Theorem isumspliti 8477

Description: Split off the first N terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2007.)

Hypotheses

Ref	Expression
isumsplit.1	|- M e. ZZ
isumsplit.2	|- N e. (ZZ>=` M)
isumsplit.3	|- F:(ZZ>=` M)-->CC
isumsplit.4	|- E.x(<.M, + >. seq F) ~~> x

Assertion

Ref	Expression
isumspliti	|- sum_k e. (ZZ>=` M)(F` k) = (sum_k e. (M...N)(F` k) + sum_k e. (ZZ>=` (N + 1))(F` k))

Distinct variable groups:   k,F,x   k,M,x   k,N,x

Proof of Theorem isumspliti

Step	Hyp	Ref	Expression
1		isumsplit.2	. . . . . . . 8 |- N e. (ZZ>=` M)
2		isumsplit.1	. . . . . . . . 9 |- M e. ZZ
3	2	eluz1i 7591	. . . . . . . 8 |- (N e. (ZZ>=` M) <-> (N e. ZZ /\ M <_ N))
4	1, 3	mpbi 206	. . . . . . 7 |- (N e. ZZ /\ M <_ N)
5	4	simpli 347	. . . . . 6 |- N e. ZZ
6		peano2z 7375	. . . . . 6 |- (N e. ZZ -> (N + 1) e. ZZ)
7	5, 6	ax-mp 7	. . . . 5 |- (N + 1) e. ZZ
8		isumsplit.3	. . . . . . . . 9 |- F:(ZZ>=` M)-->CC
9		fvex 4689	. . . . . . . . 9 |- (ZZ>=` M) e. _V
10		fex 4595	. . . . . . . . 9 |- ((F:(ZZ>=` M)-->CC /\ (ZZ>=` M) e. _V) -> F e. _V)
11	8, 9, 10	mp2an 761	. . . . . . . 8 |- F e. _V
12		sumex 8241	. . . . . . . 8 |- sum_k e. (ZZ>=` M)(F` k) e. _V
13		isumsplit.4	. . . . . . . . 9 |- E.x(<.M, + >. seq F) ~~> x
14	11	isumclim2 8460	. . . . . . . . 9 |- ((M e. ZZ /\ E.x(<.M, + >. seq F) ~~> x) -> (<.M, + >. seq F) ~~> sum_k e. (ZZ>=` M)(F` k))
15	2, 13, 14	mp2an 761	. . . . . . . 8 |- (<.M, + >. seq F) ~~> sum_k e. (ZZ>=` M)(F` k)
16	11, 12, 15, 8	clim2serzi 8405	. . . . . . 7 |- (N e. (ZZ>=` M) -> (<.(N + 1), + >. seq F) ~~> (sum_k e. (ZZ>=` M)(F` k) - ((<.M, + >. seq F)` N)))
17	1, 16	ax-mp 7	. . . . . 6 |- (<.(N + 1), + >. seq F) ~~> (sum_k e. (ZZ>=` M)(F` k) - ((<.M, + >. seq F)` N))
18	11	fsumserz 8259	. . . . . . . 8 |- (N e. (ZZ>=` M) -> sum_k e. (M...N)(F` k) = ((<.M, + >. seq F)` N))
19	1, 18	ax-mp 7	. . . . . . 7 |- sum_k e. (M...N)(F` k) = ((<.M, + >. seq F)` N)
20	19	opreq2i 4893	. . . . . 6 |- (sum_k e. (ZZ>=` M)(F` k) - sum_k e. (M...N)(F` k)) = (sum_k e. (ZZ>=` M)(F` k) - ((<.M, + >. seq F)` N))
21	17, 20	breqtrri 3362	. . . . 5 |- (<.(N + 1), + >. seq F) ~~> (sum_k e. (ZZ>=` M)(F` k) - sum_k e. (M...N)(F` k))
22		oprex 4907	. . . . . 6 |- (sum_k e. (ZZ>=` M)(F` k) - sum_k e. (M...N)(F` k)) e. _V
23	11, 22	isumclim 8457	. . . . 5 |- (((N + 1) e. ZZ /\ (<.(N + 1), + >. seq F) ~~> (sum_k e. (ZZ>=` M)(F` k) - sum_k e. (M...N)(F` k))) -> sum_k e. (ZZ>=` (N + 1))(F` k) = (sum_k e. (ZZ>=` M)(F` k) - sum_k e. (M...N)(F` k)))
