Theorem fsum1ps 8278

Description: Separate out the first term in a finite sum.

Assertion

Ref	Expression
fsum1ps	|- ((N e. (ZZ>=` M) /\ M < N /\ A.k e. (M...N)A e. CC) -> sum_k e. (M...N)A = ([_M / k]_A + sum_k e. ((M + 1)...N)A))

Distinct variable groups:   k,M   k,N

Proof of Theorem fsum1ps

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . . 7 |- (j = (M + 1) -> (M...j) = (M...(M + 1)))
2	1	raleqdv 2269	. . . . . 6 |- (j = (M + 1) -> (A.k e. (M...j)A e. CC <-> A.k e. (M...(M + 1))A e. CC))
3	1	sumeq1d 8250	. . . . . . 7 |- (j = (M + 1) -> sum_k e. (M...j)A = sum_k e. (M...(M + 1))A)
4		opreq2 4890	. . . . . . . . 9 |- (j = (M + 1) -> ((M + 1)...j) = ((M + 1)...(M + 1)))
5	4	sumeq1d 8250	. . . . . . . 8 |- (j = (M + 1) -> sum_k e. ((M + 1)...j)A = sum_k e. ((M + 1)...(M + 1))A)
6	5	opreq2d 4898	. . . . . . 7 |- (j = (M + 1) -> ([_M / k]_A + sum_k e. ((M + 1)...j)A) = ([_M / k]_A + sum_k e. ((M + 1)...(M + 1))A))
7	3, 6	eqeq12d 1899	. . . . . 6 |- (j = (M + 1) -> (sum_k e. (M...j)A = ([_M / k]_A + sum_k e. ((M + 1)...j)A) <-> sum_k e. (M...(M + 1))A = ([_M / k]_A + sum_k e. ((M + 1)...(M + 1))A)))
8	2, 7	imbi12d 688	. . . . 5 |- (j = (M + 1) -> ((A.k e. (M...j)A e. CC -> sum_k e. (M...j)A = ([_M / k]_A + sum_k e. ((M + 1)...j)A)) <-> (A.k e. (M...(M + 1))A e. CC -> sum_k e. (M...(M + 1))A = ([_M / k]_A + sum_k e. ((M + 1)...(M + 1))A))))
9		opreq2 4890	. . . . . . 7 |- (j = m -> (M...j) = (M...m))
10	9	raleqdv 2269	. . . . . 6 |- (j = m -> (A.k e. (M...j)A e. CC <-> A.k e. (M...m)A e. CC))
11	9	sumeq1d 8250	. . . . . . 7 |- (j = m -> sum_k e. (M...j)A = sum_k e. (M...m)A)
12		opreq2 4890	. . . . . . . . 9 |- (j = m -> ((M + 1)...j) = ((M + 1)...m))
13	12	sumeq1d 8250	. . . . . . . 8 |- (j = m -> sum_k e. ((M + 1)...j)A = sum_k e. ((M + 1)...m)A)
14	13	opreq2d 4898	. . . . . . 7 |- (j = m -> ([_M / k]_A + sum_k e. ((M + 1)...j)A) = ([_M / k]_A + sum_k e. ((M + 1)...m)A))
15	11, 14	eqeq12d 1899	. . . . . 6 |- (j = m -> (sum_k e. (M...j)A = ([_M / k]_A + sum_k e. ((M + 1)...j)A) <-> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A)))
16	10, 15	imbi12d 688	. . . . 5 |- (j = m -> ((A.k e. (M...j)A e. CC -> sum_k e. (M...j)A = ([_M / k]_A + sum_k e. ((M + 1)...j)A)) <-> (A.k e. (M...m)A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A))))
17		opreq2 4890	. . . . . . 7 |- (j = (m + 1) -> (M...j) = (M...(m + 1)))
18	17	raleqdv 2269	. . . . . 6 |- (j = (m + 1) -> (A.k e. (M...j)A e. CC <-> A.k e. (M...(m + 1))A e. CC))
19	17	sumeq1d 8250	. . . . . . 7 |- (j = (m + 1) -> sum_k e. (M...j)A = sum_k e. (M...(m + 1))A)
20		opreq2 4890	. . . . . . . . 9 |- (j = (m + 1) -> ((M + 1)...j) = ((M + 1)...(m + 1)))
21	20	sumeq1d 8250	. . . . . . . 8 |- (j = (m + 1) -> sum_k e. ((M + 1)...j)A = sum_k e. ((M + 1)...(m + 1))A)
22	21	opreq2d 4898	. . . . . . 7 |- (j = (m + 1) -> ([_M / k]_A + sum_k e. ((M + 1)...j)A) = ([_M / k]_A + sum_k e. ((M + 1)...(m + 1))A))
