P. 108 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10701-10800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rpexpcl 10701	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 10702	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 10703	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 10704	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 10705	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( N e. ZZ -> ( -u 1 ^ N ) e. ZZ )
Theorem	expclzaplem 10706*	Closure law for integer exponentiation. Lemma for expclzap 10707 and expap0i 10714. (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 10707	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 10708	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 10709	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 10710	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 10711	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 10712	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 10713 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 10713	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 10714	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 10715	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 10716	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 10717	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
|- ( N e. NN -> ( 0 ^ N ) = 0 )
Theorem	expge0 10718	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 10719	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 10720	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 10721	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 10722	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 10723	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 10724	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 10725	Lemma for expaddzap 10726. (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 10726	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 10727	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 10728	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 10729	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 10730	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 10731	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 10732	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 10733	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 10734	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 10735	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 10736	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 10737	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 10738	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 10739	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 10740	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 10741	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
|- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
Theorem	sqsubswap 10742	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 10743	Closure of square. (Contributed by NM, 10-Aug-1999.)
|- ( A e. CC -> ( A ^ 2 ) e. CC )
Theorem	sqmul 10744	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 10745	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 10746	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 10747	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 10748	A number is nonzero iff its square is nonzero. See also sqap0 10749 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 10749	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 10750	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
|- ( A e. RR -> ( A ^ 2 ) e. RR )
Theorem	sqgt0ap 10751	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 10752	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 10753	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 10754	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> ( A ^ 2 ) e. QQ )
Theorem	sq11 10755	The square function is one-to-one for nonnegative reals. Also see sq11ap 10850 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 10756	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 10757	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 10758	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 10759	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
|- ( A e. RR -> 0 <_ ( A ^ 2 ) )
Theorem	zsqcl2 10760	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 10761	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 10762	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
|- A e. CC   =>    |- ( A ^ 2 ) = ( A x. A )
Theorem	sqcli 10763	Closure of square. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( A ^ 2 ) e. CC
Theorem	sqeq0i 10764	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 10765	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 10766	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 10767	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
|- A e. RR   =>    |- ( A ^ 2 ) e. RR
Theorem	sqgt0api 10768	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 10769	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
|- A e. RR   =>    |- 0 <_ ( A ^ 2 )
Theorem	lt2sqi 10770	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 10771	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 10772	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 10773	The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
|- ( 0 ^ 2 ) = 0
Theorem	sq0i 10774	If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
|- ( A = 0 -> ( A ^ 2 ) = 0 )
Theorem	sq0id 10775	If a number is zero, its square is zero. Deduction form of sq0i 10774. Converse of sqeq0d 10815. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A = 0 )   =>    |- ( ph -> ( A ^ 2 ) = 0 )
Theorem	sq1 10776	The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
|- ( 1 ^ 2 ) = 1
Theorem	neg1sqe1 10777	-u 1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( -u 1 ^ 2 ) = 1
Theorem	sq2 10778	The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
|- ( 2 ^ 2 ) = 4
Theorem	sq3 10779	The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
|- ( 3 ^ 2 ) = 9
Theorem	sq4e2t8 10780	The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.)
|- ( 4 ^ 2 ) = ( 2 x. 8 )
Theorem	cu2 10781	The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
|- ( 2 ^ 3 ) = 8
Theorem	irec 10782	The reciprocal of _i. (Contributed by NM, 11-Oct-1999.)
|- ( 1 / _i ) = -u _i
Theorem	i2 10783	_i squared. (Contributed by NM, 6-May-1999.)
|- ( _i ^ 2 ) = -u 1
Theorem	i3 10784	_i cubed. (Contributed by NM, 31-Jan-2007.)
|- ( _i ^ 3 ) = -u _i
Theorem	i4 10785	_i to the fourth power. (Contributed by NM, 31-Jan-2007.)
|- ( _i ^ 4 ) = 1
Theorem	nnlesq 10786	A positive integer is less than or equal to its square. For general integers, see zzlesq 10851. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)
|- ( N e. NN -> N <_ ( N ^ 2 ) )
Theorem	iexpcyc 10787	Taking _i to the K-th power is the same as using the K mod 4 -th power instead, by i4 10785. (Contributed by Mario Carneiro, 7-Jul-2014.)
|- ( K e. ZZ -> ( _i ^ ( K mod 4 ) ) = ( _i ^ K ) )
Theorem	expnass 10788	A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)
|- ( ( 3 ^ 3 ) ^ 3 ) < ( 3 ^ ( 3 ^ 3 ) )
Theorem	subsq 10789	Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( A + B ) x. ( A - B ) ) )
Theorem	subsq2 10790	Express the difference of the squares of two numbers as a polynomial in the difference of the numbers. (Contributed by NM, 21-Feb-2008.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( ( A - B ) ^ 2 ) + ( ( 2 x. B ) x. ( A - B ) ) ) )
Theorem	binom2i 10791	The square of a binomial. (Contributed by NM, 11-Aug-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) )
Theorem	subsqi 10792	Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)
|- A e. CC   &    |- B e. CC   =>    |- ( ( A ^ 2 ) - ( B ^ 2 ) ) = ( ( A + B ) x. ( A - B ) )
Theorem	qsqeqor 10793	The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> ( A = B \/ A = -u B ) ) )
Theorem	binom2 10794	The square of a binomial. (Contributed by FL, 10-Dec-2006.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )
Theorem	binom21 10795	Special case of binom2 10794 where B = 1. (Contributed by Scott Fenton, 11-May-2014.)
|- ( A e. CC -> ( ( A + 1 ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. A ) ) + 1 ) )
Theorem	binom2sub 10796	Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) )
Theorem	binom2sub1 10797	Special case of binom2sub 10796 where B = 1. (Contributed by AV, 2-Aug-2021.)
|- ( A e. CC -> ( ( A - 1 ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. A ) ) + 1 ) )
Theorem	binom2subi 10798	Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)
|- A e. CC   &    |- B e. CC   =>    |- ( ( A - B ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) )
Theorem	mulbinom2 10799	The square of a binomial with factor. (Contributed by AV, 19-Jul-2021.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( C x. A ) + B ) ^ 2 ) = ( ( ( ( C x. A ) ^ 2 ) + ( ( 2 x. C ) x. ( A x. B ) ) ) + ( B ^ 2 ) ) )
Theorem	binom3 10800	The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 3 ) = ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) )