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Theorem List for Metamath Proof Explorer - 20701-20800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorematan0 20701 The arctangent of zero is zero. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan

Theorematandmcj 20702 The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan arctan

Theorematancj 20703 The arctangent function distributes under conjugation. (The condition that is necessary because the branch cuts are chosen so that the negative imaginary line "agrees with" neighboring values with negative real part, while the positive imaginary line agrees with values with positive real part. This makes atanneg 20700 true unconditionally but messes up conjugation symmetry, and it is impossible to have both in a single-valued function. The claim is true on the imaginary line between and , though.) (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan arctan arctan

Theorematanrecl 20704 The arctangent function is real for all real inputs. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan

Theoremefiatan 20705 Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.)
arctan arctan

Theorematanlogaddlem 20706 Lemma for atanlogadd 20707. (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan

Theorematanlogadd 20707 The rule is not always true on the complexes, but it is true when the arguments of and sum to within the interval , so there are some cases such as this one with and which are true unconditionally. This result can also be stated as " is analytic". (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan

Theorematanlogsublem 20708 Lemma for atanlogsub 20709. (Contributed by Mario Carneiro, 4-Apr-2015.)
arctan

Theorematanlogsub 20709 A variation on atanlogadd 20707, to show that under more limited conditions. (Contributed by Mario Carneiro, 4-Apr-2015.)
arctan

Theoremefiatan2 20710 Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan arctan

Theorem2efiatan 20711 Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 2-Apr-2015.)
arctan arctan

Theoremtanatan 20712 The arctangent function is an inverse to . (Contributed by Mario Carneiro, 2-Apr-2015.)
arctan arctan

Theorematandmtan 20713 The tangent function has range contained in the domain of the arctangent. (Contributed by Mario Carneiro, 31-Mar-2015.)
arctan

Theoremcosatan 20714 The cosine of an arctangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan arctan

Theoremcosatanne0 20715 The arctangent function has range contained in the domain of the tangent. (Contributed by Mario Carneiro, 3-Apr-2015.)
arctan arctan

Theorematantan 20716 The arctangent function is an inverse to . (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan

Theorematantanb 20717 Relationship between tangent and arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan arctan

Theorematanbndlem 20718 Lemma for atanbnd 20719. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan

Theorematanbnd 20719 The arctangent function is bounded by on the reals. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan

Theorematanord 20720 The arctangent function is strictly increasing. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan arctan

Theorematan1 20721 The arctangent of is . (Contributed by Mario Carneiro, 2-Apr-2015.)
arctan

Theorembndatandm 20722 A point in the open unit disk is in the domain of the arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan

Theorematans 20723* The "domain of continuity" of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theorematans2 20724* It suffices to show that and are in the continuity domain of to show that is in the continuity domain of arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theorematansopn 20725* The domain of continuity of the arctangent is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
fld

Theorematansssdm 20726* The domain of continuity of the arctangent is a subset of the actual domain of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
arctan

Theoremressatans 20727* The real number line is a subset of the domain of continuity of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremdvatan 20728* The derivative of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
arctan

Theorematancn 20729* The arctangent is a continuous function. (Contributed by Mario Carneiro, 7-Apr-2015.)
arctan

Theorematantayl 20730* The Taylor series for arctan. (Contributed by Mario Carneiro, 1-Apr-2015.)
arctan

Theorematantayl2 20731* The Taylor series for arctan. (Contributed by Mario Carneiro, 1-Apr-2015.)
arctan

Theorematantayl3 20732* The Taylor series for arctan. (Contributed by Mario Carneiro, 7-Apr-2015.)
arctan

