P. 120 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11901-12000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	climsqz2 11901*	Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> G e. W )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) <_ ( F ` k ) )   &    |- ( ( ph /\ k e. Z ) -> A <_ ( G ` k ) )   =>    |- ( ph -> G ~~> A )
Theorem	clim2ser 11902*	The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ph -> seq M ( + , F ) ~~> A )   =>    |- ( ph -> seq ( N + 1 ) ( + , F ) ~~> ( A - ( seq M ( + , F ) ` N ) ) )
Theorem	clim2ser2 11903*	The limit of an infinite series with an initial segment added. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ph -> seq ( N + 1 ) ( + , F ) ~~> A )   =>    |- ( ph -> seq M ( + , F ) ~~> ( A + ( seq M ( + , F ) ` N ) ) )
Theorem	iserex 11904*	An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   =>    |- ( ph -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )
Theorem	isermulc2 11905*	Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. CC )   &    |- ( ph -> seq M ( + , F ) ~~> A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( C x. ( F ` k ) ) )   =>    |- ( ph -> seq M ( + , G ) ~~> ( C x. A ) )
Theorem	climlec2 11906*	Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> F ~~> B )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> A <_ ( F ` k ) )   =>    |- ( ph -> A <_ B )
Theorem	iserle 11907*	Comparison of the limits of two infinite series. (Contributed by Paul Chapman, 12-Nov-2007.) (Revised by Mario Carneiro, 3-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> seq M ( + , F ) ~~> A )   &    |- ( ph -> seq M ( + , G ) ~~> B )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) )   =>    |- ( ph -> A <_ B )
Theorem	iserge0 11908*	The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> seq M ( + , F ) ~~> A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) )   =>    |- ( ph -> 0 <_ A )
Theorem	climub 11909*	The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ph -> F ~~> A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )   =>    |- ( ph -> ( F ` N ) <_ A )
Theorem	climserle 11910*	The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ph -> seq M ( + , F ) ~~> A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) )   =>    |- ( ph -> ( seq M ( + , F ) ` N ) <_ A )
Theorem	iser3shft 11911*	Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Jim Kingdon, 17-Oct-2022.)
|- ( ph -> F e. V )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ~~> A <-> seq ( M + N ) ( .+ , ( F shift N ) ) ~~> A ) )
Theorem	climcau 11912*	A converging sequence of complex numbers is a Cauchy sequence. The converse would require excluded middle or a different definition of Cauchy sequence (for example, fixing a rate of convergence as in climcvg1n 11915). Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.)
|- Z = ( ZZ>= ` M )   =>    |- ( ( M e. ZZ /\ F e. dom ~~> ) -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x )
Theorem	climrecvg1n 11913*	A Cauchy sequence of real numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within C / n of the nth term, where C is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( abs ` ( ( F ` k ) - ( F ` n ) ) ) < ( C / n ) )   =>    |- ( ph -> F e. dom ~~> )
Theorem	climcvg1nlem 11914*	Lemma for climcvg1n 11915. We construct sequences of the real and imaginary parts of each term of F, show those converge, and use that to show that F converges. (Contributed by Jim Kingdon, 24-Aug-2021.)
|- ( ph -> F : NN --> CC )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( abs ` ( ( F ` k ) - ( F ` n ) ) ) < ( C / n ) )   &    |- G = ( x e. NN |-> ( Re ` ( F ` x ) ) )   &    |- H = ( x e. NN |-> ( Im ` ( F ` x ) ) )   &    |- J = ( x e. NN |-> ( _i x. ( H ` x ) ) )   =>    |- ( ph -> F e. dom ~~> )
Theorem	climcvg1n 11915*	A Cauchy sequence of complex numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within C / n of the nth term, where C is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
|- ( ph -> F : NN --> CC )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( abs ` ( ( F ` k ) - ( F ` n ) ) ) < ( C / n ) )   =>    |- ( ph -> F e. dom ~~> )
Theorem	climcaucn 11916*	A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 11912 but adds the part that ( F ` k ) is complex. (Contributed by Jim Kingdon, 24-Aug-2021.)
|- Z = ( ZZ>= ` M )   =>    |- ( ( M e. ZZ /\ F e. dom ~~> ) -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) )
Theorem	serf0 11917*	If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   =>    |- ( ph -> F ~~> 0 )
4.9.2  Finite and infinite sums
Syntax	csu 11918	Extend class notation to include finite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is a commonly used word in comments.)
