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	fz1f1o 11901*	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	nnf1o 11902	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 11903*	Lemma for sumrbdc 11905. (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 11904*	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 11905*	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 11906*	Lemma for summodc 11909. (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 11907*	Lemma for summodc 11909. (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 11908*	Lemma for summodc 11909. (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 11909*	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 11910*	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 11911*	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 11912*	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 11913*	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 11914	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 11915*	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 11916*	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 11917*	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 11918*	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 11919*	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 11920*	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 11921*	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 11922*	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 11923*	A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 11938 and fsump1 11946, which should make our notation clear and from which, along with closure fsumcl 11926, 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 11924*	- 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 11925*	- 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 11926*	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 11927*	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 11928*	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 11929*	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 11930*	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 11931*	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 11932*	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 11933*	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 11934*	Split a sum into two parts. A version of fsumsplit 11933 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 11935*	A sum of a singleton is the term. A version of sumsn 11937 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 11936*	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 11937*	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 11938*	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 11939*	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 11940*	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 11941*	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 11942*	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 11943*	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 11944*	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 11945*	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 11946*	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 11947*	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 11948*	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 11949*	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 11950*	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 11951*	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 11952*	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 11953*	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 11954*	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 11955*	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 11956*	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 11957*	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 11958*	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 11959*	Optimized version of fsump1 11946 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 11960*	Lemma for fsum2d 11961- 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 )
Theorem	fsum2d 11961*	Write a double sum as a sum over a two-dimensional region. Note that B ( j ) is a function of j. (Contributed by Mario Carneiro, 27-Apr-2014.)
|- ( 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 -> sum_ j e. A sum_ k e. B C = sum_ z e. U_ j e. A ( { j } X. B ) D )
Theorem	fsumxp 11962*	Combine two sums into a single sum over the cartesian product. (Contributed by Mario Carneiro, 23-Apr-2014.)
|- ( z = <. j , k >. -> D = C )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> sum_ j e. A sum_ k e. B C = sum_ z e. ( A X. B ) D )
Theorem	fsumcnv 11963*	Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)
|- ( x = <. j , k >. -> B = D )   &    |- ( y = <. k , j >. -> C = D )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> Rel A )   &    |- ( ( ph /\ x e. A ) -> B e. CC )   =>    |- ( ph -> sum_ x e. A B = sum_ y e. `' A C )
Theorem	fisumcom2 11964*	Interchange order of summation. Note that B ( j ) and D ( k ) are not necessarily constant expressions. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.) (Proof shortened by JJ, 2-Aug-2021.)
|- ( ph -> A e. Fin )   &    |- ( ph -> C e. Fin )   &    |- ( ( ph /\ j e. A ) -> B e. Fin )   &    |- ( ( ph /\ k e. C ) -> D e. Fin )   &    |- ( ph -> ( ( j e. A /\ k e. B ) <-> ( k e. C /\ j e. D ) ) )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> E e. CC )   =>    |- ( ph -> sum_ j e. A sum_ k e. B E = sum_ k e. C sum_ j e. D E )
Theorem	fsumcom 11965*	Interchange order of summation. (Contributed by NM, 15-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> sum_ j e. A sum_ k e. B C = sum_ k e. B sum_ j e. A C )
Theorem	fsum0diaglem 11966*	Lemma for fisum0diag 11967. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
|- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( k e. ( 0 ... N ) /\ j e. ( 0 ... ( N - k ) ) ) )
Theorem	fisum0diag 11967*	Two ways to express "the sum of A ( j , k ) over the triangular region M <_ j, M <_ k, j + k <_ N". (Contributed by NM, 31-Dec-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