Theoremleibpilem1 20733 Lemma for leibpi 20735. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremleibpilem2 20734* The Leibniz formula for . (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremleibpi 20735 The Leibniz formula for . This proof depends on three main facts: (1) the series is convergent, because it is an alternating series (iseralt 12433). (2) Using leibpilem2 20734 to rewrite the series as a power series, it is the special case of the Taylor series for arctan (atantayl2 20731). (3) Although we cannot directly plug into atantayl2 20731, Abel's theorem (abelth2 20311) says that the limit along any sequence converging to , such as , of the power series converges to the power series extended to , and then since arctan is continuous at (atancn 20729) we get the desired result. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremleibpisum 20736 The Leibniz formula for . This version of leibpi 20735 looks nicer but does not assert that the series is convergent so is not as practically useful. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremlog2cnv 20737 Using the Taylor series for arctan , produce a rapidly convergent series for . (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremlog2tlbnd 20738* Bound the error term in the series of log2cnv 20737. (Contributed by Mario Carneiro, 7-Apr-2015.)

13.3.8  The Birthday Problem

Theoremlog2ublem1 20739 Lemma for log2ub 20742. The proof of log2ub 20742, which is simply the evaluation of log2tlbnd 20738 for , takes the form of the addition of five fractions and showing this is less than another fraction. We could just perform exact arithmetic on these fractions, get a large rational number, and just multiply everything to verify the claim, but as anyone who uses decimal numbers for this task knows, it is often better to pick a common denominator (usually a large power of ) and work with the closest approximations of the form for some integer instead. It turns out that for our purposes it is sufficient to take , which is also nice because it shares many factors in common with the fractions in question. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremlog2ublem2 20740* Lemma for log2ub 20742. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremlog2ublem3 20741 Lemma for log2ub 20742. In decimal, this is a proof that the first four terms of the series for is less than . (Contributed by Mario Carneiro, 17-Apr-2015.)
;;;;

Theoremlog2ub 20742 is less than . If written in decimal, this is because 0.693147... is less than 253/365 = 0.693151... , so this is a very tight bound, at five decimal places. (Contributed by Mario Carneiro, 7-Apr-2015.)
;; ;;

Theorembirthdaylem1 20743* Lemma for birthday 20746. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theorembirthdaylem2 20744* For general and , count the fraction of injective functions from to . (Contributed by Mario Carneiro, 7-May-2015.)

Theorembirthdaylem3 20745* For general and , upper-bound the fraction of injective functions from to . (Contributed by Mario Carneiro, 17-Apr-2015.)

Theorembirthday 20746* The Birthday Problem. There is a more than even chance that out of 23 people in a room, at least two of them have the same birthday. Mathematically, this is asserting that for and , fewer than half of the set of all functions from to are injective. (Contributed by Mario Carneiro, 17-Apr-2015.)
;       ;;

13.3.9  Areas in R^2

Syntaxcarea 20747 Area of regions in the complex plane.
area

Definitiondf-area 20748* Define the area of a subset of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremdmarea 20749* The domain of the area function is the set of finitely measurable subsets of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremareambl 20750 The fibers of a measurable region are finitely meaurable subsets of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremareass 20751 A measurable region is a subset of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremdfarea 20752* Rewrite df-area 20748 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareaf 20753 Area meaurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareacl 20754 The area of a measurable region is a real number. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareage0 20755 The area of a measurable region is greater than or equal to zero. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareaval 20756* The area of a measurable region is greater than or equal to zero. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

13.3.10  More miscellaneous converging sequences

Theoremrlimcnp 20757* Relate a limit of a real-valued sequence at infinity to the continuity of the function at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
fld       t

Theoremrlimcnp2 20758* Relate a limit of a real-valued sequence at infinity to the continuity of the function at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
fld       t

Theoremrlimcnp3 20759* Relate a limit of a real-valued sequence at infinity to the continuity of the function at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)
fld       t

Theoremxrlimcnp 20760* Relate a limit of a real-valued sequence at infinity to the continuity of the corresponding extended real function at . Since any limit can be written in the form on the left side of the implication, this shows that real limits are a special case of topological continuity at a point. (Contributed by Mario Carneiro, 8-Sep-2015.)
fld       ordTop t

Theoremefrlim 20761* The limit of the sequence is the exponential function. This is often taken as an alternate definition of the exponential function (see also dfef2 20762). (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremdfef2 20762* The limit of the sequence as goes to is . This is another common definition of . (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremcxplim 20763* A power to a negative exponent goes to zero as the base becomes large. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Mario Carneiro, 18-May-2016.)

