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Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremo1cxp 20101* An eventually bounded function taken to a nonnegative power is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxp2limlem 20102* A linear factor grows slower than any exponential with base greater than . (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxp2lim 20103* Any power grows slower than any exponential with base greater than . (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremcxploglim 20104* The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremcxploglim2 20105* Every power of the logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremdivsqrsumlem 20106* Lemma for divsqrsum 20108 and divsqrsum2 20109. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdivsqrsumf 20107* The function used in divsqrsum 20108 is a real function. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremdivsqrsum 20108* The sum is asymptotic to with a finite limit . (In fact, this limit is .) (Contributed by Mario Carneiro, 9-May-2016.)

Theoremdivsqrsum2 20109* A bound on the distance of the sum from its asymptotic value . (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdivsqrsumo1 20110* The sum has the asymptotic expansion , for some . (Contributed by Mario Carneiro, 10-May-2016.)

13.3.10  Inequality of arithmetic and geometric means

Theoremcvxcl 20111* Closure of a 0-1 linear combination in a convex set. (Contributed by Mario Carneiro, 21-Jun-2015.)

Theoremscvxcvx 20112* A strictly convex function is convex. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremjensenlem1 20113* Lemma for jensen 20115. (Contributed by Mario Carneiro, 4-Jun-2016.)
fld g                             fld g        fld g

Theoremjensenlem2 20114* Lemma for jensen 20115. (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 20115* 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 20116 Lemma for amgm 20117. (Contributed by Mario Carneiro, 21-Jun-2015.)
mulGrpfld                            g fld g

Theoremamgm 20117 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

13.3.11  Euler-Mascheroni constant

Syntaxcem 20118 The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)

Definitiondf-em 20119 Define the Euler-Macheroni constant, 0.577... . This is the limit of the series , with a proof that the limit exists in emcl 20128. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremlogdifbnd 20120 Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)

Theorememcllem1 20121* Lemma for emcl 20128. The series and are sequences of real numbers that approach from above and below, respectively. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem2 20122* Lemma for emcl 20128. is increasing, and is decreasing. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem3 20123* Lemma for emcl 20128. The function is the difference between and . (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem4 20124* Lemma for emcl 20128. The difference between series and tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem5 20125* Lemma for emcl 20128. The partial sums of the series , which is used in the definition df-em 20119, is in fact the same as . (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem6 20126* Lemma for emcl 20128. By the previous lemmas, and must approach a common limit, which is by definition. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememcllem7 20127* Lemma for emcl 20128 and harmonicbnd 20129. Derive bounds on as and . (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)

Theorememcl 20128 Closure and bounds for the Euler-Macheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremharmonicbnd 20129* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremharmonicbnd2 20130* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theorememre 20131 The Euler-Macheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theorememgt0 20132 The Euler-Macheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremharmonicbnd3 20133* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmoniclbnd 20134* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmonicubnd 20135* A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremharmonicbnd4 20136* The asymptotic behavior of . (Contributed by Mario Carneiro, 14-May-2016.)

Theoremfsumharmonic 20137* 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 20138 The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in are and . (Note that from prmdiveq 12728, is the modular inverse of in . (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremwilthlem2 20139* Lemma for wilth 20141: 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 20138 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 20140* Lemma for wilth 20141. Here we round out the argument of wilthlem2 20139 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 20141 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 20140 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 20142* Lemma for fta 20149: "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 20143* Lemma for fta 20149. 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 20144* Lemma for fta 20149. There exists a global minimum of the function . The proof uses a circle of radius where is the value coming from ftalem1 20142; 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 20145* Lemma for fta 20149: Closure of the auxiliary variables for ftalem5 20146. (Contributed by Mario Carneiro, 20-Sep-2014.)
coeff       deg       Poly

Theoremftalem5 20146* Lemma for fta 20149: Main proof. We have already shifted the minimum found in ftalem3 20144 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 20147* Lemma for fta 20149: Discharge the auxiliary variables in ftalem5 20146. (Contributed by Mario Carneiro, 20-Sep-2014.)
coeff       deg       Poly

Theoremftalem7 20148* Lemma for fta 20149. Shift the minimum away from zero by a change of variables. (Contributed by Mario Carneiro, 14-Sep-2014.)
coeff       deg       Poly

Theoremfta 20149* 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

13.4.3  The Basel problem (ζ(2) = π2/6)

Theorembasellem1 20150 Lemma for basel 20159. Closure of the sequence of roots. (Contributed by Mario Carneiro, 30-Jul-2014.)

Theorembasellem2 20151* Lemma for basel 20159. Show that is a polynomial of degree , and compute its coefficient function. (Contributed by Mario Carneiro, 30-Jul-2014.)
Poly deg coeff

Theorembasellem3 20152* Lemma for basel 20159. Using the binomial theorem and de Moivre's formula, we have the identity , so taking imaginary parts yields , where . (Contributed by Mario Carneiro, 30-Jul-2014.)

