Theorem isumsplit 14572

Description: Split off the first N terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)

Hypotheses

Ref	Expression
isumsplit.1	|- Z = ( ZZ>= ` M )
isumsplit.2	|- W = ( ZZ>= ` N )
isumsplit.3	|- ( ph -> N e. Z )
isumsplit.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
isumsplit.5	|- ( ( ph /\ k e. Z ) -> A e. CC )
isumsplit.6	|- ( ph -> seq M ( + , F ) e. dom ~~> )

Assertion

Ref	Expression
isumsplit	|- ( ph -> sum_ k e. Z A = ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. W A ) )

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

Allowed substitution hint:   A( k)

Proof of Theorem isumsplit

Dummy variables j m x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumsplit.1	. 2 |- Z = ( ZZ>= ` M )
2		isumsplit.3	. . . 4 |- ( ph -> N e. Z )
3	2, 1	syl6eleq 2711	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
4		eluzel2 11692	. . 3 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
5	3, 4	syl 17	. 2 |- ( ph -> M e. ZZ )
6		isumsplit.4	. 2 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
7		isumsplit.5	. 2 |- ( ( ph /\ k e. Z ) -> A e. CC )
8		isumsplit.2	. . 3 |- W = ( ZZ>= ` N )
9		eluzelz 11697	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
10	3, 9	syl 17	. . 3 |- ( ph -> N e. ZZ )
11		uzss 11708	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
12	3, 11	syl 17	. . . . . . 7 |- ( ph -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
13	12, 8, 1	3sstr4g 3646	. . . . . 6 |- ( ph -> W C_ Z )
14	13	sselda 3603	. . . . 5 |- ( ( ph /\ k e. W ) -> k e. Z )
15	14, 6	syldan 487	. . . 4 |- ( ( ph /\ k e. W ) -> ( F ` k ) = A )
16	14, 7	syldan 487	. . . 4 |- ( ( ph /\ k e. W ) -> A e. CC )
17		isumsplit.6	. . . . 5 |- ( ph -> seq M ( + , F ) e. dom ~~> )
18	6, 7	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
19	1, 2, 18	iserex 14387	. . . . 5 |- ( ph -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )
20	17, 19	mpbid 222	. . . 4 |- ( ph -> seq N ( + , F ) e. dom ~~> )
21	8, 10, 15, 16, 20	isumclim2 14489	. . 3 |- ( ph -> seq N ( + , F ) ~~> sum_ k e. W A )
22		fzfid 12772	. . . 4 |- ( ph -> ( M ... ( N - 1 ) ) e. Fin )
23		elfzuz 12338	. . . . . 6 |- ( k e. ( M ... ( N - 1 ) ) -> k e. ( ZZ>= ` M ) )
24	23, 1	syl6eleqr 2712	. . . . 5 |- ( k e. ( M ... ( N - 1 ) ) -> k e. Z )
25	24, 7	sylan2 491	. . . 4 |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> A e. CC )
26	22, 25	fsumcl 14464	. . 3 |- ( ph -> sum_ k e. ( M ... ( N - 1 ) ) A e. CC )
27	14, 18	syldan 487	. . . . 5 |- ( ( ph /\ k e. W ) -> ( F ` k ) e. CC )
28	8, 10, 27	serf 12829	. . . 4 |- ( ph -> seq N ( + , F ) : W --> CC )
29	28	ffvelrnda 6359	. . 3 |- ( ( ph /\ j e. W ) -> ( seq N ( + , F ) ` j ) e. CC )
30	5	zred 11482	. . . . . . . . . . . 12 |- ( ph -> M e. RR )
31	30	ltm1d 10956	. . . . . . . . . . 11 |- ( ph -> ( M - 1 ) < M )
32		peano2zm 11420	. . . . . . . . . . . . 13 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
33	5, 32	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( M - 1 ) e. ZZ )
34		fzn 12357	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ ( M - 1 ) e. ZZ ) -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
35	5, 33, 34	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
36	31, 35	mpbid 222	. . . . . . . . . 10 |- ( ph -> ( M ... ( M - 1 ) ) = (/) )
