Theorem fsumser 14461

Description: A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition of follows as fsum1 14476 and fsump1i 14500, which should make our notation clear and from which, along with closure fsumcl 14464, 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 2689	. . . . . 6 |- ( m = k -> ( m e. ( M ... N ) <-> k e. ( M ... N ) ) )
2		fveq2 6191	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
3	1, 2	ifbieq1d 4109	. . . . 5 |- ( m = k -> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) = if ( k e. ( M ... N ) , ( F ` k ) , 0 ) )
4		eqid 2622	. . . . 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 6201	. . . . . 6 |- ( F ` k ) e. _V
6		c0ex 10034	. . . . . 6 |- 0 e. _V
7	5, 6	ifex 4156	. . . . 5 |- if ( k e. ( M ... N ) , ( F ` k ) , 0 ) e. _V
8	3, 4, 7	fvmpt 6282	. . . 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 4116	. . . 4 |- ( ph -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = if ( k e. ( M ... N ) , A , 0 ) )
11	8, 10	sylan9eqr 2678	. . 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 3624	. . . 4 |- ( M ... N ) C_ ( M ... N )
15	14	a1i 11	. . 3 |- ( ph -> ( M ... N ) C_ ( M ... N ) )
16	11, 12, 13, 15	fsumsers 14459	. 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 12338	. . . . . 6 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
18	17, 8	syl 17	. . . . 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 4092	. . . . 5 |- ( k e. ( M ... N ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = ( F ` k ) )
20	18, 19	eqtrd 2656	. . . 4 |- ( k e. ( M ... N ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = ( F ` k ) )
21	20	adantl 482	. . 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 12825	. 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 2656	1 |- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , F ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   + caddc 9939   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  isumclim3  14490  seqabs  14546  cvgcmpce  14550  isumsplit  14572  climcndslem1  14581  climcndslem2  14582  climcnds  14583  trireciplem  14594  geolim  14601  geo2lim  14606  mertenslem2  14617  mertens  14618  efcvgfsum  14816  effsumlt  14841  prmreclem6  15625  prmrec  15626  ovollb2lem  23256  ovoliunlem1  23270  ovoliun2  23274  ovolscalem1  23281  ovolicc2lem4  23288  uniioovol  23347  uniioombllem3  23353  uniioombllem6  23356  mtest  24158  mtestbdd  24159  psercn2  24177  pserdvlem2  24182  abelthlem6  24190  logfac  24347  emcllem5  24726  lgamcvg2  24781  basellem8  24814  prmorcht  24904  pclogsum  24940  dchrisumlem2  25179  dchrmusum2  25183  dchrvmasumiflem1  25190  dchrisum0re  25202  dchrisum0lem1b  25204  dchrisum0lem2a  25206  dchrisum0lem2  25207  esumpcvgval  30140  esumcvg  30148  esumcvgsum  30150  knoppcnlem11  32493  fsumsermpt  39811  sumnnodd  39862  fourierdlem112  40435  sge0isum  40644  sge0seq  40663