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Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorematantan 20801 The arctangent function is an inverse to . (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan

Theorematantanb 20802 Relationship between tangent and arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan arctan

Theorematanbndlem 20803 Lemma for atanbnd 20804. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan

Theorematanbnd 20804 The arctangent function is bounded by on the reals. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan

Theorematanord 20805 The arctangent function is strictly increasing. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan arctan

Theorematan1 20806 The arctangent of is . (Contributed by Mario Carneiro, 2-Apr-2015.)
arctan

Theorembndatandm 20807 A point in the open unit disk is in the domain of the arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.)
arctan

Theorematans 20808* The "domain of continuity" of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theorematans2 20809* 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 20810* The domain of continuity of the arctangent is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
fld

Theorematansssdm 20811* 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 20812* The real number line is a subset of the domain of continuity of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremdvatan 20813* The derivative of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
arctan

Theorematancn 20814* The arctangent is a continuous function. (Contributed by Mario Carneiro, 7-Apr-2015.)
arctan

Theorematantayl 20815* The Taylor series for arctan. (Contributed by Mario Carneiro, 1-Apr-2015.)
arctan

Theorematantayl2 20816* The Taylor series for arctan. (Contributed by Mario Carneiro, 1-Apr-2015.)
arctan

Theorematantayl3 20817* The Taylor series for arctan. (Contributed by Mario Carneiro, 7-Apr-2015.)
arctan

Theoremleibpilem1 20818 Lemma for leibpi 20820. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremleibpilem2 20819* The Leibniz formula for . (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremleibpi 20820 The Leibniz formula for . This proof depends on three main facts: (1) the series is convergent, because it is an alternating series (iseralt 12516). (2) Using leibpilem2 20819 to rewrite the series as a power series, it is the special case of the Taylor series for arctan (atantayl2 20816). (3) Although we cannot directly plug into atantayl2 20816, Abel's theorem (abelth2 20396) 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 20814) we get the desired result. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremleibpisum 20821 The Leibniz formula for . This version of leibpi 20820 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 20822 Using the Taylor series for arctan , produce a rapidly convergent series for . (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremlog2tlbnd 20823* Bound the error term in the series of log2cnv 20822. (Contributed by Mario Carneiro, 7-Apr-2015.)

13.3.8  The Birthday Problem

Theoremlog2ublem1 20824 Lemma for log2ub 20827. The proof of log2ub 20827, which is simply the evaluation of log2tlbnd 20823 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 20825* Lemma for log2ub 20827. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremlog2ublem3 20826 Lemma for log2ub 20827. 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 20827 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 20828* Lemma for birthday 20831. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theorembirthdaylem2 20829* For general and , count the fraction of injective functions from to . (Contributed by Mario Carneiro, 7-May-2015.)

Theorembirthdaylem3 20830* For general and , upper-bound the fraction of injective functions from to . (Contributed by Mario Carneiro, 17-Apr-2015.)

Theorembirthday 20831* 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 20832 Area of regions in the complex plane.
area

Definitiondf-area 20833* Define the area of a subset of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremdmarea 20834* The domain of the area function is the set of finitely measurable subsets of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremareambl 20835 The fibers of a measurable region are finitely meaurable subsets of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremareass 20836 A measurable region is a subset of . (Contributed by Mario Carneiro, 21-Jun-2015.)
area

Theoremdfarea 20837* Rewrite df-area 20833 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareaf 20838 Area meaurement is a function whose values are nonnegative reals. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareacl 20839 The area of a measurable region is a real number. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareage0 20840 The area of a measurable region is greater than or equal to zero. (Contributed by Mario Carneiro, 21-Jun-2015.)
area area

Theoremareaval 20841* 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 20842* 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 20843* 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 20844* 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 20845* 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 20846* The limit of the sequence is the exponential function. This is often taken as an alternate definition of the exponential function (see also dfef2 20847). (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremdfef2 20847* The limit of the sequence as goes to is . This is another common definition of . (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremcxplim 20848* 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 20849 The inverse square root function converges to zero. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremrlimcxp 20850* Any power to a positive exponent of a converging sequence also converges. (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremo1cxp 20851* An eventually bounded function taken to a nonnegative power is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxp2limlem 20852* A linear factor grows slower than any exponential with base greater than . (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxp2lim 20853* Any power grows slower than any exponential with base greater than . (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremcxploglim 20854* The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremcxploglim2 20855* Every power of the logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremdivsqrsumlem 20856* Lemma for divsqrsum 20858 and divsqrsum2 20859. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdivsqrsumf 20857* The function used in divsqrsum 20858 is a real function. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremdivsqrsum 20858* The sum is asymptotic to with a finite limit . (In fact, this limit is .) (Contributed by Mario Carneiro, 9-May-2016.)

Theoremdivsqrsum2 20859* A bound on the distance of the sum from its asymptotic value . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdivsqrsumo1 20860* 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 20861* Closure of a 0-1 linear combination in a convex set. (Contributed by Mario Carneiro, 21-Jun-2015.)

