Theorem fsumshft 8291

Description: Index shift of a finite sum.

Assertion

Ref	Expression
fsumshft	|- ((N e. (ZZ>=` M) /\ K e. ZZ /\ A.j e. (M...N)A e. CC) -> sum_j e. (M...N)A = sum_k e. ((M + K)...(N + K))[_(k - K) / j]_A)

Distinct variable groups:   A,k   j,k,K   j,M,k   j,N,k

Proof of Theorem fsumshft

Step	Hyp	Ref	Expression
1		0z 7355	. . . 4 |- 0 e. ZZ
2		fsumrev 8289	. . . 4 |- ((N e. (ZZ>=` M) /\ 0 e. ZZ /\ A.j e. (M...N)A e. CC) -> sum_j e. (M...N)A = sum_m e. ((0 - N)...(0 - M))[_(0 - m) / j]_A)
3	1, 2	mp3an2 1179	. . 3 |- ((N e. (ZZ>=` M) /\ A.j e. (M...N)A e. CC) -> sum_j e. (M...N)A = sum_m e. ((0 - N)...(0 - M))[_(0 - m) / j]_A)
4	3	3adant2 895	. 2 |- ((N e. (ZZ>=` M) /\ K e. ZZ /\ A.j e. (M...N)A e. CC) -> sum_j e. (M...N)A = sum_m e. ((0 - N)...(0 - M))[_(0 - m) / j]_A)
5		uzneg 7598	. . . . 5 |- (N e. (ZZ>=` M) -> -uM e. (ZZ>=` -uN))
6		df-neg 6513	. . . . . 6 |- -uM = (0 - M)
7		df-neg 6513	. . . . . . 7 |- -uN = (0 - N)
8	7	fveq2i 4684	. . . . . 6 |- (ZZ>=` -uN) = (ZZ>=` (0 - N))
9	6, 8	eleq12i 1962	. . . . 5 |- (-uM e. (ZZ>=` -uN) <-> (0 - M) e. (ZZ>=` (0 - N)))
10	5, 9	sylib 215	. . . 4 |- (N e. (ZZ>=` M) -> (0 - M) e. (ZZ>=` (0 - N)))
11	10	3ad2ant1 897	. . 3 |- ((N e. (ZZ>=` M) /\ K e. ZZ /\ A.j e. (M...N)A e. CC) -> (0 - M) e. (ZZ>=` (0 - N)))
12		simp2 877	. . 3 |- ((N e. (ZZ>=` M) /\ K e. ZZ /\ A.j e. (M...N)A e. CC) -> K e. ZZ)
13		eluzel2 7593	. . . . . . 7 |- (N e. (ZZ>=` M) -> M e. ZZ)
14		eluzelz 7592	. . . . . . 7 |- (N e. (ZZ>=` M) -> N e. ZZ)
15		fzrevral 7698	. . . . . . . 8 |- ((M e. ZZ /\ N e. ZZ /\ 0 e. ZZ) -> (A.j e. (M...N)A e. CC <-> A.m e. ((0 - N)...(0 - M))[(0 - m) / j]A e. CC))
16	1, 15	mp3an3 1180	. . . . . . 7 |- ((M e. ZZ /\ N e. ZZ) -> (A.j e. (M...N)A e. CC <-> A.m e. ((0 - N)...(0 - M))[(0 - m) / j]A e. CC))
17	13, 14, 16	syl11anc 524	. . . . . 6 |- (N e. (ZZ>=` M) -> (A.j e. (M...N)A e. CC <-> A.m e. ((0 - N)...(0 - M))[(0 - m) / j]A e. CC))
18		oprex 4907	. . . . . . . 8 |- (0 - m) e. _V
19		sbcel1g 2556	. . . . . . . 8 |- ((0 - m) e. _V -> ([(0 - m) / j]A e. CC <-> [_(0 - m) / j]_A e. CC))
20	18, 19	ax-mp 7	. . . . . . 7 |- ([(0 - m) / j]A e. CC <-> [_(0 - m) / j]_A e. CC)
21	20	ralbii 2127	. . . . . 6 |- (A.m e. ((0 - N)...(0 - M))[(0 - m) / j]A e. CC <-> A.m e. ((0 - N)...(0 - M))[_(0 - m) / j]_A e. CC)
22	17, 21	syl6bb 595	. . . . 5 |- (N e. (ZZ>=` M) -> (A.j e. (M...N)A e. CC <-> A.m e. ((0 - N)...(0 - M))[_(0 - m) / j]_A e. CC))