23	19, 22	eqeq12d 1899	. . . . . 6 |- (j = (m + 1) -> (sum_k e. (M...j)A = ([_M / k]_A + sum_k e. ((M + 1)...j)A) <-> sum_k e. (M...(m + 1))A = ([_M / k]_A + sum_k e. ((M + 1)...(m + 1))A)))
24	18, 23	imbi12d 688	. . . . 5 |- (j = (m + 1) -> ((A.k e. (M...j)A e. CC -> sum_k e. (M...j)A = ([_M / k]_A + sum_k e. ((M + 1)...j)A)) <-> (A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...(m + 1))A = ([_M / k]_A + sum_k e. ((M + 1)...(m + 1))A))))
25		opreq2 4890	. . . . . . 7 |- (j = N -> (M...j) = (M...N))
26	25	raleqdv 2269	. . . . . 6 |- (j = N -> (A.k e. (M...j)A e. CC <-> A.k e. (M...N)A e. CC))
27	25	sumeq1d 8250	. . . . . . 7 |- (j = N -> sum_k e. (M...j)A = sum_k e. (M...N)A)
28		opreq2 4890	. . . . . . . . 9 |- (j = N -> ((M + 1)...j) = ((M + 1)...N))
29	28	sumeq1d 8250	. . . . . . . 8 |- (j = N -> sum_k e. ((M + 1)...j)A = sum_k e. ((M + 1)...N)A)
30	29	opreq2d 4898	. . . . . . 7 |- (j = N -> ([_M / k]_A + sum_k e. ((M + 1)...j)A) = ([_M / k]_A + sum_k e. ((M + 1)...N)A))
31	27, 30	eqeq12d 1899	. . . . . 6 |- (j = N -> (sum_k e. (M...j)A = ([_M / k]_A + sum_k e. ((M + 1)...j)A) <-> sum_k e. (M...N)A = ([_M / k]_A + sum_k e. ((M + 1)...N)A)))
32	26, 31	imbi12d 688	. . . . 5 |- (j = N -> ((A.k e. (M...j)A e. CC -> sum_k e. (M...j)A = ([_M / k]_A + sum_k e. ((M + 1)...j)A)) <-> (A.k e. (M...N)A e. CC -> sum_k e. (M...N)A = ([_M / k]_A + sum_k e. ((M + 1)...N)A))))
33		fsump1s 8273	. . . . . . . 8 |- ((M e. (ZZ>=` M) /\ A.k e. (M...(M + 1))A e. CC) -> sum_k e. (M...(M + 1))A = (sum_k e. (M...M)A + [_(M + 1) / k]_A))
34		uzid 7596	. . . . . . . 8 |- (M e. ZZ -> M e. (ZZ>=` M))
35	33, 34	sylan 497	. . . . . . 7 |- ((M e. ZZ /\ A.k e. (M...(M + 1))A e. CC) -> sum_k e. (M...(M + 1))A = (sum_k e. (M...M)A + [_(M + 1) / k]_A))
36		fzssp1 7679	. . . . . . . . . . . . . . 15 |- ((M e. ZZ /\ M e. ZZ) -> (M...M) C_ (M...(M + 1)))
37	36	anidms 480	. . . . . . . . . . . . . 14 |- (M e. ZZ -> (M...M) C_ (M...(M + 1)))
38	37	sseld 2619	. . . . . . . . . . . . 13 |- (M e. ZZ -> (k e. (M...M) -> k e. (M...(M + 1))))
39	38	imim1d 33	. . . . . . . . . . . 12 |- (M e. ZZ -> ((k e. (M...(M + 1)) -> A e. CC) -> (k e. (M...M) -> A e. CC)))
40	39	ralimdv2 2173	. . . . . . . . . . 11 |- (M e. ZZ -> (A.k e. (M...(M + 1))A e. CC -> A.k e. (M...M)A e. CC))
41	40	imp 377	. . . . . . . . . 10 |- ((M e. ZZ /\ A.k e. (M...(M + 1))A e. CC) -> A.k e. (M...M)A e. CC)
42		fsum1s 8269	. . . . . . . . . 10 |- ((M e. ZZ /\ A.k e. (M...M)A e. CC) -> sum_k e. (M...M)A = [_M / k]_A)
43	41, 42	syldan 516	. . . . . . . . 9 |- ((M e. ZZ /\ A.k e. (M...(M + 1))A e. CC) -> sum_k e. (M...M)A = [_M / k]_A)
44	43	eqcomd 1889	. . . . . . . 8 |- ((M e. ZZ /\ A.k e. (M...(M + 1))A e. CC) -> [_M / k]_A = sum_k e. (M...M)A)
45		peano2z 7375	. . . . . . . . . . . . . 14 |- (M e. ZZ -> (M + 1) e. ZZ)
46		fzp1ss 7680	. . . . . . . . . . . . . 14 |- ((M e. ZZ /\ (M + 1) e. ZZ) -> ((M + 1)...(M + 1)) C_ (M...(M + 1)))
