P. 111 of Theorem List - Intuitionistic Logic Explorer

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

Theorem	expn1ap0 11001	A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ #

Theorem	expcllem 11002*	Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
$/\$ $/\$

Theorem	expcl2lemap 11003*	Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
$/\$ $/\$ # $/\$ # $/\$

Theorem	nnexpcl 11004	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	nn0expcl 11005	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
$/\$

Theorem	zexpcl 11006	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	qexpcl 11007	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	reexpcl 11008	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
$/\$

Theorem	expcl 11009	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
$/\$

Theorem	rpexpcl 11010	Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
$/\$

Theorem	reexpclzap 11011	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$

Theorem	qexpclz 11012	Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
$/\$ $/\$

Theorem	m1expcl2 11013	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)


Theorem	m1expcl 11014	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)


Theorem	expclzaplem 11015*	Closure law for integer exponentiation. Lemma for expclzap 11016 and expap0i 11023. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$ #

Theorem	expclzap 11016	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$

Theorem	nn0expcli 11017	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	nn0sqcl 11018	The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)


Theorem	expm1t 11019	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
$/\$

Theorem	1exp 11020	Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)


Theorem	expap0 11021	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 11022 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 11022	Positive integer exponentiation is 0 iff its base is 0. (Contributed by NM, 23-Feb-2005.)
$/\$

Theorem	expap0i 11023	Integer exponentiation is apart from zero if its base is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ #

Theorem	expgt0 11024	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 11025	Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
$/\$ # $/\$

Theorem	0exp 11026	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)


Theorem	expge0 11027	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 11028	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 11029	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 11030	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 11031	Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$ # $/\$

Theorem	exprecap 11032	Integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$

Theorem	expadd 11033	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 11034	Lemma for expaddzap 11035. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$ $/\$

Theorem	expaddzap 11035	Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expmul 11036	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 11037	Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	m1expeven 11038	Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)


Theorem	expsubap 11039	Exponent subtraction law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expp1zap 11040	Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$

Theorem	expm1ap 11041	Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$

Theorem	expdivap 11042	Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ $/\$ # $/\$

Theorem	ltexp2a 11043	Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	leexp2a 11044	Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
$/\$ $/\$

Theorem	leexp2r 11045	Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	leexp1a 11046	Weak base ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
$/\$ $/\$ $/\$ $/\$

Theorem	exple1 11047	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 11048	An upper bound on when . (Contributed by NM, 19-Dec-2005.)
$/\$ $/\$

Theorem	sqval 11049	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)


Theorem	sqneg 11050	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)


Theorem	sqsubswap 11051	Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
$/\$

Theorem	sqcl 11052	Closure of square. (Contributed by NM, 10-Aug-1999.)


Theorem	sqmul 11053	Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
$/\$

Theorem	sqeq0 11054	A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)


Theorem	sqdivap 11055	Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ $/\$ #

Theorem	sqdividap 11056	The square of a complex number apart from zero divided by itself equals that number. (Contributed by AV, 19-Jul-2021.)
$/\$ #

Theorem	sqne0 11057	A number is nonzero iff its square is nonzero. See also sqap0 11058 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)


Theorem	sqap0 11058	A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
# #

Theorem	resqcl 11059	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)


Theorem	sqgt0ap 11060	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ #

Theorem	nnsqcl 11061	The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)


Theorem	zsqcl 11062	Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)


Theorem	qsqcl 11063	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)


Theorem	sq11 11064	The square function is one-to-one for nonnegative reals. Also see sq11ap 11160 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 11065	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)
$/\$ $/\$ $/\$

Theorem	le2sq 11066	The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)
$/\$ $/\$ $/\$

Theorem	le2sq2 11067	The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)
$/\$ $/\$ $/\$

Theorem	sqge0 11068	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)


Theorem	zsqcl2 11069	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 11070	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 11071	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)


Theorem	sqcli 11072	Closure of square. (Contributed by NM, 2-Aug-1999.)


Theorem	sqeq0i 11073	A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)


Theorem	sqmuli 11074	Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)


Theorem	sqdivapi 11075	Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theorem	resqcli 11076	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)


Theorem	sqgt0api 11077	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theorem	sqge0i 11078	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)


Theorem	lt2sqi 11079	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)
$/\$

Theorem	le2sqi 11080	The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)
$/\$

Theorem	sq11i 11081	The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)
$/\$

Theorem	sq0 11082	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)


Theorem	sq0i 11083	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)


Theorem	sq0id 11084	If a number is zero, its square is zero. Deduction form of sq0i 11083. Converse of sqeq0d 11125. (Contributed by David Moews, 28-Feb-2017.)


Theorem	sq1 11085	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)


Theorem	neg1sqe1 11086	squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)


Theorem	sq2 11087	The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)


Theorem	sq3 11088	The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)


Theorem	sq4e2t8 11089	The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)


Theorem	cu2 11090	The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)


Theorem	irec 11091	The reciprocal of . (Contributed by NM, 11-Oct-1999.)


Theorem	i2 11092	squared. (Contributed by NM, 6-May-1999.)


Theorem	i3 11093	cubed. (Contributed by NM, 31-Jan-2007.)


Theorem	i4 11094	to the fourth power. (Contributed by NM, 31-Jan-2007.)


Theorem	nnlesq 11095	A positive integer is less than or equal to its square. For general integers, see zzlesq 11161. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)


Theorem	iexpcyc 11096	Taking to the -th power is the same as using the -th power instead, by i4 11094. (Contributed by Mario Carneiro, 7-Jul-2014.)


Theorem	expnass 11097	A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)


Theorem	subsq 11098	Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
$/\$

Theorem	subsq2 11099	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 11100	The square of a binomial. (Contributed by NM, 11-Aug-1999.)

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