class sum_ k e. A B
Definition	df-sumdc 11919*	Define the sum of a series with an index set of integers A. The variable k is normally a free variable in B, i.e., B can be thought of as B ( k ). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. In both cases we have an if expression so that we only need B to be defined where k e. A. In the infinite case, we also require that the indexing set be a decidable subset of an upperset of integers (that is, membership of integers in it is decidable). These two methods of summation produce the same result on their common region of definition (i.e., finite sets of integers). Examples: sum_ k e. { 1 , 2 , 4 } k means 1 + 2 + 4 = 7, and sum_ k e. NN ( 1 / ( 2 ^ k ) ) = 1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 12088). (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 21-May-2023.)
|- sum_ k e. A B = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) ) )
Theorem	sumeq1 11920	Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
|- ( A = B -> sum_ k e. A C = sum_ k e. B C )
Theorem	nfsum1 11921	Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
|- F/_ k A   =>    |- F/_ k sum_ k e. A B
Theorem	nfsum 11922	Bound-variable hypothesis builder for sum: if x is (effectively) not free in A and B, it is not free in sum_ k e. A B. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x sum_ k e. A B
Theorem	sumdc 11923*	Decidability of a subset of upper integers. (Contributed by Jim Kingdon, 1-Jan-2022.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> A. x e. ( ZZ>= ` M )DECID x e. A )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> DECID N e. A )
Theorem	sumeq2 11924*	Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
|- ( A. k e. A B = C -> sum_ k e. A B = sum_ k e. A C )
Theorem	cbvsum 11925	Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
|- ( j = k -> B = C )   &    |- F/_ k A   &    |- F/_ j A   &    |- F/_ k B   &    |- F/_ j C   =>    |- sum_ j e. A B = sum_ k e. A C
Theorem	cbvsumv 11926*	Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
|- ( j = k -> B = C )   =>    |- sum_ j e. A B = sum_ k e. A C
Theorem	cbvsumi 11927*	Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
|- F/_ k B   &    |- F/_ j C   &    |- ( j = k -> B = C )   =>    |- sum_ j e. A B = sum_ k e. A C
Theorem	sumeq1i 11928*	Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
|- A = B   =>    |- sum_ k e. A C = sum_ k e. B C
Theorem	sumeq2i 11929*	Equality inference for sum. (Contributed by NM, 3-Dec-2005.)
|- ( k e. A -> B = C )   =>    |- sum_ k e. A B = sum_ k e. A C
Theorem	sumeq12i 11930*	Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
|- A = B   &    |- ( k e. A -> C = D )   =>    |- sum_ k e. A C = sum_ k e. B D
Theorem	sumeq1d 11931*	Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)
|- ( ph -> A = B )   =>    |- ( ph -> sum_ k e. A C = sum_ k e. B C )
Theorem	sumeq2d 11932*	Equality deduction for sum. Note that unlike sumeq2dv 11933, k may occur in ph. (Contributed by NM, 1-Nov-2005.)
|- ( ph -> A. k e. A B = C )   =>    |- ( ph -> sum_ k e. A B = sum_ k e. A C )
Theorem	sumeq2dv 11933*	Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- ( ( ph /\ k e. A ) -> B = C )   =>    |- ( ph -> sum_ k e. A B = sum_ k e. A C )
Theorem	sumeq2ad 11934*	Equality deduction for sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> B = C )   =>    |- ( ph -> sum_ k e. A B = sum_ k e. A C )
Theorem	sumeq2sdv 11935*	Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)
|- ( ph -> B = C )   =>    |- ( ph -> sum_ k e. A B = sum_ k e. A C )
Theorem	2sumeq2dv 11936*	Equality deduction for double sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- ( ( ph /\ j e. A /\ k e. B ) -> C = D )   =>    |- ( ph -> sum_ j e. A sum_ k e. B C = sum_ j e. A sum_ k e. B D )
Theorem	sumeq12dv 11937*	Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
|- ( ph -> A = B )   &    |- ( ( ph /\ k e. A ) -> C = D )   =>    |- ( ph -> sum_ k e. A C = sum_ k e. B D )
Theorem	sumeq12rdv 11938*	Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)
|- ( ph -> A = B )   &    |- ( ( ph /\ k e. B ) -> C = D )   =>    |- ( ph -> sum_ k e. A C = sum_ k e. B D )
Theorem	sumfct 11939*	A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 18-Sep-2022.)
