Theorem fsumcllem 6903

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 3908	. . . . 5 |- (j = M -> (M...j) = (M...M))
2	1	raleq1d 1765	. . . 4 |- (j = M -> (A.k e. (M...j)A e. C <-> A.k e. (M...M)A e. C))
3	1	sumeq1d 6879	. . . . 5 |- (j = M -> sum_k e. (M...j)A = sum_k e. (M...M)A)
4	3	eleq1d 1516	. . . 4 |- (j = M -> (sum_k e. (M...j)A e. C <-> sum_k e. (M...M)A e. C))
5	2, 4	imbi12d 624	. . 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 3908	. . . . 5 |- (j = m -> (M...j) = (M...m))
7	6	raleq1d 1765	. . . 4 |- (j = m -> (A.k e. (M...j)A e. C <-> A.k e. (M...m)A e. C))
8	6	sumeq1d 6879	. . . . 5 |- (j = m -> sum_k e. (M...j)A = sum_k e. (M...m)A)
9	8	eleq1d 1516	. . . 4 |- (j = m -> (sum_k e. (M...j)A e. C <-> sum_k e. (M...m)A e. C))
10	7, 9	imbi12d 624	. . 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 3908	. . . . 5 |- (j = (m + 1) -> (M...j) = (M...(m + 1)))
12	11	raleq1d 1765	. . . 4 |- (j = (m + 1) -> (A.k e. (M...j)A e. C <-> A.k e. (M...(m + 1))A e. C))
13	11	sumeq1d 6879	. . . . 5 |- (j = (m + 1) -> sum_k e. (M...j)A = sum_k e. (M...(m + 1))A)
14	13	eleq1d 1516	. . . 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 624	. . 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 3908	. . . . 5 |- (j = N -> (M...j) = (M...N))
17	16	raleq1d 1765	. . . 4 |- (j = N -> (A.k e. (M...j)A e. C <-> A.k e. (M...N)A e. C))
18	16	sumeq1d 6879	. . . . 5 |- (j = N -> sum_k e. (M...j)A = sum_k e. (M...N)A)
19	18	eleq1d 1516	. . . 4 |- (j = N -> (sum_k e. (M...j)A e. C <-> sum_k e. (M...N)A e. C))
20	17, 19	imbi12d 624	. . 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 6898	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)A e. C) -> sum_k e. (M...M)A = [_M / k]_A)
22		ra4csbela 2013	. . . . . 6 |- ((M e. (M...M) /\ A.k e. (M...M)A e. C) -> [_M / k]_A e. C)
23		elfz3t 6374	. . . . . 6 |- (M e. ZZ -> M e. (M...M))
24	22, 23	sylan 448	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)A e. C) -> [_M / k]_A e. C)
25	21, 24	eqeltrd 1524	. . . 4 |- ((M e. ZZ /\ A.k e. (M...M)A e. C) -> sum_k e. (M...M)A e. C)
26	25	ex 373	. . 3 |- (M e. ZZ -> (A.k e. (M...M)A e. C -> sum_k e. (M...M)A e. C))
27		fsump1s 6902	. . . . . 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 394	. . . . 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		fsumcllem.1	. . . . . . 7 |- ((x e. C /\ y e. C) -> (x + y) e. C)
30	29	caoprcl 3992	. . . . . 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)
31		fzssp1t 6389	. . . . . . . . . . . 12 |- ((M e. ZZ /\ m e. ZZ) -> (M...m) (_ (M...(m + 1)))
32		eluzel2 6307	. . . . . . . . . . . 12 |- (m e. (ZZ>` M) -> M e. ZZ)
33		eluzelz 6306	. . . . . . . . . . . 12 |- (m e. (ZZ>` M) -> m e. ZZ)
34	31, 32, 33	sylanc 471	. . . . . . . . . . 11 |- (m e. (ZZ>` M) -> (M...m) (_ (M...(m + 1)))
35	34	sseld 2038	. . . . . . . . . 10 |- (m e. (ZZ>` M) -> (k e. (M...m) -> k e. (M...(m + 1))))
36	35	imim1d 28	. . . . . . . . 9 |- (m e. (ZZ>` M) -> ((k e. (M...(m + 1)) -> A e. C) -> (k e. (M...m) -> A e. C)))
37	36	r19.20dv2 1687	. . . . . . . 8 |- (m e. (ZZ>` M) -> (A.k e. (M...(m + 1))A e. C -> A.k e. (M...m)A e. C))
38	37	imim1d 28	. . . . . . 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)))
39	38	imp32 363	. . . . . 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)
40		ra4csbela 2013	. . . . . . . 8 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. C) -> [_(m + 1) / k]_A e. C)
41		peano2uz 6330	. . . . . . . . 9 |- (m e. (ZZ>` M) -> (m + 1) e. (ZZ>` M))
42		eluzfz2t 6372	. . . . . . . . 9 |- ((m + 1) e. (ZZ>` M) -> (m + 1) e. (M...(m + 1)))
43	41, 42	syl 10	. . . . . . . 8 |- (m e. (ZZ>` M) -> (m + 1) e. (M...(m + 1)))
44	40, 43	sylan 448	. . . . . . 7 |- ((m e. (ZZ>` M) /\ A.k e. (M...(m + 1))A e. C) -> [_(m + 1) / k]_A e. C)
45	44	adantrl 394	. . . . . 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)
46	30, 39, 45	sylanc 471	. . . . 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 1524	. . . 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 377	. . 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 6333	. 2 |- (N e. (ZZ>` M) -> (A.k e. (M...N)A e. C -> sum_k e. (M...N)A e. C))
50	49	imp 350	1 |- ((N e. (ZZ>` M) /\ A.k e. (M...N)A e. C) ->