Theorem isumsplit 11454

Description: Split off the first N terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 21-Oct-2022.)

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	eleqtrdi 2263	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
4		eluzel2 9492	. . 3 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
5	3, 4	syl 14	. 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 9496	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
10	3, 9	syl 14	. . 3 |- ( ph -> N e. ZZ )
11		uzss 9507	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
12	3, 11	syl 14	. . . . . . 7 |- ( ph -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
13	12, 8, 1	3sstr4g 3190	. . . . . 6 |- ( ph -> W C_ Z )
14	13	sselda 3147	. . . . 5 |- ( ( ph /\ k e. W ) -> k e. Z )
15	14, 6	syldan 280	. . . 4 |- ( ( ph /\ k e. W ) -> ( F ` k ) = A )
16	14, 7	syldan 280	. . . 4 |- ( ( ph /\ k e. W ) -> A e. CC )
17		isumsplit.6	. . . . 5 |- ( ph -> seq M ( + , F ) e. dom ~~> )
18	6, 7	eqeltrd 2247	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
19	1, 2, 18	iserex 11302	. . . . 5 |- ( ph -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )
20	17, 19	mpbid 146	. . . 4 |- ( ph -> seq N ( + , F ) e. dom ~~> )
21	8, 10, 15, 16, 20	isumclim2 11385	. . 3 |- ( ph -> seq N ( + , F ) ~~> sum_ k e. W A )
22		peano2zm 9250	. . . . . 6 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
23	10, 22	syl 14	. . . . 5 |- ( ph -> ( N - 1 ) e. ZZ )
24	5, 23	fzfigd 10387	. . . 4 |- ( ph -> ( M ... ( N - 1 ) ) e. Fin )
25		elfzuz 9977	. . . . . 6 |- ( k e. ( M ... ( N - 1 ) ) -> k e. ( ZZ>= ` M ) )
26	25, 1	eleqtrrdi 2264	. . . . 5 |- ( k e. ( M ... ( N - 1 ) ) -> k e. Z )
27	26, 7	sylan2 284	. . . 4 |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> A e. CC )
28	24, 27	fsumcl 11363	. . 3 |- ( ph -> sum_ k e. ( M ... ( N - 1 ) ) A e. CC )
29	14, 18	syldan 280	. . . . 5 |- ( ( ph /\ k e. W ) -> ( F ` k ) e. CC )
30	8, 10, 29	serf 10430	. . . 4 |- ( ph -> seq N ( + , F ) : W --> CC )
31	30	ffvelrnda 5631	. . 3 |- ( ( ph /\ j e. W ) -> ( seq N ( + , F ) ` j ) e. CC )
32	5	zred 9334	. . . . . . . . . . . 12 |- ( ph -> M e. RR )
33	32	ltm1d 8848	. . . . . . . . . . 11 |- ( ph -> ( M - 1 ) < M )
34		peano2zm 9250	. . . . . . . . . . . . 13 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
35	5, 34	syl 14	. . . . . . . . . . . 12 |- ( ph -> ( M - 1 ) e. ZZ )
36		fzn 9998	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ ( M - 1 ) e. ZZ ) -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
37	5, 35, 36	syl2anc 409	. . . . . . . . . . 11 |- ( ph -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
38	33, 37	mpbid 146	. . . . . . . . . 10 |- ( ph -> ( M ... ( M - 1 ) ) = (/) )
39	38	sumeq1d 11329	. . . . . . . . 9 |- ( ph -> sum_ k e. ( M ... ( M - 1 ) ) A = sum_ k e. (/) A )
40	39	adantr 274	. . . . . . . 8 |- ( ( ph /\ j e. W ) -> sum_ k e. ( M ... ( M - 1 ) ) A = sum_ k e. (/) A )
41		sum0 11351	. . . . . . . 8 |- sum_ k e. (/) A = 0
42	40, 41	eqtrdi 2219	. . . . . . 7 |- ( ( ph /\ j e. W ) -> sum_ k e. ( M ... ( M - 1 ) ) A = 0 )
43	42	oveq1d 5868	. . . . . 6 |- ( ( ph /\ j e. W ) -> ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) = ( 0 + ( seq M ( + , F ) ` j ) ) )
44	13	sselda 3147	. . . . . . . 8 |- ( ( ph /\ j e. W ) -> j e. Z )
45	1, 5, 18	serf 10430	. . . . . . . . 9 |- ( ph -> seq M ( + , F ) : Z --> CC )
46	45	ffvelrnda 5631	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
47	44, 46	syldan 280	. . . . . . 7 |- ( ( ph /\ j e. W ) -> ( seq M ( + , F ) ` j ) e. CC )
48	47	addid2d 8069	. . . . . 6 |- ( ( ph /\ j e. W ) -> ( 0 + ( seq M ( + , F ) ` j ) ) = ( seq M ( + , F ) ` j ) )
49	43, 48	eqtr2d 2204	. . . . 5 |- ( ( ph /\ j e. W ) -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) )
50		oveq1 5860	. . . . . . . . 9 |- ( N = M -> ( N - 1 ) = ( M - 1 ) )
51	50	oveq2d 5869	. . . . . . . 8 |- ( N = M -> ( M ... ( N - 1 ) ) = ( M ... ( M - 1 ) ) )
52	51	sumeq1d 11329	. . . . . . 7 |- ( N = M -> sum_ k e. ( M ... ( N - 1 ) ) A = sum_ k e. ( M ... ( M - 1 ) ) A )
53		seqeq1 10404	. . . . . . . 8 |- ( N = M -> seq N ( + , F ) = seq M ( + , F ) )
54	53	fveq1d 5498	. . . . . . 7 |- ( N = M -> ( seq N ( + , F ) ` j ) = ( seq M ( + , F ) ` j ) )
55	52, 54	oveq12d 5871	. . . . . 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 ) ) )
56	55	eqeq2d 2182	. . . . 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 ) ) ) )