|- ( A. k e. A B e. CC -> sum_ j e. A ( ( k e. A |-> B ) ` j ) = sum_ k e. A B )
Theorem	fz1f1o 11940*	A lemma for working with finite sums. (Contributed by Mario Carneiro, 22-Apr-2014.)
|- ( A e. Fin -> ( A = (/) \/ ( (♯ ` A ) e. NN /\ E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) ) )
Theorem	fzf1o 11941*	A finite set can be enumerated by integers starting at one. (Contributed by Jim Kingdon, 4-Apr-2026.)
|- ( A e. Fin -> E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
Theorem	nnf1o 11942	Lemma for sum and product theorems. (Contributed by Jim Kingdon, 15-Aug-2022.)
|- ( ph -> ( M e. NN /\ N e. NN ) )   &    |- ( ph -> F : ( 1 ... M ) -1-1-onto-> A )   &    |- ( ph -> G : ( 1 ... N ) -1-1-onto-> A )   =>    |- ( ph -> N = M )
Theorem	sumrbdclem 11943*	Lemma for sumrbdc 11945. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )
Theorem	fsum3cvg 11944*	The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 12-Nov-2022.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> A C_ ( M ... N ) )   =>    |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )
Theorem	sumrbdc 11945*	Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> A C_ ( ZZ>= ` N ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> DECID k e. A )   =>    |- ( ph -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )
Theorem	summodclem3 11946*	Lemma for summodc 11949. (Contributed by Mario Carneiro, 29-Mar-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> ( M e. NN /\ N e. NN ) )   &    |- ( ph -> f : ( 1 ... M ) -1-1-onto-> A )   &    |- ( ph -> K : ( 1 ... N ) -1-1-onto-> A )   &    |- G = ( n e. NN |-> if ( n <_ M , [_ ( f ` n ) / k ]_ B , 0 ) )   &    |- H = ( n e. NN |-> if ( n <_ N , [_ ( K ` n ) / k ]_ B , 0 ) )   =>    |- ( ph -> ( seq 1 ( + , G ) ` M ) = ( seq 1 ( + , H ) ` N ) )
Theorem	summodclem2a 11947*	Lemma for summodc 11949. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )   &    |- H = ( n e. NN |-> if ( n <_ N , [_ ( K ` n ) / k ]_ B , 0 ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> f : ( 1 ... N ) -1-1-onto-> A )   &    |- ( ph -> K Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )   =>    |- ( ph -> seq M ( + , F ) ~~> ( seq 1 ( + , G ) ` N ) )
Theorem	summodclem2 11948*	Lemma for summodc 11949. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )   =>    |- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )
Theorem	summodc 11949*	A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )   &    |- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )   =>    |- ( ph -> E* x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , G ) ` m ) ) ) )
Theorem	zsumdc 11950*	Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A C_ Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = if ( k e. A , B , 0 ) )   &    |- ( ph -> A. x e. Z DECID x e. A )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> sum_ k e. A B = ( ~~> ` seq M ( + , F ) ) )
Theorem	isum 11951*	Series sum with an upper integer index set (i.e. an infinite series). (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 7-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ( ph /\ k e. Z ) -> B e. CC )   =>    |- ( ph -> sum_ k e. Z B = ( ~~> ` seq M ( + , F ) ) )
Theorem	fsumgcl 11952*	Closure for a function used to describe a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
|- ( k = ( F ` n ) -> B = C )   &    |- ( ph -> M e. NN )   &    |- ( ph -> F : ( 1 ... M ) -1-1-onto-> A )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ n e. ( 1 ... M ) ) -> ( G ` n ) = C )   =>    |- ( ph -> A. n e. ( 1 ... M ) ( G ` n ) e. CC )
Theorem	fsum3 11953*	The value of a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
|- ( k = ( F ` n ) -> B = C )   &    |- ( ph -> M e. NN )   &    |- ( ph -> F : ( 1 ... M ) -1-1-onto-> A )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ n e. ( 1 ... M ) ) -> ( G ` n ) = C )   =>    |- ( ph -> sum_ k e. A B = ( seq 1 ( + , ( n e. NN |-> if ( n <_ M , ( G ` n ) , 0 ) ) ) ` M ) )
Theorem	sum0 11954	Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
|- sum_ k e. (/) A = 0
Theorem	isumz 11955*	Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
|- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) \/ A e. Fin ) -> sum_ k e. A 0 = 0 )
Theorem	fsumf1o 11956*	Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.)
