Theorem iserzshft2i 8367

Description: Index shift of an infinite series. (Contributed by Paul Chapman, 19-Nov-2007.)

Hypotheses

Ref	Expression
iserzshft2.1	|- F e. _V
iserzshft2.2	|- G e. _V
iserzshft2.3	|- M e. ZZ
iserzshft2.4	|- K e. ZZ

Assertion

Ref	Expression
iserzshft2i	|- ((A e. B /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) -> ((<.M, + >. seq F) ~~> A <-> (<.(M + K), + >. seq G) ~~> A))

Distinct variable groups:   k,F   k,G   k,K   k,M

Proof of Theorem iserzshft2i

Step	Hyp	Ref	Expression
1		climcl 8238	. . . 4 |- ((A e. B /\ (<.M, + >. seq F) ~~> A) -> A e. CC)
2	1	anim1i 361	. . 3 |- (((A e. B /\ (<.M, + >. seq F) ~~> A) /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) -> (A e. CC /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))))
3	2	an1rs 547	. 2 |- (((A e. B /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) /\ (<.M, + >. seq F) ~~> A) -> (A e. CC /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))))
4		climcl 8238	. . . 4 |- ((A e. B /\ (<.(M + K), + >. seq G) ~~> A) -> A e. CC)
5	4	anim1i 361	. . 3 |- (((A e. B /\ (<.(M + K), + >. seq G) ~~> A) /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) -> (A e. CC /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))))
6	5	an1rs 547	. 2 |- (((A e. B /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) /\ (<.(M + K), + >. seq G) ~~> A) -> (A e. CC /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))))
7		breq2 3342	. . . . . 6 |- (A = if(A e. CC, A, 0) -> ((<.M, + >. seq F) ~~> A <-> (<.M, + >. seq F) ~~> if(A e. CC, A, 0)))
8		breq2 3342	. . . . . 6 |- (A = if(A e. CC, A, 0) -> ((<.(M + K), + >. seq G) ~~> A <-> (<.(M + K), + >. seq G) ~~> if(A e. CC, A, 0)))
9	7, 8	bibi12d 691	. . . . 5 |- (A = if(A e. CC, A, 0) -> (((<.M, + >. seq F) ~~> A <-> (<.(M + K), + >. seq G) ~~> A) <-> ((<.M, + >. seq F) ~~> if(A e. CC, A, 0) <-> (<.(M + K), + >. seq G) ~~> if(A e. CC, A, 0))))
10	9	imbi2d 674	. . . 4 |- (A = if(A e. CC, A, 0) -> ((A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) -> ((<.M, + >. seq F) ~~> A <-> (<.(M + K), + >. seq G) ~~> A)) <-> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) -> ((<.M, + >. seq F) ~~> if(A e. CC, A, 0) <-> (<.(M + K), + >. seq G) ~~> if(A e. CC, A, 0)))))
11		oprex 4907	. . . . . 6 |- (<.M, + >. seq F) e. _V
12		oprex 4907	. . . . . 6 |- (<.(M + K), + >. seq G) e. _V
13		iserzshft2.3	. . . . . 6 |- M e. ZZ
14		iserzshft2.4	. . . . . 6 |- K e. ZZ
15	11, 12, 13, 14	climshft2i 8366	. . . . 5 |- ((if(A e. CC, A, 0) e. CC /\ A.m e. (ZZ>=` M)((<.(M + K), + >. seq G)` (m + K)) = ((<.M, + >. seq F)` m)) -> ((<.M, + >. seq F) ~~> if(A e. CC, A, 0) <-> (<.(M + K), + >. seq G) ~~> if(A e. CC, A, 0)))
16		0cn 6481	. . . . . 6 |- 0 e. CC
17	16	elimel 3025	. . . . 5 |- if(A e. CC, A, 0) e. CC
18		fsumshft 8291	. . . . . . . . . . 11 |- ((m e. (ZZ>=` M) /\ K e. ZZ /\ A.k e. (M...m)(F` k) e. CC) -> sum_k e. (M...m)(F` k) = sum_n e. ((M + K)...(m + K))[_(n - K) / k]_(F` k))
19	14, 18	mp3an2 1179	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...m)(F` k) e. CC) -> sum_k e. (M...m)(F` k) = sum_n e. ((M + K)...(m + K))[_(n - K) / k]_(F` k))
20		r19.26 2219	. . . . . . . . . . . 12 |- (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) <-> (A.k e. (ZZ>=` M)(F` k) e. CC /\ A.k e. (ZZ>=` M)(G` (k + K)) = (F` k)))
21	20	simplbi 349	. . . . . . . . . . 11 |- (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) -> A.k e. (ZZ>=` M)(F` k) e. CC)