24	7, 21, 23	mp2an 761	. . . 4 |- sum_k e. (ZZ>=` (N + 1))(F` k) = (sum_k e. (ZZ>=` M)(F` k) - sum_k e. (M...N)(F` k))
25	24	eqcomi 1888	. . 3 |- (sum_k e. (ZZ>=` M)(F` k) - sum_k e. (M...N)(F` k)) = sum_k e. (ZZ>=` (N + 1))(F` k)
26	11	isumcl 8470	. . . . 5 |- ((M e. ZZ /\ E.x(<.M, + >. seq F) ~~> x) -> sum_k e. (ZZ>=` M)(F` k) e. CC)
27	2, 13, 26	mp2an 761	. . . 4 |- sum_k e. (ZZ>=` M)(F` k) e. CC
28		elfzuz 7658	. . . . . . 7 |- (k e. (M...N) -> k e. (ZZ>=` M))
29	8	ffvelrni 4788	. . . . . . 7 |- (k e. (ZZ>=` M) -> (F` k) e. CC)
30	28, 29	syl 12	. . . . . 6 |- (k e. (M...N) -> (F` k) e. CC)
31	30	rgen 2159	. . . . 5 |- A.k e. (M...N)(F` k) e. CC
32		fsumcl 8275	. . . . 5 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(F` k) e. CC) -> sum_k e. (M...N)(F` k) e. CC)
33	1, 31, 32	mp2an 761	. . . 4 |- sum_k e. (M...N)(F` k) e. CC
34		breq2 3342	. . . . . . 7 |- (x = (sum_k e. (ZZ>=` M)(F` k) - sum_k e. (M...N)(F` k)) -> ((<.(N + 1), + >. seq F) ~~> x <-> (<.(N + 1), + >. seq F) ~~> (sum_k e. (ZZ>=` M)(F` k) - sum_k e. (M...N)(F` k))))
35	22, 34	cla4ev 2371	. . . . . 6 |- ((<.(N + 1), + >. seq F) ~~> (sum_k e. (ZZ>=` M)(F` k) - sum_k e. (M...N)(F` k)) -> E.x(<.(N + 1), + >. seq F) ~~> x)
36	21, 35	ax-mp 7	. . . . 5 |- E.x(<.(N + 1), + >. seq F) ~~> x
37	11	isumcl 8470	. . . . 5 |- (((N + 1) e. ZZ /\ E.x(<.(N + 1), + >. seq F) ~~> x) -> sum_k e. (ZZ>=` (N + 1))(F` k) e. CC)
38	7, 36, 37	mp2an 761	. . . 4 |- sum_k e. (ZZ>=` (N + 1))(F` k) e. CC
39	27, 33, 38	subaddi 6528	. . 3 |- ((sum_k e. (ZZ>=` M)(F` k) - sum_k e. (M...N)(F` k)) = sum_k e. (ZZ>=` (N + 1))(F` k) <-> (sum_k e. (M...N)(F` k) + sum_k e. (ZZ>=` (N + 1))(F` k)) = sum_k e. (ZZ>=` M)(F` k))
40	25, 39	mpbi 206	. 2 |- (sum_k e. (M...N)(F` k) + sum_k e. (ZZ>=` (N + 1))(F` k)) = sum_k e. (ZZ>=` M)(F` k)
41	40	eqcomi 1888	1 |- sum_k e. (ZZ>=` M)(F` k) = (sum_k e. (M...N)(F` k) + sum_k e. (ZZ>=` (N + 1))(F` k))

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  _Vcvv 2292  <.cop 3046   class class class wbr 3338  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  1c1 6387   + caddc 6389   - cmin 6445   <_ cle 6448  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637   seq cseqz 7774   ~~> cli 8234  sum_csu 8239

This theorem is referenced by:  isum0spliti 8478  fsumltisumii 15822  fsumleisumii 15825

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240

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