Theorem isumsplit 12042

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 2322	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
4		eluzel2 9750	. . 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 9755	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
10	3, 9	syl 14	. . 3 |- ( ph -> N e. ZZ )
11		uzss 9767	. . . . . . . 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 3268	. . . . . 6 |- ( ph -> W C_ Z )
14	13	sselda 3225	. . . . 5 |- ( ( ph /\ k e. W ) -> k e. Z )
15	14, 6	syldan 282	. . . 4 |- ( ( ph /\ k e. W ) -> ( F ` k ) = A )
16	14, 7	syldan 282	. . . 4 |- ( ( ph /\ k e. W ) -> A e. CC )
17		isumsplit.6	. . . . 5 |- ( ph -> seq M ( + , F ) e. dom ~~> )
18	6, 7	eqeltrd 2306	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
19	1, 2, 18	iserex 11890	. . . . 5 |- ( ph -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )
20	17, 19	mpbid 147	. . . 4 |- ( ph -> seq N ( + , F ) e. dom ~~> )
21	8, 10, 15, 16, 20	isumclim2 11973	. . 3 |- ( ph -> seq N ( + , F ) ~~> sum_ k e. W A )
22		peano2zm 9507	. . . . . 6 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
23	10, 22	syl 14	. . . . 5 |- ( ph -> ( N - 1 ) e. ZZ )
24	5, 23	fzfigd 10683	. . . 4 |- ( ph -> ( M ... ( N - 1 ) ) e. Fin )
25		elfzuz 10246	. . . . . 6 |- ( k e. ( M ... ( N - 1 ) ) -> k e. ( ZZ>= ` M ) )
26	25, 1	eleqtrrdi 2323	. . . . 5 |- ( k e. ( M ... ( N - 1 ) ) -> k e. Z )
27	26, 7	sylan2 286	. . . 4 |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> A e. CC )
28	24, 27	fsumcl 11951	. . 3 |- ( ph -> sum_ k e. ( M ... ( N - 1 ) ) A e. CC )
29	14, 18	syldan 282	. . . . 5 |- ( ( ph /\ k e. W ) -> ( F ` k ) e. CC )
30	8, 10, 29	serf 10735	. . . 4 |- ( ph -> seq N ( + , F ) : W --> CC )
31	30	ffvelcdmda 5778	. . 3 |- ( ( ph /\ j e. W ) -> ( seq N ( + , F ) ` j ) e. CC )
32	5	zred 9592	. . . . . . . . . . . 12 |- ( ph -> M e. RR )
33	32	ltm1d 9102	. . . . . . . . . . 11 |- ( ph -> ( M - 1 ) < M )
34		peano2zm 9507	. . . . . . . . . . . . 13 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
35	5, 34	syl 14	. . . . . . . . . . . 12 |- ( ph -> ( M - 1 ) e. ZZ )
36		fzn 10267	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ ( M - 1 ) e. ZZ ) -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
37	5, 35, 36	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
38	33, 37	mpbid 147	. . . . . . . . . 10 |- ( ph -> ( M ... ( M - 1 ) ) = (/) )
39	38	sumeq1d 11917	. . . . . . . . 9 |- ( ph -> sum_ k e. ( M ... ( M - 1 ) ) A = sum_ k e. (/) A )
40	39	adantr 276	. . . . . . . 8 |- ( ( ph /\ j e. W ) -> sum_ k e. ( M ... ( M - 1 ) ) A = sum_ k e. (/) A )
41		sum0 11939	. . . . . . . 8 |- sum_ k e. (/) A = 0
42	40, 41	eqtrdi 2278	. . . . . . 7 |- ( ( ph /\ j e. W ) -> sum_ k e. ( M ... ( M - 1 ) ) A = 0 )
43	42	oveq1d 6028	. . . . . 6 |- ( ( ph /\ j e. W ) -> ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) = ( 0 + ( seq M ( + , F ) ` j ) ) )
44	13	sselda 3225	. . . . . . . 8 |- ( ( ph /\ j e. W ) -> j e. Z )
45	1, 5, 18	serf 10735	. . . . . . . . 9 |- ( ph -> seq M ( + , F ) : Z --> CC )
46	45	ffvelcdmda 5778	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
47	44, 46	syldan 282	. . . . . . 7 |- ( ( ph /\ j e. W ) -> ( seq M ( + , F ) ` j ) e. CC )
48	47	addlidd 8319	. . . . . 6 |- ( ( ph /\ j e. W ) -> ( 0 + ( seq M ( + , F ) ` j ) ) = ( seq M ( + , F ) ` j ) )
49	43, 48	eqtr2d 2263	. . . . 5 |- ( ( ph /\ j e. W ) -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) )
50		oveq1 6020	. . . . . . . . 9 |- ( N = M -> ( N - 1 ) = ( M - 1 ) )
51	50	oveq2d 6029	. . . . . . . 8 |- ( N = M -> ( M ... ( N - 1 ) ) = ( M ... ( M - 1 ) ) )
52	51	sumeq1d 11917	. . . . . . 7 |- ( N = M -> sum_ k e. ( M ... ( N - 1 ) ) A = sum_ k e. ( M ... ( M - 1 ) ) A )
53		seqeq1 10702	. . . . . . . 8 |- ( N = M -> seq N ( + , F ) = seq M ( + , F ) )
54	53	fveq1d 5637	. . . . . . 7 |- ( N = M -> ( seq N ( + , F ) ` j ) = ( seq M ( + , F ) ` j ) )
55	52, 54	oveq12d 6031	. . . . . 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 2241	. . . . 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 157	. . . 4 |- ( ( ph /\ j e. W ) -> ( N = M -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( N - 1 ) ) A + ( seq N ( + , F ) ` j ) ) ) )
