Theorem isum1p 11803

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 2205	. . 3 |- ( ZZ>= ` ( M + 1 ) ) = ( ZZ>= ` ( M + 1 ) )
3		isum1p.3	. . . . . 6 |- ( ph -> M e. ZZ )
4		uzid 9662	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
5	3, 4	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
6		peano2uz 9704	. . . . 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 2299	. . 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 11802	. 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 9496	. . . . . . 7 |- ( ph -> M e. CC )
14		ax-1cn 8018	. . . . . . 7 |- 1 e. CC
15		pncan 8278	. . . . . . 7 |- ( ( M e. CC /\ 1 e. CC ) -> ( ( M + 1 ) - 1 ) = M )
16	13, 14, 15	sylancl 413	. . . . . 6 |- ( ph -> ( ( M + 1 ) - 1 ) = M )
17	16	oveq2d 5960	. . . . 5 |- ( ph -> ( M ... ( ( M + 1 ) - 1 ) ) = ( M ... M ) )
18	17	sumeq1d 11677	. . . 4 |- ( ph -> sum_ k e. ( M ... ( ( M + 1 ) - 1 ) ) A = sum_ k e. ( M ... M ) A )
19		elfzuz 10143	. . . . . . 7 |- ( k e. ( M ... M ) -> k e. ( ZZ>= ` M ) )
20	19, 1	eleqtrrdi 2299	. . . . . 6 |- ( k e. ( M ... M ) -> k e. Z )
21	20, 9	sylan2 286	. . . . 5 |- ( ( ph /\ k e. ( M ... M ) ) -> ( F ` k ) = A )
22	21	sumeq2dv 11679	. . . 4 |- ( ph -> sum_ k e. ( M ... M ) ( F ` k ) = sum_ k e. ( M ... M ) A )
23		fveq2 5576	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
24	23	eleq1d 2274	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. CC <-> ( F ` M ) e. CC ) )
25	9, 10	eqeltrd 2282	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
26	25	ralrimiva 2579	. . . . . 6 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
27	5, 1	eleqtrrdi 2299	. . . . . 6 |- ( ph -> M e. Z )
28	24, 26, 27	rspcdva 2882	. . . . 5 |- ( ph -> ( F ` M ) e. CC )
29	23	fsum1 11723	. . . . 5 |- ( ( M e. ZZ /\ ( F ` M ) e. CC ) -> sum_ k e. ( M ... M ) ( F ` k ) = ( F ` M ) )
30	3, 28, 29	syl2anc 411	. . . 4 |- ( ph -> sum_ k e. ( M ... M ) ( F ` k ) = ( F ` M ) )
31	18, 22, 30	3eqtr2d 2244	. . 3 |- ( ph -> sum_ k e. ( M ... ( ( M + 1 ) - 1 ) ) A = ( F ` M ) )
32	31	oveq1d 5959	. 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 2238	1 |- ( ph -> sum_ k e. Z A = ( ( F ` M ) + sum_ k e. ( ZZ>= ` ( M + 1 ) ) A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   dom cdm 4675   ` cfv 5271  (class class class)co 5944   CCcc 7923   1c1 7926   + caddc 7928   - cmin 8243   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130   seqcseq 10592   ~~> cli 11589   sum_csu 11664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-frec 6477  df-1o 6502  df-oadd 6506  df-er 6620  df-en 6828  df-dom 6829  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-fz 10131  df-fzo 10265  df-seqfrec 10593  df-exp 10684  df-ihash 10921  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590  df-sumdc 11665

This theorem is referenced by:  isumnn0nn  11804  efsep  12002