P. 106 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	seq3caopr3 10501*	Lemma for seq3caopr2 10502. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   &    |- ( ( ph /\ n e. ( M..^ N ) ) -> ( ( ( seq M ( .+ , F ) ` n ) Q ( seq M ( .+ , G ) ` n ) ) .+ ( ( F ` ( n + 1 ) ) Q ( G ` ( n + 1 ) ) ) ) = ( ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) Q ( ( seq M ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) )
Theorem	seq3caopr2 10502*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   &    |- ( ( ph /\ ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ) -> ( ( x Q z ) .+ ( y Q w ) ) = ( ( x .+ y ) Q ( z .+ w ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) )
Theorem	seq3caopr 10503*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( 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 /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( 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	iseqf1olemkle 10504*	Lemma for seq3f1o 10524. (Contributed by Jim Kingdon, 21-Aug-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   =>    |- ( ph -> K <_ ( `' J ` K ) )
Theorem	iseqf1olemklt 10505*	Lemma for seq3f1o 10524. (Contributed by Jim Kingdon, 21-Aug-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   &    |- ( ph -> K =/= ( `' J ` K ) )   =>    |- ( ph -> K < ( `' J ` K ) )
Theorem	iseqf1olemqcl 10506	Lemma for seq3f1o 10524. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   =>    |- ( ph -> if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) e. ( M ... N ) )
Theorem	iseqf1olemqval 10507*	Lemma for seq3f1o 10524. Value of the function Q. (Contributed by Jim Kingdon, 28-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   =>    |- ( ph -> ( Q ` A ) = if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) )
Theorem	iseqf1olemnab 10508*	Lemma for seq3f1o 10524. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   =>    |- ( ph -> -. ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) )
Theorem	iseqf1olemab 10509*	Lemma for seq3f1o 10524. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ph -> A e. ( K ... ( `' J ` K ) ) )   &    |- ( ph -> B e. ( K ... ( `' J ` K ) ) )   =>    |- ( ph -> A = B )
Theorem	iseqf1olemnanb 10510*	Lemma for seq3f1o 10524. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ph -> -. A e. ( K ... ( `' J ` K ) ) )   &    |- ( ph -> -. B e. ( K ... ( `' J ` K ) ) )   =>    |- ( ph -> A = B )
Theorem	iseqf1olemqf 10511*	Lemma for seq3f1o 10524. Domain and codomain of Q. (Contributed by Jim Kingdon, 26-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   =>    |- ( ph -> Q : ( M ... N ) --> ( M ... N ) )
Theorem	iseqf1olemmo 10512*	Lemma for seq3f1o 10524. Showing that Q is one-to-one. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   =>    |- ( ph -> A = B )
Theorem	iseqf1olemqf1o 10513*	Lemma for seq3f1o 10524. Q is a permutation of ( M ... N ). Q is formed from the constant portion of J, followed by the single element K (at position K), followed by the rest of J (with the K deleted and the elements before K moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   =>    |- ( ph -> Q : ( M ... N ) -1-1-onto-> ( M ... N ) )
Theorem	iseqf1olemqk 10514*	Lemma for seq3f1o 10524. Q is constant for one more position than J is. (Contributed by Jim Kingdon, 21-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   =>    |- ( ph -> A. x e. ( M ... K ) ( Q ` x ) = x )
Theorem	iseqf1olemjpcl 10515*	Lemma for seq3f1o 10524. A closure lemma involving J and P. (Contributed by Jim Kingdon, 29-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( [_ J / f ]_ P ` x ) e. S )
Theorem	iseqf1olemqpcl 10516*	Lemma for seq3f1o 10524. A closure lemma involving Q and P. (Contributed by Jim Kingdon, 29-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( [_ Q / f ]_ P ` x ) e. S )
Theorem	iseqf1olemfvp 10517*	Lemma for seq3f1o 10524. (Contributed by Jim Kingdon, 30-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> T : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> ( [_ T / f ]_ P ` A ) = ( G ` ( T ` A ) ) )
Theorem	seq3f1olemqsumkj 10518*	Lemma for seq3f1o 10524. Q gives the same sum as J in the range ( K ... ( `' J ` K ) ). (Contributed by Jim Kingdon, 29-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 )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   &    |- ( ph -> K =/= ( `' J ` K ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> ( seq K ( .+ , [_ J / f ]_ P ) ` ( `' J ` K ) ) = ( seq K ( .+ , [_ Q / f ]_ P ) ` ( `' J ` K ) ) )
Theorem	seq3f1olemqsumk 10519*	Lemma for seq3f1o 10524. Q gives the same sum as J in the range ( K ... N ). (Contributed by Jim Kingdon, 22-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 )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   &    |- ( ph -> K =/= ( `' J ` K ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> ( seq K ( .+ , [_ J / f ]_ P ) ` N ) = ( seq K ( .+ , [_ Q / f ]_ P ) ` N ) )
Theorem	seq3f1olemqsum 10520*	Lemma for seq3f1o 10524. Q gives the same sum as J. (Contributed by Jim Kingdon, 21-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 )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   &    |- ( ph -> K =/= ( `' J ` K ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- P = ( x e. ( ZZ>= ` M ) |-> if ( x <_ N , ( G ` ( f ` x ) ) , ( G ` M ) ) )   =>    |- ( ph -> ( seq M ( .+ , [_ J / f ]_ P ) ` N ) = ( seq M ( .+ , [_ Q / f ]_ P ) ` N ) )
Theorem	seq3f1olemstep 10521*	Lemma for seq3f1o 10524. Given a permutation which is constant up to a point, supply a new one which is constant for one more position. (Contributed by Jim Kingdon, 19-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 )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   &    |- ( ph -> ( seq M ( .+ , [_ J / f ]_ P ) ` N ) = ( seq M ( .+ , L ) ` N ) )   &    |- 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 ... K ) ( f ` x ) = x /\ ( seq M ( .+ , P ) ` N ) = ( seq M ( .+ , L ) ` N ) ) )
