Theorem isumsplit 12115

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 2324	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
4		eluzel2 9804	. . 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 9809	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
10	3, 9	syl 14	. . 3 |- ( ph -> N e. ZZ )
11		uzss 9821	. . . . . . . 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 3271	. . . . . 6 |- ( ph -> W C_ Z )
14	13	sselda 3228	. . . . 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 2308	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
19	1, 2, 18	iserex 11962	. . . . 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 12046	. . 3 |- ( ph -> seq N ( + , F ) ~~> sum_ k e. W A )
22		peano2zm 9561	. . . . . 6 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
23	10, 22	syl 14	. . . . 5 |- ( ph -> ( N - 1 ) e. ZZ )
24	5, 23	fzfigd 10739	. . . 4 |- ( ph -> ( M ... ( N - 1 ) ) e. Fin )
25		elfzuz 10301	. . . . . 6 |- ( k e. ( M ... ( N - 1 ) ) -> k e. ( ZZ>= ` M ) )
26	25, 1	eleqtrrdi 2325	. . . . 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 12024	. . 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 10791	. . . 4 |- ( ph -> seq N ( + , F ) : W --> CC )
31	30	ffvelcdmda 5790	. . 3 |- ( ( ph /\ j e. W ) -> ( seq N ( + , F ) ` j ) e. CC )
32	5	zred 9646	. . . . . . . . . . . 12 |- ( ph -> M e. RR )
33	32	ltm1d 9154	. . . . . . . . . . 11 |- ( ph -> ( M - 1 ) < M )
34		peano2zm 9561	. . . . . . . . . . . . 13 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
35	5, 34	syl 14	. . . . . . . . . . . 12 |- ( ph -> ( M - 1 ) e. ZZ )
36		fzn 10322	. . . . . . . . . . . 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 11989	. . . . . . . . 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 12012	. . . . . . . 8 |- sum_ k e. (/) A = 0
42	40, 41	eqtrdi 2280	. . . . . . 7 |- ( ( ph /\ j e. W ) -> sum_ k e. ( M ... ( M - 1 ) ) A = 0 )
43	42	oveq1d 6043	. . . . . 6 |- ( ( ph /\ j e. W ) -> ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) = ( 0 + ( seq M ( + , F ) ` j ) ) )
44	13	sselda 3228	. . . . . . . 8 |- ( ( ph /\ j e. W ) -> j e. Z )
45	1, 5, 18	serf 10791	. . . . . . . . 9 |- ( ph -> seq M ( + , F ) : Z --> CC )
46	45	ffvelcdmda 5790	. . . . . . . 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 8371	. . . . . 6 |- ( ( ph /\ j e. W ) -> ( 0 + ( seq M ( + , F ) ` j ) ) = ( seq M ( + , F ) ` j ) )
49	43, 48	eqtr2d 2265	. . . . 5 |- ( ( ph /\ j e. W ) -> ( seq M ( + , F ) ` j ) = ( sum_ k e. ( M ... ( M - 1 ) ) A + ( seq M ( + , F ) ` j ) ) )
50		oveq1 6035	. . . . . . . . 9 |- ( N = M -> ( N - 1 ) = ( M - 1 ) )
51	50	oveq2d 6044	. . . . . . . 8 |- ( N = M -> ( M ... ( N - 1 ) ) = ( M ... ( M - 1 ) ) )
52	51	sumeq1d 11989	. . . . . . 7 |- ( N = M -> sum_ k e. ( M ... ( N - 1 ) ) A = sum_ k e. ( M ... ( M - 1 ) ) A )
53		seqeq1 10758	. . . . . . . 8 |- ( N = M -> seq N ( + , F ) = seq M ( + , F ) )
54	53	fveq1d 5650	. . . . . . 7 |- ( N = M -> ( seq N ( + , F ) ` j ) = ( seq M ( + , F ) ` j ) )
55	52, 54	oveq12d 6046	. . . . . 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 2243	. . . . 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 8200	. . . . . . . 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 8205	. . . . . . . 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 9647	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
65		ax-1cn 8168	. . . . . . . . . . . . 13 |- 1 e. CC
66		npcan 8430	. . . . . . . . . . . . 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 2237	. . . . . . . . . . 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 5652	. . . . . . . . 9 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ZZ>= ` N ) = ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
71	8, 70	eqtrid 2276	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> W = ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
72	62, 71	eleqtrd 2310	. . . . . . 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 9824	. . . . . . . 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 2298	. . . . . . . . . 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 2308	. . . . . . 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 10796	. . . . . 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 12021	. . . . . . 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 10761	. . . . . . . 8 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> seq N ( + , F ) = seq ( ( N - 1 ) + 1 ) ( + , F ) )
85	84	fveq1d 5650	. . . . . . 7 |- ( ( ( ph /\ j e. W ) /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq N ( + , F ) ` j ) = ( seq ( ( N - 1 ) + 1 ) ( + , F ) ` j ) )
86	83, 85	oveq12d 6046	. . . . . 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 2267	. . . . 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 9834	. . . . . 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 11953	. 2 |- ( ph -> seq M ( + , F ) ~~> ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. W A ) )
94	1, 5, 6, 7, 93	isumclim 12045	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 2202   C_ wss 3201   (/)c0 3496   class class class wbr 4093   dom cdm 4731   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   < clt 8256   - cmin 8392   ZZcz 9523   ZZ>=cuz 9799   ...cfz 10288   seqcseq 10755   ~~> cli 11901   sum_csu 11976

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-seqfrec 10756  df-exp 10847  df-ihash 11084  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902  df-sumdc 11977

This theorem is referenced by:  isum1p  12116  geolim2  12136  mertenslem2  12160  mertensabs  12161  effsumlt  12316  eirraplem  12401  trilpolemeq1  16755  trilpolemlt1  16756  nconstwlpolemgt0  16780