22		elfzuz 7658	. . . . . . . . . . . . 13 |- (k e. (M...m) -> k e. (ZZ>=` M))
23	22	imim1i 19	. . . . . . . . . . . 12 |- ((k e. (ZZ>=` M) -> (F` k) e. CC) -> (k e. (M...m) -> (F` k) e. CC))
24	23	ralimi2 2165	. . . . . . . . . . 11 |- (A.k e. (ZZ>=` M)(F` k) e. CC -> A.k e. (M...m)(F` k) e. CC)
25	21, 24	syl 12	. . . . . . . . . 10 |- (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) -> A.k e. (M...m)(F` k) e. CC)
26	19, 25	sylan2 500	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) -> sum_k e. (M...m)(F` k) = sum_n e. ((M + K)...(m + K))[_(n - K) / k]_(F` k))
27		npcan 6559	. . . . . . . . . . . . . . . . 17 |- ((n e. CC /\ K e. CC) -> ((n - K) + K) = n)
28		eluzelz 7592	. . . . . . . . . . . . . . . . . 18 |- (n e. (ZZ>=` (M + K)) -> n e. ZZ)
29		zcn 7349	. . . . . . . . . . . . . . . . . 18 |- (n e. ZZ -> n e. CC)
30	28, 29	syl 12	. . . . . . . . . . . . . . . . 17 |- (n e. (ZZ>=` (M + K)) -> n e. CC)
31	14	zrei 7350	. . . . . . . . . . . . . . . . . 18 |- K e. RR
32	31	recni 6467	. . . . . . . . . . . . . . . . 17 |- K e. CC
33	27, 30, 32	sylancl 525	. . . . . . . . . . . . . . . 16 |- (n e. (ZZ>=` (M + K)) -> ((n - K) + K) = n)
34	33	fveq2d 4685	. . . . . . . . . . . . . . 15 |- (n e. (ZZ>=` (M + K)) -> (G` ((n - K) + K)) = (G` n))
35	34	adantl 424	. . . . . . . . . . . . . 14 |- ((A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) /\ n e. (ZZ>=` (M + K))) -> (G` ((n - K) + K)) = (G` n))
36	13, 14	eluzsubi 7606	. . . . . . . . . . . . . . . . 17 |- (n e. (ZZ>=` (M + K)) -> (n - K) e. (ZZ>=` M))
37		fveq2 4681	. . . . . . . . . . . . . . . . . . . 20 |- (k = (n - K) -> (F` k) = (F` (n - K)))
38	37	eleq1d 1963	. . . . . . . . . . . . . . . . . . 19 |- (k = (n - K) -> ((F` k) e. CC <-> (F` (n - K)) e. CC))
39		opreq1 4889	. . . . . . . . . . . . . . . . . . . . 21 |- (k = (n - K) -> (k + K) = ((n - K) + K))
40	39	fveq2d 4685	. . . . . . . . . . . . . . . . . . . 20 |- (k = (n - K) -> (G` (k + K)) = (G` ((n - K) + K)))
41	40, 37	eqeq12d 1899	. . . . . . . . . . . . . . . . . . 19 |- (k = (n - K) -> ((G` (k + K)) = (F` k) <-> (G` ((n - K) + K)) = (F` (n - K))))
42	38, 41	anbi12d 690	. . . . . . . . . . . . . . . . . 18 |- (k = (n - K) -> (((F` k) e. CC /\ (G` (k + K)) = (F` k)) <-> ((F` (n - K)) e. CC /\ (G` ((n - K) + K)) = (F` (n - K)))))
43	42	rcla4v 2376	. . . . . . . . . . . . . . . . 17 |- ((n - K) e. (ZZ>=` M) -> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) -> ((F` (n - K)) e. CC /\ (G` ((n - K) + K)) = (F` (n - K)))))
44	36, 43	syl 12	. . . . . . . . . . . . . . . 16 |- (n e. (ZZ>=` (M + K)) -> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) -> ((F` (n - K)) e. CC /\ (G` ((n - K) + K)) = (F` (n - K)))))
45	44	impcom 378	. . . . . . . . . . . . . . 15 |- ((A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) /\ n e. (ZZ>=` (M + K))) -> ((F` (n - K)) e. CC /\ (G` ((n - K) + K)) = (F` (n - K))))
46	45	simprd 352	. . . . . . . . . . . . . 14 |- ((A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) /\ n e. (ZZ>=` (M + K))) -> (G` ((n - K) + K)) = (F` (n - K)))
47	35, 46	eqtr3d 1927	. . . . . . . . . . . . 13 |- ((A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) /\ n e. (ZZ>=` (M + K))) -> (G` n) = (F` (n - K)))
48		oprex 4907	. . . . . . . . . . . . . 14 |- (n - K) e. _V
49		ax-17 1317	. . . . . . . . . . . . . 14 |- (x e. (F` (n - K)) -> A.k x e. (F` (n - K)))
50	48, 49, 37	csbief 2576	. . . . . . . . . . . . 13 |- [_(n - K) / k]_(F` k) = (F` (n - K))
