem	sqr2re 6931	The square root of 2 exists and is a real number.
|- (sqr` 2) e. RR
Imaginary and complex number properties
Theorem	irec 6932	The reciprocal of i.
|- (1 / i) = -ui
Theorem	i2 6933	i squared.
|- (i^2) = -u1
Theorem	i3 6934	i cubed.
|- (i^3) = -ui
Theorem	i4 6935	i to the fourth power.
|- (i^4) = 1
Theorem	inelr 6936	The imaginary unit i is not a real number.
|- -. i e. RR
Theorem	crulem 6937	Lemma for crui 6938.
Theorem	crui 6938	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	cru 6939	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	crne0i 6940	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	crmuli 6941	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	crreczi 6942	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 6943	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 6944	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 6945	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 6946	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 6947 for a further refinement.
|- ((NN (. ZZ /\ ZZ (. QQ) /\ (QQ (. RR /\ RR (. CC))
Theorem	nthruz 6947	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 6946.
|- (NN (. NN0 /\ NN0 (. ZZ)
Real and imaginary parts; conjugate; absolute value
Syntax	cre 6948	Extend class notation to include real part of a complex number.
class Re
Syntax	cim 6949	Extend class notation to include imaginary part of a complex number.
class Im
Syntax	ccj 6950	Extend class notation to include complex conjugate function.
class *
Syntax	cabs 6951	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs
Definition	df-re 6952	Define a function whose value is the real part of a complex number. See reval 6956 for its value, recli 6966 for its closure, and replim 6962 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 6953	Define a function whose value is the imaginary part of a complex number. See imval 6957 for its value, imcli 6967 for its closure, and replim 6962 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 6954	Define the complex conjugate function. See cjcli 6968 for its closure and cjval 6964 for its value.
|- * = {<.x, y>. | (x e. CC /\ y = ((Re` x) - (i x. (Im` x))))}
Definition	df-abs 6955	Define the function for the absolute value (modulus) of a complex number. See abscli 7041 for its closure and absval 6963 or absval2i 7043 for its value.
|- abs = {<.x, y>. | (x e. CC /\ y = (sqr` (x x. (*` x))))}
Theorem	reval 6956	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	imval 6957	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	recl 6958	The real part of a complex number is real.
|- (A e. CC -> (Re` A) e. RR)
Theorem	imcl 6959	The imaginary part of a complex number is real.
|- (A e. CC -> (Im` A) e. RR)
Theorem	ref 6960	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- Re:CC-->RR
Theorem	imf 6961	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- Im:CC-->RR
Theorem	replim 6962	Reconstruct a complex number from its real and imaginary parts.
|- (A e. CC -> A = ((Re` A) + (i x. (Im` A))))
Theorem	absval 6963	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	cjval 6964	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	cjcl 6965	The conjugate of a complex number is a complex number (closure law).
|- (A e. CC -> (*` A) e. CC)
Theorem	recli 6966	The real part of a complex number is real (closure law).
|- A e. CC   =>   |- (Re` A) e. RR
Theorem	imcli 6967	The imaginary part of a complex number is real (closure law).
|- A e. CC   =>   |- (Im` A) e. RR
Theorem	cjcli 6968	Closure law for complex conjugate.
|- A e. CC   =>   |- (*` A) e. CC
Theorem	replimi 6969	Construct a complex number from its real and imaginary parts.
|- A e. CC   =>   |- A = ((Re` A) + (i x. (Im` A)))
Theorem	crre 6970	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	crim 6971	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	crrei 6972	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	crimi 6973	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	imre 6974	The imaginary part of a complex number in terms of the real part function.
|- (A e. CC -> (Im` A) = (Re` (-ui x. A)))
Theorem	reim0 6975	The imaginary part of a real number is 0.
|- (A e. RR -> (Im` A) = 0)
Theorem	reim0b 6976	A number is real iff its imaginary part is 0.
|- (A e. CC -> (A e. RR <-> (Im` A) = 0))
Theorem	rereb 6977	A number is real iff it equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133.
|- (A e. CC -> (A e. RR <-> (Re` A) = A))
Theorem	mulre 6978	A product with a nonzero real multiplier is real iff the multiplicand is real.
|- ((A e. CC /\ B e. RR /\ B =/= 0) -> (A e. RR <-> (B x. A) e. RR))
Theorem	cjcji 6979	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	reim0bi 6980	A number is real iff its imaginary part is 0.
|- A e. CC   =>   |- (A e. RR <-> (Im` A) = 0)
Theorem	rerebi 6981	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	cjrebi 6982	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	recji 6983	Real part of a complex conjugate.
|- A e. CC   =>   |- (Re` (*` A)) = (Re` A)
Theorem	imcji 6984	Imaginary part of a complex conjugate.
|- A e. CC   =>   |- (Im` (*` A)) = -u(Im` A)
Theorem	readdi 6985	Real part distributes over addition.
|- A e. CC   &   |- B e. CC   =>   |- (Re` (A + B)) = ((Re` A) + (Re` B))
Theorem	imaddi 6986	Imaginary part distributes over addition.
|- A e. CC   &   |- B e. CC   =>   |- (Im` (A + B)) = ((Im` A) + (Im` B))
Theorem	remuli 6987	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	immuli 6988	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	cjaddi 6989	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	cjmuli 6990	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	ipcni 6991	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	cjmulrcli 6992	A complex number times its conjugate is real.
|- A e. CC   =>   |- (A x. (*` A)) e. RR
Theorem	cjmulvali 6993	A complex number times its conjugate.
|- A e. CC   =>   |- (A x. (*` A)) = (((Re` A)^2) + ((Im` A)^2))
Theorem	cjmulge0i 6994	A complex number times its conjugate is nonnegative.
|- A e. CC   =>   |- 0 <_ (A x. (*` A))
Theorem	renegi 6995	Real part of negative.
|- A e. CC   =>   |- (Re` -uA) = -u(Re` A)
Theorem	negrebi 6996	The negative of a real is real.
|- A e. CC   =>   |- (-uA e. RR <-> A e. RR)
Theorem	imnegi 6997	Imaginary part of negative.
|- A e. CC   =>   |- (Im` -uA) = -u(Im` A)
Theorem	cjnegi 6998	Complex conjugate of negative.
|- A e. CC   =>   |- (*` -uA) = -u(*` A)
Theorem	addcji 6999	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	rere 7000	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)