P. 104 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 10301-10400 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	zexpcl 10301	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	qexpcl 10302	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
$/\$

Theorem	reexpcl 10303	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
$/\$

Theorem	expcl 10304	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
$/\$

Theorem	rpexpcl 10305	Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
$/\$

Theorem	reexpclzap 10306	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$

Theorem	qexpclz 10307	Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
$/\$ $/\$

Theorem	m1expcl2 10308	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)


Theorem	m1expcl 10309	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)


Theorem	expclzaplem 10310*	Closure law for integer exponentiation. Lemma for expclzap 10311 and expap0i 10318. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$ #

Theorem	expclzap 10311	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
$/\$ # $/\$

Theorem	nn0expcli 10312	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	nn0sqcl 10313	The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)


Theorem	expm1t 10314	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
$/\$

Theorem	1exp 10315	Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)


Theorem	expap0 10316	Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 10317 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 10317	Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)
$/\$

Theorem	expap0i 10318	Integer exponentiation is apart from zero if its mantissa is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ #

Theorem	expgt0 10319	Nonnegative integer exponentiation with a positive mantissa is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$

Theorem	expnegzap 10320	Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
$/\$ # $/\$

Theorem	0exp 10321	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)


Theorem	expge0 10322	Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$

Theorem	expge1 10323	Nonnegative integer exponentiation with a mantissa greater than or equal to 1 is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$

Theorem	expgt1 10324	Positive integer exponentiation with a mantissa greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$

Theorem	mulexp 10325	Positive 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 10326	Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$ # $/\$

Theorem	exprecap 10327	Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$

Theorem	expadd 10328	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 10329	Lemma for expaddzap 10330. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$ $/\$

Theorem	expaddzap 10330	Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expmul 10331	Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)
$/\$ $/\$

Theorem	expmulzap 10332	Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	m1expeven 10333	Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)


Theorem	expsubap 10334	Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$ $/\$

Theorem	expp1zap 10335	Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$

Theorem	expm1ap 10336	Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ # $/\$

Theorem	expdivap 10337	Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ $/\$ # $/\$

Theorem	ltexp2a 10338	Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	leexp2a 10339	Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
$/\$ $/\$

Theorem	leexp2r 10340	Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	leexp1a 10341	Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
$/\$ $/\$ $/\$ $/\$

Theorem	exple1 10342	Nonnegative integer exponentiation with a mantissa between 0 and 1 inclusive is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007.) (Revised by Mario Carneiro, 5-Jun-2014.)
$/\$ $/\$ $/\$

Theorem	expubnd 10343	An upper bound on when . (Contributed by NM, 19-Dec-2005.)
$/\$ $/\$

Theorem	sqval 10344	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)


Theorem	sqneg 10345	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)


Theorem	sqsubswap 10346	Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
$/\$

Theorem	sqcl 10347	Closure of square. (Contributed by NM, 10-Aug-1999.)


Theorem	sqmul 10348	Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
$/\$

Theorem	sqeq0 10349	A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)


Theorem	sqdivap 10350	Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ $/\$ #

Theorem	sqne0 10351	A number is nonzero iff its square is nonzero. See also sqap0 10352 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)


Theorem	sqap0 10352	A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
# #

Theorem	resqcl 10353	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)


Theorem	sqgt0ap 10354	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
$/\$ #

Theorem	nnsqcl 10355	The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)


Theorem	zsqcl 10356	Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)


Theorem	qsqcl 10357	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)


Theorem	sq11 10358	The square function is one-to-one for nonnegative reals. Also see sq11ap 10451 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 10359	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)
$/\$ $/\$ $/\$

Theorem	le2sq 10360	The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)
$/\$ $/\$ $/\$

Theorem	le2sq2 10361	The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)
$/\$ $/\$ $/\$

Theorem	sqge0 10362	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)


Theorem	zsqcl2 10363	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 10364	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 10365	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)


Theorem	sqcli 10366	Closure of square. (Contributed by NM, 2-Aug-1999.)


Theorem	sqeq0i 10367	A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)


Theorem	sqmuli 10368	Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)


Theorem	sqdivapi 10369	Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theorem	resqcli 10370	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)


Theorem	sqgt0api 10371	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
#

Theorem	sqge0i 10372	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)


Theorem	lt2sqi 10373	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)
$/\$

Theorem	le2sqi 10374	The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)
$/\$

Theorem	sq11i 10375	The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)
$/\$

Theorem	sq0 10376	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)


Theorem	sq0i 10377	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)


Theorem	sq0id 10378	If a number is zero, its square is zero. Deduction form of sq0i 10377. Converse of sqeq0d 10416. (Contributed by David Moews, 28-Feb-2017.)


Theorem	sq1 10379	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)


Theorem	neg1sqe1 10380	squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)


Theorem	sq2 10381	The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)


Theorem	sq3 10382	The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)


Theorem	sq4e2t8 10383	The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)


Theorem	cu2 10384	The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)


Theorem	irec 10385	The reciprocal of . (Contributed by NM, 11-Oct-1999.)


Theorem	i2 10386	squared. (Contributed by NM, 6-May-1999.)


Theorem	i3 10387	cubed. (Contributed by NM, 31-Jan-2007.)


Theorem	i4 10388	to the fourth power. (Contributed by NM, 31-Jan-2007.)


Theorem	nnlesq 10389	A positive integer is less than or equal to its square. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)


Theorem	iexpcyc 10390	Taking to the -th power is the same as using the -th power instead, by i4 10388. (Contributed by Mario Carneiro, 7-Jul-2014.)


Theorem	expnass 10391	A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)


Theorem	subsq 10392	Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
$/\$

Theorem	subsq2 10393	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 10394	The square of a binomial. (Contributed by NM, 11-Aug-1999.)


Theorem	subsqi 10395	Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)


Theorem	binom2 10396	The square of a binomial. (Contributed by FL, 10-Dec-2006.)
$/\$

Theorem	binom21 10397	Special case of binom2 10396 where . (Contributed by Scott Fenton, 11-May-2014.)


Theorem	binom2sub 10398	Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)
$/\$

Theorem	binom2sub1 10399	Special case of binom2sub 10398 where . (Contributed by AV, 2-Aug-2021.)


Theorem	binom2subi 10400	Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)

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