P. 117 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 11601-11700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	bdtri 11601	Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
$/\$ $/\$ $/\$ $/\$ inf inf inf

Theorem	mul0inf 11602	Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11423 and mulap0bd 8743 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.)
$/\$ inf

Theorem	mingeb 11603	Equivalence of and being equal to the minimum of two reals. (Contributed by Jim Kingdon, 14-Oct-2024.)
$/\$ inf

Theorem	2zinfmin 11604	Two ways to express the minimum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 14-Oct-2024.)
$/\$ inf

4.8.7 The maximum of two extended reals

Theorem	xrmaxleim 11605	Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
$/\$

Theorem	xrmaxiflemcl 11606	Lemma for xrmaxif 11612. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$

Theorem	xrmaxifle 11607	An upper bound for in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)
$/\$

Theorem	xrmaxiflemab 11608	Lemma for xrmaxif 11612. A variation of xrmaxleim 11605- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 26-Apr-2023.)


Theorem	xrmaxiflemlub 11609	Lemma for xrmaxif 11612. A least upper bound for . (Contributed by Jim Kingdon, 28-Apr-2023.)
$\/$

Theorem	xrmaxiflemcom 11610	Lemma for xrmaxif 11612. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$

Theorem	xrmaxiflemval 11611*	Lemma for xrmaxif 11612. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$ $/\$ $/\$

Theorem	xrmaxif 11612	Maximum of two extended reals in terms of expressions. (Contributed by Jim Kingdon, 26-Apr-2023.)
$/\$

Theorem	xrmaxcl 11613	The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$

Theorem	xrmax1sup 11614	An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
$/\$

Theorem	xrmax2sup 11615	An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
$/\$

Theorem	xrmaxrecl 11616	The maximum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
$/\$

Theorem	xrmaxleastlt 11617	The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.)
$/\$ $/\$ $/\$ $\/$

Theorem	xrltmaxsup 11618	The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.)
$/\$ $/\$ $\/$

Theorem	xrmaxltsup 11619	Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)
$/\$ $/\$ $/\$

Theorem	xrmaxlesup 11620	Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 10-May-2023.)
$/\$ $/\$ $/\$

Theorem	xrmaxaddlem 11621	Lemma for xrmaxadd 11622. The case where is real. (Contributed by Jim Kingdon, 11-May-2023.)
$/\$ $/\$

Theorem	xrmaxadd 11622	Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)
$/\$ $/\$

4.8.8 The minimum of two extended reals

Theorem	xrnegiso 11623	Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
$/\$

Theorem	infxrnegsupex 11624*	The infimum of a set of extended reals is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)
$/\$ inf

Theorem	xrnegcon1d 11625	Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)


Theorem	xrminmax 11626	Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.)
$/\$ inf

Theorem	xrmincl 11627	The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.)
$/\$ inf

Theorem	xrmin1inf 11628	The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.)
$/\$ inf

Theorem	xrmin2inf 11629	The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.)
$/\$ inf

Theorem	xrmineqinf 11630	The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim Kingdon, 3-May-2023.)
$/\$ $/\$ inf

Theorem	xrltmininf 11631	Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)
$/\$ $/\$ inf $/\$

Theorem	xrlemininf 11632	Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 4-May-2023.)
$/\$ $/\$ inf $/\$

Theorem	xrminltinf 11633	Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.)
$/\$ $/\$ inf $\/$

Theorem	xrminrecl 11634	The minimum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
$/\$ inf inf

Theorem	xrminrpcl 11635	The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.)
$/\$ inf

Theorem	xrminadd 11636	Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)
$/\$ $/\$ inf inf

Theorem	xrbdtri 11637	Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
$/\$ $/\$ $/\$ $/\$ $/\$ inf inf inf

Theorem	iooinsup 11638	Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
$/\$ $/\$ $/\$ inf

4.9 Elementary limits and convergence

4.9.1 Limits

Syntax	cli 11639	Extend class notation with convergence relation for limits.


