P. 248 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 24701-24800 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	o1cxp 24701*	An eventually bounded function taken to a nonnegative power is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
$/\$

Theorem	cxp2limlem 24702*	A linear factor grows slower than any exponential with base greater than . (Contributed by Mario Carneiro, 15-Sep-2014.)
$/\$

Theorem	cxp2lim 24703*	Any power grows slower than any exponential with base greater than . (Contributed by Mario Carneiro, 18-Sep-2014.)
$/\$ $/\$

Theorem	cxploglim 24704*	The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.)


Theorem	cxploglim2 24705*	Every power of the logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 20-May-2016.)
$/\$

Theorem	divsqrtsumlem 24706*	Lemma for divsqrsum 24708 and divsqrtsum2 24709. (Contributed by Mario Carneiro, 18-May-2016.)
$/\$ $/\$ $/\$

Theorem	divsqrsumf 24707*	The function used in divsqrsum 24708 is a real function. (Contributed by Mario Carneiro, 12-May-2016.)


Theorem	divsqrsum 24708*	The sum is asymptotic to with a finite limit . (In fact, this limit is .) (Contributed by Mario Carneiro, 9-May-2016.)


Theorem	divsqrtsum2 24709*	A bound on the distance of the sum from its asymptotic value . (Contributed by Mario Carneiro, 18-May-2016.)
$/\$

Theorem	divsqrtsumo1 24710*	The sum has the asymptotic expansion , for some . (Contributed by Mario Carneiro, 10-May-2016.)


14.3.12 Inequality of arithmetic and geometric means

Theorem	cvxcl 24711*	Closure of a 0-1 linear combination in a convex set. (Contributed by Mario Carneiro, 21-Jun-2015.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	scvxcvx 24712*	A strictly convex function is convex. (Contributed by Mario Carneiro, 20-Jun-2015.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	jensenlem1 24713*	Lemma for jensen 24715. (Contributed by Mario Carneiro, 4-Jun-2016.)
$/\$ $/\$ ℂ_fld _g $/\$ $/\$ $/\$ ℂ_fld _g ℂ_fld _g

Theorem	jensenlem2 24714*	Lemma for jensen 24715. (Contributed by Mario Carneiro, 21-Jun-2015.)
$/\$ $/\$ ℂ_fld _g $/\$ $/\$ $/\$ ℂ_fld _g ℂ_fld _g ℂ_fld _g ℂ_fld _g ℂ_fld _g ℂ_fld _g $/\$ ℂ_fld _g ℂ_fld _g

Theorem	jensen 24715*	Jensen's inequality, a finite extension of the definition of convexity (the last hypothesis). (Contributed by Mario Carneiro, 21-Jun-2015.) (Proof shortened by AV, 27-Jul-2019.)
$/\$ $/\$ ℂ_fld _g $/\$ $/\$ $/\$ ℂ_fld _g ℂ_fld _g $/\$ ℂ_fld _g ℂ_fld _g ℂ_fld _g ℂ_fld _g

Theorem	amgmlem 24716	Lemma for amgm 24717. (Contributed by Mario Carneiro, 21-Jun-2015.)
mulGrpℂ_fld _g ℂ_fld _g

Theorem	amgm 24717	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 ℂ_fld _g calculates the group sum in the additive group (i.e. the sum of the elements). This is Metamath 100 proof #38. (Contributed by Mario Carneiro, 20-Jun-2015.)
mulGrpℂ_fld $/\$ $/\$ _g ℂ_fld _g

14.3.13 Euler-Mascheroni constant

Syntax	cem 24718	The Euler-Mascheroni constant. (The label abbreviates Euler-Mascheroni.)


Definition	df-em 24719	Define the Euler-Mascheroni constant, 0.577... . This is the limit of the series , with a proof that the limit exists in emcl 24729. (Contributed by Mario Carneiro, 11-Jul-2014.)


Theorem	logdifbnd 24720	Bound on the difference of logs. (Contributed by Mario Carneiro, 23-May-2016.)


Theorem	logdiflbnd 24721	Lower bound on the difference of logs. (Contributed by Mario Carneiro, 3-Jul-2017.)


Theorem	emcllem1 24722*	Lemma for emcl 24729. The series and are sequences of real numbers that approach from above and below, respectively. (Contributed by Mario Carneiro, 11-Jul-2014.)
$/\$

Theorem	emcllem2 24723*	Lemma for emcl 24729. is increasing, and is decreasing. (Contributed by Mario Carneiro, 11-Jul-2014.)
$/\$

Theorem	emcllem3 24724*	Lemma for emcl 24729. The function is the difference between and . (Contributed by Mario Carneiro, 11-Jul-2014.)


