P. 115 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 11401-11500 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	dfabsmax 11401	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 11402	Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 10-Feb-2022.)
$/\$ $/\$ $/\$

Theorem	max0addsup 11403	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 11404*	Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.)
$/\$ $/\$

Theorem	rexico 11405*	Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.)
$/\$

Theorem	maxclpr 11406	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 9389 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 11407	The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.)
$/\$

Theorem	zmaxcl 11408	The maximum of two integers is an integer. (Contributed by Jim Kingdon, 27-Sep-2022.)
$/\$

Theorem	nn0maxcl 11409	The maximum of two nonnegative integers is a nonnegative integer. (Contributed by Jim Kingdon, 28-Oct-2025.)
$/\$

Theorem	2zsupmax 11410	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 11411*	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 11412*	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 11413	The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.)
inf inf

Theorem	minmax 11414	Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 8-Feb-2021.)
$/\$ inf

Theorem	mincl 11415	The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
$/\$ inf

Theorem	min1inf 11416	The minimum of two numbers is less than or equal to the first. (Contributed by Jim Kingdon, 8-Feb-2021.)
$/\$ inf

Theorem	min2inf 11417	The minimum of two numbers is less than or equal to the second. (Contributed by Jim Kingdon, 9-Feb-2021.)
$/\$ inf

Theorem	lemininf 11418	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 11419	Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.)
$/\$ $/\$ inf $/\$

Theorem	minabs 11420	The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.)
$/\$ inf

Theorem	minclpr 11421	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 9389 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 11422	The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
$/\$ inf

Theorem	bdtrilem 11423	Lemma for bdtri 11424. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.)
$/\$ $/\$ $/\$ $/\$

Theorem	bdtri 11424	Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
$/\$ $/\$ $/\$ $/\$ inf inf inf

Theorem	mul0inf 11425	Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11246 and mulap0bd 8703 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.)
$/\$ inf

Theorem	mingeb 11426	Equivalence of and being equal to the minimum of two reals. (Contributed by Jim Kingdon, 14-Oct-2024.)
$/\$ inf

Theorem	2zinfmin 11427	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 11428	Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
$/\$

Theorem	xrmaxiflemcl 11429	Lemma for xrmaxif 11435. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$

Theorem	xrmaxifle 11430	An upper bound for in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)
$/\$

Theorem	xrmaxiflemab 11431	Lemma for xrmaxif 11435. A variation of xrmaxleim 11428- 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 11432	Lemma for xrmaxif 11435. A least upper bound for . (Contributed by Jim Kingdon, 28-Apr-2023.)
$\/$

Theorem	xrmaxiflemcom 11433	Lemma for xrmaxif 11435. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$

Theorem	xrmaxiflemval 11434*	Lemma for xrmaxif 11435. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$ $/\$ $/\$

Theorem	xrmaxif 11435	Maximum of two extended reals in terms of expressions. (Contributed by Jim Kingdon, 26-Apr-2023.)
$/\$

Theorem	xrmaxcl 11436	The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.)
$/\$

Theorem	xrmax1sup 11437	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 11438	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 11439	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 11440	The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.)
$/\$ $/\$ $/\$ $\/$

Theorem	xrltmaxsup 11441	The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.)
$/\$ $/\$ $\/$

Theorem	xrmaxltsup 11442	Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)
$/\$ $/\$ $/\$

Theorem	xrmaxlesup 11443	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 11444	Lemma for xrmaxadd 11445. The case where is real. (Contributed by Jim Kingdon, 11-May-2023.)
$/\$ $/\$

Theorem	xrmaxadd 11445	Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)
$/\$ $/\$

4.8.8 The minimum of two extended reals

Theorem	xrnegiso 11446	Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
$/\$

Theorem	infxrnegsupex 11447*	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 11448	Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)


Theorem	xrminmax 11449	Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.)
$/\$ inf

Theorem	xrmincl 11450	The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.)
$/\$ inf

Theorem	xrmin1inf 11451	The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.)
$/\$ inf

Theorem	xrmin2inf 11452	The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.)
$/\$ inf

Theorem	xrmineqinf 11453	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 11454	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 11455	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 11456	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 11457	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 11458	The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.)
$/\$ inf

Theorem	xrminadd 11459	Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)
$/\$ $/\$ inf inf

Theorem	xrbdtri 11460	Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
$/\$ $/\$ $/\$ $/\$ $/\$ inf inf inf

Theorem	iooinsup 11461	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 11462	Extend class notation with convergence relation for limits.


Definition	df-clim 11463*	Define the limit relation for complex number sequences. See clim 11465 for its relational expression. (Contributed by NM, 28-Aug-2005.)
$/\$ $/\$

Theorem	climrel 11464	The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)


Theorem	clim 11465*	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 11466	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 11467*	Express the predicate: The limit of complex number sequence is , or converges to , with more general quantifier restrictions than clim 11465. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$

Theorem	clim2c 11468*	Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	clim0 11469*	Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	clim0c 11470*	Express the predicate converges to . (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	climi 11471*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$

Theorem	climi2 11472*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climi0 11473*	Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climconst 11474*	An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climconst2 11475	A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	climz 11476	The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)


Theorem	climuni 11477	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 11478	The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)


Theorem	climdm 11479	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 11480*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)


Theorem	climreu 11481*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)


Theorem	climmo 11482*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)


Theorem	climeq 11483*	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 11484*	Exhibit a function with the same convergence properties as the not-quite-function . (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	2clim 11485*	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 11486	A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)
$/\$

Theorem	climres 11487	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 11488	A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
$/\$

Theorem	serclim0 11489	The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)


Theorem	climshft2 11490*	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 11491*	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 11492*	Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	climcn2 11493*	Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	addcn2 11494*	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 14906 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)
$/\$ $/\$ $/\$

Theorem	subcn2 11495*	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 11496*	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 11497*	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 2196. (Contributed by Mario Carneiro, 9-Feb-2014.) Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
inf $/\$ # $/\$ #

Theorem	cn1lem 11498*	A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$ $/\$

Theorem	abscn2 11499*	The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$

Theorem	cjcn2 11500*	The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
$/\$

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