Definition	df-clim 11640*	Define the limit relation for complex number sequences. See clim 11642 for its relational expression. (Contributed by NM, 28-Aug-2005.)
$/\$ $/\$

Theorem	climrel 11641	The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)


Theorem	clim 11642*	Express the predicate: The limit of complex number sequence is , or converges to . This means that for any real , no matter how small, there always exists an integer such that the absolute difference of any later complex number in the sequence and the limit is less than . (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
$/\$ $/\$ $/\$

Theorem	climcl 11643	Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)


Theorem	clim2 11644*	Express the predicate: The limit of complex number sequence is , or converges to , with more general quantifier restrictions than clim 11642. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$

Theorem	clim2c 11645*	Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	clim0 11646*	Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	clim0c 11647*	Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	climi 11648*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	climi2 11649*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climi0 11650*	Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climconst 11651*	An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climconst2 11652	A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climz 11653	The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)


Theorem	climuni 11654	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.)
$/\$

Theorem	fclim 11655	The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)


Theorem	climdm 11656	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.)


Theorem	climeu 11657*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)


Theorem	climreu 11658*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)


Theorem	climmo 11659*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)


Theorem	climeq 11660*	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.)
$/\$

Theorem	climmpt 11661*	Exhibit a function with the same convergence properties as the not-quite-function . (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	2clim 11662*	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.)
$/\$

Theorem	climshftlemg 11663	A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)
$/\$

Theorem	climres 11664	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.)
$/\$

Theorem	climshft 11665	A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	serclim0 11666	The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)


Theorem	climshft2 11667*	A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.)
$/\$

Theorem	climabs0 11668*	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.)
$/\$ $/\$

Theorem	climcn1 11669*	Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	climcn2 11670*	Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	addcn2 11671*	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 and df-cncf are not yet available to us. See addcncntop 15084 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$

Theorem	subcn2 11672*	Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$

Theorem	mulcn2 11673*	Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$

Theorem	reccn2ap 11674*	The reciprocal function is continuous. The class is just for convenience in writing the proof and typically would be passed in as an instance of eqid 2206. (Contributed by Mario Carneiro, 9-Feb-2014.) Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
inf $/\$ # $/\$ #

Theorem	cn1lem 11675*	A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$ $/\$

Theorem	abscn2 11676*	The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$

Theorem	cjcn2 11677*	The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$

Theorem	recn2 11678*	The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$

Theorem	imcn2 11679*	The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$

Theorem	climcn1lem 11680*	The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$ $/\$ $/\$

Theorem	climabs 11681*	Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
$/\$ $/\$

Theorem	climcj 11682*	Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
$/\$ $/\$

Theorem	climre 11683*	Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
$/\$ $/\$

Theorem	climim 11684*	Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
$/\$ $/\$

Theorem	climrecl 11685*	The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.)
$/\$

Theorem	climge0 11686*	A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.)
$/\$ $/\$

Theorem	climadd 11687*	Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$

Theorem	climmul 11688*	Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
$/\$ $/\$ $/\$

Theorem	climsub 11689*	Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
$/\$ $/\$ $/\$

Theorem	climaddc1 11690*	Limit of a constant added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
$/\$ $/\$

Theorem	climaddc2 11691*	Limit of a constant added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
$/\$ $/\$

Theorem	climmulc2 11692*	Limit of a sequence multiplied by a constant . Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
$/\$ $/\$

Theorem	climsubc1 11693*	Limit of a constant subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$ $/\$

Theorem	climsubc2 11694*	Limit of a constant minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
$/\$ $/\$

Theorem	climle 11695*	Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
$/\$ $/\$ $/\$

Theorem	climsqz 11696*	Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	climsqz2 11697*	Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	clim2ser 11698*	The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
$/\$

Theorem	clim2ser2 11699*	The limit of an infinite series with an initial segment added. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
$/\$

Theorem	iserex 11700*	An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.)
$/\$

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