Theorem	emcllem4 24725*	Lemma for emcl 24729. The difference between series and tends to zero. (Contributed by Mario Carneiro, 11-Jul-2014.)


Theorem	emcllem5 24726*	Lemma for emcl 24729. The partial sums of the series , which is used in the definition df-em 24719, is in fact the same as . (Contributed by Mario Carneiro, 11-Jul-2014.)


Theorem	emcllem6 24727*	Lemma for emcl 24729. By the previous lemmas, and must approach a common limit, which is by definition. (Contributed by Mario Carneiro, 11-Jul-2014.)
$/\$

Theorem	emcllem7 24728*	Lemma for emcl 24729 and harmonicbnd 24730. Derive bounds on as and . (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)
$/\$ $/\$

Theorem	emcl 24729	Closure and bounds for the Euler-Mascheroni constant. (Contributed by Mario Carneiro, 11-Jul-2014.)


Theorem	harmonicbnd 24730*	A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 9-Apr-2016.)


Theorem	harmonicbnd2 24731*	A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)


Theorem	emre 24732	The Euler-Mascheroni constant is a real number. (Contributed by Mario Carneiro, 11-Jul-2014.)


Theorem	emgt0 24733	The Euler-Mascheroni constant is positive. (Contributed by Mario Carneiro, 11-Jul-2014.)


Theorem	harmonicbnd3 24734*	A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)


Theorem	harmoniclbnd 24735*	A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)


Theorem	harmonicubnd 24736*	A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016.)
$/\$

Theorem	harmonicbnd4 24737*	The asymptotic behavior of . (Contributed by Mario Carneiro, 14-May-2016.)


Theorem	fsumharmonic 24738*	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.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

14.3.14 Zeta function

Syntax	czeta 24739	The Riemann zeta function.


Definition	df-zeta 24740*	Define the Riemann zeta function. This definition uses a series expansion of the alternating zeta function ~? zetaalt that is convergent everywhere except , but going from the alternating zeta function to the regular zeta function requires dividing by , which has zeroes other than . To extract the correct value of the zeta function at these points, we extend the divided alternating zeta function by continuity. (Contributed by Mario Carneiro, 18-Jul-2014.)
$\$ $\$

Theorem	zetacvg 24741*	The zeta series is convergent. (Contributed by Mario Carneiro, 18-Jul-2014.)
$/\$

14.3.15 Gamma function

Syntax	clgam 24742	Logarithm of the Gamma function.


Syntax	cgam 24743	The Gamma function.


Syntax	cigam 24744	The inverse Gamma function.
1/

Definition	df-lgam 24745*	Define the log-Gamma function. We can work with this form of the gamma function a bit easier than the equivalent expression for the gamma function itself, and moreover this function is not actually equal to because the branch cuts are placed differently (we do have , though). This definition is attributed to Euler, and unlike the usual integral definition is defined on the entire complex plane except the nonpositive integers $\$ , where the function has simple poles. (Contributed by Mario Carneiro, 12-Jul-2014.)
$\$ $\$

Definition	df-gam 24746	Define the Gamma function. See df-lgam 24745 for more information about the reason for this definition in terms of the log-gamma function. (Contributed by Mario Carneiro, 12-Jul-2014.)


Definition	df-igam 24747	Define the inverse Gamma function, which is defined everywhere, unlike the Gamma function itself. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/ $\$

Theorem	eldmgm 24748	Elementhood in the set of non-nonpositive integers. (Contributed by Mario Carneiro, 12-Jul-2014.)
$\$ $\$ $/\$

Theorem	dmgmaddn0 24749	If is not a nonpositive integer, then is nonzero for any nonnegative integer . (Contributed by Mario Carneiro, 12-Jul-2014.)
$\$ $\$ $/\$

Theorem	dmlogdmgm 24750	If is in the continuous domain of the logarithm, then it is in the domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
$\$ $\$ $\$

Theorem	rpdmgm 24751	A positive real number is in the domain of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	dmgmn0 24752	If is not a nonpositive integer, then is nonzero. (Contributed by Mario Carneiro, 3-Jul-2017.)
$\$ $\$

Theorem	dmgmaddnn0 24753	If is not a nonpositive integer and is a nonnegative integer, then is also not a nonpositive integer. (Contributed by Mario Carneiro, 6-Jul-2017.)
$\$ $\$ $\$ $\$

Theorem	dmgmdivn0 24754	Lemma for lgamf 24768. (Contributed by Mario Carneiro, 3-Jul-2017.)
$\$ $\$

Theorem	lgamgulmlem1 24755*	Lemma for lgamgulm 24761. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$ $\$ $\$

Theorem	lgamgulmlem2 24756*	Lemma for lgamgulm 24761. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$

Theorem	lgamgulmlem3 24757*	Lemma for lgamgulm 24761. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$

