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**Theorem List for Intuitionistic Logic Explorer -** 11901-12000 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	maxleb 11901	Equivalence of and being equal to the maximum of two reals. Lemma 3.12 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)
$/\$

Theorem	dfabsmax 11902	Absolute value of a real number in terms of maximum. Definition 3.13 of [Geuvers], p. 11. (Contributed by BJ and Jim Kingdon, 21-Dec-2021.)


Theorem	maxltsup 11903	Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 10-Feb-2022.)
$/\$ $/\$ $/\$

Theorem	max0addsup 11904	The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.)


Theorem	rexanre 11905*	Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)
$/\$ $/\$

Theorem	rexico 11906*	Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)
$/\$

Theorem	maxclpr 11907	The maximum of two real numbers is one of those numbers if and only if dichotomy ( $\/$ ) holds. For example, this can be combined with zletric 9621 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 1-Feb-2022.)
$/\$ $\/$

Theorem	rpmaxcl 11908	The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.)
$/\$

Theorem	zmaxcl 11909	The maximum of two integers is an integer. (Contributed by Jim Kingdon, 27-Sep-2022.)
$/\$

Theorem	nn0maxcl 11910	The maximum of two nonnegative integers is a nonnegative integer. (Contributed by Jim Kingdon, 28-Oct-2025.)
$/\$

Theorem	2zsupmax 11911	Two ways to express the maximum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 22-Jan-2023.)
$/\$

Theorem	fimaxre2 11912*	A nonempty finite set of real numbers has an upper bound. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.)
$/\$

Theorem	negfi 11913*	The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.)
$/\$

4.8.6 The minimum of two real numbers

Theorem	mincom 11914	The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.)
inf inf

Theorem	minmax 11915	Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 8-Feb-2021.)
$/\$ inf

Theorem	mincl 11916	The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
$/\$ inf

Theorem	min1inf 11917	The minimum of two numbers is less than or equal to the first. (Contributed by Jim Kingdon, 8-Feb-2021.)
$/\$ inf

Theorem	min2inf 11918	The minimum of two numbers is less than or equal to the second. (Contributed by Jim Kingdon, 9-Feb-2021.)
$/\$ inf

Theorem	lemininf 11919	Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007.)
$/\$ $/\$ inf $/\$

Theorem	ltmininf 11920	Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.)
$/\$ $/\$ inf $/\$

Theorem	minabs 11921	The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.)
$/\$ inf

Theorem	minclpr 11922	The minimum of two real numbers is one of those numbers if and only if dichotomy ( $\/$ ) holds. For example, this can be combined with zletric 9621 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 23-May-2023.)
$/\$ inf $\/$

Theorem	rpmincl 11923	The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
$/\$ inf

Theorem	bdtrilem 11924	Lemma for bdtri 11925. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.)
$/\$ $/\$ $/\$ $/\$

Theorem	bdtri 11925	Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
$/\$ $/\$ $/\$ $/\$ inf inf inf

Theorem	mul0inf 11926	Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11747 and mulap0bd 8931 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.)
$/\$ inf

Theorem	mingeb 11927	Equivalence of and being equal to the minimum of two reals. (Contributed by Jim Kingdon, 14-Oct-2024.)
$/\$ inf

Theorem	2zinfmin 11928	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 11929	Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
$/\$

Theorem	xrmaxiflemcl 11930	Lemma for xrmaxif 11936. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$

Theorem	xrmaxifle 11931	An upper bound for in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)
$/\$

Theorem	xrmaxiflemab 11932	Lemma for xrmaxif 11936. A variation of xrmaxleim 11929- 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 11933	Lemma for xrmaxif 11936. A least upper bound for . (Contributed by Jim Kingdon, 28-Apr-2023.)
$\/$

Theorem	xrmaxiflemcom 11934	Lemma for xrmaxif 11936. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$

Theorem	xrmaxiflemval 11935*	Lemma for xrmaxif 11936. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$ $/\$ $/\$

Theorem	xrmaxif 11936	Maximum of two extended reals in terms of expressions. (Contributed by Jim Kingdon, 26-Apr-2023.)
$/\$

Theorem	xrmaxcl 11937	The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$

Theorem	xrmax1sup 11938	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 11939	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 11940	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 11941	The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.)
$/\$ $/\$ $/\$ $\/$

Theorem	xrltmaxsup 11942	The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.)
$/\$ $/\$ $\/$

Theorem	xrmaxltsup 11943	Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)
$/\$ $/\$ $/\$

Theorem	xrmaxlesup 11944	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 11945	Lemma for xrmaxadd 11946. The case where is real. (Contributed by Jim Kingdon, 11-May-2023.)
$/\$ $/\$

Theorem	xrmaxadd 11946	Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)
$/\$ $/\$

4.8.8 The minimum of two extended reals

Theorem	xrnegiso 11947	Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
$/\$

Theorem	infxrnegsupex 11948*	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 11949	Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)


Theorem	xrminmax 11950	Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.)
$/\$ inf

Theorem	xrmincl 11951	The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.)
$/\$ inf

Theorem	xrmin1inf 11952	The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.)
$/\$ inf

Theorem	xrmin2inf 11953	The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.)
$/\$ inf

Theorem	xrmineqinf 11954	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 11955	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 11956	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 11957	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 11958	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 11959	The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.)
$/\$ inf

Theorem	xrminadd 11960	Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)
$/\$ $/\$ inf inf

Theorem	xrbdtri 11961	Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
$/\$ $/\$ $/\$ $/\$ $/\$ inf inf inf

Theorem	iooinsup 11962	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 11963	Extend class notation with convergence relation for limits.


Definition	df-clim 11964*	Define the limit relation for complex number sequences. See clim 11966 for its relational expression. (Contributed by NM, 28-Aug-2005.)
$/\$ $/\$

Theorem	climrel 11965	The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)


Theorem	clim 11966*	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 11967	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 11968*	Express the predicate: The limit of complex number sequence is , or converges to , with more general quantifier restrictions than clim 11966. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$

Theorem	clim2c 11969*	Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	clim0 11970*	Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	clim0c 11971*	Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	climi 11972*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	climi2 11973*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climi0 11974*	Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climconst 11975*	An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climconst2 11976	A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climz 11977	The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)


Theorem	climuni 11978	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 11979	The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)


Theorem	climdm 11980	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 11981*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)


Theorem	climreu 11982*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)


Theorem	climmo 11983*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)


Theorem	climeq 11984*	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 11985*	Exhibit a function with the same convergence properties as the not-quite-function . (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	2clim 11986*	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 11987	A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)
$/\$

Theorem	climres 11988	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 11989	A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	serclim0 11990	The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)


Theorem	climshft2 11991*	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 11992*	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 11993*	Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	climcn2 11994*	Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	addcn2 11995*	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 15427 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$

Theorem	subcn2 11996*	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 11997*	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 11998*	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 2232. (Contributed by Mario Carneiro, 9-Feb-2014.) Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
inf $/\$ # $/\$ #

Theorem	cn1lem 11999*	A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$ $/\$

Theorem	abscn2 12000*	The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$

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