|- ( ( ph /\ ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) ) -> A e. CC )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> sum_ j e. ( 0 ... N ) sum_ k e. ( 0 ... ( N - j ) ) A = sum_ k e. ( 0 ... N ) sum_ j e. ( 0 ... ( N - k ) ) A )
Theorem	mptfzshft 11968*	1-1 onto function in maps-to notation which shifts a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) : ( ( M + K ) ... ( N + K ) ) -1-1-onto-> ( M ... N ) )
Theorem	fsumrev 11969*	Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( K - k ) -> A = B )   =>    |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( K - N ) ... ( K - M ) ) B )
Theorem	fsumshft 11970*	Index shift of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) (Proof shortened by AV, 8-Sep-2019.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( k - K ) -> A = B )   =>    |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M + K ) ... ( N + K ) ) B )
Theorem	fsumshftm 11971*	Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( k + K ) -> A = B )   =>    |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )
Theorem	fisumrev2 11972*	Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 13-Apr-2016.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( ( M + N ) - k ) -> A = B )   =>    |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( M ... N ) B )
Theorem	fisum0diag2 11973*	Two ways to express "the sum of A ( j , k ) over the triangular region 0 <_ j, 0 <_ k, j + k <_ N". (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( x = k -> B = A )   &    |- ( x = ( k - j ) -> B = C )   &    |- ( ( ph /\ ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) ) -> A e. CC )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> sum_ j e. ( 0 ... N ) sum_ k e. ( 0 ... ( N - j ) ) A = sum_ k e. ( 0 ... N ) sum_ j e. ( 0 ... k ) C )
Theorem	fsummulc2 11974*	A finite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> C e. CC )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( C x. sum_ k e. A B ) = sum_ k e. A ( C x. B ) )
Theorem	fsummulc1 11975*	A finite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> C e. CC )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( sum_ k e. A B x. C ) = sum_ k e. A ( B x. C ) )
Theorem	fsumdivapc 11976*	A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> C e. CC )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( sum_ k e. A B / C ) = sum_ k e. A ( B / C ) )
Theorem	fsumneg 11977*	Negation of a finite sum. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> sum_ k e. A -u B = -u sum_ k e. A B )
Theorem	fsumsub 11978*	Split a finite sum over a subtraction. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-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	fsum2mul 11979*	Separate the nested sum of the product C ( j ) x. D ( k ). (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ( ph /\ j e. A ) -> C e. CC )   &    |- ( ( ph /\ k e. B ) -> D e. CC )   =>    |- ( ph -> sum_ j e. A sum_ k e. B ( C x. D ) = ( sum_ j e. A C x. sum_ k e. B D ) )
Theorem	fsumconst 11980*	The sum of constant terms ( k is not free in B). (Contributed by NM, 24-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ( A e. Fin /\ B e. CC ) -> sum_ k e. A B = ( (♯ ` A ) x. B ) )
Theorem	fsumdifsnconst 11981*	The sum of constant terms ( k is not free in C) over an index set excluding a singleton. (Contributed by AV, 7-Jan-2022.)
|- ( ( A e. Fin /\ B e. A /\ C e. CC ) -> sum_ k e. ( A \ { B } ) C = ( ( (♯ ` A ) - 1 ) x. C ) )
Theorem	modfsummodlem1 11982*	Lemma for modfsummod 11984. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
|- ( A. k e. ( A u. { z } ) B e. ZZ -> [_ z / k ]_ B e. ZZ )
Theorem	modfsummodlemstep 11983*	Induction step for modfsummod 11984. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by Jim Kingdon, 12-Oct-2022.)
|- ( ph -> A e. Fin )   &    |- ( ph -> N e. NN )   &    |- ( ph -> A. k e. ( A u. { z } ) B e. ZZ )   &    |- ( ph -> -. z e. A )   &    |- ( ph -> ( sum_ k e. A B mod N ) = ( sum_ k e. A ( B mod N ) mod N ) )   =>    |- ( ph -> ( sum_ k e. ( A u. { z } ) B mod N ) = ( sum_ k e. ( A u. { z } ) ( B mod N ) mod N ) )
Theorem	modfsummod 11984*	A finite sum modulo a positive integer equals the finite sum of their summands modulo the positive integer, modulo the positive integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
|- ( ph -> N e. NN )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A. k e. A B e. ZZ )   =>    |- ( ph -> ( sum_ k e. A B mod N ) = ( sum_ k e. A ( B mod N ) mod N ) )
Theorem	fsumge0 11985*	If all of the terms of a finite sum are nonnegative, so is the sum. (Contributed by NM, 26-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   =>    |- ( ph -> 0 <_ sum_ k e. A B )
Theorem	fsumlessfi 11986*	A shorter sum of nonnegative terms is no greater than a longer one. (Contributed by NM, 26-Dec-2005.) (Revised by Jim Kingdon, 12-Oct-2022.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   &    |- ( ph -> C C_ A )   &    |- ( ph -> C e. Fin )   =>    |- ( ph -> sum_ k e. C B <_ sum_ k e. A B )
Theorem	fsumge1 11987*	A sum of nonnegative numbers is greater than or equal to any one of its terms. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 4-Jun-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   &    |- ( k = M -> B = C )   &    |- ( ph -> M e. A )   =>    |- ( ph -> C <_ sum_ k e. A B )
Theorem	fsum00 11988*	A sum of nonnegative numbers is zero iff all terms are zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   =>    |- ( ph -> ( sum_ k e. A B = 0 <-> A. k e. A B = 0 ) )
Theorem	fsumle 11989*	If all of the terms of finite sums compare, so do the sums. (Contributed by NM, 11-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> C e. RR )   &    |- ( ( ph /\ k e. A ) -> B <_ C )   =>    |- ( ph -> sum_ k e. A B <_ sum_ k e. A C )
Theorem	fsumlt 11990*	If every term in one finite sum is less than the corresponding term in another, then the first sum is less than the second. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> A =/= (/) )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> C e. RR )   &    |- ( ( ph /\ k e. A ) -> B < C )   =>    |- ( ph -> sum_ k e. A B < sum_ k e. A C )
Theorem	fsumabs 11991*	Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( abs ` sum_ k e. A B ) <_ sum_ k e. A ( abs ` B ) )
Theorem	telfsumo 11992*	Sum of a telescoping series, using half-open intervals. (Contributed by Mario Carneiro, 2-May-2016.)
