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	sqrlem15 6701	Lemma for square root theorem.
Theorem	sqrlem16 6702	Lemma for square root theorem.
Theorem	sqrlem17 6703	Lemma for square root theorem.
Theorem	sqrlem18 6704	Lemma for square root theorem.
Theorem	sqrlem19 6705	Lemma for square root theorem.
Theorem	sqrlem20 6706	Lemma for square root theorem.
Theorem	sqrlem21 6707	Lemma for square root theorem.
Theorem	sqrlem22 6708	Lemma for square root theorem.
Theorem	sqrlem23 6709	Lemma for square root theorem.
Theorem	sqrlem24 6710	Lemma for square root closure.
Theorem	sqrgt0i 6711	The square root of a positive real is positive.
|- A e. RR   &   |- 0 < A   =>   |- 0 < (sqr` A)
Theorem	sqrlem26 6712	Lemma for square root theorem.
Theorem	sqrth 6713	Square root theorem. Theorem I.35 of [Apostol] p. 29. (A bit of trivia: This theorem was added to the database before the number 2 was defined and before exponents were defined. Thus you will see (1 + 1) and (x x. x) throughout its lemmas.)
|- A e. RR   =>   |- (0 <_ A -> ((sqr` A) x. (sqr` A)) = A)
Theorem	sqrcl 6714	The square root of a nonnegative real is a real.
|- A e. RR   =>   |- (0 <_ A -> (sqr` A) e. RR)
Theorem	sqrgt0 6715	The square root of a positive real is positive.
|- A e. RR   =>   |- (0 < A -> 0 < (sqr` A))
Theorem	sqrge0 6716	The square root of a nonnegative real is nonnegative.
|- A e. RR   =>   |- (0 <_ A -> 0 <_ (sqr` A))
Theorem	sqr11 6717	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 6718	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 6719	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 6720	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 6721	Square root is monotonic.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> (A <_ B <-> (sqr` A) <_ (sqr` B)))
Theorem	sqrlt 6722	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 6723	Square root of square.
|- A e. RR   =>   |- (0 <_ A -> (sqr` (A x. A)) = A)
Theorem	sqrclt 6724	The square root of a nonnegative real is a real.
|- ((A e. RR /\ 0 <_ A) -> (sqr` A) e. RR)
Theorem	sqrgt0t 6725	The square root of a positive real is positive.
|- ((A e. RR /\ 0 < A) -> 0 < (sqr` A))
Theorem	sqrge0t 6726	The square root of a nonnegative real is nonnegative.
|- ((A e. RR /\ 0 <_ A) -> 0 <_ (sqr` A))
Theorem	sqrlet 6727	Square root is monotonic.
|- (((A e. RR /\ B e. RR) /\ (0 <_ A /\ 0 <_ B)) -> (A <_ B <-> (sqr` A) <_ (sqr` B)))
Theorem	sqr00t 6728	A square root is zero iff its argument is 0.
|- ((A e. RR /\ 0 <_ A) -> ((sqr` A) = 0 <-> A = 0))
Theorem	rpsqrclt 6729	The square root of a positive real is a postive real.
|- (A e. RR+ -> (sqr` A) e. RR+)
Theorem	sqr1 6730	The square root of 1 is 1.
|- (sqr` 1) = 1
Theorem	sqr4 6731	The square root of 4 is 2.
|- (sqr` 4) = 2
Theorem	sqr9 6732	The square root of 9 is 3.
|- (sqr` 9) = 3
Theorem	sqr2gt1lt2 6733	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 6734	Square root of square.
|- A e. RR   =>   |- (0 <_ A -> (sqr` (A^2)) = A)
Theorem	sqsqr 6735	Square of square root.
|- A e. RR   =>   |- (0 <_ A -> ((sqr` A)^2) = A)
Theorem	sqrsqt 6736	Square root of square.
|- ((A e. RR /\ 0 <_ A) -> (sqr` (A^2)) = A)
Theorem	sqsqrt 6737	Square of square root.
|- ((A e. RR /\ 0 <_ A) -> ((sqr` A)^2) = A)
Irrationality of square root of 2
Theorem	sqr2irrlem1 6738	Lemma for irrationality of square root of 2. Technical lemma used to simplify the main induction step.
Theorem	sqr2irrlem2 6739	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.
Theorem	sqr2irrlem3 6740	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 6741	Lemma for irrationality of square root of 2.
Theorem	sqr2irrlem5 6742	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.
Theorem	sqr2irr 6743	The square root of 2 is irrational.
|- (sqr` 2) e/ QQ
Theorem	sqr2re 6744	The square root of 2 exists and is a real number.
|- (sqr` 2) e. RR
Imaginary and complex number properties
Theorem	irec 6745	The reciprocal of i.
|- (1 / i) = -ui
Theorem	i2 6746	i squared.
|- (i^2) = -u1
Theorem	i3 6747	i cubed.
|- (i^3) = -ui
Theorem	i4 6748	i to the fourth power.
|- (i^4) = 1
Theorem	inelr 6749	The imaginary unit i is not a real number.
|- -. i e. RR
Theorem	crulem 6750	Lemma for cru 6751.
Theorem	cru 6751	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 6752	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 6753	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 6754	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 6755	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 6756	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 6757	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 6758	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 6759	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 6760 for a further refinement.
|- ((NN (. ZZ /\ ZZ (. QQ) /\ (QQ (. RR /\ RR (. CC))
Theorem	nthruz 6760	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 6759.
|- (NN (. NN0 /\ NN0 (. ZZ)
Real and imaginary parts; conjugate; absolute value
Syntax	cre 6761	Extend class notation to include real part of a complex number.
class Re
Syntax	cim 6762	Extend class notation to include imaginary part of a complex number.
class Im
Syntax	ccj 6763	Extend class notation to include complex conjugate function.
class *
Syntax	cabs 6764	Extend class notation to include a function for the absolute value (modulus) of a complex number.
class abs
Definition	df-re 6765	Define a function whose value is the real part of a complex number. See revalt 6769 for its value, recl 6779 for its closure, and replimt 6775 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 6766	Define a function whose value is the imaginary part of a complex number. See imvalt 6770 for its value, imcl 6780 for its closure, and replimt 6775 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 6767	Define the complex conjugate function. See cjcl 6781 for its closure and cjvalt 6777 for its value.
|- * = {<.x, y>. | (x e. CC /\ y = ((Re` x) - (i x. (Im` x))))}
Definition	df-abs 6768	Define the function for the absolute value (modulus) of a complex number. See abscl 6852 for its closure and absvalt 6776 or absval2 6854 for its value.
|- abs = {<.x, y>. | (x e. CC /\ y = (sqr` (x x. (*` x))))}
Theorem	revalt 6769	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 6770	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 6771	The real part of a complex number is real.
|- (A e. CC -> (Re` A) e. RR)
Theorem	imclt 6772	The imaginary part of a complex number is real.
|- (A e. CC -> (Im` A) e. RR)
Theorem	ref 6773	Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- Re:CC-->RR
Theorem	imf 6774	Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.)
|- Im:CC-->RR
Theorem	replimt 6775	Reconstruct a complex number from its real and imaginary parts.
|- (A e. CC -> A = ((