Theorem fsumser 13554

Description: A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition of follows as fsum1 13566 and fsump1i 13586, which should make our notation clear and from which, along with closure fsumcl 13557, 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 2454	. . . . . 6 |- ( m = k -> ( m e. ( M ... N ) <-> k e. ( M ... N ) ) )
2		fveq2 5774	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
3	1, 2	ifbieq1d 3880	. . . . 5 |- ( m = k -> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) = if ( k e. ( M ... N ) , ( F ` k ) , 0 ) )
4		eqid 2382	. . . . 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 5784	. . . . . 6 |- ( F ` k ) e. _V
6		c0ex 9501	. . . . . 6 |- 0 e. _V
7	5, 6	ifex 3925	. . . . 5 |- if ( k e. ( M ... N ) , ( F ` k ) , 0 ) e. _V
8	3, 4, 7	fvmpt 5857	. . . 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 3887	. . . 4 |- ( ph -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = if ( k e. ( M ... N ) , A , 0 ) )
11	8, 10	sylan9eqr 2445	. . 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 3436	. . . 4 |- ( M ... N ) C_ ( M ... N )
15	14	a1i 11	. . 3 |- ( ph -> ( M ... N ) C_ ( M ... N ) )
16	11, 12, 13, 15	fsumsers 13552	. 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 11605	. . . . . 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 3863	. . . . 5 |- ( k e. ( M ... N ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = ( F ` k ) )
20	18, 19	eqtrd 2423	. . . 4 |- ( k e. ( M ... N ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = ( F ` k ) )
21	20	adantl 464	. . 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 12034	. 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 2423	1 |- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , F ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1399   e. wcel 1826   C_ wss 3389   ifcif 3857   |-> cmpt 4425   ` cfv 5496  (class class class)co 6196   CCcc 9401   0cc0 9403   + caddc 9406   ZZ>=cuz 11001   ...cfz 11593   seqcseq 12010   sum_csu 13510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-fzo 11718  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-sum 13511

This theorem is referenced by:  isumclim3  13576  seqabs  13630  cvgcmpce  13634  isumsplit  13654  climcndslem1  13663  climcndslem2  13664  climcnds  13665  trireciplem  13675  geolim  13681  geo2lim  13686  mertenslem2  13696  mertens  13697  efcvgfsum  13823  effsumlt  13848  prmreclem6  14441  prmrec  14442  ovollb2lem  21984  ovoliunlem1  21998  ovoliun2  22002  ovolscalem1  22009  ovolicc2lem4  22016  uniioovol  22073  uniioombllem3  22079  uniioombllem6  22082  mtest  22884  mtestbdd  22885  psercn2  22903  pserdvlem2  22908  abelthlem6  22916  logfac  23073  emcllem5  23446  basellem8  23478  prmorcht  23569  pclogsum  23607  dchrisumlem2  23792  dchrmusum2  23796  dchrvmasumiflem1  23803  dchrisum0re  23815  dchrisum0lem1b  23817  dchrisum0lem2a  23819  dchrisum0lem2  23820  esumpcvgval  28226  esumcvg  28234  esumcvgsum  28236  lgamcvg2  28786  sumnnodd  31802  fourierdlem112  32167