Theorem fsumcllem 8274

Description: - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.)

Hypothesis

Ref	Expression
fsumcllem.1	|- ((x e. C /\ y e. C) -> (x + y) e. C)

Assertion

Ref	Expression
fsumcllem	|- ((N e. (ZZ>=` M) /\ A.k e. (M...N)A e. C) -> sum_k e. (M...N)A e. C)

Distinct variable groups:   x,y,A   x,k,y,C   k,M,x,y   k,N

Proof of Theorem fsumcllem

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . 5 |- (j = M -> (M...j) = (M...M))
2	1	raleqdv 2269	. . . 4 |- (j = M -> (A.k e. (M...j)A e. C <-> A.k e. (M...M)A e. C))
3	1	sumeq1d 8250	. . . . 5 |- (j = M -> sum_k e. (M...j)A = sum_k e. (M...M)A)
4	3	eleq1d 1963	. . . 4 |- (j = M -> (sum_k e. (M...j)A e. C <-> sum_k e. (M...M)A e. C))
5	2, 4	imbi12d 688	. . 3 |- (j = M -> ((A.k e. (M...j)A e. C -> sum_k e. (M...j)A e. C) <-> (A.k e. (M...M)A e. C -> sum_k e. (M...M)A e. C)))
6		opreq2 4890	. . . . 5 |- (j = m -> (M...j) = (M...m))
7	6	raleqdv 2269	. . . 4 |- (j = m -> (A.k e. (M...j)A e. C <-> A.k e. (M...m)A e. C))
8	6	sumeq1d 8250	. . . . 5 |- (j = m -> sum_k e. (M...j)A = sum_k e. (M...m)A)
9	8	eleq1d 1963	. . . 4 |- (j = m -> (sum_k e. (M...j)A e. C <-> sum_k e. (M...m)A e. C))
10	7, 9	imbi12d 688	. . 3 |- (j = m -> ((A.k e. (M...j)A e. C -> sum_k e. (M...j)A e. C) <-> (A.k e. (M...m)A e. C -> sum_k e. (M...m)A e. C)))
11		opreq2 4890	. . . . 5 |- (j = (m + 1) -> (M...j) = (M...(m + 1)))
12	11	raleqdv 2269	. . . 4 |- (j = (m + 1) -> (A.k e. (M...j)A e. C <-> A.k e. (M...(m + 1))A e. C))
13	11	sumeq1d 8250	. . . . 5 |- (j = (m + 1) -> sum_k e. (M...j)A = sum_k e. (M...(m + 1))A)
14	13	eleq1d 1963	. . . 4 |- (j = (m + 1) -> (sum_k e. (M...j)A e. C <-> sum_k e. (M...(m + 1))A e. C))
15	12, 14	imbi12d 688	. . 3 |- (j = (m + 1) -> ((A.k e. (M...j)A e. C -> sum_k e. (M...j)A e. C) <-> (A.k e. (M...(m + 1))A e. C -> sum_k e. (M...(m + 1))A e. C)))
16		opreq2 4890	. . . . 5 |- (j = N -> (M...j) = (M...N))
17	16	raleqdv 2269	. . . 4 |- (j = N -> (A.k e. (M...j)A e. C <-> A.k e. (M...N)A e. C))
18	16	sumeq1d 8250	. . . . 5 |- (j = N -> sum_k e. (M...j)A = sum_k e. (M...N)A)
19	18	eleq1d 1963	. . . 4 |- (j = N -> (sum_k e. (M...j)A e. C <-> sum_k e. (M...N)A e. C))
20	17, 19	imbi12d 688	. . 3 |- (j = N -> ((A.k e. (M...j)A e. C -> sum_k e. (M...j)A e. C) <-> (A.k e. (M...N)A e. C -> sum_k e. (M...N)A e. C)))
21		fsum1s 8269	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)A e. C) -> sum_k e. (M...M)A = [_M / k]_A)
22		ra4csbela 2587	. . . . . 6 |- ((M e. (M...M) /\ A.k e. (M...M)A e. C) -> [_M / k]_A e. C)
23		elfz3 7661	. . . . . 6 |- (M e. ZZ -> M e. (M...M))
24	22, 23	sylan 497	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)A e. C) -> [_M / k]_A e. C)
25	21, 24	eqeltrd 1971	. . . 4 |- ((M e. ZZ /\ A.k e. (M...M)A e. C) -> sum_k e. (M...M)A e. C)
26	25	ex 402	. . 3 |- (M e. ZZ -> (A.k e. (M...M)A e. C -> sum_k e. (M...M)A e. C))
27		fsump1s 8273	. . . . . 6 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. C) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
28	27	adantrl 430	. . . . 5 |- ((m e. (ZZ>=` M) /\ ((A.k e. (M...m)A e. C -> sum_k e. (M...m)A e. C) /\ A.k e. (M...(m + 1))A e. C)) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
