mmtheorems67 - Metamath Proof Explorer

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**Statement List for Metamath Proof Explorer -** 6601-6700 - Page 67 of 107
Type	Label	Description
Statement

Theorem	bernneq 6601	Bernoulli's inequality, due to Johan Bernoulli (1667-1748).
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Theorem	bernneq2 6602	Variation of Bernoulli's inequality bernneq 6601.
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Theorem	expnbndt 6603	Exponentiation with a mantissa greater than 1 has no upper bound.
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Discriminant

Theorem	discrlem1 6604	Lemma for discriminant theorem.

Theorem	discrlem2 6605	Lemma for discriminant theorem.

Theorem	discrlem3 6606	Lemma for discriminant theorem.

Theorem	discrlem 6607	If a quadratic polynomial with real coefficients is nonnegative for all values, then its discriminant is non-positive. The antecedent is redundant but simplifies the proof.


More natural number properties

Theorem	nnsqcl 6608	The square of a natural number is a natural number.


Theorem	nnlesq 6609	A natural number is less than or equal to its square.


Theorem	nnesq 6610	A natural number is even iff its square is even.


Ordered pair theorem for nonnegative integers

Theorem	nn0le2msqt 6611	The square function on nonnegative integers is monotonic. (Contributed by Raph Levien, 10-Dec-2002.)


Theorem	nn0opthlem1 6612	A rather pretty lemma for nn0opth 6614. (Contributed by Raph Levien, 10-Dec-2002.)


Theorem	nn0opthlem2 6613	Lemma for nn0opth 6614.

Theorem	nn0opth 6614	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers and by . If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 2412 that works for any set. (Contributed by Raph Levien, 10-Dec-2002.)
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Theorem	nn0opth2 6615	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opth 6614.
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Theorem	nn0opth2t 6616	An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opth 6614.
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Square root

Syntax	csqr 6617	Extend class notation to include positive square root of a positive real number.


Definition	df-sqr 6618	Define a function whose value is the square root of a nonnegative real number. The square root of is the supremum of all reals whose square is less than . See sqrcl 6648 for its closure, sqrval 6619 for its value, sqrsq 6668 and sqsqr 6669 for its relationship to squares, and sqr11 6651 for uniqueness.
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Theorem	sqrval 6619	Value of square root function.
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Theorem	sqr0 6620	Square root of zero.


Theorem	sqrlem1 6621	Lemma for square root theorem.

Theorem	sqrlem2 6622	Lemma for square root theorem.

Theorem	sqrlem3 6623	Lemma for square root theorem.

Theorem	sqrlem4 6624	Lemma for square root theorem.

Theorem	sqrlem5 6625	Lemma for square root theorem.

Theorem	sqrlem6 6626	Lemma for square root theorem.

Theorem	sqrlem7 6627	Lemma for square root theorem.

Theorem	sqrlem8 6628	Lemma for square root theorem.

Theorem	sqrlem9 6629	Lemma for square root theorem.

Theorem	sqrlem10 6630	Lemma for square root theorem.

Theorem	sqrlem11 6631	Lemma for square root theorem.

Theorem	sqrlem12 6632	Lemma for square root theorem.

Theorem	sqrlem13 6633	Lemma for square root theorem.

Theorem	sqrlem14 6634	Lemma for square root theorem.

Theorem	sqrlem15 6635	Lemma for square root theorem.

Theorem	sqrlem16 6636	Lemma for square root theorem.

Theorem	sqrlem17 6637	Lemma for square root theorem.

Theorem	sqrlem18 6638	Lemma for square root theorem.

Theorem	sqrlem19 6639	Lemma for square root theorem.

Theorem	sqrlem20 6640	Lemma for square root theorem.

Theorem	sqrlem21 6641	Lemma for square root theorem.

Theorem	sqrlem22 6642	Lemma for square root theorem.

Theorem	sqrlem23 6643	Lemma for square root theorem.

Theorem	sqrlem24 6644	Lemma for square root closure.

Theorem	sqrgt0i 6645	The square root of a positive real is positive.


Theorem	sqrlem26 6646	Lemma for square root theorem.

Theorem	sqrth 6647	Square root theorem. Theorem I.35 of [Apostol] p. 29. (A bit of trivia: This theorem was added to the database before the number was defined and before exponents were defined. Thus you will see and throughout its lemmas.)


Theorem	sqrcl 6648	The square root of a nonnegative real is a real.


Theorem	sqrgt0 6649	The square root of a positive real is positive.


Theorem	sqrge0 6650	The square root of a nonnegative real is nonnegative.


Theorem	sqr11 6651	The square root function is one-to-one.
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Theorem	sqrmuli 6652	Square root distributes over multiplication.


Theorem	sqrmul 6653	Square root distributes over multiplication.
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Theorem	sqrmsq2 6654	Relationship between square root and squares.
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Theorem	sqrle 6655	Square root is monotonic.
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Theorem	sqrlt 6656	Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005.)
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Theorem	sqrmsq 6657	Square root of square.


Theorem	sqrclt 6658	The square root of a nonnegative real is a real.
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Theorem	sqrgt0t 6659	The square root of a positive real is positive.
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Theorem	sqrge0t 6660	The square root of a nonnegative real is nonnegative.
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Theorem	sqrlet 6661	Square root is monotonic.
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Theorem	sqr00t 6662	A square root is zero iff its argument is 0.
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Theorem	rpsqrclt 6663	The square root of a positive real is a postive real.


Theorem	sqr1 6664	The square root of 1 is 1.


Theorem	sqr4 6665	The square root of 4 is 2.


Theorem	sqr9 6666	The square root of 9 is 3.


Theorem	sqr2gt1lt2 6667	The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.)
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Theorem	sqrsq 6668	Square root of square.


Theorem	sqsqr 6669	Square of square root.


Theorem	sqrsqt 6670	Square root of square.
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Theorem	sqsqrt 6671	Square of square root.
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Irrationality of square root of 2

Theorem	sqr2irrlem1 6672	Lemma for irrationality of square root of 2. Technical lemma used to simplify the main induction step.

Theorem	sqr2irrlem2 6673	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.

Theorem	sqr2irrlem3 6674	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.


Theorem	sqr2irrlem4 6675	Lemma for irrationality of square root of 2.

Theorem	sqr2irrlem5 6676	Lemma for irrationality of square root of 2. Eliminates hypotheses with weak deduction theorem.

Theorem	sqr2irr 6677	The square root of 2 is irrational.


Theorem	sqr2re 6678	The square root of 2 exists and is a real number.


Imaginary and complex number properties

Theorem	irec 6679	The reciprocal of .


Theorem	i2 6680	squared.


Theorem	i3 6681	cubed.


Theorem	i4 6682	to the fourth power.


Theorem	inelr 6683	The imaginary unit is not a real number.


Theorem	crulem 6684	Lemma for cru 6685.

Theorem	cru 6685	The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130.