Theoremscvxcvx 20862* A strictly convex function is convex. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremjensenlem1 20863* Lemma for jensen 20865. (Contributed by Mario Carneiro, 4-Jun-2016.)
fld g                             fld g        fld g

Theoremjensenlem2 20864* Lemma for jensen 20865. (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 20865* 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 20866 Lemma for amgm 20867. (Contributed by Mario Carneiro, 21-Jun-2015.)
mulGrpfld                            g fld g

Theoremamgm 20867 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 20868 The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)

Definitiondf-em 20869 Define the Euler-Macheroni constant, 0.577... . This is the limit of the series , with a proof that the limit exists in emcl 20879. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremlogdifbnd 20870 Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)

Theoremlogdiflbnd 20871 Lower bound on the difference of logs. (Contributed by Mario Carneiro, 3-Jul-2017.)

Theorememcllem1 20872* Lemma for emcl 20879. The series and are sequences of real numbers that approach from above and below, respectively. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem2 20873* Lemma for emcl 20879. is increasing, and is decreasing. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem3 20874* Lemma for emcl 20879. The function is the difference between and . (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem4 20875* Lemma for emcl 20879. The difference between series and tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem5 20876* Lemma for emcl 20879. The partial sums of the series , which is used in the definition df-em 20869, is in fact the same as . (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem6 20877* Lemma for emcl 20879. By the previous lemmas, and must approach a common limit, which is by definition. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem7 20878* Lemma for emcl 20879 and harmonicbnd 20880. Derive bounds on as and . (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)

Theorememcl 20879 Closure and bounds for the Euler-Macheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremharmonicbnd 20880* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremharmonicbnd2 20881* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theorememre 20882 The Euler-Macheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememgt0 20883 The Euler-Macheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremharmonicbnd3 20884* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmoniclbnd 20885* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmonicubnd 20886* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmonicbnd4 20887* The asymptotic behavior of . (Contributed by Mario Carneiro, 14-May-2016.)

Theoremfsumharmonic 20888* Bound a finite sum based on the harmonic series, where the "strong" bound only applies asymptotically, and there is a "weak" bound for the remaining values. (Contributed by Mario Carneiro, 18-May-2016.)

13.4  Basic number theory

13.4.1  Wilson's theorem

Theoremwilthlem1 20889 The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in are and . (Note that from prmdiveq 13213, is the modular inverse of in . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremwilthlem2 20890* Lemma for wilth 20892: induction step. The "hand proof" version of this theorem works by writing out the list of all numbers from to in pairs such that a number is paired with its inverse. Every number has a unique inverse different from itself except and , and so each pair multiplies to , and and multiply to , so the full product is equal to . Here we make this precise by doing the product pair by pair.

The induction hypothesis says that every subset of that is closed under inverse (i.e. all pairs are matched up) and contains multiplies to . Given such a set, we take out one element . If there are no such elements, then which forms the base case. Otherwise, is also closed under inverse and contains , so the induction hypothesis says that this equals ; and the remaining two elements are either equal to each other, in which case wilthlem1 20889 gives that or , and we've already excluded the second case, so the product gives ; or and their product is . In either case the accumulated product is unaffected. (Contributed by Mario Carneiro, 24-Jan-2015.)

mulGrpfld                            g        g

Theoremwilthlem3 20891* Lemma for wilth 20892. Here we round out the argument of wilthlem2 20890 with the final step of the induction. The induction argument shows that every subset of that is closed under inverse and contains multiplies to , and clearly itself is such a set. Thus, the product of all the elements is , and all that is left is to translate the group sum notation (which we used for its unordered summing capabilities) into an ordered sequence to match the definition of the factorial. (Contributed by Mario Carneiro, 24-Jan-2015.)
mulGrpfld

Theoremwilth 20892 Wilson's theorem. A number is prime iff it is greater or equal to and is congruent to , , or alternatively if divides . In this part of the proof we show the relatively simple reverse implication; see wilthlem3 20891 for the forward implication. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by Fan Zheng, 16-Jun-2016.)

13.4.2  The Fundamental Theorem of Algebra

Theoremftalem1 20893* Lemma for fta 20900: "growth lemma". There exists some such that is arbitrarily close in proportion to its dominant term. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremftalem2 20894* Lemma for fta 20900. There exists some such that has magnitude greater than outside the closed ball B(0,r). (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremftalem3 20895* Lemma for fta 20900. There exists a global minimum of the function . The proof uses a circle of radius where is the value coming from ftalem1 20893; since this is a compact set, the minimum on this disk is achieved, and this must then be the global minimum. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly                     fld

Theoremftalem4 20896* Lemma for fta 20900: Closure of the auxiliary variables for ftalem5 20897. (Contributed by Mario Carneiro, 20-Sep-2014.)
coeff       deg       Poly

Theoremftalem5 20897* Lemma for fta 20900: Main proof. We have already shifted the minimum found in ftalem3 20895 to zero by a change of variables, and now we show that the minimum value is zero. Expanding in a series about the minimum value, let be the lowest term in the polynomial that is nonzero, and let be a -th root of . Then an evaluation of where is a sufficiently small positive number yields for the first term and for the -th term, and all higher terms are bounded because is small. Thus, , in contradiction to our choice of as the minimum. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremftalem6 20898* Lemma for fta 20900: Discharge the auxiliary variables in ftalem5 20897. (Contributed by Mario Carneiro, 20-Sep-2014.)
coeff       deg       Poly

Theoremftalem7 20899* Lemma for fta 20900. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremfta 20900* The Fundamental Theorem of Algebra. Any polynomial with positive degree (i.e. non-constant) has a root. (Contributed by Mario Carneiro, 15-Sep-2014.)
Poly deg

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