29		eluzel2 7593	. . . . . . . . . . . 12 |- (m e. (ZZ>=` M) -> M e. ZZ)
30		eluzelz 7592	. . . . . . . . . . . 12 |- (m e. (ZZ>=` M) -> m e. ZZ)
31		fzssp1 7679	. . . . . . . . . . . 12 |- ((M e. ZZ /\ m e. ZZ) -> (M...m) C_ (M...(m + 1)))
32	29, 30, 31	syl11anc 524	. . . . . . . . . . 11 |- (m e. (ZZ>=` M) -> (M...m) C_ (M...(m + 1)))
33	32	sseld 2619	. . . . . . . . . 10 |- (m e. (ZZ>=` M) -> (k e. (M...m) -> k e. (M...(m + 1))))
34	33	imim1d 33	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> ((k e. (M...(m + 1)) -> A e. C) -> (k e. (M...m) -> A e. C)))
35	34	ralimdv2 2173	. . . . . . . 8 |- (m e. (ZZ>=` M) -> (A.k e. (M...(m + 1))A e. C -> A.k e. (M...m)A e. C))
36	35	imim1d 33	. . . . . . 7 |- (m e. (ZZ>=` M) -> ((A.k e. (M...m)A e. C -> sum_k e. (M...m)A e. C) -> (A.k e. (M...(m + 1))A e. C -> sum_k e. (M...m)A e. C)))
37	36	imp32 390	. . . . . 6 |- ((m e. (ZZ>=` M) /\ ((A.k e. (M...m)A e. C -> sum_k e. (M...m)A e. C) /\ A.k e. (M...(m + 1))A e. C)) -> sum_k e. (M...m)A e. C)
38		ra4csbela 2587	. . . . . . . 8 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. C) -> [_(m + 1) / k]_A e. C)
39		peano2uz 7616	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> (m + 1) e. (ZZ>=` M))
40		eluzfz2 7659	. . . . . . . . 9 |- ((m + 1) e. (ZZ>=` M) -> (m + 1) e. (M...(m + 1)))
41	39, 40	syl 12	. . . . . . . 8 |- (m e. (ZZ>=` M) -> (m + 1) e. (M...(m + 1)))
42	38, 41	sylan 497	. . . . . . 7 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. C) -> [_(m + 1) / k]_A e. C)
43	42	adantrl 430	. . . . . 6 |- ((m e. (ZZ>=` M) /\ ((A.k e. (M...m)A e. C -> sum_k e. (M...m)A e. C) /\ A.k e. (M...(m + 1))A e. C)) -> [_(m + 1) / k]_A e. C)
44		fsumcllem.1	. . . . . . 7 |- ((x e. C /\ y e. C) -> (x + y) e. C)
45	44	caoprcl 4985	. . . . . 6 |- ((sum_k e. (M...m)A e. C /\ [_(m + 1) / k]_A e. C) -> (sum_k e. (M...m)A + [_(m + 1) / k]_A) e. C)
46	37, 43, 45	syl11anc 524	. . . . 5 |- ((m e. (ZZ>=` M) /\ ((A.k e. (M...m)A e. C -> sum_k e. (M...m)A e. C) /\ A.k e. (M...(m + 1))A e. C)) -> (sum_k e. (M...m)A + [_(m + 1) / k]_A) e. C)
47	28, 46	eqeltrd 1971	. . . 4 |- ((m e. (ZZ>=` M) /\ ((A.k e. (M...m)A e. C -> sum_k e. (M...m)A e. C) /\ A.k e. (M...(m + 1))A e. C)) -> sum_k e. (M...(m + 1))A e. C)
48	47	exp32 408	. . 3 |- (m e. (ZZ>=` M) -> ((A.k e. (M...m)A e. C -> sum_k e. (M...m)A e. C) -> (A.k e. (M...(m + 1))A e. C -> sum_k e. (M...(m + 1))A e. C)))
49	5, 10, 15, 20, 26, 48	uzind4 7619	. 2 |- (N e. (ZZ>=` M) -> (A.k e. (M...N)A e. C -> sum_k e. (M...N)A e. C))
50	49	imp 377	1 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)A e. C) -> sum_k e. (M...N)A e. C)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  [_csb 2540   C_ wss 2593  ` cfv 3998  (class class class)co 4884  1c1 6387   + caddc 6389  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239

This theorem is referenced by:  fsumcl 8275  fsumrecl 8277  eirrlem2 8652

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  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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