23	22	biimpa 460	. . . 4 |- ((N e. (ZZ>=` M) /\ A.j e. (M...N)A e. CC) -> A.m e. ((0 - N)...(0 - M))[_(0 - m) / j]_A e. CC)
24	23	3adant2 895	. . 3 |- ((N e. (ZZ>=` M) /\ K e. ZZ /\ A.j e. (M...N)A e. CC) -> A.m e. ((0 - N)...(0 - M))[_(0 - m) / j]_A e. CC)
25		fsumrev 8289	. . 3 |- (((0 - M) e. (ZZ>=` (0 - N)) /\ K e. ZZ /\ A.m e. ((0 - N)...(0 - M))[_(0 - m) / j]_A e. CC) -> sum_m e. ((0 - N)...(0 - M))[_(0 - m) / j]_A = sum_k e. ((K - (0 - M))...(K - (0 - N)))[_(K - k) / m]_[_(0 - m) / j]_A)
26	11, 12, 24, 25	syl111anc 1100	. 2 |- ((N e. (ZZ>=` M) /\ K e. ZZ /\ A.j e. (M...N)A e. CC) -> sum_m e. ((0 - N)...(0 - M))[_(0 - m) / j]_A = sum_k e. ((K - (0 - M))...(K - (0 - N)))[_(K - k) / m]_[_(0 - m) / j]_A)
27		subneg 6554	. . . . . . . . . 10 |- ((K e. CC /\ M e. CC) -> (K - -uM) = (K + M))
28		addcom 6458	. . . . . . . . . 10 |- ((K e. CC /\ M e. CC) -> (K + M) = (M + K))
29	27, 28	eqtrd 1925	. . . . . . . . 9 |- ((K e. CC /\ M e. CC) -> (K - -uM) = (M + K))
30	6	opreq2i 4893	. . . . . . . . 9 |- (K - -uM) = (K - (0 - M))
31	29, 30	syl5eqr 1942	. . . . . . . 8 |- ((K e. CC /\ M e. CC) -> (K - (0 - M)) = (M + K))
32	31	adantrr 431	. . . . . . 7 |- ((K e. CC /\ (M e. CC /\ N e. CC)) -> (K - (0 - M)) = (M + K))
33		subneg 6554	. . . . . . . . . 10 |- ((K e. CC /\ N e. CC) -> (K - -uN) = (K + N))
34		addcom 6458	. . . . . . . . . 10 |- ((K e. CC /\ N e. CC) -> (K + N) = (N + K))
35	33, 34	eqtrd 1925	. . . . . . . . 9 |- ((K e. CC /\ N e. CC) -> (K - -uN) = (N + K))
36	7	opreq2i 4893	. . . . . . . . 9 |- (K - -uN) = (K - (0 - N))
37	35, 36	syl5eqr 1942	. . . . . . . 8 |- ((K e. CC /\ N e. CC) -> (K - (0 - N)) = (N + K))
38	37	adantrl 430	. . . . . . 7 |- ((K e. CC /\ (M e. CC /\ N e. CC)) -> (K - (0 - N)) = (N + K))
39	32, 38	opreq12d 4900	. . . . . 6 |- ((K e. CC /\ (M e. CC /\ N e. CC)) -> ((K - (0 - M))...(K - (0 - N))) = ((M + K)...(N + K)))
40		zcn 7349	. . . . . 6 |- (K e. ZZ -> K e. CC)
41		zcn 7349	. . . . . . . 8 |- (M e. ZZ -> M e. CC)
42	13, 41	syl 12	. . . . . . 7 |- (N e. (ZZ>=` M) -> M e. CC)
43		zcn 7349	. . . . . . . 8 |- (N e. ZZ -> N e. CC)
44	14, 43	syl 12	. . . . . . 7 |- (N e. (ZZ>=` M) -> N e. CC)
45	42, 44	jca 310	. . . . . 6 |- (N e. (ZZ>=` M) -> (M e. CC /\ N e. CC))
46	39, 40, 45	syl2an 503	. . . . 5 |- ((K e. ZZ /\ N e. (ZZ>=` M)) -> ((K - (0 - M))...(K - (0 - N))) = ((M + K)...(N + K)))
47	46	ancoms 484	. . . 4 |- ((N e. (ZZ>=` M) /\ K e. ZZ) -> ((K - (0 - M))...(K - (0 - N))) = ((M + K)...(N + K)))