58		addcl 8147	. . . . . . . 8 |- ( ( k e. CC /\ m e. CC ) -> ( k + m ) e. CC )
59	58	adantl 277	. . . . . . 7 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( k e. CC /\ m e. CC ) ) -> ( k + m ) e. CC )
60		addass 8152	. . . . . . . 8 |- ( ( k e. CC /\ m e. CC /\ x e. CC ) -> ( ( k + m ) + x ) = ( k + ( m + x ) ) )
61	60	adantl 277	. . . . . . 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 528	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> j e. W )
63		simpll 527	. . . . . . . . . . 11 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ph )
64	10	zcnd 9593	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
65		ax-1cn 8115	. . . . . . . . . . . . 13 |- 1 e. CC
66		npcan 8378	. . . . . . . . . . . . 13 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
67	64, 65, 66	sylancl 413	. . . . . . . . . . . 12 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
68	67	eqcomd 2235	. . . . . . . . . . 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 5639	. . . . . . . . 9 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ZZ>= ` N ) = ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
71	8, 70	eqtrid 2274	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> W = ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
72	62, 71	eleqtrd 2308	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> j e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
73	5	adantr 276	. . . . . . . 8 |- ( ( ph /\ j e. W ) -> M e. ZZ )
74		eluzp1m1 9770	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
75	73, 74	sylan 283	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
76	1	eleq2i 2296	. . . . . . . . . 10 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
77	76, 6	sylan2br 288	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = A )
78	63, 77	sylan 283	. . . . . . . 8 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = A )
79	76, 7	sylan2br 288	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> A e. CC )
80	63, 79	sylan 283	. . . . . . . 8 |- ( ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> A e. CC )
81	78, 80	eqeltrd 2306	. . . . . . 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 10740	. . . . . 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 11948	. . . . . . 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 10705	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> seq N ( + , F ) = seq ( ( N - 1 ) + 1 ) ( + , F ) )
85	84	fveq1d 5637	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq N ( + , F ) ` j ) = ( seq ( ( N - 1 ) + 1 ) ( + , F ) ` j ) )
86	83, 85	oveq12d 6031	. . . . . 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 2265	. . . . 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 115	. . . 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 9780	. . . . . 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 276	. . . 4 |- ( ( ph /\ j e. W ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
92	57, 88, 91	mpjaod 723	. . 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 11881	. 2 |- ( ph -> seq M ( + , F ) ~~> ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. W A ) )
94	1, 5, 6, 7, 93	isumclim 11972	1 |- ( ph -> sum_ k e. Z A = ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. W A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   C_ wss 3198   (/)c0 3492   class class class wbr 4086   dom cdm 4723   ` cfv 5324  (class class class)co 6013   CCcc 8020   0cc0 8022   1c1 8023   + caddc 8025   < clt 8204   - cmin 8340   ZZcz 9469   ZZ>=cuz 9745   ...cfz 10233   seqcseq 10699   ~~> cli 11829   sum_csu 11904

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-seqfrec 10700  df-exp 10791  df-ihash 11028  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-clim 11830  df-sumdc 11905

This theorem is referenced by:  isum1p  12043  geolim2  12063  mertenslem2  12087  mertensabs  12088  effsumlt  12243  eirraplem  12328  trilpolemeq1  16580  trilpolemlt1  16581  nconstwlpolemgt0  16604