P. 233 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 23201-23300 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	ulmcl 23201	Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)


Theorem	ulmf 23202*	Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)


Theorem	ulmpm 23203	Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)


Theorem	ulmf2 23204	Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.)
$/\$

Theorem	ulm2 23205*	Simplify ulmval 23200 when and are known to be functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
$/\$ $/\$ $/\$

Theorem	ulmi 23206*	The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.)
$/\$ $/\$ $/\$

Theorem	ulmclm 23207*	A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015.)
$/\$

Theorem	ulmres 23208	A sequence of functions converges iff the tail of the sequence converges (for any finite cutoff). (Contributed by Mario Carneiro, 24-Mar-2015.)


Theorem	ulmshftlem 23209*	Lemma for ulmshft 23210. (Contributed by Mario Carneiro, 24-Mar-2015.)


Theorem	ulmshft 23210*	A sequence of functions converges iff the shifted sequence converges. (Contributed by Mario Carneiro, 24-Mar-2015.)


Theorem	ulm0 23211	Every function converges uniformly on the empty set. (Contributed by Mario Carneiro, 3-Mar-2015.)
$/\$

Theorem	ulmuni 23212	An sequence of functions uniformly converges to at most one limit. (Contributed by Mario Carneiro, 5-Jul-2017.)
$/\$

Theorem	ulmdm 23213	Two ways to express that a function has a limit. (The expression is sometimes useful as a shorthand for "the unique limit of the function "). (Contributed by Mario Carneiro, 5-Jul-2017.)


Theorem	ulmcaulem 23214*	Lemma for ulmcau 23215 and ulmcau2 23216: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 13397. (Contributed by Mario Carneiro, 1-Mar-2015.)


Theorem	ulmcau 23215*	A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every there is a such that for all the functions and are uniformly within of each other on . This is the four-quantifier version, see ulmcau2 23216 for the more conventional five-quantifier version. (Contributed by Mario Carneiro, 1-Mar-2015.)


Theorem	ulmcau2 23216*	A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every there is a such that for all the functions and are uniformly within of each other on . (Contributed by Mario Carneiro, 1-Mar-2015.)


Theorem	ulmss 23217*	A uniform limit of functions is still a uniform limit if restricted to a subset. (Contributed by Mario Carneiro, 3-Mar-2015.)
$/\$

Theorem	ulmbdd 23218*	A uniform limit of bounded functions is bounded. (Contributed by Mario Carneiro, 27-Feb-2015.)
$/\$

Theorem	ulmcn 23219	A uniform limit of continuous functions is continuous. (Contributed by Mario Carneiro, 27-Feb-2015.)


Theorem	ulmdvlem1 23220*	Lemma for ulmdv 23223. (Contributed by Mario Carneiro, 3-Mar-2015.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	ulmdvlem2 23221*	Lemma for ulmdv 23223. (Contributed by Mario Carneiro, 8-May-2015.)
$/\$ $/\$

Theorem	ulmdvlem3 23222*	Lemma for ulmdv 23223. (Contributed by Mario Carneiro, 8-May-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
$/\$ $/\$

Theorem	ulmdv 23223*	If is a sequence of differentiable functions on which converge pointwise to , and the derivatives of converge uniformly to , then is differentiable with derivative . (Contributed by Mario Carneiro, 27-Feb-2015.)
$/\$

Theorem	mtest 23224*	The Weierstrass M-test. If is a sequence of functions which are uniformly bounded by the convergent sequence , then the series generated by the sequence converges uniformly. (Contributed by Mario Carneiro, 3-Mar-2015.)
$/\$ $/\$ $/\$

Theorem	mtestbdd 23225*	Given the hypotheses of the Weierstrass M-test, the convergent function of the sequence is uniformly bounded. (Contributed by Mario Carneiro, 9-Jul-2017.)
$/\$ $/\$ $/\$

Theorem	mbfulm 23226	A uniform limit of measurable functions is measurable. (This is just a corollary of the fact that a pointwise limit of measurable functions is measurable, see mbflim 22503.) (Contributed by Mario Carneiro, 18-Mar-2015.)
MblFn MblFn

Theorem	iblulm 23227	A uniform limit of integrable functions is integrable. (Contributed by Mario Carneiro, 3-Mar-2015.)


Theorem	itgulm 23228*	A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)


Theorem	itgulm2 23229*	A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)
$/\$ $/\$ $/\$

14.2.3 Power series

Theorem	pserval 23230*	Value of the function that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)


Theorem	pserval2 23231*	Value of the function that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
$/\$

Theorem	psergf 23232*	The sequence of terms in the infinite sequence defining a power series for fixed . (Contributed by Mario Carneiro, 26-Feb-2015.)


