Theorem isumsplit 12202

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 2327	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
4		eluzel2 9876	. . 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 9881	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
10	3, 9	syl 14	. . 3 |- ( ph -> N e. ZZ )
11		uzss 9893	. . . . . . . 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 3285	. . . . . 6 |- ( ph -> W C_ Z )
14	13	sselda 3242	. . . . 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 2311	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
19	1, 2, 18	iserex 12049	. . . . 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 12133	. . 3 |- ( ph -> seq N ( + , F ) ~~> sum_ k e. W A )
22		peano2zm 9632	. . . . . 6 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
23	10, 22	syl 14	. . . . 5 |- ( ph -> ( N - 1 ) e. ZZ )
24	5, 23	fzfigd 10817	. . . 4 |- ( ph -> ( M ... ( N - 1 ) ) e. Fin )
25		elfzuz 10374	. . . . . 6 |- ( k e. ( M ... ( N - 1 ) ) -> k e. ( ZZ>= ` M ) )
26	25, 1	eleqtrrdi 2328	. . . . 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 12111	. . 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 10869	. . . 4 |- ( ph -> seq N ( + , F ) : W --> CC )
31	30	ffvelcdmda 5817	. . 3 |- ( ( ph /\ j e. W ) -> ( seq N ( + , F ) ` j ) e. CC )
32	5	zred 9718	. . . . . . . . . . . 12 |- ( ph -> M e. RR )
33	32	ltm1d 9223	. . . . . . . . . . 11 |- ( ph -> ( M - 1 ) < M )
34		peano2zm 9632	. . . . . . . . . . . . 13 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
35	5, 34	syl 14	. . . . . . . . . . . 12 |- ( ph -> ( M - 1 ) e. ZZ )
36		fzn 10396	. . . . . . . . . . . 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 12076	. . . . . . . . 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 12099	. . . . . . . 8 |- sum_ k e. (/) A = 0
42	40, 41	eqtrdi 2283	. . . . . . 7 |- ( ( ph /\ j e. W ) -> sum_ k e. ( M ... ( M - 1 ) ) A = 0 )
43	42	oveq1d 6073	. . . . . 6 |- ( ( ph /\ j e. W ) -> ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) = ( 0 + ( seq M ( + , F ) ` j ) ) )
44	13	sselda 3242	. . . . . . . 8 |- ( ( ph /\ j e. W ) -> j e. Z )
45	1, 5, 18	serf 10869	. . . . . . . . 9 |- ( ph -> seq M ( + , F ) : Z --> CC )
46	45	ffvelcdmda 5817	. . . . . . . 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 8439	. . . . . 6 |- ( ( ph /\ j e. W ) -> ( 0 + ( seq M ( + , F ) ` j ) ) = ( seq M ( + , F ) ` j ) )
49	43, 48	eqtr2d 2268	. . . . 5 |- ( ( ph /\ j e. W ) -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) )
50		oveq1 6065	. . . . . . . . 9 |- ( N = M -> ( N - 1 ) = ( M - 1 ) )
51	50	oveq2d 6074	. . . . . . . 8 |- ( N = M -> ( M ... ( N - 1 ) ) = ( M ... ( M - 1 ) ) )
52	51	sumeq1d 12076	. . . . . . 7 |- ( N = M -> sum_ k e. ( M ... ( N - 1 ) ) A = sum_ k e. ( M ... ( M - 1 ) ) A )
53		seqeq1 10836	. . . . . . . 8 |- ( N = M -> seq N ( + , F ) = seq M ( + , F ) )
54	53	fveq1d 5677	. . . . . . 7 |- ( N = M -> ( seq N ( + , F ) ` j ) = ( seq M ( + , F ) ` j ) )
55	52, 54	oveq12d 6076	. . . . . 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 2246	. . . . 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 8268	. . . . . . . 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 8273	. . . . . . . 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 529	. . . . . . . 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 9719	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
65		ax-1cn 8236	. . . . . . . . . . . . 13 |- 1 e. CC
66		npcan 8498	. . . . . . . . . . . . 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 2240	. . . . . . . . . . 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 5679	. . . . . . . . 9 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ZZ>= ` N ) = ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
71	8, 70	eqtrid 2279	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> W = ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
72	62, 71	eleqtrd 2313	. . . . . . 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 9896	. . . . . . . 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 2301	. . . . . . . . . 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 2311	. . . . . . 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 10874	. . . . . 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 12108	. . . . . . 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 10839	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> seq N ( + , F ) = seq ( ( N - 1 ) + 1 ) ( + , F ) )
85	84	fveq1d 5677	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq N ( + , F ) ` j ) = ( seq ( ( N - 1 ) + 1 ) ( + , F ) ` j ) )
86	83, 85	oveq12d 6076	. . . . . 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 2270	. . . . 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 9906	. . . . . 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 726	. . 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 12040	. 2 |- ( ph -> seq M ( + , F ) ~~> ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. W A ) )
94	1, 5, 6, 7, 93	isumclim 12132	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 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   C_ wss 3214   (/)c0 3512   class class class wbr 4114   dom cdm 4754   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   - cmin 8460   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833   ~~> cli 11988   sum_csu 12063

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064

This theorem is referenced by:  isum1p  12203  geolim2  12223  mertenslem2  12247  mertensabs  12248  effsumlt  12403  eirraplem  12488  trilpolemeq1  16950  trilpolemlt1  16951  nconstwlpolemgt0  16976