37	36	sumeq1d 14431	. . . . . . . . 9 |- ( ph -> sum_ k e. ( M ... ( M - 1 ) ) A = sum_ k e. (/) A )
38	37	adantr 481	. . . . . . . 8 |- ( ( ph /\ j e. W ) -> sum_ k e. ( M ... ( M - 1 ) ) A = sum_ k e. (/) A )
39		sum0 14452	. . . . . . . 8 |- sum_ k e. (/) A = 0
40	38, 39	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ j e. W ) -> sum_ k e. ( M ... ( M - 1 ) ) A = 0 )
41	40	oveq1d 6665	. . . . . 6 |- ( ( ph /\ j e. W ) -> ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) = ( 0 + ( seq M ( + , F ) ` j ) ) )
42	13	sselda 3603	. . . . . . . 8 |- ( ( ph /\ j e. W ) -> j e. Z )
43	1, 5, 18	serf 12829	. . . . . . . . 9 |- ( ph -> seq M ( + , F ) : Z --> CC )
44	43	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
45	42, 44	syldan 487	. . . . . . 7 |- ( ( ph /\ j e. W ) -> ( seq M ( + , F ) ` j ) e. CC )
46	45	addid2d 10237	. . . . . 6 |- ( ( ph /\ j e. W ) -> ( 0 + ( seq M ( + , F ) ` j ) ) = ( seq M ( + , F ) ` j ) )
47	41, 46	eqtr2d 2657	. . . . 5 |- ( ( ph /\ j e. W ) -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) )
48		oveq1 6657	. . . . . . . . 9 |- ( N = M -> ( N - 1 ) = ( M - 1 ) )
49	48	oveq2d 6666	. . . . . . . 8 |- ( N = M -> ( M ... ( N - 1 ) ) = ( M ... ( M - 1 ) ) )
50	49	sumeq1d 14431	. . . . . . 7 |- ( N = M -> sum_ k e. ( M ... ( N - 1 ) ) A = sum_ k e. ( M ... ( M - 1 ) ) A )
51		seqeq1 12804	. . . . . . . 8 |- ( N = M -> seq N ( + , F ) = seq M ( + , F ) )
52	51	fveq1d 6193	. . . . . . 7 |- ( N = M -> ( seq N ( + , F ) ` j ) = ( seq M ( + , F ) ` j ) )
53	50, 52	oveq12d 6668	. . . . . 6 |- ( N = M -> ( sum_ k e. ( M ... ( N - 1 ) ) A + ( seq N ( + , F ) ` j ) ) = ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) )
54	53	eqeq2d 2632	. . . . 5 |- ( N = M -> ( ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( N - 1 ) ) A + ( seq N ( + , F ) ` j ) ) <-> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) ) )
55	47, 54	syl5ibrcom 237	. . . 4 |- ( ( ph /\ j e. W ) -> ( N = M -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( N - 1 ) ) A + ( seq N ( + , F ) ` j ) ) ) )
56		addcl 10018	. . . . . . . 8 |- ( ( k e. CC /\ m e. CC ) -> ( k + m ) e. CC )
57	56	adantl 482	. . . . . . 7 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( k e. CC /\ m e. CC ) ) -> ( k + m ) e. CC )
58		addass 10023	. . . . . . . 8 |- ( ( k e. CC /\ m e. CC /\ x e. CC ) -> ( ( k + m ) + x ) = ( k + ( m + x ) ) )
59	58	adantl 482	. . . . . . 7 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( k e. CC /\ m e. CC /\ x e. CC ) ) -> ( ( k + m ) + x ) = ( k + ( m + x ) ) )
60		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> j e. W )
61		simpll 790	. . . . . . . . . . 11 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ph )
62	10	zcnd 11483	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
63		ax-1cn 9994	. . . . . . . . . . . . 13 |- 1 e. CC
64		npcan 10290	. . . . . . . . . . . . 13 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
65	62, 63, 64	sylancl 694	. . . . . . . . . . . 12 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
66	65	eqcomd 2628	. . . . . . . . . . 11 |- ( ph -> N = ( ( N - 1 ) + 1 ) )
67	61, 66	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> N = ( ( N - 1 ) + 1 ) )
68	67	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ZZ>= ` N ) = ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
