P. 121 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12001-12100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	climconst2 12001	A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- ( ZZ>= ` M ) C_ Z   &    |- Z e. _V   =>    |- ( ( A e. CC /\ M e. ZZ ) -> ( Z X. { A } ) ~~> A )
Theorem	climz 12002	The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- ( ZZ X. { 0 } ) ~~> 0
Theorem	climuni 12003	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
|- ( ( F ~~> A /\ F ~~> B ) -> A = B )
Theorem	fclim 12004	The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ~~> : dom ~~> --> CC
Theorem	climdm 12005	Two ways to express that a function has a limit. (The expression ( ~~> ` F ) is sometimes useful as a shorthand for "the unique limit of the function F"). (Contributed by Mario Carneiro, 18-Mar-2014.)
|- ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) )
Theorem	climeu 12006*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
|- ( F ~~> A -> E! x F ~~> x )
Theorem	climreu 12007*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
|- ( F ~~> A -> E! x e. CC F ~~> x )
Theorem	climmo 12008*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)
|- E* x F ~~> x
Theorem	climeq 12009*	Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> F e. V )   &    |- ( ph -> G e. W )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( G ` k ) )   =>    |- ( ph -> ( F ~~> A <-> G ~~> A ) )
Theorem	climmpt 12010*	Exhibit a function G with the same convergence properties as the not-quite-function F. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- G = ( k e. Z |-> ( F ` k ) )   =>    |- ( ( M e. ZZ /\ F e. V ) -> ( F ~~> A <-> G ~~> A ) )
Theorem	2clim 12011*	If two sequences converge to each other, they converge to the same limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> G e. V )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )   &    |- ( ph -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - ( G ` k ) ) ) < x )   &    |- ( ph -> F ~~> A )   =>    |- ( ph -> G ~~> A )
Theorem	climshftlemg 12012	A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)
|- ( ( M e. ZZ /\ F e. V ) -> ( F ~~> A -> ( F shift M ) ~~> A ) )
Theorem	climres 12013	A function restricted to upper integers converges iff the original function converges. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- ( ( M e. ZZ /\ F e. V ) -> ( ( F |` ( ZZ>= ` M ) ) ~~> A <-> F ~~> A ) )
Theorem	climshft 12014	A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- ( ( M e. ZZ /\ F e. V ) -> ( ( F shift M ) ~~> A <-> F ~~> A ) )
Theorem	serclim0 12015	The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
|- ( M e. ZZ -> seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 )
Theorem	climshft2 12016*	A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> K e. ZZ )   &    |- ( ph -> F e. W )   &    |- ( ph -> G e. X )   &    |- ( ( ph /\ k e. Z ) -> ( G ` ( k + K ) ) = ( F ` k ) )   =>    |- ( ph -> ( F ~~> A <-> G ~~> A ) )
Theorem	climabs0 12017*	Convergence to zero of the absolute value is equivalent to convergence to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ph -> G e. W )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )   =>    |- ( ph -> ( F ~~> 0 <-> G ~~> 0 ) )
Theorem	climcn1 12018*	Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. B )   &    |- ( ( ph /\ z e. B ) -> ( F ` z ) e. CC )   &    |- ( ph -> G ~~> A )   &    |- ( ph -> H e. W )   &    |- ( ( ph /\ x e. RR+ ) -> E. y e. RR+ A. z e. B ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. B )   &    |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( F ` ( G ` k ) ) )   =>    |- ( ph -> H ~~> ( F ` A ) )
Theorem	climcn2 12019*	Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. C )   &    |- ( ph -> B e. D )   &    |- ( ( ph /\ ( u e. C /\ v e. D ) ) -> ( u F v ) e. CC )   &    |- ( ph -> G ~~> A )   &    |- ( ph -> H ~~> B )   &    |- ( ph -> K e. W )   &    |- ( ( ph /\ x e. RR+ ) -> E. y e. RR+ E. z e. RR+ A. u e. C A. v e. D ( ( ( abs ` ( u - A ) ) < y /\ ( abs ` ( v - B ) ) < z ) -> ( abs ` ( ( u F v ) - ( A F B ) ) ) < x ) )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. C )   &    |- ( ( ph /\ k e. Z ) -> ( H ` k ) e. D )   &    |- ( ( ph /\ k e. Z ) -> ( K ` k ) = ( ( G ` k ) F ( H ` k ) ) )   =>    |- ( ph -> K ~~> ( A F B ) )
Theorem	addcn2 12020*	Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (We write out the definition directly because df-cn and df-cncf are not yet available to us. See addcncntop 15553 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u + v ) - ( B + C ) ) ) < A ) )
Theorem	subcn2 12021*	Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) )
Theorem	mulcn2 12022*	Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u x. v ) - ( B x. C ) ) ) < A ) )
Theorem	reccn2ap 12023*	The reciprocal function is continuous. The class T is just for convenience in writing the proof and typically would be passed in as an instance of eqid 2234. (Contributed by Mario Carneiro, 9-Feb-2014.) Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
|- T = (inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) x. ( ( abs ` A ) / 2 ) )   =>    |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> E. y e. RR+ A. z e. { w e. CC | w # 0 } ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) )
Theorem	cn1lem 12024*	A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
|- F : CC --> CC   &    |- ( ( z e. CC /\ A e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) )   =>    |- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
Theorem	abscn2 12025*	The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
|- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( abs ` z ) - ( abs ` A ) ) ) < x ) )
Theorem	cjcn2 12026*	The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
|- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( * ` z ) - ( * ` A ) ) ) < x ) )
Theorem	recn2 12027*	The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
|- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( Re ` z ) - ( Re ` A ) ) ) < x ) )
Theorem	imcn2 12028*	The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
|- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( Im ` z ) - ( Im ` A ) ) ) < x ) )
Theorem	climcn1lem 12029*	The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> G e. W )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- H : CC --> CC   &    |- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( H ` z ) - ( H ` A ) ) ) < x ) )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( H ` ( F ` k ) ) )   =>    |- ( ph -> G ~~> ( H ` A ) )
Theorem	climabs 12030*	Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> G e. W )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )   =>    |- ( ph -> G ~~> ( abs ` A ) )
