Theorem isum1p 11455

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 2170	. . 3 |- ( ZZ>= ` ( M + 1 ) ) = ( ZZ>= ` ( M + 1 ) )
3		isum1p.3	. . . . . 6 |- ( ph -> M e. ZZ )
4		uzid 9501	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
5	3, 4	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
6		peano2uz 9542	. . . . 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 2264	. . 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 11454	. 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 9335	. . . . . . 7 |- ( ph -> M e. CC )
14		ax-1cn 7867	. . . . . . 7 |- 1 e. CC
15		pncan 8125	. . . . . . 7 |- ( ( M e. CC /\ 1 e. CC ) -> ( ( M + 1 ) - 1 ) = M )
16	13, 14, 15	sylancl 411	. . . . . 6 |- ( ph -> ( ( M + 1 ) - 1 ) = M )
17	16	oveq2d 5869	. . . . 5 |- ( ph -> ( M ... ( ( M + 1 ) - 1 ) ) = ( M ... M ) )
18	17	sumeq1d 11329	. . . 4 |- ( ph -> sum_ k e. ( M ... ( ( M + 1 ) - 1 ) ) A = sum_ k e. ( M ... M ) A )
19		elfzuz 9977	. . . . . . 7 |- ( k e. ( M ... M ) -> k e. ( ZZ>= ` M ) )
20	19, 1	eleqtrrdi 2264	. . . . . 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 11331	. . . 4 |- ( ph -> sum_ k e. ( M ... M ) ( F ` k ) = sum_ k e. ( M ... M ) A )
23		fveq2 5496	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
24	23	eleq1d 2239	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. CC <-> ( F ` M ) e. CC ) )
25	9, 10	eqeltrd 2247	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
26	25	ralrimiva 2543	. . . . . 6 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
27	5, 1	eleqtrrdi 2264	. . . . . 6 |- ( ph -> M e. Z )
28	24, 26, 27	rspcdva 2839	. . . . 5 |- ( ph -> ( F ` M ) e. CC )
29	23	fsum1 11375	. . . . 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 2209	. . 3 |- ( ph -> sum_ k e. ( M ... ( ( M + 1 ) - 1 ) ) A = ( F ` M ) )
32	31	oveq1d 5868	. 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 2203	1 |- ( ph -> sum_ k e. Z A = ( ( F ` M ) + sum_ k e. ( ZZ>= ` ( M + 1 ) ) A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   dom cdm 4611   ` cfv 5198  (class class class)co 5853   CCcc 7772   1c1 7775   + caddc 7777   - cmin 8090   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   seqcseq 10401   ~~> cli 11241   sum_csu 11316

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317

This theorem is referenced by:  isumnn0nn  11456  efsep  11654