Theorem	lgamgulmlem4 24758*	Lemma for lgamgulm 24761. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$

Theorem	lgamgulmlem5 24759*	Lemma for lgamgulm 24761. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$ $/\$ $/\$

Theorem	lgamgulmlem6 24760*	The series is uniformly convergent on the compact region , which describes a circle of radius with holes of size around the poles of the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
$/\$ $/\$

Theorem	lgamgulm 24761*	The series is uniformly convergent on the compact region , which describes a circle of radius with holes of size around the poles of the gamma function. (Contributed by Mario Carneiro, 3-Jul-2017.)
$/\$

Theorem	lgamgulm2 24762*	Rewrite the limit of the sequence in terms of the log-Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
$/\$ $/\$

Theorem	lgambdd 24763*	The log-Gamma function is bounded on the region . (Contributed by Mario Carneiro, 9-Jul-2017.)
$/\$

Theorem	lgamucov 24764*	The regions used in the proof of lgamgulm 24761 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
$/\$ $\$ $\$ ℂ_fld

Theorem	lgamucov2 24765*	The regions used in the proof of lgamgulm 24761 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
$/\$ $\$ $\$

Theorem	lgamcvglem 24766*	Lemma for lgamf 24768 and lgamcvg 24780. (Contributed by Mario Carneiro, 8-Jul-2017.)
$/\$ $\$ $\$ $/\$

Theorem	lgamcl 24767	The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 8-Jul-2017.)
$\$ $\$

Theorem	lgamf 24768	The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)
$\$ $\$

Theorem	gamf 24769	The Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)
$\$ $\$

Theorem	gamcl 24770	The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
$\$ $\$

Theorem	eflgam 24771	The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
$\$ $\$

Theorem	gamne0 24772	The Gamma function is never zero. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	igamval 24773	Value of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/ $\$

Theorem	igamz 24774	Value of the inverse Gamma function on nonpositive integers. (Contributed by Mario Carneiro, 16-Jul-2017.)
$\$ 1/

Theorem	igamgam 24775	Value of the inverse Gamma function in terms of the Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
$\$ $\$ 1/

Theorem	igamlgam 24776	Value of the inverse Gamma function in terms of the log-Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
$\$ $\$ 1/

Theorem	igamf 24777	Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/

Theorem	igamcl 24778	Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/

Theorem	gamigam 24779	The Gamma function is the inverse of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
$\$ $\$ 1/

Theorem	lgamcvg 24780*	The series converges to . (Contributed by Mario Carneiro, 6-Jul-2017.)
$\$ $\$

Theorem	lgamcvg2 24781*	The series converges to . (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	gamcvg 24782*	The pointwise exponential of the series converges to . (Contributed by Mario Carneiro, 6-Jul-2017.)
$\$ $\$

Theorem	lgamp1 24783	The functional equation of the (log) Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	gamp1 24784	The functional equation of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	gamcvg2lem 24785*	Lemma for gamcvg2 24786. (Contributed by Mario Carneiro, 10-Jul-2017.)
$\$ $\$

Theorem	gamcvg2 24786*	An infinite product expression for the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	regamcl 24787	The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
$\$ $\$

Theorem	relgamcl 24788	The log-Gamma function is real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)


Theorem	rpgamcl 24789	The log-Gamma function is positive real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)


Theorem	lgam1 24790	The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)


Theorem	gam1 24791	The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)


Theorem	facgam 24792	The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)


Theorem	gamfac 24793	The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)


14.4 Basic number theory

14.4.1 Wilson's theorem

Theorem	wilthlem1 24794	The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in are and . (Note that from prmdiveq 15491, is the modular inverse of in . (Contributed by Mario Carneiro, 24-Jan-2015.)
$/\$ $\/$

Theorem	wilthlem2 24795*	Lemma for wilth 24797: 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 24794 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.) (Proof shortened by AV, 27-Jul-2019.)
mulGrpℂ_fld $/\$ _g _g

Theorem	wilthlem3 24796*	Lemma for wilth 24797. Here we round out the argument of wilthlem2 24795 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.) (Proof shortened by AV, 27-Jul-2019.)
mulGrpℂ_fld $/\$

Theorem	wilth 24797	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 24796 for the forward implication. This is Metamath 100 proof #51. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
$/\$

Theorem	wilthimp 24798	The forward implication of Wilson's theorem wilth 24797 (see wilthlem3 24796), expressed using the modulo operation: For any prime we have (mod ), see theorem 5.24 in [ApostolNT] p. 116. (Contributed by AV, 21-Jul-2021.)


14.4.2 The Fundamental Theorem of Algebra

Theorem	ftalem1 24799*	Lemma for fta 24806: "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

Theorem	ftalem2 24800*	Lemma for fta 24806. 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

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26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42551

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