P. 109 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10801-10900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	seq3caopr2 10801*	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	seqcaopr2g 10802*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)
|- ( ( 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. ( M ... N ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F e. W )   &    |- ( ph -> G e. X )   &    |- ( ph -> H e. Y )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) )
Theorem	seq3caopr 10803*	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	seqcaoprg 10804*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-May-2014.)
|- ( ( 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. ( M ... N ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) .+ ( G ` k ) ) )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F e. W )   &    |- ( ph -> G e. X )   &    |- ( ph -> H e. Y )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , G ) ` N ) ) )
Theorem	iseqf1olemkle 10805*	Lemma for seq3f1o 10825. (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 10806*	Lemma for seq3f1o 10825. (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 10807	Lemma for seq3f1o 10825. (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 10808*	Lemma for seq3f1o 10825. 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 10809*	Lemma for seq3f1o 10825. (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 10810*	Lemma for seq3f1o 10825. (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 10811*	Lemma for seq3f1o 10825. (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 10812*	Lemma for seq3f1o 10825. 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 10813*	Lemma for seq3f1o 10825. 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 10814*	Lemma for seq3f1o 10825. 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 10815*	Lemma for seq3f1o 10825. 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 10816*	Lemma for seq3f1o 10825. 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 10817*	Lemma for seq3f1o 10825. 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 10818*	Lemma for seq3f1o 10825. (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 10819*	Lemma for seq3f1o 10825. 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 10820*	Lemma for seq3f1o 10825. 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 10821*	Lemma for seq3f1o 10825. 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 10822*	Lemma for seq3f1o 10825. 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 10823*	Lemma for seq3f1o 10825. 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 10824*	Lemma for seq3f1o 10825. 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 10825*	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 10826*	Lemma for seqf1og 10829. (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 10827*	Lemma for seqf1og 10829. (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 10828*	Lemma for seqf1og 10829. (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 10829*	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 10830*	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 10831*	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 10832*	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 10833*	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 10834*	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 10835*	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 10836*	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 10837*	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 10838*	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 10839*	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 10840*	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 10841	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 10842	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 10843*	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 10844*	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 10845*	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 10846	Extend class notation to include exponentiation of a complex number to an integer power.
class ^
Definition	df-exp 10847*	Define exponentiation to nonnegative integer powers. For example, ( 5 ^ 2 ) = 2 5 (see ex-exp 16424). This definition is not meant to be used directly; instead, exp0 10851 and expp1 10854 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 10852). 4-Jun-2014: The definition was extended to include negative integer exponents. For example, ( -u 3 ^ -u 2 ) = ( 1 / 9 ) (ex-exp 16424). 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 10848	Lemma for exp3val 10849. 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 10849	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 10850	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 10851	Value of a complex number raised to the 0th power. Note that under our definition, 0 ^ 0 = 1 (0exp0e1 10852) , 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 10852	The zeroth power of zero equals one. See comment of exp0 10851. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 0 ^ 0 ) = 1
Theorem	exp1 10853	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 10854	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 10855	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 10856	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 10857	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 10858*	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 10859*	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 10860	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. NN /\ N e. NN0 ) -> ( A ^ N ) e. NN )
Theorem	nn0expcl 10861	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. NN0 /\ N e. NN0 ) -> ( A ^ N ) e. NN0 )
Theorem	zexpcl 10862	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. ZZ /\ N e. NN0 ) -> ( A ^ N ) e. ZZ )
Theorem	qexpcl 10863	Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
|- ( ( A e. QQ /\ N e. NN0 ) -> ( A ^ N ) e. QQ )
Theorem	reexpcl 10864	Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. RR /\ N e. NN0 ) -> ( A ^ N ) e. RR )
Theorem	expcl 10865	Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
Theorem	rpexpcl 10866	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 10867	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 10868	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 10869	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 10870	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( N e. ZZ -> ( -u 1 ^ N ) e. ZZ )
Theorem	expclzaplem 10871*	Closure law for integer exponentiation. Lemma for expclzap 10872 and expap0i 10879. (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 10872	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 10873	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 10874	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 10875	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 10876	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 10877	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 10878 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 10878	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 10879	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 10880	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 10881	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 10882	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
|- ( N e. NN -> ( 0 ^ N ) = 0 )
Theorem	expge0 10883	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 10884	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 10885	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 10886	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 10887	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 10888	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 10889	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 10890	Lemma for expaddzap 10891. (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 10891	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 10892	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 10893	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 10894	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 10895	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 10896	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 10897	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 10898	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 10899	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 10900	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 ) )