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**Theorem List for Intuitionistic Logic Explorer -** 11001-11100 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	0exp0e1 11001	The zeroth power of zero equals one. See comment of exp0 11000. (Contributed by David A. Wheeler, 8-Dec-2018.)


Theorem	exp1 11002	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 11003	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 11004	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ # $/\$

Theorem	expineg2 11005	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expn1ap0 11006	A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ #

Theorem	expcllem 11007*	Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
$/\$ $/\$

Theorem	expcl2lemap 11008*	Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ $/\$ # $/\$ # $/\$

Theorem	nnexpcl 11009	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	nn0expcl 11010	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
$/\$

Theorem	zexpcl 11011	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	qexpcl 11012	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	reexpcl 11013	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
$/\$

Theorem	expcl 11014	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
$/\$

Theorem	rpexpcl 11015	Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
$/\$

Theorem	reexpclzap 11016	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$

Theorem	qexpclz 11017	Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
$/\$ $/\$

Theorem	m1expcl2 11018	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)


Theorem	m1expcl 11019	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)


Theorem	expclzaplem 11020*	Closure law for integer exponentiation. Lemma for expclzap 11021 and expap0i 11028. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$ #

Theorem	expclzap 11021	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$

Theorem	nn0expcli 11022	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	nn0sqcl 11023	The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)


Theorem	expm1t 11024	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
$/\$

Theorem	1exp 11025	Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)


Theorem	expap0 11026	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 11027 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 11027	Positive integer exponentiation is 0 iff its base is 0. (Contributed by NM, 23-Feb-2005.)
$/\$

Theorem	expap0i 11028	Integer exponentiation is apart from zero if its base is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ #

Theorem	expgt0 11029	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 11030	Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
$/\$ # $/\$

Theorem	0exp 11031	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)


Theorem	expge0 11032	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 11033	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 11034	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 11035	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 11036	Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$ # $/\$

Theorem	exprecap 11037	Integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$

Theorem	expadd 11038	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 11039	Lemma for expaddzap 11040. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$ $/\$

Theorem	expaddzap 11040	Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expmul 11041	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 11042	Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	m1expeven 11043	Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)


Theorem	expsubap 11044	Exponent subtraction law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expp1zap 11045	Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$

Theorem	expm1ap 11046	Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$

Theorem	expdivap 11047	Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ $/\$ # $/\$

Theorem	ltexp2a 11048	Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	leexp2a 11049	Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
$/\$ $/\$

Theorem	leexp2r 11050	Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	leexp1a 11051	Weak base ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
$/\$ $/\$ $/\$ $/\$

Theorem	exple1 11052	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 11053	An upper bound on when . (Contributed by NM, 19-Dec-2005.)
$/\$ $/\$

Theorem	sqval 11054	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)


Theorem	sqneg 11055	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)


Theorem	sqsubswap 11056	Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
$/\$

Theorem	sqcl 11057	Closure of square. (Contributed by NM, 10-Aug-1999.)


Theorem	sqmul 11058	Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
$/\$

Theorem	sqeq0 11059	A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)


Theorem	sqdivap 11060	Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ $/\$ #

Theorem	sqdividap 11061	The square of a complex number apart from zero divided by itself equals that number. (Contributed by AV, 19-Jul-2021.)
$/\$ #

Theorem	sqne0 11062	A number is nonzero iff its square is nonzero. See also sqap0 11063 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)


Theorem	sqap0 11063	A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
# #

Theorem	resqcl 11064	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)


Theorem	sqgt0ap 11065	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ #

Theorem	nnsqcl 11066	The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)


Theorem	zsqcl 11067	Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)


Theorem	qsqcl 11068	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)


Theorem	sq11 11069	The square function is one-to-one for nonnegative reals. Also see sq11ap 11165 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 11070	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)
$/\$ $/\$ $/\$

Theorem	le2sq 11071	The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)
$/\$ $/\$ $/\$

Theorem	le2sq2 11072	The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)
$/\$ $/\$ $/\$

Theorem	sqge0 11073	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)


Theorem	zsqcl2 11074	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 11075	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 11076	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)


Theorem	sqcli 11077	Closure of square. (Contributed by NM, 2-Aug-1999.)


Theorem	sqeq0i 11078	A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)


Theorem	sqmuli 11079	Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)


Theorem	sqdivapi 11080	Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theorem	resqcli 11081	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)


Theorem	sqgt0api 11082	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theorem	sqge0i 11083	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)


Theorem	lt2sqi 11084	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)
$/\$

Theorem	le2sqi 11085	The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)
$/\$

Theorem	sq11i 11086	The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)
$/\$

Theorem	sq0 11087	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)


Theorem	sq0i 11088	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)


Theorem	sq0id 11089	If a number is zero, its square is zero. Deduction form of sq0i 11088. Converse of sqeq0d 11130. (Contributed by David Moews, 28-Feb-2017.)


Theorem	sq1 11090	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)


Theorem	neg1sqe1 11091	squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)


Theorem	sq2 11092	The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)


Theorem	sq3 11093	The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)


Theorem	sq4e2t8 11094	The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)


Theorem	cu2 11095	The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)


Theorem	irec 11096	The reciprocal of . (Contributed by NM, 11-Oct-1999.)


Theorem	i2 11097	squared. (Contributed by NM, 6-May-1999.)


Theorem	i3 11098	cubed. (Contributed by NM, 31-Jan-2007.)


Theorem	i4 11099	to the fourth power. (Contributed by NM, 31-Jan-2007.)


Theorem	nnlesq 11100	A positive integer is less than or equal to its square. For general integers, see zzlesq 11166. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)

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