Theorem	seq3f1olemp 10522*	Lemma for seq3f1o 10524. 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 10523*	Lemma for seq3f1o 10524. 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 10524*	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	ser3add 10525*	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 10526*	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 10527*	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 10528*	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 10529*	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 10530*	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 10531*	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 10532*	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	seq3distr 10533*	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 10534	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 10535	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 10536*	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 10537*	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 10538*	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 10539	Extend class notation to include exponentiation of a complex number to an integer power.
class ^
Definition	df-exp 10540*	Define exponentiation to nonnegative integer powers. For example, ( 5 ^ 2 ) = 2 5 (see ex-exp 14883). This definition is not meant to be used directly; instead, exp0 10544 and expp1 10547 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 10545). 4-Jun-2014: The definition was extended to include negative integer exponents. For example, ( -u 3 ^ -u 2 ) = ( 1 / 9 ) (ex-exp 14883). 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 10541	Lemma for exp3val 10542. 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 10542	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 10543	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 10544	Value of a complex number raised to the 0th power. Note that under our definition, 0 ^ 0 = 1 (0exp0e1 10545) , 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 10545	The zeroth power of zero equals one. See comment of exp0 10544. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 0 ^ 0 ) = 1
Theorem	exp1 10546	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 10547	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 10548	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 10549	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 10550	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 10551*	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 10552*	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 10553	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. NN /\ N e. NN0 ) -> ( A ^ N ) e. NN )
Theorem	nn0expcl 10554	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. NN0 /\ N e. NN0 ) -> ( A ^ N ) e. NN0 )
Theorem	zexpcl 10555	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. ZZ /\ N e. NN0 ) -> ( A ^ N ) e. ZZ )
Theorem	qexpcl 10556	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. QQ /\ N e. NN0 ) -> ( A ^ N ) e. QQ )
Theorem	reexpcl 10557	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. RR /\ N e. NN0 ) -> ( A ^ N ) e. RR )
Theorem	expcl 10558	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
Theorem	rpexpcl 10559	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 10560	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 10561	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 10562	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 10563	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( N e. ZZ -> ( -u 1 ^ N ) e. ZZ )
Theorem	expclzaplem 10564*	Closure law for integer exponentiation. Lemma for expclzap 10565 and expap0i 10572. (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 10565	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 10566	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 10567	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 10568	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 10569	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 10570	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 10571 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 10571	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 10572	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 10573	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 10574	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 10575	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
|- ( N e. NN -> ( 0 ^ N ) = 0 )
Theorem	expge0 10576	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 10577	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 10578	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 10579	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 10580	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 10581	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 10582	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 10583	Lemma for expaddzap 10584. (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 10584	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 10585	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 10586	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 10587	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 10588	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 10589	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 10590	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 10591	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 10592	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 10593	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 10594	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 10595	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 10596	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 10597	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 10598	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 10599	The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
|- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
Theorem	sqsubswap 10600	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 ) )