|- ( k = G -> B = D )   &    |- ( ph -> C e. Fin )   &    |- ( ph -> F : C -1-1-onto-> A )   &    |- ( ( ph /\ n e. C ) -> ( F ` n ) = G )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> sum_ k e. A B = sum_ n e. C D )
Theorem	isumss 11957*	Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 21-Sep-2022.)
|- ( ph -> A C_ B )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ( ph /\ k e. ( B \ A ) ) -> C = 0 )   &    |- ( ph -> A. j e. ( ZZ>= ` M )DECID j e. A )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> B C_ ( ZZ>= ` M ) )   &    |- ( ph -> A. j e. ( ZZ>= ` M )DECID j e. B )   =>    |- ( ph -> sum_ k e. A C = sum_ k e. B C )
Theorem	fisumss 11958*	Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 23-Sep-2022.)
|- ( ph -> A C_ B )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ( ph /\ k e. ( B \ A ) ) -> C = 0 )   &    |- ( ph -> A. j e. B DECID j e. A )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> sum_ k e. A C = sum_ k e. B C )
Theorem	isumss2 11959*	Change the index set of a sum by adding zeroes. The nonzero elements are in the contained set A and the added zeroes compose the rest of the containing set B which needs to be summable. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Jim Kingdon, 24-Sep-2022.)
|- ( ph -> A C_ B )   &    |- ( ph -> A. j e. B DECID j e. A )   &    |- ( ph -> A. k e. A C e. CC )   &    |- ( ph -> ( ( M e. ZZ /\ B C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. B ) \/ B e. Fin ) )   =>    |- ( ph -> sum_ k e. A C = sum_ k e. B if ( k e. A , C , 0 ) )
Theorem	fsum3cvg2 11960*	The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = if ( k e. A , B , 0 ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> A C_ ( M ... N ) )   =>    |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )
Theorem	fsumsersdc 11961*	Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Jim Kingdon, 5-May-2023.)
|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = if ( k e. A , B , 0 ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> A C_ ( M ... N ) )   =>    |- ( ph -> sum_ k e. A B = ( seq M ( + , F ) ` N ) )
Theorem	fsum3cvg3 11962*	A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A C_ Z )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> seq M ( + , F ) e. dom ~~> )
Theorem	fsum3ser 11963*	A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 11978 and fsump1 11986, which should make our notation clear and from which, along with closure fsumcl 11966, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 1-Oct-2022.)
|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = A )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> A e. CC )   =>    |- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , F ) ` N ) )
Theorem	fsumcl2lem 11964*	- Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by Mario Carneiro, 3-Jun-2014.)
|- ( ph -> S C_ CC )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. S )   &    |- ( ph -> A =/= (/) )   =>    |- ( ph -> sum_ k e. A B e. S )
Theorem	fsumcllem 11965*	- Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 3-Jun-2014.)
|- ( ph -> S C_ CC )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. S )   &    |- ( ph -> 0 e. S )   =>    |- ( ph -> sum_ k e. A B e. S )
Theorem	fsumcl 11966*	Closure of a finite sum of complex numbers A ( k ). (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> sum_ k e. A B e. CC )
Theorem	fsumrecl 11967*	Closure of a finite sum of reals. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   =>    |- ( ph -> sum_ k e. A B e. RR )
Theorem	fsumzcl 11968*	Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. ZZ )   =>    |- ( ph -> sum_ k e. A B e. ZZ )
Theorem	fsumnn0cl 11969*	Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. NN0 )   =>    |- ( ph -> sum_ k e. A B e. NN0 )
Theorem	fsumrpcl 11970*	Closure of a finite sum of positive reals. (Contributed by Mario Carneiro, 3-Jun-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> A =/= (/) )   &    |- ( ( ph /\ k e. A ) -> B e. RR+ )   =>    |- ( ph -> sum_ k e. A B e. RR+ )
Theorem	fsumzcl2 11971*	A finite sum with integer summands is an integer. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
|- ( ( A e. Fin /\ A. k e. A B e. ZZ ) -> sum_ k e. A B e. ZZ )
Theorem	fsumadd 11972*	The sum of two finite sums. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   =>    |- ( ph -> sum_ k e. A ( B + C ) = ( sum_ k e. A B + sum_ k e. A C ) )
Theorem	fsumsplit 11973*	Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.)
|- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> U = ( A u. B ) )   &    |- ( ph -> U e. Fin )   &    |- ( ( ph /\ k e. U ) -> C e. CC )   =>    |- ( ph -> sum_ k e. U C = ( sum_ k e. A C + sum_ k e. B C ) )
Theorem	fsumsplitf 11974*	Split a sum into two parts. A version of fsumsplit 11973 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> U = ( A u. B ) )   &    |- ( ph -> U e. Fin )   &    |- ( ( ph /\ k e. U ) -> C e. CC )   =>    |- ( ph -> sum_ k e. U C = ( sum_ k e. A C + sum_ k e. B C ) )
Theorem	sumsnf 11975*	A sum of a singleton is the term. A version of sumsn 11977 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/_ k B   &    |- ( k = M -> A = B )   =>    |- ( ( M e. V /\ B e. CC ) -> sum_ k e. { M } A = B )
Theorem	fsumsplitsn 11976*	Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- F/_ k D   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. V )   &    |- ( ph -> -. B e. A )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( k = B -> C = D )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> sum_ k e. ( A u. { B } ) C = ( sum_ k e. A C + D ) )
Theorem	sumsn 11977*	A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
|- ( k = M -> A = B )   =>    |- ( ( M e. V /\ B e. CC ) -> sum_ k e. { M } A = B )
Theorem	fsum1 11978*	The finite sum of A ( k ) from k = M to M (i.e. a sum with only one term) is B i.e. A ( M ). (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
|- ( k = M -> A = B )   =>    |- ( ( M e. ZZ /\ B e. CC ) -> sum_ k e. ( M ... M ) A = B )
Theorem	sumpr 11979*	A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.)
|- ( k = A -> C = D )   &    |- ( k = B -> C = E )   &    |- ( ph -> ( D e. CC /\ E e. CC ) )   &    |- ( ph -> ( A e. V /\ B e. W ) )   &    |- ( ph -> A =/= B )   =>    |- ( ph -> sum_ k e. { A , B } C = ( D + E ) )
Theorem	sumtp 11980*	A sum over a triple is the sum of the elements. (Contributed by AV, 24-Jul-2020.)
|- ( k = A -> D = E )   &    |- ( k = B -> D = F )   &    |- ( k = C -> D = G )   &    |- ( ph -> ( E e. CC /\ F e. CC /\ G e. CC ) )   &    |- ( ph -> ( A e. V /\ B e. W /\ C e. X ) )   &    |- ( ph -> A =/= B )   &    |- ( ph -> A =/= C )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> sum_ k e. { A , B , C } D = ( ( E + F ) + G ) )
Theorem	sumsns 11981*	A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
|- ( ( M e. V /\ [_ M / k ]_ A e. CC ) -> sum_ k e. { M } A = [_ M / k ]_ A )
Theorem	fsumm1 11982*	Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   &    |- ( k = N -> A = B )   =>    |- ( ph -> sum_ k e. ( M ... N ) A = ( sum_ k e. ( M ... ( N - 1 ) ) A + B ) )
Theorem	fzosump1 11983*	Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 13-Apr-2016.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   &    |- ( k = N -> A = B )   =>    |- ( ph -> sum_ k e. ( M..^ ( N + 1 ) ) A = ( sum_ k e. ( M..^ N ) A + B ) )
Theorem	fsum1p 11984*	Separate out the first term in a finite sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   &    |- ( k = M -> A = B )   =>    |- ( ph -> sum_ k e. ( M ... N ) A = ( B + sum_ k e. ( ( M + 1 ) ... N ) A ) )
Theorem	fsumsplitsnun 11985*	Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 17-Dec-2021.)
