Theorem isum1p 11389

Description: The infinite sum of a converging infinite series equals the first term plus the infinite sum of the rest of it. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)

Hypotheses

Ref	Expression
isum1p.1	|- Z = ( ZZ>= ` M )
isum1p.3	|- ( ph -> M e. ZZ )
isum1p.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
isum1p.5	|- ( ( ph /\ k e. Z ) -> A e. CC )
isum1p.6	|- ( ph -> seq M ( + , F ) e. dom ~~> )

Assertion

Ref	Expression
isum1p	|- ( ph -> sum_ k e. Z A = ( ( F ` M ) + sum_ k e. ( ZZ>= ` ( M + 1 ) ) A ) )

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

Allowed substitution hint:   A( k)

Proof of Theorem isum1p

Step	Hyp	Ref	Expression
1		isum1p.1	. . 3 |- Z = ( ZZ>= ` M )
2		eqid 2157	. . 3 |- ( ZZ>= ` ( M + 1 ) ) = ( ZZ>= ` ( M + 1 ) )
3		isum1p.3	. . . . . 6 |- ( ph -> M e. ZZ )
4		uzid 9453	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
5	3, 4	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
6		peano2uz 9494	. . . . 5 |- ( M e. ( ZZ>= ` M ) -> ( M + 1 ) e. ( ZZ>= ` M ) )
7	5, 6	syl 14	. . . 4 |- ( ph -> ( M + 1 ) e. ( ZZ>= ` M ) )
8	7, 1	eleqtrrdi 2251	. . 3 |- ( ph -> ( M + 1 ) e. Z )
9		isum1p.4	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
10		isum1p.5	. . 3 |- ( ( ph /\ k e. Z ) -> A e. CC )
11		isum1p.6	. . 3 |- ( ph -> seq M ( + , F ) e. dom ~~> )
12	1, 2, 8, 9, 10, 11	isumsplit 11388	. 2 |- ( ph -> sum_ k e. Z A = ( sum_ k e. ( M ... ( ( M + 1 ) - 1 ) ) A + sum_ k e. ( ZZ>= ` ( M + 1 ) ) A ) )
13	3	zcnd 9287	. . . . . . 7 |- ( ph -> M e. CC )
14		ax-1cn 7825	. . . . . . 7 |- 1 e. CC
15		pncan 8081	. . . . . . 7 |- ( ( M e. CC /\ 1 e. CC ) -> ( ( M + 1 ) - 1 ) = M )
16	13, 14, 15	sylancl 410	. . . . . 6 |- ( ph -> ( ( M + 1 ) - 1 ) = M )
17	16	oveq2d 5840	. . . . 5 |- ( ph -> ( M ... ( ( M + 1 ) - 1 ) ) = ( M ... M ) )
18	17	sumeq1d 11263	. . . 4 |- ( ph -> sum_ k e. ( M ... ( ( M + 1 ) - 1 ) ) A = sum_ k e. ( M ... M ) A )
19		elfzuz 9924	. . . . . . 7 |- ( k e. ( M ... M ) -> k e. ( ZZ>= ` M ) )
20	19, 1	eleqtrrdi 2251	. . . . . 6 |- ( k e. ( M ... M ) -> k e. Z )
21	20, 9	sylan2 284	. . . . 5 |- ( ( ph /\ k e. ( M ... M ) ) -> ( F ` k ) = A )
22	21	sumeq2dv 11265	. . . 4 |- ( ph -> sum_ k e. ( M ... M ) ( F ` k ) = sum_ k e. ( M ... M ) A )
23		fveq2 5468	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
24	23	eleq1d 2226	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. CC <-> ( F ` M ) e. CC ) )
25	9, 10	eqeltrd 2234	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
26	25	ralrimiva 2530	. . . . . 6 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
27	5, 1	eleqtrrdi 2251	. . . . . 6 |- ( ph -> M e. Z )
28	24, 26, 27	rspcdva 2821	. . . . 5 |- ( ph -> ( F ` M ) e. CC )
29	23	fsum1 11309	. . . . 5 |- ( ( M e. ZZ /\ ( F ` M ) e. CC ) -> sum_ k e. ( M ... M ) ( F ` k ) = ( F ` M ) )
30	3, 28, 29	syl2anc 409	. . . 4 |- ( ph -> sum_ k e. ( M ... M ) ( F ` k ) = ( F ` M ) )
31	18, 22, 30	3eqtr2d 2196	. . 3 |- ( ph -> sum_ k e. ( M ... ( ( M + 1 ) - 1 ) ) A = ( F ` M ) )
32	31	oveq1d 5839	. 2 |- ( ph -> ( sum_ k e. ( M ... ( ( M + 1 ) - 1 ) ) A + sum_ k e. ( ZZ>= ` ( M + 1 ) ) A ) = ( ( F ` M ) + sum_ k e. ( ZZ>= ` ( M + 1 ) ) A ) )
33	12, 32	eqtrd 2190	1 |- ( ph -> sum_ k e. Z A = ( ( F ` M ) + sum_ k e. ( ZZ>= ` ( M + 1 ) ) A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   dom cdm 4586   ` cfv 5170  (class class class)co 5824   CCcc 7730   1c1 7733   + caddc 7735   - cmin 8046   ZZcz 9167   ZZ>=cuz 9439   ...cfz 9912   seqcseq 10344   ~~> cli 11175   sum_csu 11250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-frec 6338  df-1o 6363  df-oadd 6367  df-er 6480  df-en 6686  df-dom 6687  df-fin 6688  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-fz 9913  df-fzo 10042  df-seqfrec 10345  df-exp 10419  df-ihash 10650  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-clim 11176  df-sumdc 11251

This theorem is referenced by:  isumnn0nn  11390  efsep  11588