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	seq3f1olemp 10901*	Lemma for seq3f1o 10903. Existence of a constant permutation of ( M ... N ) which leads to the same sum as the permutation F itself. (Contributed by Jim Kingdon, 18-Aug-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- L = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( F ` x ) ) , ( G ` M ) ) )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> E. f ( f : ( M ... N ) -1-1-onto-> ( M ... N ) /\ A. x e. ( M ... N ) ( f ` x ) = x /\ ( seq M ( .+ , P ) ` N ) = ( seq M ( .+ , L ) ` N ) ) )
Theorem	seq3f1oleml 10902*	Lemma for seq3f1o 10903. This is more or less the result, but stated in terms of F and G without H. L and H may differ in terms of what happens to terms after N. The terms after N don't matter for the value at N but we need some definition given the way our theorems concerning seq work. (Contributed by Jim Kingdon, 17-Aug-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- L = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( F ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> ( seq M ( .+ , L ) ` N ) = ( seq M ( .+ , G ) ` N ) )
Theorem	seq3f1o 10903*	Rearrange a sum via an arbitrary bijection on ( M ... N ). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 3-Nov-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( H ` x ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( G ` ( F ` k ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( seq M ( .+ , G ) ` N ) )
Theorem	seqf1oglem2a 10904*	Lemma for seqf1og 10907. (Contributed by Mario Carneiro, 24-Apr-2016.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> C C_ S )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> G : A --> C )   &    |- ( ph -> K e. A )   &    |- ( ph -> ( M ... N ) C_ A )   &    |- ( ph -> A e. W )   =>    |- ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` N ) ) = ( ( seq M ( .+ , G ) ` N ) .+ ( G ` K ) ) )
Theorem	seqf1oglem1 10905*	Lemma for seqf1og 10907. (Contributed by Mario Carneiro, 26-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> C C_ S )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) )   &    |- ( ph -> G : ( M ... ( N + 1 ) ) --> C )   &    |- J = ( k e. ( M ... N ) |-> ( F ` if ( k < K , k , ( k + 1 ) ) ) )   &    |- K = ( `' F ` ( N + 1 ) )   =>    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
Theorem	seqf1oglem2 10906*	Lemma for seqf1og 10907. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> C C_ S )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) )   &    |- ( ph -> G : ( M ... ( N + 1 ) ) --> C )   &    |- J = ( k e. ( M ... N ) |-> ( F ` if ( k < K , k , ( k + 1 ) ) ) )   &    |- K = ( `' F ` ( N + 1 ) )   &    |- ( ph -> A. g A. f ( ( f : ( M ... N ) -1-1-onto-> ( M ... N ) /\ g : ( M ... N ) --> C ) -> ( seq M ( .+ , ( g o. f ) ) ` N ) = ( seq M ( .+ , g ) ` N ) ) )   =>    |- ( ph -> ( seq M ( .+ , ( G o. F ) ) ` ( N + 1 ) ) = ( seq M ( .+ , G ) ` ( N + 1 ) ) )
Theorem	seqf1og 10907*	Rearrange a sum via an arbitrary bijection on ( M ... N ). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 29-Aug-2025.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> C C_ S )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( G ` x ) e. C )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( G ` ( F ` k ) ) )   &    |- ( ph -> G e. W )   &    |- ( ph -> H e. X )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( seq M ( .+ , G ) ` N ) )
Theorem	ser3add 10908*	The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 4-Oct-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) + ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( + , H ) ` N ) = ( ( seq M ( + , F ) ` N ) + ( seq M ( + , G ) ` N ) ) )
Theorem	ser3sub 10909*	The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) - ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( + , H ) ` N ) = ( ( seq M ( + , F ) ` N ) - ( seq M ( + , G ) ` N ) ) )
Theorem	seq3id3 10910*	A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a .+ -idempotent sums (or " .+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- ( ph -> ( Z .+ Z ) = Z )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z )   &    |- ( ph -> Z e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = Z )
Theorem	seq3id 10911*	Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for .+) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- ( ( ph /\ x e. S ) -> ( Z .+ x ) = x )   &    |- ( ph -> Z e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> ( F ` N ) e. S )   &    |- ( ( ph /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` N ) ) = seq N ( .+ , F ) )
Theorem	seq3id2 10912*	The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for .+) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 12-Nov-2022.)
