P. 110 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fser0const 10901*	Simplifying an expression which turns out just to be a constant zero sequence. (Contributed by Jim Kingdon, 16-Sep-2022.)
|- Z = ( ZZ>= ` M )   =>    |- ( N e. Z -> ( n e. Z |-> if ( n <_ N , ( ( Z X. { 0 } ) ` n ) , 0 ) ) = ( Z X. { 0 } ) )
Theorem	ser3ge0 10902*	A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> 0 <_ ( F ` k ) )   =>    |- ( ph -> 0 <_ ( seq M ( + , F ) ` N ) )
Theorem	ser3le 10903*	Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. RR )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) <_ ( G ` k ) )   =>    |- ( ph -> ( seq M ( + , F ) ` N ) <_ ( seq M ( + , G ) ` N ) )
4.6.6  Integer powers
Syntax	cexp 10904	Extend class notation to include exponentiation of a complex number to an integer power.
class ^
Definition	df-exp 10905*	Define exponentiation to nonnegative integer powers. For example, ( 5 ^ 2 ) = 2 5 (see ex-exp 16512). This definition is not meant to be used directly; instead, exp0 10909 and expp1 10912 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that 0 ^ 0 = 1 per the convention of Definition 10-4.1 of [Gleason] p. 134 (see 0exp0e1 10910). 4-Jun-2014: The definition was extended to include negative integer exponents. For example, ( -u 3 ^ -u 2 ) = ( 1 / 9 ) (ex-exp 16512). The case x = 0 , y < 0 gives the value ( 1 / 0 ), so we will avoid this case in our theorems. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.)
|- ^ = ( x e. CC , y e. ZZ |-> if ( y = 0 , 1 , if ( 0 < y , ( seq 1 ( x. , ( NN X. { x } ) ) ` y ) , ( 1 / ( seq 1 ( x. , ( NN X. { x } ) ) ` -u y ) ) ) ) )
Theorem	exp3vallem 10906	Lemma for exp3val 10907. 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.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) # 0 )
Theorem	exp3val 10907	Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
|- ( ( A e. CC /\ N e. ZZ /\ ( A # 0 \/ 0 <_ N ) ) -> ( A ^ N ) = if ( N = 0 , 1 , if ( 0 < N , ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u N ) ) ) ) )
Theorem	expnnval 10908	Value of exponentiation to positive integer powers. (Contributed by Mario Carneiro, 4-Jun-2014.)
|- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
Theorem	exp0 10909	Value of a complex number raised to the 0th power. Note that under our definition, 0 ^ 0 = 1 (0exp0e1 10910) , 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.)
|- ( A e. CC -> ( A ^ 0 ) = 1 )
Theorem	0exp0e1 10910	The zeroth power of zero equals one. See comment of exp0 10909. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 0 ^ 0 ) = 1
Theorem	exp1 10911	Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)
|- ( A e. CC -> ( A ^ 1 ) = A )
Theorem	expp1 10912	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.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
Theorem	expnegap0 10913	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
|- ( ( A e. CC /\ A # 0 /\ N e. NN0 ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
Theorem	expineg2 10914	Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
Theorem	expn1ap0 10915	A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
|- ( ( A e. CC /\ A # 0 ) -> ( A ^ -u 1 ) = ( 1 / A ) )
Theorem	expcllem 10916*	Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
|- F C_ CC   &    |- ( ( x e. F /\ y e. F ) -> ( x x. y ) e. F )   &    |- 1 e. F   =>    |- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F )
Theorem	expcl2lemap 10917*	Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
|- F C_ CC   &    |- ( ( x e. F /\ y e. F ) -> ( x x. y ) e. F )   &    |- 1 e. F   &    |- ( ( x e. F /\ x # 0 ) -> ( 1 / x ) e. F )   =>    |- ( ( A e. F /\ A # 0 /\ B e. ZZ ) -> ( A ^ B ) e. F )
Theorem	nnexpcl 10918	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. NN /\ N e. NN0 ) -> ( A ^ N ) e. NN )
Theorem	nn0expcl 10919	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. NN0 /\ N e. NN0 ) -> ( A ^ N ) e. NN0 )
Theorem	zexpcl 10920	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. ZZ /\ N e. NN0 ) -> ( A ^ N ) e. ZZ )
Theorem	qexpcl 10921	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. QQ /\ N e. NN0 ) -> ( A ^ N ) e. QQ )
Theorem	reexpcl 10922	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. RR /\ N e. NN0 ) -> ( A ^ N ) e. RR )
Theorem	expcl 10923	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
Theorem	rpexpcl 10924	Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
|- ( ( A e. RR+ /\ N e. ZZ ) -> ( A ^ N ) e. RR+ )
Theorem	reexpclzap 10925	Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
|- ( ( A e. RR /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. RR )
Theorem	qexpclz 10926	Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
|- ( ( A e. QQ /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) e. QQ )
Theorem	m1expcl2 10927	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } )
Theorem	m1expcl 10928	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( N e. ZZ -> ( -u 1 ^ N ) e. ZZ )
Theorem	expclzaplem 10929*	Closure law for integer exponentiation. Lemma for expclzap 10930 and expap0i 10937. (Contributed by Jim Kingdon, 9-Jun-2020.)
|- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. { z e. CC | z # 0 } )
Theorem	expclzap 10930	Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
|- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. CC )
Theorem	nn0expcli 10931	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
|- A e. NN0   &    |- N e. NN0   =>    |- ( A ^ N ) e. NN0
Theorem	nn0sqcl 10932	The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
|- ( A e. NN0 -> ( A ^ 2 ) e. NN0 )
Theorem	expm1t 10933	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
|- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( ( A ^ ( N - 1 ) ) x. A ) )
Theorem	1exp 10934	Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
|- ( N e. ZZ -> ( 1 ^ N ) = 1 )
Theorem	expap0 10935	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 10936 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.)
|- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) # 0 <-> A # 0 ) )
Theorem	expeq0 10936	Positive integer exponentiation is 0 iff its base is 0. (Contributed by NM, 23-Feb-2005.)
|- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) = 0 <-> A = 0 ) )
Theorem	expap0i 10937	Integer exponentiation is apart from zero if its base is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
|- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) # 0 )
Theorem	expgt0 10938	A positive real raised to an integer power is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
|- ( ( A e. RR /\ N e. ZZ /\ 0 < A ) -> 0 < ( A ^ N ) )
Theorem	expnegzap 10939	Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
|- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
Theorem	0exp 10940	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
|- ( N e. NN -> ( 0 ^ N ) = 0 )
Theorem	expge0 10941	A nonnegative real raised to a nonnegative integer is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
|- ( ( A e. RR /\ N e. NN0 /\ 0 <_ A ) -> 0 <_ ( A ^ N ) )
Theorem	expge1 10942	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.)
|- ( ( A e. RR /\ N e. NN0 /\ 1 <_ A ) -> 1 <_ ( A ^ N ) )
Theorem	expgt1 10943	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.)
|- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < ( A ^ N ) )
Theorem	mulexp 10944	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.)
|- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
Theorem	mulexpzap 10945	Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) /\ N e. ZZ ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) )
Theorem	exprecap 10946	Integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
|- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( ( 1 / A ) ^ N ) = ( 1 / ( A ^ N ) ) )
Theorem	expadd 10947	Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)
|- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
Theorem	expaddzaplem 10948	Lemma for expaddzap 10949. (Contributed by Jim Kingdon, 10-Jun-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. RR /\ -u M e. NN ) /\ N e. NN0 ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
Theorem	expaddzap 10949	Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M + N ) ) = ( ( A ^ M ) x. ( A ^ N ) ) )
Theorem	expmul 10950	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.)
|- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) )
Theorem	expmulzap 10951	Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) )
Theorem	m1expeven 10952	Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
|- ( N e. ZZ -> ( -u 1 ^ ( 2 x. N ) ) = 1 )
Theorem	expsubap 10953	Exponent subtraction law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M - N ) ) = ( ( A ^ M ) / ( A ^ N ) ) )
Theorem	expp1zap 10954	Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
|- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
Theorem	expm1ap 10955	Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
|- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ ( N - 1 ) ) = ( ( A ^ N ) / A ) )
Theorem	expdivap 10956	Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
|- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ N e. NN0 ) -> ( ( A / B ) ^ N ) = ( ( A ^ N ) / ( B ^ N ) ) )
Theorem	ltexp2a 10957	Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
|- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) < ( A ^ N ) )
Theorem	leexp2a 10958	Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
|- ( ( A e. RR /\ 1 <_ A /\ N e. ( ZZ>= ` M ) ) -> ( A ^ M ) <_ ( A ^ N ) )
Theorem	leexp2r 10959	Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
|- ( ( ( A e. RR /\ M e. NN0 /\ N e. ( ZZ>= ` M ) ) /\ ( 0 <_ A /\ A <_ 1 ) ) -> ( A ^ N ) <_ ( A ^ M ) )
Theorem	leexp1a 10960	Weak base ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
|- ( ( ( A e. RR /\ B e. RR /\ N e. NN0 ) /\ ( 0 <_ A /\ A <_ B ) ) -> ( A ^ N ) <_ ( B ^ N ) )
Theorem	exple1 10961	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.)
|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> ( A ^ N ) <_ 1 )
Theorem	expubnd 10962	An upper bound on A ^ N when 2 <_ A. (Contributed by NM, 19-Dec-2005.)
