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Theorem List for Metamath Proof Explorer - 12301-12400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrlimss 12301 Domain closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremrlimcl 12302 Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremclim2 12303* Express the predicate: The limit of complex number sequence is , or converges to , with more general quantifier restrictions than clim 12293. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclim2c 12304* Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclim0 12305* Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclim0c 12306* Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrlim0 12307* Express the predicate converges to . (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)

Theoremrlim0lt 12308* Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)

Theoremclimi 12309* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclimi2 12310* Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclimi0 12311* Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrlimi 12312* Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 28-Feb-2015.)

Theoremrlimi2 12313* Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremello1 12314* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremello12 12315* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremello12r 12316* Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1f 12317 An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1dm 12318 An eventually upper bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1bdd 12319* The defining property of an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremello1mpt 12320* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremello1mpt2 12321* Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremello1d 12322* Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1bdd2 12323* If an eventually bounded function is bounded on every interval by a function , then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremlo1bddrp 12324* Refine o1bdd2 12340 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)

Theoremelo1 12325* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremelo12 12326* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremelo12r 12327* Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremo1f 12328 An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremo1dm 12329 An eventually bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremo1bdd 12330* The defining property of an eventually bounded function. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremlo1o1 12331 A function is eventually bounded iff its absolute value is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1o12 12332* A function is eventually bounded iff its absolute value is eventually upper bounded. (This function is useful for converting theorems about to .) (Contributed by Mario Carneiro, 26-May-2016.)

Theoremelo1mpt 12333* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Theoremelo1mpt2 12334* Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 12-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Theoremelo1d 12335* Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Theoremo1lo1 12336* A real function is eventually bounded iff it is eventually lower bounded and eventually upper bounded. (Contributed by Mario Carneiro, 25-May-2016.)

Theoremo1lo12 12337* A lower bounded real function is eventually bounded iff it is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremo1lo1d 12338* A real eventually bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremicco1 12339* Derive eventual boundedness from separate upper and lower eventual bounds. (Contributed by Mario Carneiro, 15-Apr-2016.)

Theoremo1bdd2 12340* If an eventually bounded function is bounded on every interval by a function , then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Theoremo1bddrp 12341* Refine o1bdd2 12340 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)

Theoremclimconst 12342* An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrlimconst 12343* A constant sequence converges to its value. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremrlimclim1 12344 Forward direction of rlimclim 12345. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremrlimclim 12345 A sequence on an upper integer set converges in the real sense iff it converges in the integer sense. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremclimrlim2 12346* Produce a real limit from an integer limit, where the real function is only dependent on the integer part of . (Contributed by Mario Carneiro, 2-May-2016.)

Theoremclimconst2 12347 A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclimz 12348 The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrlimuni 12349 A real function whose domain is unbounded above converges to at most one limit. (Contributed by Mario Carneiro, 8-May-2016.)

Theoremrlimdm 12350 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, 8-May-2016.)

Theoremclimuni 12351 An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)

Theoremfclim 12352 The limit relation is function-like, and with range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremclimdm 12353 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, 18-Mar-2014.)

Theoremclimeu 12354* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)

Theoremclimreu 12355* An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)

Theoremclimmo 12356* An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)

Theoremrlimres 12357 The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremlo1res 12358 The restriction of an eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremo1res 12359 The restriction of an eventually bounded function is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Theoremrlimres2 12360* The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremlo1res2 12361* The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremo1res2 12362* The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremlo1resb 12363 The restriction of a function to an unbounded-above interval is eventually upper bounded iff the original is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremrlimresb 12364 The restriction of a function to an unbounded-above interval converges iff the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremo1resb 12365 The restriction of a function to an unbounded-above interval is eventually bounded iff the original is eventually bounded. (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremclimeq 12366* Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremlo1eq 12367* Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremrlimeq 12368* Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremo1eq 12369* Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremclimmpt 12370* Exhibit a function with the same convergence properties as the not-quite-function . (Contributed by Mario Carneiro, 31-Jan-2014.)

Theorem2clim 12371* If two sequences converge to each other, they converge to the same limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)

Theoremclimmpt2 12372* Relate an integer limit on a not-quite-function to a real limit. (Contributed by Mario Carneiro, 17-Sep-2014.)

Theoremclimshftlem 12373 A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremclimres 12374 A function restricted to upper integers converges iff the original function converges. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclimshft 12375 A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremserclim0 12376 The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)

Theoremrlimcld2 12377* If is a closed set in the topology of the complexes (stated here in basic form), and all the elements of the sequence lie in , then the limit of the sequence also lies in . (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimrege0 12378* The limit of a sequence of complexes with nonnegative real part has nonnegative real part. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimrecl 12379* The limit of a real sequence is real. (Contributed by Mario Carneiro, 9-May-2016.)

Theoremrlimge0 12380* The limit of a sequence of nonnegative reals is nonnegative. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremclimshft2 12381* A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.)

Theoremclimrecl 12382* The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.)

Theoremclimge0 12383* A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.)

Theoremclimabs0 12384* Convergence to zero of the absolute value is equivalent to convergence to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremo1co 12385* Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremo1compt 12386* Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremrlimcn1 12387* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 17-Sep-2014.)

Theoremrlimcn1b 12388* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimcn2 12389* Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014.)

Theoremclimcn1 12390* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremclimcn2 12391* Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremaddcn2 12392* Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (We write out the definition directly because df-cn 17296 and df-cncf 18913 are not yet available to us. See addcn 18900 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremsubcn2 12393* Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremmulcn2 12394* Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremreccn2 12395* The reciprocal function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) (Revised by Mario Carneiro, 22-Sep-2014.)

Theoremcn1lem 12396* A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremabscn2 12397* The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremcjcn2 12398* The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremrecn2 12399* The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremimcn2 12400* The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

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