P. 114 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11301-11400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	climuni 11301	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 11302	The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ~~> : dom ~~> --> CC
Theorem	climdm 11303	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 11304*	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 11305*	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 11306*	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 11307*	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 11308*	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 11309*	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 11310	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 11311	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 11312	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 11313	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 11314*	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 11315*	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 11316*	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 11317*	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 11318*	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 14055 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 11319*	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 11320*	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 11321*	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 2177. (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 11322*	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 11323*	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 11324*	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 11325*	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 11326*	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 11327*	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 11328*	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 11329*	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 11330*	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 11331*	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 11332*	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 11333*	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 11334*	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 11335*	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 11336*	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 11337*	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 11338*	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 11339*	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 11340*	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 11341*	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 11342*	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 11343*	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 11344*	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 11345*	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 11346*	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 11347*	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 11348*	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 11349*	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 11350*	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 11351*	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 11352*	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 11353*	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 11354*	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 11355*	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 11358). 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 11356*	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 11357*	Lemma for climcvg1n 11358. 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 11358*	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 11359*	A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 11355 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 11360*	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 )
4.8.2  Finite and infinite sums
Syntax	csu 11361	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 11362*	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 11530). (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 11363	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 11364	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 11365	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 11366*	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 11367*	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 11368	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 11369*	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 11370*	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 11371*	Equality inference for sum. (Contributed by NM, 2-Jan-2006.)
|- A = B   =>    |- sum_ k e. A C = sum_ k e. B C
Theorem	sumeq2i 11372*	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 11373*	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 11374*	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 11375*	Equality deduction for sum. Note that unlike sumeq2dv 11376, 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 11376*	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 11377*	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 11378*	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 11379*	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 11380*	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 11381*	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 11382*	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 11383*	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 11384	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 11385*	Lemma for sumrbdc 11387. (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 11386*	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 11387*	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 11388*	Lemma for summodc 11391. (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 11389*	Lemma for summodc 11391. (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 11390*	Lemma for summodc 11391. (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 11391*	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 11392*	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 11393*	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 11394*	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 11395*	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 11396	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 11397*	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 11398*	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 11399*	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 11400*	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 )