P. 107 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10601-10700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	seq3homo 10601*	Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 10-Oct-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( H ` ( x .+ y ) ) = ( ( H ` x ) Q ( H ` y ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( H ` ( F ` x ) ) = ( G ` x ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   =>    |- ( ph -> ( H ` ( seq M ( .+ , F ) ` N ) ) = ( seq M ( Q , G ) ` N ) )
Theorem	seq3z 10602*	If the operation .+ has an absorbing element Z (a.k.a. zero element), then any sequence containing a Z evaluates to Z. (Contributed by Mario Carneiro, 27-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = Z )   &    |- ( ( ph /\ x e. S ) -> ( x .+ Z ) = Z )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> ( F ` K ) = Z )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = Z )
Theorem	seqfeq3 10603*	Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)
|- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )   =>    |- ( ph -> seq M ( .+ , F ) = seq M ( Q , F ) )
Theorem	seqhomog 10604*	Apply a homomorphism to a sequence. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( H ` ( x .+ y ) ) = ( ( H ` x ) Q ( H ` y ) ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( H ` ( F ` x ) ) = ( G ` x ) )   &    |- ( ph -> F e. V )   &    |- ( ph -> G e. W )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> Q e. Y )   =>    |- ( ph -> ( H ` ( seq M ( .+ , F ) ` N ) ) = ( seq M ( Q , G ) ` N ) )
Theorem	seqfeq4g 10605*	Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. S )   &    |- ( ph -> F e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> Q e. X )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq M ( Q , F ) ` N ) )
Theorem	seq3distr 10606*	The distributive property for series. (Contributed by Jim Kingdon, 10-Oct-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( C T ( x .+ y ) ) = ( ( C T x ) .+ ( C T y ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) = ( C T ( G ` x ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x T y ) e. S )   &    |- ( ph -> C e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( C T ( seq M ( .+ , G ) ` N ) ) )
Theorem	ser0 10607	The value of the partial sums in a zero-valued infinite series. (Contributed by Mario Carneiro, 31-Aug-2013.) (Revised by Mario Carneiro, 15-Dec-2014.)
|- Z = ( ZZ>= ` M )   =>    |- ( N e. Z -> ( seq M ( + , ( Z X. { 0 } ) ) ` N ) = 0 )
Theorem	ser0f 10608	A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> seq M ( + , ( Z X. { 0 } ) ) = ( Z X. { 0 } ) )
Theorem	fser0const 10609*	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 10610*	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 10611*	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 10612	Extend class notation to include exponentiation of a complex number to an integer power.
class ^
Definition	df-exp 10613*	Define exponentiation to nonnegative integer powers. For example, ( 5 ^ 2 ) = 2 5 (see ex-exp 15289). This definition is not meant to be used directly; instead, exp0 10617 and expp1 10620 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 10618). 4-Jun-2014: The definition was extended to include negative integer exponents. For example, ( -u 3 ^ -u 2 ) = ( 1 / 9 ) (ex-exp 15289). 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 10614	Lemma for exp3val 10615. 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 10615	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 10616	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 10617	Value of a complex number raised to the 0th power. Note that under our definition, 0 ^ 0 = 1 (0exp0e1 10618) , 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 10618	The zeroth power of zero equals one. See comment of exp0 10617. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 0 ^ 0 ) = 1
Theorem	exp1 10619	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 10620	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 10621	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 10622	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 10623	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 10624*	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 10625*	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 10626	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. NN /\ N e. NN0 ) -> ( A ^ N ) e. NN )
Theorem	nn0expcl 10627	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. NN0 /\ N e. NN0 ) -> ( A ^ N ) e. NN0 )
Theorem	zexpcl 10628	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. ZZ /\ N e. NN0 ) -> ( A ^ N ) e. ZZ )
Theorem	qexpcl 10629	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. QQ /\ N e. NN0 ) -> ( A ^ N ) e. QQ )
Theorem	reexpcl 10630	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. RR /\ N e. NN0 ) -> ( A ^ N ) e. RR )
Theorem	expcl 10631	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
Theorem	rpexpcl 10632	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 10633	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 10634	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 10635	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 10636	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( N e. ZZ -> ( -u 1 ^ N ) e. ZZ )
Theorem	expclzaplem 10637*	Closure law for integer exponentiation. Lemma for expclzap 10638 and expap0i 10645. (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 10638	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 10639	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 10640	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 10641	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 10642	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 10643	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 10644 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 10644	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 10645	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 10646	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 10647	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 10648	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
|- ( N e. NN -> ( 0 ^ N ) = 0 )
Theorem	expge0 10649	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 10650	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 10651	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 10652	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 10653	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 10654	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 10655	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 10656	Lemma for expaddzap 10657. (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 10657	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 10658	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 10659	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 10660	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 10661	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 10662	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 10663	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 10664	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 10665	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 10666	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 10667	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 10668	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 10669	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 10670	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 10671	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 10672	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
|- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
Theorem	sqsubswap 10673	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 10674	Closure of square. (Contributed by NM, 10-Aug-1999.)
|- ( A e. CC -> ( A ^ 2 ) e. CC )
Theorem	sqmul 10675	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 10676	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 10677	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 10678	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 10679	A number is nonzero iff its square is nonzero. See also sqap0 10680 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 10680	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 10681	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
|- ( A e. RR -> ( A ^ 2 ) e. RR )
Theorem	sqgt0ap 10682	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 10683	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 10684	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 10685	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> ( A ^ 2 ) e. QQ )
Theorem	sq11 10686	The square function is one-to-one for nonnegative reals. Also see sq11ap 10781 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 10687	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 10688	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 10689	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 10690	A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
|- ( A e. RR -> 0 <_ ( A ^ 2 ) )
Theorem	zsqcl2 10691	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 10692	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 10693	Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
|- A e. CC   =>    |- ( A ^ 2 ) = ( A x. A )
Theorem	sqcli 10694	Closure of square. (Contributed by NM, 2-Aug-1999.)
|- A e. CC   =>    |- ( A ^ 2 ) e. CC
Theorem	sqeq0i 10695	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 10696	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 10697	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 10698	Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
|- A e. RR   =>    |- ( A ^ 2 ) e. RR
Theorem	sqgt0api 10699	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 10700	A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
|- A e. RR   =>    |- 0 <_ ( A ^ 2 )