Theorem	radcnvlem1 23233*	Lemma for radcnvlt1 23238, radcnvle 23240. If is a point closer to zero than and the power series converges at , then it converges absolutely at , even if the terms in the sequence are multiplied by . (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	radcnvlem2 23234*	Lemma for radcnvlt1 23238, radcnvle 23240. If is a point closer to zero than and the power series converges at , then it converges absolutely at . (Contributed by Mario Carneiro, 26-Feb-2015.)


Theorem	radcnvlem3 23235*	Lemma for radcnvlt1 23238, radcnvle 23240. If is a point closer to zero than and the power series converges at , then it converges at . (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	radcnv0 23236*	Zero is always a convergent point for any power series. (Contributed by Mario Carneiro, 26-Feb-2015.)


Theorem	radcnvcl 23237*	The radius of convergence of an infinite series is a nonnegative extended real number. (Contributed by Mario Carneiro, 26-Feb-2015.)


Theorem	radcnvlt1 23238*	If is within the open disk of radius centered at zero, then the infinite series converges absolutely at , and also converges when the series is multiplied by . (Contributed by Mario Carneiro, 26-Feb-2015.)
$/\$

Theorem	radcnvlt2 23239*	If is within the open disk of radius centered at zero, then the infinite series converges at . (Contributed by Mario Carneiro, 26-Feb-2015.)


Theorem	radcnvle 23240*	If is a convergent point of the infinite series, then is within the closed disk of radius centered at zero. Or, by contraposition, the series diverges at any point strictly more than from the origin. (Contributed by Mario Carneiro, 26-Feb-2015.)


Theorem	dvradcnv 23241*	The radius of convergence of the (formal) derivative of the power series is at least as large as the radius of convergence of . (In fact they are equal, but we don't have as much use for the negative side of this claim.) (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	pserulm 23242*	If is a region contained in a circle of radius , then the sequence of partial sums of the infinite series converges uniformly on . (Contributed by Mario Carneiro, 26-Feb-2015.)


Theorem	psercn2 23243*	Since by pserulm 23242 the series converges uniformly, it is also continuous by ulmcn 23219. (Contributed by Mario Carneiro, 3-Mar-2015.)


Theorem	psercnlem2 23244*	Lemma for psercn 23246. (Contributed by Mario Carneiro, 18-Mar-2015.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	psercnlem1 23245*	Lemma for psercn 23246. (Contributed by Mario Carneiro, 18-Mar-2015.)
$/\$ $/\$ $/\$

Theorem	psercn 23246*	An infinite series converges to a continuous function on the open disk of radius , where is the radius of convergence of the series. (Contributed by Mario Carneiro, 4-Mar-2015.)


Theorem	pserdvlem1 23247*	Lemma for pserdv 23249. (Contributed by Mario Carneiro, 7-May-2015.)
$/\$ $/\$ $/\$

Theorem	pserdvlem2 23248*	Lemma for pserdv 23249. (Contributed by Mario Carneiro, 7-May-2015.)
$/\$

Theorem	pserdv 23249*	The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	pserdv2 23250*	The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	abelthlem1 23251*	Lemma for abelth 23261. (Contributed by Mario Carneiro, 1-Apr-2015.)


Theorem	abelthlem2 23252*	Lemma for abelth 23261. The peculiar region , known as a Stolz angle , is a teardrop-shaped subset of the closed unit ball containing . Indeed, except for itself, the rest of the Stolz angle is enclosed in the open unit ball. (Contributed by Mario Carneiro, 31-Mar-2015.)
$/\$ $\$

Theorem	abelthlem3 23253*	Lemma for abelth 23261. (Contributed by Mario Carneiro, 31-Mar-2015.)
$/\$

Theorem	abelthlem4 23254*	Lemma for abelth 23261. (Contributed by Mario Carneiro, 31-Mar-2015.)