Theoremsqrlim 20764 The inverse square root function converges to zero. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremrlimcxp 20765* Any power to a positive exponent of a converging sequence also converges. (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremo1cxp 20766* An eventually bounded function taken to a nonnegative power is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxp2limlem 20767* A linear factor grows slower than any exponential with base greater than . (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxp2lim 20768* Any power grows slower than any exponential with base greater than . (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremcxploglim 20769* The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremcxploglim2 20770* Every power of the logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremdivsqrsumlem 20771* Lemma for divsqrsum 20773 and divsqrsum2 20774. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdivsqrsumf 20772* The function used in divsqrsum 20773 is a real function. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremdivsqrsum 20773* The sum is asymptotic to with a finite limit . (In fact, this limit is .) (Contributed by Mario Carneiro, 9-May-2016.)

Theoremdivsqrsum2 20774* A bound on the distance of the sum from its asymptotic value . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdivsqrsumo1 20775* The sum has the asymptotic expansion , for some . (Contributed by Mario Carneiro, 10-May-2016.)

13.3.11  Inequality of arithmetic and geometric means

Theoremcvxcl 20776* Closure of a 0-1 linear combination in a convex set. (Contributed by Mario Carneiro, 21-Jun-2015.)

Theoremscvxcvx 20777* A strictly convex function is convex. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremjensenlem1 20778* Lemma for jensen 20780. (Contributed by Mario Carneiro, 4-Jun-2016.)
fld g                             fld g        fld g

Theoremjensenlem2 20779* Lemma for jensen 20780. (Contributed by Mario Carneiro, 21-Jun-2015.)
fld g                             fld g        fld g               fld g        fld g fld g        fld g fld g fld g

Theoremjensen 20780* Jensen's inequality, a finite extension of the definition of convexity (the last hypothesis). (Contributed by Mario Carneiro, 21-Jun-2015.)
fld g               fld g fld g fld g fld g fld g fld g

Theoremamgmlem 20781 Lemma for amgm 20782. (Contributed by Mario Carneiro, 21-Jun-2015.)
mulGrpfld                            g fld g

Theoremamgm 20782 Inequality of arithmetic and geometric means. Here g calculates the group sum within the multiplicative monoid of the complex numbers (or in other words, it multiplies the elements together), and fld g calculates the group sum in the additive group (i.e. the sum of the elements). (Contributed by Mario Carneiro, 20-Jun-2015.)
mulGrpfld       g fld g

13.3.12  Euler-Mascheroni constant

Syntaxcem 20783 The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)

Definitiondf-em 20784 Define the Euler-Macheroni constant, 0.577... . This is the limit of the series , with a proof that the limit exists in emcl 20794. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremlogdifbnd 20785 Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)

Theoremlogdiflbnd 20786 Lower bound on the difference of logs. (Contributed by Mario Carneiro, 3-Jul-2017.)

Theorememcllem1 20787* Lemma for emcl 20794. The series and are sequences of real numbers that approach from above and below, respectively. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem2 20788* Lemma for emcl 20794. is increasing, and is decreasing. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem3 20789* Lemma for emcl 20794. The function is the difference between and . (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem4 20790* Lemma for emcl 20794. The difference between series and tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem5 20791* Lemma for emcl 20794. The partial sums of the series , which is used in the definition df-em 20784, is in fact the same as . (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem6 20792* Lemma for emcl 20794. By the previous lemmas, and must approach a common limit, which is by definition. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem7 20793* Lemma for emcl 20794 and harmonicbnd 20795. Derive bounds on as and . (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)

Theorememcl 20794 Closure and bounds for the Euler-Macheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremharmonicbnd 20795* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremharmonicbnd2 20796* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theorememre 20797 The Euler-Macheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememgt0 20798 The Euler-Macheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremharmonicbnd3 20799* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmoniclbnd 20800* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

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