51	47, 50	syl6eqr 1946	. . . . . . . . . . . 12 |- ((A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) /\ n e. (ZZ>=` (M + K))) -> (G` n) = [_(n - K) / k]_(F` k))
52		elfzuz 7658	. . . . . . . . . . . 12 |- (n e. ((M + K)...(m + K)) -> n e. (ZZ>=` (M + K)))
53	51, 52	sylan2 500	. . . . . . . . . . 11 |- ((A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) /\ n e. ((M + K)...(m + K))) -> (G` n) = [_(n - K) / k]_(F` k))
54	53	sumeq2dv 8252	. . . . . . . . . 10 |- (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) -> sum_n e. ((M + K)...(m + K))(G` n) = sum_n e. ((M + K)...(m + K))[_(n - K) / k]_(F` k))
55	54	adantl 424	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) -> sum_n e. ((M + K)...(m + K))(G` n) = sum_n e. ((M + K)...(m + K))[_(n - K) / k]_(F` k))
56	26, 55	eqtr4d 1928	. . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) -> sum_k e. (M...m)(F` k) = sum_n e. ((M + K)...(m + K))(G` n))
57		iserzshft2.1	. . . . . . . . . 10 |- F e. _V
58	57	fsumserz 8259	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> sum_k e. (M...m)(F` k) = ((<.M, + >. seq F)` m))
59	58	adantr 425	. . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) -> sum_k e. (M...m)(F` k) = ((<.M, + >. seq F)` m))
60	13, 14	eluzaddi 7605	. . . . . . . . . 10 |- (m e. (ZZ>=` M) -> (m + K) e. (ZZ>=` (M + K)))
61		iserzshft2.2	. . . . . . . . . . 11 |- G e. _V
62	61	fsumserz 8259	. . . . . . . . . 10 |- ((m + K) e. (ZZ>=` (M + K)) -> sum_n e. ((M + K)...(m + K))(G` n) = ((<.(M + K), + >. seq G)` (m + K)))
63	60, 62	syl 12	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> sum_n e. ((M + K)...(m + K))(G` n) = ((<.(M + K), + >. seq G)` (m + K)))
64	63	adantr 425	. . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) -> sum_n e. ((M + K)...(m + K))(G` n) = ((<.(M + K), + >. seq G)` (m + K)))
65	56, 59, 64	3eqtr3rd 1936	. . . . . . 7 |- ((m e. (ZZ>=` M) /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) -> ((<.(M + K), + >. seq G)` (m + K)) = ((<.M, + >. seq F)` m))
66	65	expcom 403	. . . . . 6 |- (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) -> (m e. (ZZ>=` M) -> ((<.(M + K), + >. seq G)` (m + K)) = ((<.M, + >. seq F)` m)))
67	66	r19.21aiv 2175	. . . . 5 |- (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) -> A.m e. (ZZ>=` M)((<.(M + K), + >. seq G)` (m + K)) = ((<.M, + >. seq F)` m))
68	15, 17, 67	sylancr 526	. . . 4 |- (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) -> ((<.M, + >. seq F) ~~> if(A e. CC, A, 0) <-> (<.(M + K), + >. seq G) ~~> if(A e. CC, A, 0)))
69	10, 68	dedth 3011	. . 3 |- (A e. CC -> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k)) -> ((<.M, + >. seq F) ~~> A <-> (<.(M + K), + >. seq G) ~~> A)))
70	69	imp 377	. 2 |- ((A e. CC /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) -> ((<.M, + >. seq F) ~~> A <-> (<.(M + K), + >. seq G) ~~> A))
71	3, 6, 70	pm5.21nd 744	1 |- ((A e. B /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` (k + K)) = (F` k))) -> ((<.M, + >. seq F) ~~> A <-> (<.(M + K), + >. seq G) ~~> A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  [_csb 2540  ifcif 2982  <.cop 3046   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386   + caddc 6389   - cmin 6445  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637   seq cseqz 7774   ~~> cli 8234  sum_csu 8239

This theorem is referenced by:  ef1tllem 8643  fsumltisumi 15823  iserzshft2 15829

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-clim 8235  df-sum 8240

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