mmtheorems68 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6701-6800 - Page 68 of 108

Type	Label	Description
Statement
Theorem	sqrgt0 6701	The square root of a positive real is positive.
|- A e. RR   =>   |- (0 < A -> 0 < (sqr` A))
Theorem	sqrge0 6702	The square root of a nonnegative real is nonnegative.
|- A e. RR   =>   |- (0 <_ A -> 0 <_ (sqr` A))
Theorem	sqr11 6703	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	sqrmuli 6704	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	sqrmul 6705	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	sqrmsq2 6706	Relationship between square root and squares.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> ((sqr` A) = B <-> A = (B x. B)))
Theorem	sqrle 6707	Square root is monotonic.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> (A <_ B <-> (sqr` A) <_ (sqr` B)))
Theorem	sqrlt 6708	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	sqrmsq 6709	Square root of square.
|- A e. RR   =>   |- (0 <_ A -> (sqr` (A x. A)) = A)
Theorem	sqrclt 6710	The square root of a nonnegative real is a real.
|- ((A e. RR /\ 0 <_ A) -> (sqr` A) e. RR)
Theorem	sqrgt0t 6711	The square root of a positive real is positive.
|- ((A e. RR /\ 0 < A) -> 0 < (sqr` A))
Theorem	sqrge0t 6712	The square root of a nonnegative real is nonnegative.
|- ((A e. RR /\ 0 <_ A) -> 0 <_ (sqr` A))
Theorem	sqrlet 6713	Square root is monotonic.
|- (((A e. RR /\ B e. RR) /\ (0 <_ A /\ 0 <_ B)) -> (A <_ B <-> (sqr` A) <_ (sqr` B)))
Theorem	sqr00t 6714	A square root is zero iff its argument is 0.
|- ((A e. RR /\ 0 <_ A) -> ((sqr` A) = 0 <-> A = 0))
Theorem	rpsqrclt 6715	The square root of a positive real is a postive real.
|- (A e. RR+ -> (sqr` A) e. RR+)
Theorem	sqr1 6716	The square root of 1 is 1.
|- (sqr` 1) = 1
Theorem	sqr4 6717	The square root of 4 is 2.
|- (sqr` 4) = 2
Theorem	sqr9 6718	The square root of 9 is 3.
|- (sqr` 9) = 3
Theorem	sqr2gt1lt2 6719	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	sqrsq 6720	Square root of square.
|- A e. RR   =>   |- (0 <_ A -> (sqr` (A^2)) = A)
Theorem	sqsqr 6721	Square of square root.
|- A e. RR   =>   |- (0 <_ A -> ((sqr` A)^2) = A)
Theorem	sqrsqt 6722	Square root of square.
|- ((A e. RR /\ 0 <_ A) -> (sqr` (A^2)) = A)
Theorem	sqsqrt 6723	Square of square root.
|- ((A e. RR /\ 0 <_ A) -> ((sqr` A)^2) = A)
Irrationality of square root of 2
Theorem	sqr2irrlem1 6724	Lemma for irrationality of square root of 2. Technical lemma used to simplify the main induction step.
Theorem	sqr2irrlem2 6725	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.
Theorem	sqr2irrlem3 6726	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 6727	Lemma for irrationality of square root of 2.
Theorem	sqr2irrlem5 6728	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.
Theorem	sqr2irr 6729	The square root of 2 is irrational.
|- (sqr` 2) e/ QQ
Theorem	sqr2re 6730	The square root of 2 exists and is a real number.
|- (sqr` 2) e. RR
Imaginary and complex number properties
Theorem	irec 6731	The reciprocal of i.
|- (1 / i) = -ui
Theorem	i2 6732	i squared.
|- (i^2) = -u1
Theorem	i3 6733	i cubed.
|- (i^3) = -ui
Theorem	i4 6734	i to the fourth power.
|- (i^4) = 1
Theorem	inelr 6735	The imaginary unit i is not a real number.
|- -. i e. RR
Theorem	crulem 6736	Lemma for cru 6737.
Theorem	cru 6737	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 6738	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	crne0 6739	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 6740	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 6741	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 6742	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 6743	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 6744	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 6745	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 6746 for a further refinement.
|- ((NN (. ZZ /\ ZZ (. QQ) /\ (QQ (. RR /\ RR (. CC))
Theorem	nthruz 6746	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 6745.
|- (NN (. NN0 /\ NN0 (. ZZ)
Real and imaginary parts; conjugate; absolute value
Syntax	cre 6747	Extend class notation to include real part of a complex number.
class Re
Syntax	cim 6748	Extend class notation to include imaginary part of a complex number.
class Im
Syntax	ccj 6749	Extend class notation to include complex conjugate function.
class *
Syntax	cabs 6750	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs
Definition	df-re 6751	Define a function whose value is the real part of a complex number. See revalt 6755 for its value, recl 6765 for its closure, and replimt 6761 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 6752	Define a function whose value is the imaginary part of a complex number. See imvalt 6756 for its value, imcl 6766 for its closure, and replimt 6761 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 6753	Define the complex conjugate function. See cjcl 6767 for its closure and cjvalt 6763 for its value.
|- * = {<.x, y>. | (x e. CC /\ y = ((Re` x) - (i x. (Im` x))))}
Definition	df-abs 6754	Define the function for the absolute value (modulus) of a complex number. See abscl 6839 for its closure and absvalt 6762 or absval2 6841 for its value.
|- abs = {<.x, y>. | (x e. CC /\ y = (sqr` (x x. (*` x))))}
Theorem	revalt 6755	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 6756	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 6757	The real part of a complex number is real.
|- (A e. CC -> (Re` A) e. RR)
Theorem	imclt 6758	The imaginary part of a complex number is real.
|- (A e. CC -> (Im` A) e. RR)
Theorem	ref 6759	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- Re:CC-->RR
Theorem	imf 6760	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- Im:CC-->RR
Theorem	replimt 6761	Reconstruct a complex number from its real and imaginary parts.
|- (A e. CC -> A = ((Re` A) + (i x. (Im` A))))
Theorem	absvalt 6762	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 6763	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 6764	The conjugate of a complex number is a complex number (closure law).
|- (A e. CC -> (*` A) e. CC)
Theorem	recl 6765	The real part of a complex number is real (closure law).
|- A e. CC   =>   |- (Re` A) e. RR
Theorem	imcl 6766	The imaginary part of a complex number is real (closure law).
|- A e. CC   =>   |- (Im` A) e. RR
Theorem	cjcl 6767	Closure law for complex conjugate.
|-