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	exp3vallem 10801	Lemma for exp3val 10802. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingdon, 7-Jun-2020.)
# #

Theorem	exp3val 10802	Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
$/\$ $/\$ # $\/$

Theorem	expnnval 10803	Value of exponentiation to positive integer powers. (Contributed by Mario Carneiro, 4-Jun-2014.)
$/\$

Theorem	exp0 10804	Value of a complex number raised to the 0th power. Note that under our definition, (0exp0e1 10805) , following the convention used by Gleason. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)


Theorem	0exp0e1 10805	The zeroth power of zero equals one. See comment of exp0 10804. (Contributed by David A. Wheeler, 8-Dec-2018.)


Theorem	exp1 10806	Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)


Theorem	expp1 10807	Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)
$/\$

Theorem	expnegap0 10808	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ # $/\$

Theorem	expineg2 10809	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expn1ap0 10810	A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ #

Theorem	expcllem 10811*	Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
$/\$ $/\$

Theorem	expcl2lemap 10812*	Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ $/\$ # $/\$ # $/\$

Theorem	nnexpcl 10813	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	nn0expcl 10814	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
$/\$

Theorem	zexpcl 10815	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	qexpcl 10816	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	reexpcl 10817	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
$/\$

Theorem	expcl 10818	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
$/\$

Theorem	rpexpcl 10819	Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
$/\$

Theorem	reexpclzap 10820	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$

Theorem	qexpclz 10821	Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
$/\$ $/\$

Theorem	m1expcl2 10822	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)


Theorem	m1expcl 10823	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)


Theorem	expclzaplem 10824*	Closure law for integer exponentiation. Lemma for expclzap 10825 and expap0i 10832. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$ #

Theorem	expclzap 10825	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$

Theorem	nn0expcli 10826	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	nn0sqcl 10827	The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)


Theorem	expm1t 10828	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
$/\$

Theorem	1exp 10829	Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)


Theorem	expap0 10830	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 10831 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 10831	Positive integer exponentiation is 0 iff its base is 0. (Contributed by NM, 23-Feb-2005.)
$/\$

Theorem	expap0i 10832	Integer exponentiation is apart from zero if its base is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ #

Theorem	expgt0 10833	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 10834	Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
$/\$ # $/\$

Theorem	0exp 10835	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)


Theorem	expge0 10836	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 10837	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 10838	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 10839	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 10840	Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$ # $/\$

Theorem	exprecap 10841	Integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$

Theorem	expadd 10842	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 10843	Lemma for expaddzap 10844. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$ $/\$

Theorem	expaddzap 10844	Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expmul 10845	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 10846	Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	m1expeven 10847	Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)


Theorem	expsubap 10848	Exponent subtraction law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expp1zap 10849	Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$

Theorem	expm1ap 10850	Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$

Theorem	expdivap 10851	Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ $/\$ # $/\$

Theorem	ltexp2a 10852	Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	leexp2a 10853	Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
$/\$ $/\$

Theorem	leexp2r 10854	Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	leexp1a 10855	Weak base ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
$/\$ $/\$ $/\$ $/\$

Theorem	exple1 10856	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 10857	An upper bound on when . (Contributed by NM, 19-Dec-2005.)
$/\$ $/\$

Theorem	sqval 10858	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)


Theorem	sqneg 10859	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)


Theorem	sqsubswap 10860	Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
$/\$

Theorem	sqcl 10861	Closure of square. (Contributed by NM, 10-Aug-1999.)


Theorem	sqmul 10862	Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
$/\$

Theorem	sqeq0 10863	A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)


Theorem	sqdivap 10864	Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ $/\$ #

Theorem	sqdividap 10865	The square of a complex number apart from zero divided by itself equals that number. (Contributed by AV, 19-Jul-2021.)
$/\$ #

Theorem	sqne0 10866	A number is nonzero iff its square is nonzero. See also sqap0 10867 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)


Theorem	sqap0 10867	A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
# #

Theorem	resqcl 10868	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)


Theorem	sqgt0ap 10869	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ #

Theorem	nnsqcl 10870	The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)


Theorem	zsqcl 10871	Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)


Theorem	qsqcl 10872	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)


Theorem	sq11 10873	The square function is one-to-one for nonnegative reals. Also see sq11ap 10968 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 10874	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)
$/\$ $/\$ $/\$

Theorem	le2sq 10875	The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)
$/\$ $/\$ $/\$

Theorem	le2sq2 10876	The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)
$/\$ $/\$ $/\$

Theorem	sqge0 10877	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)


Theorem	zsqcl2 10878	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 10879	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 10880	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)


Theorem	sqcli 10881	Closure of square. (Contributed by NM, 2-Aug-1999.)


Theorem	sqeq0i 10882	A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)


Theorem	sqmuli 10883	Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)


Theorem	sqdivapi 10884	Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theorem	resqcli 10885	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)


Theorem	sqgt0api 10886	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theorem	sqge0i 10887	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)


Theorem	lt2sqi 10888	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)
$/\$

Theorem	le2sqi 10889	The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)
$/\$

Theorem	sq11i 10890	The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)
$/\$

Theorem	sq0 10891	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)


Theorem	sq0i 10892	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)


Theorem	sq0id 10893	If a number is zero, its square is zero. Deduction form of sq0i 10892. Converse of sqeq0d 10933. (Contributed by David Moews, 28-Feb-2017.)


Theorem	sq1 10894	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)


Theorem	neg1sqe1 10895	squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)


Theorem	sq2 10896	The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)


Theorem	sq3 10897	The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)


Theorem	sq4e2t8 10898	The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)


Theorem	cu2 10899	The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)


Theorem	irec 10900	The reciprocal of . (Contributed by NM, 11-Oct-1999.)

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