P. 109 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 10801-10900 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	expn1ap0 10801	A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ #

Theorem	expcllem 10802*	Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
$/\$ $/\$

Theorem	expcl2lemap 10803*	Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ $/\$ # $/\$ # $/\$

Theorem	nnexpcl 10804	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	nn0expcl 10805	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
$/\$

Theorem	zexpcl 10806	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	qexpcl 10807	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	reexpcl 10808	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
$/\$

Theorem	expcl 10809	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
$/\$

Theorem	rpexpcl 10810	Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
$/\$

Theorem	reexpclzap 10811	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$

Theorem	qexpclz 10812	Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
$/\$ $/\$

Theorem	m1expcl2 10813	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)


Theorem	m1expcl 10814	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)


Theorem	expclzaplem 10815*	Closure law for integer exponentiation. Lemma for expclzap 10816 and expap0i 10823. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$ #

Theorem	expclzap 10816	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$

Theorem	nn0expcli 10817	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	nn0sqcl 10818	The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)


Theorem	expm1t 10819	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
$/\$

Theorem	1exp 10820	Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)


Theorem	expap0 10821	Positive integer exponentiation is apart from zero iff its base is apart from zero. That it is easier to prove this first, and then prove expeq0 10822 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." (Remark of [Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # #

Theorem	expeq0 10822	Positive integer exponentiation is 0 iff its base is 0. (Contributed by NM, 23-Feb-2005.)
$/\$

Theorem	expap0i 10823	Integer exponentiation is apart from zero if its base is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ #

Theorem	expgt0 10824	A positive real raised to an integer power is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$

Theorem	expnegzap 10825	Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
$/\$ # $/\$

Theorem	0exp 10826	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)


Theorem	expge0 10827	A nonnegative real raised to a nonnegative integer is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$

Theorem	expge1 10828	A real greater than or equal to 1 raised to a nonnegative integer is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$

Theorem	expgt1 10829	A real greater than 1 raised to a positive integer is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$

Theorem	mulexp 10830	Nonnegative integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.)
$/\$ $/\$

Theorem	mulexpzap 10831	Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$ # $/\$

Theorem	exprecap 10832	Integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$

Theorem	expadd 10833	Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)
$/\$ $/\$

Theorem	expaddzaplem 10834	Lemma for expaddzap 10835. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$ $/\$

Theorem	expaddzap 10835	Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expmul 10836	Product of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)
$/\$ $/\$

Theorem	expmulzap 10837	Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	m1expeven 10838	Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)


Theorem	expsubap 10839	Exponent subtraction law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expp1zap 10840	Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$

Theorem	expm1ap 10841	Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$

Theorem	expdivap 10842	Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ $/\$ # $/\$

Theorem	ltexp2a 10843	Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	leexp2a 10844	Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
$/\$ $/\$

Theorem	leexp2r 10845	Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	leexp1a 10846	Weak base ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
$/\$ $/\$ $/\$ $/\$

Theorem	exple1 10847	A real between 0 and 1 inclusive raised to a nonnegative integer is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007.) (Revised by Mario Carneiro, 5-Jun-2014.)
$/\$ $/\$ $/\$

Theorem	expubnd 10848	An upper bound on when . (Contributed by NM, 19-Dec-2005.)
$/\$ $/\$

Theorem	sqval 10849	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)


Theorem	sqneg 10850	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)


Theorem	sqsubswap 10851	Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
$/\$

Theorem	sqcl 10852	Closure of square. (Contributed by NM, 10-Aug-1999.)


Theorem	sqmul 10853	Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
$/\$

Theorem	sqeq0 10854	A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)


Theorem	sqdivap 10855	Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ $/\$ #

Theorem	sqdividap 10856	The square of a complex number apart from zero divided by itself equals that number. (Contributed by AV, 19-Jul-2021.)
$/\$ #

Theorem	sqne0 10857	A number is nonzero iff its square is nonzero. See also sqap0 10858 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)


Theorem	sqap0 10858	A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
# #

Theorem	resqcl 10859	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)


Theorem	sqgt0ap 10860	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ #

Theorem	nnsqcl 10861	The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)


Theorem	zsqcl 10862	Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)


Theorem	qsqcl 10863	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)


Theorem	sq11 10864	The square function is one-to-one for nonnegative reals. Also see sq11ap 10959 which would easily follow from this given excluded middle, but which for us is proved another way. (Contributed by NM, 8-Apr-2001.) (Proof shortened by Mario Carneiro, 28-May-2016.)
$/\$ $/\$ $/\$

Theorem	lt2sq 10865	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)
$/\$ $/\$ $/\$

Theorem	le2sq 10866	The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)
$/\$ $/\$ $/\$

Theorem	le2sq2 10867	The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)
$/\$ $/\$ $/\$

Theorem	sqge0 10868	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)


Theorem	zsqcl2 10869	The square of an integer is a nonnegative integer. (Contributed by Mario Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.)


Theorem	sumsqeq0 10870	Two real numbers are equal to 0 iff their Euclidean norm is. (Contributed by NM, 29-Apr-2005.) (Revised by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 28-May-2016.)
$/\$ $/\$

Theorem	sqvali 10871	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)


Theorem	sqcli 10872	Closure of square. (Contributed by NM, 2-Aug-1999.)


Theorem	sqeq0i 10873	A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)


Theorem	sqmuli 10874	Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)


Theorem	sqdivapi 10875	Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theorem	resqcli 10876	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)


Theorem	sqgt0api 10877	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theorem	sqge0i 10878	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)


Theorem	lt2sqi 10879	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)
$/\$

Theorem	le2sqi 10880	The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)
$/\$

Theorem	sq11i 10881	The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)
$/\$

Theorem	sq0 10882	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)


Theorem	sq0i 10883	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)


Theorem	sq0id 10884	If a number is zero, its square is zero. Deduction form of sq0i 10883. Converse of sqeq0d 10924. (Contributed by David Moews, 28-Feb-2017.)


Theorem	sq1 10885	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)


Theorem	neg1sqe1 10886	squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)


Theorem	sq2 10887	The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)


Theorem	sq3 10888	The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)


Theorem	sq4e2t8 10889	The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)


Theorem	cu2 10890	The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)


Theorem	irec 10891	The reciprocal of . (Contributed by NM, 11-Oct-1999.)


Theorem	i2 10892	squared. (Contributed by NM, 6-May-1999.)


Theorem	i3 10893	cubed. (Contributed by NM, 31-Jan-2007.)


Theorem	i4 10894	to the fourth power. (Contributed by NM, 31-Jan-2007.)


Theorem	nnlesq 10895	A positive integer is less than or equal to its square. For general integers, see zzlesq 10960. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)


Theorem	iexpcyc 10896	Taking to the -th power is the same as using the -th power instead, by i4 10894. (Contributed by Mario Carneiro, 7-Jul-2014.)


Theorem	expnass 10897	A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)


Theorem	subsq 10898	Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
$/\$

Theorem	subsq2 10899	Express the difference of the squares of two numbers as a polynomial in the difference of the numbers. (Contributed by NM, 21-Feb-2008.)
$/\$

Theorem	binom2i 10900	The square of a binomial. (Contributed by NM, 11-Aug-1999.)

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