|- ( ( ph /\ x e. S ) -> ( x .+ Z ) = x )   &    |- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> N e. ( ZZ>= ` K ) )   &    |- ( ph -> ( seq M ( .+ , F ) ` K ) e. S )   &    |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> ( F ` x ) = Z )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) )
Theorem	seq3homo 10913*	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 10914*	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 10915*	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 10916*	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 10917*	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 10918*	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 10919	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 10920	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 10921*	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 10922*	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 10923*	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 10924	Extend class notation to include exponentiation of a complex number to an integer power.
class ^
Definition	df-exp 10925*	Define exponentiation to nonnegative integer powers. For example, ( 5 ^ 2 ) = 2 5 (see ex-exp 16607). This definition is not meant to be used directly; instead, exp0 10929 and expp1 10932 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 10930). 4-Jun-2014: The definition was extended to include negative integer exponents. For example, ( -u 3 ^ -u 2 ) = ( 1 / 9 ) (ex-exp 16607). 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 10926	Lemma for exp3val 10927. 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 10927	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 10928	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 10929	Value of a complex number raised to the 0th power. Note that under our definition, 0 ^ 0 = 1 (0exp0e1 10930) , 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 10930	The zeroth power of zero equals one. See comment of exp0 10929. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 0 ^ 0 ) = 1
Theorem	exp1 10931	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 10932	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 10933	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 10934	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 10935	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 10936*	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 10937*	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 10938	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. NN /\ N e. NN0 ) -> ( A ^ N ) e. NN )
Theorem	nn0expcl 10939	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. NN0 /\ N e. NN0 ) -> ( A ^ N ) e. NN0 )
Theorem	zexpcl 10940	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. ZZ /\ N e. NN0 ) -> ( A ^ N ) e. ZZ )
Theorem	qexpcl 10941	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. QQ /\ N e. NN0 ) -> ( A ^ N ) e. QQ )
Theorem	reexpcl 10942	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. RR /\ N e. NN0 ) -> ( A ^ N ) e. RR )
Theorem	expcl 10943	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
Theorem	rpexpcl 10944	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 10945	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 10946	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 10947	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 10948	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( N e. ZZ -> ( -u 1 ^ N ) e. ZZ )
Theorem	expclzaplem 10949*	Closure law for integer exponentiation. Lemma for expclzap 10950 and expap0i 10957. (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 10950	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 10951	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 10952	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 10953	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 10954	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 10955	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 10956 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 10956	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 10957	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 10958	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 10959	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 10960	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
|- ( N e. NN -> ( 0 ^ N ) = 0 )
Theorem	expge0 10961	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 10962	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 10963	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 10964	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 10965	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 10966	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 10967	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 10968	Lemma for expaddzap 10969. (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 10969	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 10970	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 10971	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 10972	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 10973	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 10974	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 10975	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 10976	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 10977	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 10978	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 10979	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 10980	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 10981	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 10982	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 10983	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 10984	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
|- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
Theorem	sqsubswap 10985	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 10986	Closure of square. (Contributed by NM, 10-Aug-1999.)
|- ( A e. CC -> ( A ^ 2 ) e. CC )
Theorem	sqmul 10987	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 10988	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 10989	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 10990	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 10991	A number is nonzero iff its square is nonzero. See also sqap0 10992 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 10992	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 10993	Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
|- ( A e. RR -> ( A ^ 2 ) e. RR )
Theorem	sqgt0ap 10994	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 10995	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 10996	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 10997	The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
|- ( A e. QQ -> ( A ^ 2 ) e. QQ )
Theorem	sq11 10998	The square function is one-to-one for nonnegative reals. Also see sq11ap 11094 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 10999	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 11000	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 ) ) )