Theorem isum1p 11918

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 2207	. . 3 |- ( ZZ>= ` ( M + 1 ) ) = ( ZZ>= ` ( M + 1 ) )
3		isum1p.3	. . . . . 6 |- ( ph -> M e. ZZ )
4		uzid 9697	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
5	3, 4	syl 14	. . . . 5 |- ( ph -> M e. ( ZZ>= ` M ) )
6		peano2uz 9739	. . . . 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 2301	. . 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 11917	. 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 9531	. . . . . . 7 |- ( ph -> M e. CC )
14		ax-1cn 8053	. . . . . . 7 |- 1 e. CC
15		pncan 8313	. . . . . . 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 5983	. . . . 5 |- ( ph -> ( M ... ( ( M + 1 ) - 1 ) ) = ( M ... M ) )
18	17	sumeq1d 11792	. . . 4 |- ( ph -> sum_ k e. ( M ... ( ( M + 1 ) - 1 ) ) A = sum_ k e. ( M ... M ) A )
19		elfzuz 10178	. . . . . . 7 |- ( k e. ( M ... M ) -> k e. ( ZZ>= ` M ) )
20	19, 1	eleqtrrdi 2301	. . . . . 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 11794	. . . 4 |- ( ph -> sum_ k e. ( M ... M ) ( F ` k ) = sum_ k e. ( M ... M ) A )
23		fveq2 5599	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
24	23	eleq1d 2276	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. CC <-> ( F ` M ) e. CC ) )
25	9, 10	eqeltrd 2284	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
26	25	ralrimiva 2581	. . . . . 6 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
27	5, 1	eleqtrrdi 2301	. . . . . 6 |- ( ph -> M e. Z )
28	24, 26, 27	rspcdva 2889	. . . . 5 |- ( ph -> ( F ` M ) e. CC )
29	23	fsum1 11838	. . . . 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 2246	. . 3 |- ( ph -> sum_ k e. ( M ... ( ( M + 1 ) - 1 ) ) A = ( F ` M ) )
32	31	oveq1d 5982	. 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 2240	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 2178   dom cdm 4693   ` cfv 5290  (class class class)co 5967   CCcc 7958   1c1 7961   + caddc 7963   - cmin 8278   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165   seqcseq 10629   ~~> cli 11704   sum_csu 11779

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-frec 6500  df-1o 6525  df-oadd 6529  df-er 6643  df-en 6851  df-dom 6852  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-seqfrec 10630  df-exp 10721  df-ihash 10958  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-clim 11705  df-sumdc 11780

This theorem is referenced by:  isumnn0nn  11919  efsep  12117