Theorembasellem4 20153* Lemma for basel 20159. By basellem3 20152, the expression goes to zero whenever for some , so this function enumerates distinct roots of a degree- polynomial, which must therefore be all the roots by fta1 19520. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasellem5 20154* Lemma for basel 20159. Using vieta1 19524, we can calculate the sum of the roots of as the quotient of the top two coefficients, and since the function enumerates the roots, we are left with an equation that sums the function at the different roots. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theorembasellem6 20155 Lemma for basel 20159. The function goes to zero because it is bounded by . (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasellem7 20156 Lemma for basel 20159. The function for any fixed goes to . (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasellem8 20157* Lemma for basel 20159. The function of partial sums of the inverse squares is bounded below by and above by , obtained by summing the inequality over the roots of the polynomial , and applying the identity basellem5 20154. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theorembasellem9 20158* Lemma for basel 20159. Since by basellem8 20157 is bounded by two expressions that tend to , must also go to by the squeeze theorem climsqz 11991. But the series is exactly the partial sums of , so it follows that this is also the value of the infinite sum . (Contributed by Mario Carneiro, 28-Jul-2014.)

Theorembasel 20159 The sum of the inverse squares is . This is commonly known as the Basel problem, with the first known proof attributed to Euler. See http://en.wikipedia.org/wiki/Basel_problem. This particular proof approach is due to Cauchy (1821). (Contributed by Mario Carneiro, 30-Jul-2014.)

13.4.4  Number-theoretical functions

Syntaxccht 20160 Extend class notation with the first Chebyshev function.

Syntaxcvma 20161 Extend class notation with the von Mangoldt function.
Λ

Syntaxcchp 20162 Extend class notation with the second Chebyshev function.
ψ

Syntaxcppi 20163 Extend class notation with the prime Pi function.
π

Syntaxcmu 20164 Extend class notation with the Möbius function.

Syntaxcsgm 20165 Extend class notation with the divisor function.

Definitiondf-cht 20166* Define the first Chebyshev function, which adds up the logarithms of all primes less than . The symbol used to represent this function is sometimes the variant greek letter theta shown here and sometimes the greek letter psi, ψ; however, this notation can also refer to the second Chebyshev function, which adds up the logarithms of prime powers instead. See https://en.wikipedia.org/wiki/Chebyshev_function for a discussion of the two functions. (Contributed by Mario Carneiro, 15-Sep-2014.)

Definitiondf-vma 20167* Define the von Mangoldt function, which gives the logarithm of the prime at a prime power, and is zero elsewhere. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Definitiondf-chp 20168* Define the second Chebyshev function, which adds up the logarithms of the primes corresponding to the prime powers less than . (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ Λ

Definitiondf-ppi 20169 Define the prime π function, which counts the number of primes less than or equal to . (Contributed by Mario Carneiro, 15-Sep-2014.)
π

Definitiondf-mu 20170* Define the Möbius function, which is zero for non-squarefree numbers and is or for squarefree numbers according as to the number of prime divisors of the number is even or odd. (Contributed by Mario Carneiro, 22-Sep-2014.)

Definitiondf-sgm 20171* Define the divisor function, which counts the number of divisors of , to the power . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremefnnfsumcl 20172* Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)

Theoremppisval 20173 The set of primes less than expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremppisval2 20174 The set of primes less than expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremppifi 20175 The set of primes less than is a finite set. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremsgmss 20176* The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremprmdvdsfi 20177* The set of prime divisors of a number is a finite set. (Contributed by Mario Carneiro, 7-Apr-2016.)

Theoremchtf 20178 Domain and range of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremchtcl 20179 Real closure of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremchtval 20180* Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremefchtcl 20181 The Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 7-Apr-2016.)

Theoremchtge0 20182 The Chebyshev function is always positive. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremvmaval 20183* Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremisppw 20184* Two ways to say that is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremisppw2 20185* Two ways to say that is a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremvmappw 20186 Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremvmaprm 20187 Value of the von Mangoldt function at a prime. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremvmacl 20188 Closure for the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremvmaf 20189 Functionality of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremefvmacl 20190 The von Mangoldt is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremvmage0 20191 The von Mangoldt function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremchpval 20192* Value of the second Chebyshev function, or summary von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ Λ

Theoremchpf 20193 Functionality of the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremchpcl 20194 Closure for the second Chebyshev function. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremefchpcl 20195 The second Chebyshev function is closed in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremchpge0 20196 The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremppival 20197 Value of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
π

Theoremppival2 20198 Value of the prime pi function. (Contributed by Mario Carneiro, 18-Sep-2014.)
π

Theoremppival2g 20199 Value of the prime pi function. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremppif 20200 Domain and range of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
π

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