|- ( k = j -> A = B )   &    |- ( k = ( j + 1 ) -> A = C )   &    |- ( k = M -> A = D )   &    |- ( k = N -> A = E )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   =>    |- ( ph -> sum_ j e. ( M..^ N ) ( B - C ) = ( D - E ) )
Theorem	telfsumo2 11993*	Sum of a telescoping series. (Contributed by Mario Carneiro, 2-May-2016.)
|- ( k = j -> A = B )   &    |- ( k = ( j + 1 ) -> A = C )   &    |- ( k = M -> A = D )   &    |- ( k = N -> A = E )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   =>    |- ( ph -> sum_ j e. ( M..^ N ) ( C - B ) = ( E - D ) )
Theorem	telfsum 11994*	Sum of a telescoping series. (Contributed by Scott Fenton, 24-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)
|- ( k = j -> A = B )   &    |- ( k = ( j + 1 ) -> A = C )   &    |- ( k = M -> A = D )   &    |- ( k = ( N + 1 ) -> A = E )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )   =>    |- ( ph -> sum_ j e. ( M ... N ) ( B - C ) = ( D - E ) )
Theorem	telfsum2 11995*	Sum of a telescoping series. (Contributed by Mario Carneiro, 15-Jun-2014.) (Revised by Mario Carneiro, 2-May-2016.)
|- ( k = j -> A = B )   &    |- ( k = ( j + 1 ) -> A = C )   &    |- ( k = M -> A = D )   &    |- ( k = ( N + 1 ) -> A = E )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )   =>    |- ( ph -> sum_ j e. ( M ... N ) ( C - B ) = ( E - D ) )
Theorem	fsumparts 11996*	Summation by parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
|- ( k = j -> ( A = B /\ V = W ) )   &    |- ( k = ( j + 1 ) -> ( A = C /\ V = X ) )   &    |- ( k = M -> ( A = D /\ V = Y ) )   &    |- ( k = N -> ( A = E /\ V = Z ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> V e. CC )   =>    |- ( ph -> sum_ j e. ( M..^ N ) ( B x. ( X - W ) ) = ( ( ( E x. Z ) - ( D x. Y ) ) - sum_ j e. ( M..^ N ) ( ( C - B ) x. X ) ) )
Theorem	fsumrelem 11997*	Lemma for fsumre 11998, fsumim 11999, and fsumcj 12000. (Contributed by Mario Carneiro, 25-Jul-2014.) (Revised by Mario Carneiro, 27-Dec-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- F : CC --> CC   &    |- ( ( x e. CC /\ y e. CC ) -> ( F ` ( x + y ) ) = ( ( F ` x ) + ( F ` y ) ) )   =>    |- ( ph -> ( F ` sum_ k e. A B ) = sum_ k e. A ( F ` B ) )
Theorem	fsumre 11998*	The real part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( Re ` sum_ k e. A B ) = sum_ k e. A ( Re ` B ) )
Theorem	fsumim 11999*	The imaginary part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( Im ` sum_ k e. A B ) = sum_ k e. A ( Im ` B ) )
Theorem	fsumcj 12000*	The complex conjugate of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( * ` sum_ k e. A B ) = sum_ k e. A ( * ` B ) )