57	49, 56	syl5ibrcom 156	. . . 4 |- ( ( ph /\ j e. W ) -> ( N = M -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( N - 1 ) ) A + ( seq N ( + , F ) ` j ) ) ) )
58		addcl 7899	. . . . . . . 8 |- ( ( k e. CC /\ m e. CC ) -> ( k + m ) e. CC )
59	58	adantl 275	. . . . . . 7 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( k e. CC /\ m e. CC ) ) -> ( k + m ) e. CC )
60		addass 7904	. . . . . . . 8 |- ( ( k e. CC /\ m e. CC /\ x e. CC ) -> ( ( k + m ) + x ) = ( k + ( m + x ) ) )
61	60	adantl 275	. . . . . . 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 ) ) )
62		simplr 525	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> j e. W )
63		simpll 524	. . . . . . . . . . 11 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ph )
64	10	zcnd 9335	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
65		ax-1cn 7867	. . . . . . . . . . . . 13 |- 1 e. CC
66		npcan 8128	. . . . . . . . . . . . 13 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
67	64, 65, 66	sylancl 411	. . . . . . . . . . . 12 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
68	67	eqcomd 2176	. . . . . . . . . . 11 |- ( ph -> N = ( ( N - 1 ) + 1 ) )
69	63, 68	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> N = ( ( N - 1 ) + 1 ) )
70	69	fveq2d 5500	. . . . . . . . 9 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ZZ>= ` N ) = ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
71	8, 70	eqtrid 2215	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> W = ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
72	62, 71	eleqtrd 2249	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> j e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
73	5	adantr 274	. . . . . . . 8 |- ( ( ph /\ j e. W ) -> M e. ZZ )
74		eluzp1m1 9510	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
75	73, 74	sylan 281	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
76	1	eleq2i 2237	. . . . . . . . . 10 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
77	76, 6	sylan2br 286	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = A )
78	63, 77	sylan 281	. . . . . . . 8 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = A )
79	76, 7	sylan2br 286	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> A e. CC )
80	63, 79	sylan 281	. . . . . . . 8 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> A e. CC )
81	78, 80	eqeltrd 2247	. . . . . . 7 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
82	59, 61, 72, 75, 81	seq3split 10435	. . . . . 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 ) ) )
83	78, 75, 80	fsum3ser 11360	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ k e. ( M ... ( N - 1 ) ) A = ( seq M ( + , F ) ` ( N - 1 ) ) )
84	69	seqeq1d 10407	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> seq N ( + , F ) = seq ( ( N - 1 ) + 1 ) ( + , F ) )
85	84	fveq1d 5498	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq N ( + , F ) ` j ) = ( seq ( ( N - 1 ) + 1 ) ( + , F ) ` j ) )
86	83, 85	oveq12d 5871	. . . . . 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 ) ) )
87	82, 86	eqtr4d 2206	. . . . 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 ) ) )
88	87	ex 114	. . . 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 ) ) ) )
89		uzp1 9520	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
90	3, 89	syl 14	. . . . 5 |- ( ph -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
91	90	adantr 274	. . . 4 |- ( ( ph /\ j e. W ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
92	57, 88, 91	mpjaod 713	. . 3 |- ( ( ph /\ j e. W ) -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( N - 1 ) ) A + ( seq N ( + , F ) ` j ) ) )
93	8, 10, 21, 28, 17, 31, 92	climaddc2 11293	. 2 |- ( ph -> seq M ( + , F ) ~~> ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. W A ) )
94	1, 5, 6, 7, 93	isumclim 11384	1 |- ( ph -> sum_ k e. Z A = ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. W A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   /\ w3a 973   = wceq 1348   e. wcel 2141   C_ wss 3121   (/)c0 3414   class class class wbr 3989   dom cdm 4611   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   < clt 7954   - 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:  isum1p  11455  geolim2  11475  mertenslem2  11499  mertensabs  11500  effsumlt  11655  eirraplem  11739  trilpolemeq1  14072  trilpolemlt1  14073  nconstwlpolemgt0  14095