|- ( ( A e. RR /\ N e. NN0 /\ 2 <_ A ) -> ( A ^ N ) <_ ( ( 2 ^ N ) x. ( ( A - 1 ) ^ N ) ) )
Theorem	sqval 10963	Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
|- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )
Theorem	sqneg 10964	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
|- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
Theorem	sqsubswap 10965	Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) ^ 2 ) = ( ( B - A ) ^ 2 ) )
Theorem	sqcl 10966	Closure of square. (Contributed by NM, 10-Aug-1999.)
|- ( A e. CC -> ( A ^ 2 ) e. CC )
Theorem	sqmul 10967	Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) ^ 2 ) = ( ( A ^ 2 ) x. ( B ^ 2 ) ) )
Theorem	sqeq0 10968	A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
|- ( A e. CC -> ( ( A ^ 2 ) = 0 <-> A = 0 ) )
Theorem	sqdivap 10969	Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
Theorem	sqdividap 10970	The square of a complex number apart from zero divided by itself equals that number. (Contributed by AV, 19-Jul-2021.)
|- ( ( A e. CC /\ A # 0 ) -> ( ( A ^ 2 ) / A ) = A )
Theorem	sqne0 10971	A number is nonzero iff its square is nonzero. See also sqap0 10972 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)
|- ( A e. CC -> ( ( A ^ 2 ) =/= 0 <-> A =/= 0 ) )
Theorem	sqap0 10972	A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( A e. CC -> ( ( A ^ 2 ) # 0 <-> A # 0 ) )
Theorem	resqcl 10973	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
|- ( A e. RR -> ( A ^ 2 ) e. RR )
Theorem	sqgt0ap 10974	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
|- ( ( A e. RR /\ A # 0 ) -> 0 < ( A ^ 2 ) )
Theorem	nnsqcl 10975	The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( A e. NN -> ( A ^ 2 ) e. NN )
Theorem	zsqcl 10976	Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
|- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
Theorem	qsqcl 10977	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> ( A ^ 2 ) e. QQ )
Theorem	sq11 10978	The square function is one-to-one for nonnegative reals. Also see sq11ap 11073 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.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> A = B ) )
Theorem	lt2sq 10979	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A < B <-> ( A ^ 2 ) < ( B ^ 2 ) ) )
Theorem	le2sq 10980	The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A <_ B <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) )
Theorem	le2sq2 10981	The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ A <_ B ) ) -> ( A ^ 2 ) <_ ( B ^ 2 ) )
Theorem	sqge0 10982	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
|- ( A e. RR -> 0 <_ ( A ^ 2 ) )
Theorem	zsqcl2 10983	The square of an integer is a nonnegative integer. (Contributed by Mario Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. ZZ -> ( A ^ 2 ) e. NN0 )
Theorem	sumsqeq0 10984	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.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( A = 0 /\ B = 0 ) <-> ( ( A ^ 2 ) + ( B ^ 2 ) ) = 0 ) )
Theorem	sqvali 10985	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
|- A e. CC   =>    |- ( A ^ 2 ) = ( A x. A )
Theorem	sqcli 10986	Closure of square. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( A ^ 2 ) e. CC
Theorem	sqeq0i 10987	A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
|- A e. CC   =>    |- ( ( A ^ 2 ) = 0 <-> A = 0 )
Theorem	sqmuli 10988	Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( ( A x. B ) ^ 2 ) = ( ( A ^ 2 ) x. ( B ^ 2 ) )
Theorem	sqdivapi 10989	Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
|- A e. CC   &    |- B e. CC   &    |- B # 0   =>    |- ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) )
Theorem	resqcli 10990	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
|- A e. RR   =>    |- ( A ^ 2 ) e. RR
Theorem	sqgt0api 10991	The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
|- A e. RR   =>    |- ( A # 0 -> 0 < ( A ^ 2 ) )
Theorem	sqge0i 10992	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
|- A e. RR   =>    |- 0 <_ ( A ^ 2 )
Theorem	lt2sqi 10993	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> ( A < B <-> ( A ^ 2 ) < ( B ^ 2 ) ) )
Theorem	le2sqi 10994	The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> ( A <_ B <-> ( A ^ 2 ) <_ ( B ^ 2 ) ) )
Theorem	sq11i 10995	The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> A = B ) )
Theorem	sq0 10996	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
|- ( 0 ^ 2 ) = 0
Theorem	sq0i 10997	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
|- ( A = 0 -> ( A ^ 2 ) = 0 )
Theorem	sq0id 10998	If a number is zero, its square is zero. Deduction form of sq0i 10997. Converse of sqeq0d 11038. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A = 0 )   =>    |- ( ph -> ( A ^ 2 ) = 0 )
Theorem	sq1 10999	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
|- ( 1 ^ 2 ) = 1
Theorem	neg1sqe1 11000	-u 1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( -u 1 ^ 2 ) = 1