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	xrltmaxsup 12001	The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C < sup ( { A , B } , RR* , < ) <-> ( C < A \/ C < B ) ) )
Theorem	xrmaxltsup 12002	Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( sup ( { A , B } , RR* , < ) < C <-> ( A < C /\ B < C ) ) )
Theorem	xrmaxlesup 12003	Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 10-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( sup ( { A , B } , RR* , < ) <_ C <-> ( A <_ C /\ B <_ C ) ) )
Theorem	xrmaxaddlem 12004	Lemma for xrmaxadd 12005. The case where A is real. (Contributed by Jim Kingdon, 11-May-2023.)
|- ( ( A e. RR /\ B e. RR* /\ C e. RR* ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
Theorem	xrmaxadd 12005	Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
4.8.8  The minimum of two extended reals
Theorem	xrnegiso 12006	Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
|- F = ( x e. RR* |-> -e x )   =>    |- ( F Isom < , `' < ( RR* , RR* ) /\ `' F = F )
Theorem	infxrnegsupex 12007*	The infimum of a set of extended reals A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)
|- ( ph -> E. x e. RR* ( A. y e. A -. y < x /\ A. y e. RR* ( x < y -> E. z e. A z < y ) ) )   &    |- ( ph -> A C_ RR* )   =>    |- ( ph -> inf ( A , RR* , < ) = -e sup ( { z e. RR* | -e z e. A } , RR* , < ) )
Theorem	xrnegcon1d 12008	Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   =>    |- ( ph -> ( -e A = B <-> -e B = A ) )
Theorem	xrminmax 12009	Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) = -e sup ( { -e A , -e B } , RR* , < ) )
Theorem	xrmincl 12010	The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) e. RR* )
Theorem	xrmin1inf 12011	The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) <_ A )
Theorem	xrmin2inf 12012	The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) <_ B )
Theorem	xrmineqinf 12013	The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ B <_ A ) -> inf ( { A , B } , RR* , < ) = B )
Theorem	xrltmininf 12014	Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < inf ( { B , C } , RR* , < ) <-> ( A < B /\ A < C ) ) )
Theorem	xrlemininf 12015	Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 4-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ inf ( { B , C } , RR* , < ) <-> ( A <_ B /\ A <_ C ) ) )
Theorem	xrminltinf 12016	Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> (inf ( { B , C } , RR* , < ) < A <-> ( B < A \/ C < A ) ) )
Theorem	xrminrecl 12017	The minimum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
|- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR* , < ) = inf ( { A , B } , RR , < ) )
Theorem	xrminrpcl 12018	The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) e. RR+ )
Theorem	xrminadd 12019	Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> inf ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +einf ( { B , C } , RR* , < ) ) )
Theorem	xrbdtri 12020	Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
|- ( ( ( A e. RR* /\ 0 <_ A ) /\ ( B e. RR* /\ 0 <_ B ) /\ ( C e. RR* /\ 0 < C ) ) -> inf ( { ( A +e B ) , C } , RR* , < ) <_ (inf ( { A , C } , RR* , < ) +einf ( { B , C } , RR* , < ) ) )
Theorem	iooinsup 12021	Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = ( sup ( { A , C } , RR* , < ) (,)inf ( { B , D } , RR* , < ) ) )
4.9  Elementary limits and convergence
4.9.1  Limits
Syntax	cli 12022	Extend class notation with convergence relation for limits.
class ~~>
Definition	df-clim 12023*	Define the limit relation for complex number sequences. See clim 12025 for its relational expression. (Contributed by NM, 28-Aug-2005.)
|- ~~> = { <. f , y >. | ( y e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) ) }
Theorem	climrel 12024	The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Rel ~~>
Theorem	clim 12025*	Express the predicate: The limit of complex number sequence F is A, or F converges to A. This means that for any real x, no matter how small, there always exists an integer j such that the absolute difference of any later complex number in the sequence and the limit is less than x. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ph -> F e. V )   &    |- ( ( ph /\ k e. ZZ ) -> ( F ` k ) = B )   =>    |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )
Theorem	climcl 12026	Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( F ~~> A -> A e. CC )
Theorem	clim2 12027*	Express the predicate: The limit of complex number sequence F is A, or F converges to A, with more general quantifier restrictions than clim 12025. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   =>    |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )
Theorem	clim2c 12028*	Express the predicate F converges to A. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> A e. CC )   &    |- ( ( ph /\ k e. Z ) -> B e. CC )   =>    |- ( ph -> ( F ~~> A <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < x ) )
Theorem	clim0 12029*	Express the predicate F converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   =>    |- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` B ) < x ) ) )
Theorem	clim0c 12030*	Express the predicate F converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ( ph /\ k e. Z ) -> B e. CC )   =>    |- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < x ) )
Theorem	climi 12031*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. RR+ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> F ~~> A )   =>    |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < C ) )
Theorem	climi2 12032*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. RR+ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> F ~~> A )   =>    |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < C )
Theorem	climi0 12033*	Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. RR+ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> F ~~> 0 )   =>    |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < C )
Theorem	climconst 12034*	An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> F e. V )   &    |- ( ph -> A e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   =>    |- ( ph -> F ~~> A )
Theorem	climconst2 12035	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 12036	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 12037	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 12038	The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
|- ~~> : dom ~~> --> CC
Theorem	climdm 12039	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 12040*	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 12041*	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 12042*	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 12043*	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 12044*	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 12045*	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 12046	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 12047	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 12048	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 12049	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 12050*	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 12051*	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 12052*	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 12053*	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 12054*	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 15586 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 12055*	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 12056*	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 12057*	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 2238. (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 12058*	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 12059*	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 12060*	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 12061*	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 12062*	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 12063*	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 12064*	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 12065*	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 12066*	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 12067*	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 12068*	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 12069*	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 12070*	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 12071*	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 12072*	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 12073*	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 12074*	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 12075*	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 12076*	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 12077*	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 12078*	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 12079*	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 12080*	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 12081*	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 12082*	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 12083*	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 12084*	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 12085*	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 12086*	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 12087*	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 12088*	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 12089*	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 12090*	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 12091*	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 12094). 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 12092*	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 12093*	Lemma for climcvg1n 12094. 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 12094*	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 12095*	A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 12091 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 12096*	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 12097	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 12098*	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 12267). (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 12099	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 12100	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