Theorem fsumser 13311

Description: A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition of follows as fsum1 13322 and fsump1i 13340, which should make our notation clear and from which, along with closure fsumcl 13314, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)

Hypotheses

Ref	Expression
fsumser.1	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = A )
fsumser.2	|- ( ph -> N e. ( ZZ>= ` M ) )
fsumser.3	|- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )

Assertion

Ref	Expression
fsumser	|- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , F ) ` N ) )

Distinct variable groups:   k, F   k, M   k, N   ph, k

Allowed substitution hint:   A( k)

Proof of Theorem fsumser

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2523	. . . . . 6 |- ( m = k -> ( m e. ( M ... N ) <-> k e. ( M ... N ) ) )
2		fveq2 5791	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
3	1, 2	ifbieq1d 3912	. . . . 5 |- ( m = k -> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) = if ( k e. ( M ... N ) , ( F ` k ) , 0 ) )
4		eqid 2451	. . . . 5 |- ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) = ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) )
5		fvex 5801	. . . . . 6 |- ( F ` k ) e. _V
6		c0ex 9483	. . . . . 6 |- 0 e. _V
7	5, 6	ifex 3958	. . . . 5 |- if ( k e. ( M ... N ) , ( F ` k ) , 0 ) e. _V
8	3, 4, 7	fvmpt 5875	. . . 4 |- ( k e. ( ZZ>= ` M ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = if ( k e. ( M ... N ) , ( F ` k ) , 0 ) )
9		fsumser.1	. . . . 5 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = A )
10	9	ifeq1da 3919	. . . 4 |- ( ph -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = if ( k e. ( M ... N ) , A , 0 ) )
11	8, 10	sylan9eqr 2514	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = if ( k e. ( M ... N ) , A , 0 ) )
12		fsumser.2	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
13		fsumser.3	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
14		ssid 3475	. . . 4 |- ( M ... N ) C_ ( M ... N )
15	14	a1i 11	. . 3 |- ( ph -> ( M ... N ) C_ ( M ... N ) )
16	11, 12, 13, 15	fsumsers 13309	. 2 |- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ) ` N ) )
17		elfzuz 11552	. . . . . 6 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
18	17, 8	syl 16	. . . . 5 |- ( k e. ( M ... N ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = if ( k e. ( M ... N ) , ( F ` k ) , 0 ) )
19		iftrue 3897	. . . . 5 |- ( k e. ( M ... N ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = ( F ` k ) )
20	18, 19	eqtrd 2492	. . . 4 |- ( k e. ( M ... N ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = ( F ` k ) )
21	20	adantl 466	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = ( F ` k ) )
22	12, 21	seqfveq 11933	. 2 |- ( ph -> ( seq M ( + , ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ) ` N ) = ( seq M ( + , F ) ` N ) )
23	16, 22	eqtrd 2492	1 |- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , F ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   C_ wss 3428   ifcif 3891   |-> cmpt 4450   ` cfv 5518  (class class class)co 6192   CCcc 9383   0cc0 9385   + caddc 9388   ZZ>=cuz 10964   ...cfz 11540   seqcseq 11909   sum_csu 13267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-sup 7794  df-oi 7827  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-rp 11095  df-fz 11541  df-fzo 11652  df-seq 11910  df-exp 11969  df-hash 12207  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-clim 13070  df-sum 13268

This theorem is referenced by:  isumclim3  13330  seqabs  13381  cvgcmpce  13385  isumsplit  13407  climcndslem1  13416  climcndslem2  13417  climcnds  13418  trireciplem  13428  geolim  13434  geo2lim  13439  mertenslem2  13449  mertens  13450  efcvgfsum  13475  effsumlt  13499  prmreclem6  14086  prmrec  14087  ovollb2lem  21089  ovoliunlem1  21103  ovoliun2  21107  ovolscalem1  21114  ovolicc2lem4  21121  uniioovol  21177  uniioombllem3  21183  uniioombllem6  21186  mtest  21987  mtestbdd  21988  psercn2  22006  pserdvlem2  22011  abelthlem6  22019  logfac  22167  emcllem5  22511  basellem8  22543  prmorcht  22634  pclogsum  22672  dchrisumlem2  22857  dchrmusum2  22861  dchrvmasumiflem1  22868  dchrisum0re  22880  dchrisum0lem1b  22882  dchrisum0lem2a  22884  dchrisum0lem2  22885  esumpcvgval  26663  esumcvg  26671  lgamcvg2  27177