Theorem	climcj 12031*	Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> G e. W )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( * ` ( F ` k ) ) )   =>    |- ( ph -> G ~~> ( * ` A ) )
Theorem	climre 12032*	Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> G e. W )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( Re ` ( F ` k ) ) )   =>    |- ( ph -> G ~~> ( Re ` A ) )
Theorem	climim 12033*	Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> G e. W )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( Im ` ( F ` k ) ) )   =>    |- ( ph -> G ~~> ( Im ` A ) )
Theorem	climrecl 12034*	The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   =>    |- ( ph -> A e. RR )
Theorem	climge0 12035*	A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) )   =>    |- ( ph -> 0 <_ A )
Theorem	climadd 12036*	Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> H e. X )   &    |- ( ph -> G ~~> B )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) + ( G ` k ) ) )   =>    |- ( ph -> H ~~> ( A + B ) )
Theorem	climmul 12037*	Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> H e. X )   &    |- ( ph -> G ~~> B )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) )   =>    |- ( ph -> H ~~> ( A x. B ) )
Theorem	climsub 12038*	Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> H e. X )   &    |- ( ph -> G ~~> B )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) - ( G ` k ) ) )   =>    |- ( ph -> H ~~> ( A - B ) )
Theorem	climaddc1 12039*	Limit of a constant C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> C e. CC )   &    |- ( ph -> G e. W )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( ( F ` k ) + C ) )   =>    |- ( ph -> G ~~> ( A + C ) )
Theorem	climaddc2 12040*	Limit of a constant C added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> C e. CC )   &    |- ( ph -> G e. W )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( C + ( F ` k ) ) )   =>    |- ( ph -> G ~~> ( C + A ) )
Theorem	climmulc2 12041*	Limit of a sequence multiplied by a constant C. Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> C e. CC )   &    |- ( ph -> G e. W )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( C x. ( F ` k ) ) )   =>    |- ( ph -> G ~~> ( C x. A ) )
Theorem	climsubc1 12042*	Limit of a constant C subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> C e. CC )   &    |- ( ph -> G e. W )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( ( F ` k ) - C ) )   =>    |- ( ph -> G ~~> ( A - C ) )
Theorem	climsubc2 12043*	Limit of a constant C minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> C e. CC )   &    |- ( ph -> G e. W )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( C - ( F ` k ) ) )   =>    |- ( ph -> G ~~> ( C - A ) )
Theorem	climle 12044*	Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F ~~> A )   &    |- ( ph -> 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	climsqz 12045*	Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-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 ) -> ( F ` k ) <_ ( G ` k ) )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) <_ A )   =>    |- ( ph -> G ~~> A )
Theorem	climsqz2 12046*	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 12047*	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 12048*	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 12049*	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 12050*	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 12051*	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 12052*	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 12053*	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 12054*	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 12055*	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 12056*	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 12057*	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 12060). 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 12058*	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 12059*	Lemma for climcvg1n 12060. 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 12060*	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 12061*	A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 12057 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 12062*	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 12063	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 12064*	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 12233). (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 12065	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 12066	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 12067	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 12068*	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 12069*	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 12070	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 12071*	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 12072*	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 12073*	Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
|- A = B   =>    |- sum_ k e. A C = sum_ k e. B C
Theorem	sumeq2i 12074*	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 12075*	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 12076*	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 12077*	Equality deduction for sum. Note that unlike sumeq2dv 12078, 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 12078*	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 12079*	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 12080*	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 12081*	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 12082*	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 12083*	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 12084*	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 12085*	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 12086*	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 12087	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 12088*	Lemma for sumrbdc 12090. (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 12089*	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 12090*	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 12091*	Lemma for summodc 12094. (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 12092*	Lemma for summodc 12094. (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 12093*	Lemma for summodc 12094. (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 12094*	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 12095*	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 12096*	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 12097*	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 12098*	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 12099	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 12100*	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 )