Theorem	abelthlem5 23255*	Lemma for abelth 23261. (Contributed by Mario Carneiro, 1-Apr-2015.)
$/\$

Theorem	abelthlem6 23256*	Lemma for abelth 23261. (Contributed by Mario Carneiro, 2-Apr-2015.)
$\$

Theorem	abelthlem7a 23257*	Lemma for abelth 23261. (Contributed by Mario Carneiro, 8-May-2015.)
$\$ $/\$

Theorem	abelthlem7 23258*	Lemma for abelth 23261. (Contributed by Mario Carneiro, 2-Apr-2015.)
$\$

Theorem	abelthlem8 23259*	Lemma for abelth 23261. (Contributed by Mario Carneiro, 2-Apr-2015.)
$/\$

Theorem	abelthlem9 23260*	Lemma for abelth 23261. By adjusting the constant term, we can assume that the entire series converges to . (Contributed by Mario Carneiro, 1-Apr-2015.)
$/\$

Theorem	abelth 23261*	Abel's theorem. If the power series is convergent at , then it is equal to the limit from "below", along a Stolz angle (note that the case of a Stolz angle is the real line ). (Continuity on $\$ follows more generally from psercn 23246.) (Contributed by Mario Carneiro, 2-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)


Theorem	abelth2 23262*	Abel's theorem, restricted to the interval. (Contributed by Mario Carneiro, 2-Apr-2015.)


14.3 Basic trigonometry

14.3.1 The exponential, sine, and cosine functions (cont.)

Theorem	efcn 23263	The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)


Theorem	sincn 23264	Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)


Theorem	coscn 23265	Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)


Theorem	reeff1olem 23266*	Lemma for reeff1o 23267. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
$/\$

Theorem	reeff1o 23267	The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)


Theorem	reefiso 23268	The exponential function on the reals determines an isomorphism from reals onto positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) (Revised by Mario Carneiro, 11-Mar-2014.)


Theorem	efcvx 23269	The exponential function on the reals is a strictly convex function. (Contributed by Mario Carneiro, 20-Jun-2015.)
$/\$ $/\$ $/\$

Theorem	reefgim 23270	The exponential function is a group isomorphism from the group of reals under addition to the group of positive reals under multiplication. (Contributed by Mario Carneiro, 21-Jun-2015.) (Revised by Thierry Arnoux, 30-Jun-2019.)
mulGrpℂ_fld ↾_s RRfld GrpIso

14.3.2 Properties of pi = 3.14159...

Theorem	pilem1 23271	Lemma for pire 23278, pigt2lt4 23276 and sinpi 23277. (Contributed by Mario Carneiro, 9-May-2014.)
$/\$

Theorem	pilem2 23272	Lemma for pire 23278, pigt2lt4 23276 and sinpi 23277. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by AV, 14-Sep-2020.)


Theorem	pilem2OLD 23273	Lemma for pire 23278, pigt2lt4 23276 and sinpi 23277. (Contributed by Mario Carneiro, 12-Jun-2014.) Obsolete version of pilem2 23272 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)


Theorem	pilem3 23274	Lemma for pire 23278, pigt2lt4 23276 and sinpi 23277. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) (Revised by AV, 14-Sep-2020.)
$/\$

Theorem	pilem3OLD 23275	Lemma for pire 23278, pigt2lt4 23276 and sinpi 23277. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) Obsolete version of pilem3 23274 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
$/\$

Theorem	pigt2lt4 23276	is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
$/\$

Theorem	sinpi 23277	The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)


Theorem	pire 23278	is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)


Theorem	picn 23279	is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)


Theorem	pipos 23280	is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)


Theorem	pirp 23281	is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)


Theorem	negpicn 23282	is a real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)


Theorem	sinhalfpilem 23283	Lemma for sinhalfpi 23288 and coshalfpi 23289. (Contributed by Paul Chapman, 23-Jan-2008.)
$/\$

Theorem	halfpire 23284	is real. (Contributed by David Moews, 28-Feb-2017.)


Theorem	neghalfpire 23285	is real. (Contributed by David A. Wheeler, 8-Dec-2018.)


Theorem	neghalfpirx 23286	is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)


Theorem	pidiv2halves 23287	Adding to itself is (common case). See 2halves 10841. (Contributed by David A. Wheeler, 8-Dec-2018.)


Theorem	sinhalfpi 23288	The sine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)


Theorem	coshalfpi 23289	The cosine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)


Theorem	cosneghalfpi 23290	The cosine of is zero. (Contributed by David Moews, 28-Feb-2017.)


Theorem	efhalfpi 23291	The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)


Theorem	cospi 23292	The cosine of is . (Contributed by Paul Chapman, 23-Jan-2008.)


Theorem	efipi 23293	The exponential of . (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)


Theorem	eulerid 23294	Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)


Theorem	sin2pi 23295	The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)


Theorem	cos2pi 23296	The cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)


Theorem	ef2pi 23297	The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)


Theorem	ef2kpi 23298	The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)


Theorem	efper 23299	The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)
$/\$

Theorem	sinperlem 23300	Lemma for sinper 23301 and cosper 23302. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
$/\$

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