48		negsubdi2 6623	. . . . . . . . . 10 |- ((K e. CC /\ k e. CC) -> -u(K - k) = (k - K))
49		zcn 7349	. . . . . . . . . 10 |- (k e. ZZ -> k e. CC)
50	48, 40, 49	syl2an 503	. . . . . . . . 9 |- ((K e. ZZ /\ k e. ZZ) -> -u(K - k) = (k - K))
51		df-neg 6513	. . . . . . . . 9 |- -u(K - k) = (0 - (K - k))
52	50, 51	syl5eqr 1942	. . . . . . . 8 |- ((K e. ZZ /\ k e. ZZ) -> (0 - (K - k)) = (k - K))
53	52	csbeq1d 2544	. . . . . . 7 |- ((K e. ZZ /\ k e. ZZ) -> [_(0 - (K - k)) / j]_A = [_(k - K) / j]_A)
54		oprex 4907	. . . . . . . 8 |- (K - k) e. _V
55	18	ax-gen 1305	. . . . . . . 8 |- A.m(0 - m) e. _V
56		opreq2 4890	. . . . . . . . 9 |- (m = (K - k) -> (0 - m) = (0 - (K - k)))
57	56	csbco3g 2585	. . . . . . . 8 |- (((K - k) e. _V /\ A.m(0 - m) e. _V) -> [_(K - k) / m]_[_(0 - m) / j]_A = [_(0 - (K - k)) / j]_A)
58	54, 55, 57	mp2an 761	. . . . . . 7 |- [_(K - k) / m]_[_(0 - m) / j]_A = [_(0 - (K - k)) / j]_A
59	53, 58	syl5eq 1940	. . . . . 6 |- ((K e. ZZ /\ k e. ZZ) -> [_(K - k) / m]_[_(0 - m) / j]_A = [_(k - K) / j]_A)
60		elfzelz 7652	. . . . . 6 |- (k e. ((K - (0 - M))...(K - (0 - N))) -> k e. ZZ)
61	59, 60	sylan2 500	. . . . 5 |- ((K e. ZZ /\ k e. ((K - (0 - M))...(K - (0 - N)))) -> [_(K - k) / m]_[_(0 - m) / j]_A = [_(k - K) / j]_A)
62	61	adantll 428	. . . 4 |- (((N e. (ZZ>=` M) /\ K e. ZZ) /\ k e. ((K - (0 - M))...(K - (0 - N)))) -> [_(K - k) / m]_[_(0 - m) / j]_A = [_(k - K) / j]_A)
63	47, 62	sumeq12dv 8255	. . 3 |- ((N e. (ZZ>=` M) /\ K e. ZZ) -> sum_k e. ((K - (0 - M))...(K - (0 - N)))[_(K - k) / m]_[_(0 - m) / j]_A = sum_k e. ((M + K)...(N + K))[_(k - K) / j]_A)
64	63	3adant3 896	. 2 |- ((N e. (ZZ>=` M) /\ K e. ZZ /\ A.j e. (M...N)A e. CC) -> sum_k e. ((K - (0 - M))...(K - (0 - N)))[_(K - k) / m]_[_(0 - m) / j]_A = sum_k e. ((M + K)...(N + K))[_(k - K) / j]_A)
65	4, 26, 64	3eqtrd 1929	1 |- ((N e. (ZZ>=` M) /\ K e. ZZ /\ A.j e. (M...N)A e. CC) -> sum_j e. (M...N)A = sum_k e. ((M + K)...(N + K))[_(k - K) / j]_A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  _Vcvv 2292  [_csb 2540  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386   + caddc 6389   - cmin 6445  -ucneg 6446  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239

This theorem is referenced by:  fsumshftm 8292  binomlem2 8327  iserzshft2i 8367  arisumilem 8486  mettrifi2 15848

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-sum 8240

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