69	8, 68	syl5eq 2668	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> W = ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
70	60, 69	eleqtrd 2703	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> j e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
71	5	adantr 481	. . . . . . . 8 |- ( ( ph /\ j e. W ) -> M e. ZZ )
72		eluzp1m1 11711	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
73	71, 72	sylan 488	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
74		elfzuz 12338	. . . . . . . . 9 |- ( k e. ( M ... j ) -> k e. ( ZZ>= ` M ) )
75	74, 1	syl6eleqr 2712	. . . . . . . 8 |- ( k e. ( M ... j ) -> k e. Z )
76	61, 75, 18	syl2an 494	. . . . . . 7 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( M ... j ) ) -> ( F ` k ) e. CC )
77	57, 59, 70, 73, 76	seqsplit 12834	. . . . . 6 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( + , F ) ` j ) = ( ( seq M ( + , F ) ` ( N - 1 ) ) + ( seq ( ( N - 1 ) + 1 ) ( + , F ) ` j ) ) )
78	61, 24, 6	syl2an 494	. . . . . . . 8 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( M ... ( N - 1 ) ) ) -> ( F ` k ) = A )
79	61, 24, 7	syl2an 494	. . . . . . . 8 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( M ... ( N - 1 ) ) ) -> A e. CC )
80	78, 73, 79	fsumser 14461	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ k e. ( M ... ( N - 1 ) ) A = ( seq M ( + , F ) ` ( N - 1 ) ) )
81	67	seqeq1d 12807	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> seq N ( + , F ) = seq ( ( N - 1 ) + 1 ) ( + , F ) )
82	81	fveq1d 6193	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq N ( + , F ) ` j ) = ( seq ( ( N - 1 ) + 1 ) ( + , F ) ` j ) )
83	80, 82	oveq12d 6668	. . . . . 6 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( sum_ k e. ( M ... ( N - 1 ) ) A + ( seq N ( + , F ) ` j ) ) = ( ( seq M ( + , F ) ` ( N - 1 ) ) + ( seq ( ( N - 1 ) + 1 ) ( + , F ) ` j ) ) )
84	77, 83	eqtr4d 2659	. . . . 5 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( N - 1 ) ) A + ( seq N ( + , F ) ` j ) ) )
85	84	ex 450	. . . 4 |- ( ( ph /\ j e. W ) -> ( N e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( N - 1 ) ) A + ( seq N ( + , F ) ` j ) ) ) )
86		uzp1 11721	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
87	3, 86	syl 17	. . . . 5 |- ( ph -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
88	87	adantr 481	. . . 4 |- ( ( ph /\ j e. W ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
89	55, 85, 88	mpjaod 396	. . 3 |- ( ( ph /\ j e. W ) -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( N - 1 ) ) A + ( seq N ( + , F ) ` j ) ) )
90	8, 10, 21, 26, 17, 29, 89	climaddc2 14366	. 2 |- ( ph -> seq M ( + , F ) ~~> ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. W A ) )
91	1, 5, 6, 7, 90	isumclim 14488	1 |- ( ph -> sum_ k e. Z A = ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. W A ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ~~> cli 14215   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  isum1p  14573  geolim2  14602  mertenslem2  14617  mertens  14618  effsumlt  14841  eirrlem  14932  rpnnen2lem8  14950  prmreclem6  15625  aaliou3lem7  24104  abelthlem7  24192  log2tlbnd  24672  subfaclim  31170  knoppndvlem6  32508  binomcxplemnn0  38548  stirlinglem12  40302