|- ( ( A e. Fin /\ ( Z e. V /\ Z e/ A ) /\ A. k e. ( A u. { Z } ) B e. ZZ ) -> sum_ k e. ( A u. { Z } ) B = ( sum_ k e. A B + [_ Z / k ]_ B ) )
Theorem	fsump1 11986*	The addition of the next term in a finite sum of A ( k ) is the current term plus B i.e. A ( N + 1 ). (Contributed by NM, 4-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )   &    |- ( k = ( N + 1 ) -> A = B )   =>    |- ( ph -> sum_ k e. ( M ... ( N + 1 ) ) A = ( sum_ k e. ( M ... N ) A + B ) )
Theorem	isumclim 11987*	An infinite sum equals the value its series converges to. (Contributed by NM, 25-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> seq M ( + , F ) ~~> B )   =>    |- ( ph -> sum_ k e. Z A = B )
Theorem	isumclim2 11988*	A converging series converges to its infinite sum. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   =>    |- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A )
Theorem	isumclim3 11989*	The sequence of partial finite sums of a converging infinite series converges to the infinite sum of the series. Note that j must not occur in A. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. dom ~~> )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ( ph /\ j e. Z ) -> ( F ` j ) = sum_ k e. ( M ... j ) A )   =>    |- ( ph -> F ~~> sum_ k e. Z A )
Theorem	sumnul 11990*	The sum of a non-convergent infinite series evaluates to the empty set. (Contributed by Paul Chapman, 4-Nov-2007.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> -. seq M ( + , F ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. Z A = (/) )
Theorem	isumcl 11991*	The sum of a converging infinite series is a complex number. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. Z A e. CC )
Theorem	isummulc2 11992*	An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( B x. sum_ k e. Z A ) = sum_ k e. Z ( B x. A ) )
Theorem	isummulc1 11993*	An infinite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( sum_ k e. Z A x. B ) = sum_ k e. Z ( A x. B ) )
Theorem	isumdivapc 11994*	An infinite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( sum_ k e. Z A / B ) = sum_ k e. Z ( A / B ) )
Theorem	isumrecl 11995*	The sum of a converging infinite real series is a real number. (Contributed by Mario Carneiro, 24-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. RR )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. Z A e. RR )
Theorem	isumge0 11996*	An infinite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 28-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. RR )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   &    |- ( ( ph /\ k e. Z ) -> 0 <_ A )   =>    |- ( ph -> 0 <_ sum_ k e. Z A )
Theorem	isumadd 11997*	Addition of infinite sums. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = B )   &    |- ( ( ph /\ k e. Z ) -> B e. CC )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   &    |- ( ph -> seq M ( + , G ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. Z ( A + B ) = ( sum_ k e. Z A + sum_ k e. Z B ) )
Theorem	sumsplitdc 11998*	Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> ( A u. B ) C_ Z )   &    |- ( ( ph /\ k e. Z ) -> DECID k e. A )   &    |- ( ( ph /\ k e. Z ) -> DECID k e. B )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = if ( k e. A , C , 0 ) )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = if ( k e. B , C , 0 ) )   &    |- ( ( ph /\ k e. ( A u. B ) ) -> C e. CC )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   &    |- ( ph -> seq M ( + , G ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. ( A u. B ) C = ( sum_ k e. A C + sum_ k e. B C ) )
Theorem	fsump1i 11999*	Optimized version of fsump1 11986 for making sums of a concrete number of terms. (Contributed by Mario Carneiro, 23-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- N = ( K + 1 )   &    |- ( k = N -> A = B )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> ( K e. Z /\ sum_ k e. ( M ... K ) A = S ) )   &    |- ( ph -> ( S + B ) = T )   =>    |- ( ph -> ( N e. Z /\ sum_ k e. ( M ... N ) A = T ) )
Theorem	fsum2dlemstep 12000*	Lemma for fsum2d 12001- induction step. (Contributed by Mario Carneiro, 23-Apr-2014.) (Revised by Jim Kingdon, 8-Oct-2022.)
|- ( z = <. j , k >. -> D = C )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ j e. A ) -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   &    |- ( ph -> -. y e. x )   &    |- ( ph -> ( x u. { y } ) C_ A )   &    |- ( ph -> x e. Fin )   &    |- ( ps <-> sum_ j e. x sum_ k e. B C = sum_ z e. U_ j e. x ( { j } X. B ) D )   =>    |- ( ( ph /\ ps ) -> sum_ j e. ( x u. { y } ) sum_ k e. B C = sum_ z e. U_ j e. ( x u. { y } ) ( { j } X. B ) D )