47	45, 46	mpdan 768	. . . . . . . . . . . . 13 |- (M e. ZZ -> ((M + 1)...(M + 1)) C_ (M...(M + 1)))
48	47	sseld 2619	. . . . . . . . . . . 12 |- (M e. ZZ -> (k e. ((M + 1)...(M + 1)) -> k e. (M...(M + 1))))
49	48	imim1d 33	. . . . . . . . . . 11 |- (M e. ZZ -> ((k e. (M...(M + 1)) -> A e. CC) -> (k e. ((M + 1)...(M + 1)) -> A e. CC)))
50	49	ralimdv2 2173	. . . . . . . . . 10 |- (M e. ZZ -> (A.k e. (M...(M + 1))A e. CC -> A.k e. ((M + 1)...(M + 1))A e. CC))
51	50	imp 377	. . . . . . . . 9 |- ((M e. ZZ /\ A.k e. (M...(M + 1))A e. CC) -> A.k e. ((M + 1)...(M + 1))A e. CC)
52		fsum1s 8269	. . . . . . . . . 10 |- (((M + 1) e. ZZ /\ A.k e. ((M + 1)...(M + 1))A e. CC) -> sum_k e. ((M + 1)...(M + 1))A = [_(M + 1) / k]_A)
53	52, 45	sylan 497	. . . . . . . . 9 |- ((M e. ZZ /\ A.k e. ((M + 1)...(M + 1))A e. CC) -> sum_k e. ((M + 1)...(M + 1))A = [_(M + 1) / k]_A)
54	51, 53	syldan 516	. . . . . . . 8 |- ((M e. ZZ /\ A.k e. (M...(M + 1))A e. CC) -> sum_k e. ((M + 1)...(M + 1))A = [_(M + 1) / k]_A)
55	44, 54	opreq12d 4900	. . . . . . 7 |- ((M e. ZZ /\ A.k e. (M...(M + 1))A e. CC) -> ([_M / k]_A + sum_k e. ((M + 1)...(M + 1))A) = (sum_k e. (M...M)A + [_(M + 1) / k]_A))
56	35, 55	eqtr4d 1928	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...(M + 1))A e. CC) -> sum_k e. (M...(M + 1))A = ([_M / k]_A + sum_k e. ((M + 1)...(M + 1))A))
57	56	ex 402	. . . . 5 |- (M e. ZZ -> (A.k e. (M...(M + 1))A e. CC -> sum_k e. (M...(M + 1))A = ([_M / k]_A + sum_k e. ((M + 1)...(M + 1))A)))
58		fzssp1 7679	. . . . . . . . . . 11 |- ((M e. ZZ /\ m e. ZZ) -> (M...m) C_ (M...(m + 1)))
59	58	3adant3 896	. . . . . . . . . 10 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> (M...m) C_ (M...(m + 1)))
60	59	sseld 2619	. . . . . . . . 9 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> (k e. (M...m) -> k e. (M...(m + 1))))
61	60	imim1d 33	. . . . . . . 8 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> ((k e. (M...(m + 1)) -> A e. CC) -> (k e. (M...m) -> A e. CC)))
62	61	ralimdv2 2173	. . . . . . 7 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> (A.k e. (M...(m + 1))A e. CC -> A.k e. (M...m)A e. CC))
63	62	imim1d 33	. . . . . 6 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> ((A.k e. (M...m)A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A)) -> (A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A))))
64		pm3.35 386	. . . . . . . . . . 11 |- ((A.k e. (M...(m + 1))A e. CC /\ (A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A))) -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A))
65	64	ancoms 484	. . . . . . . . . 10 |- (((A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A)) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A))
66	65	opreq1d 4897	. . . . . . . . 9 |- (((A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A)) /\ A.k e. (M...(m + 1))A e. CC) -> (sum_k e. (M...m)A + [_(m + 1) / k]_A) = (([_M / k]_A + sum_k e. ((M + 1)...m)A) + [_(m + 1) / k]_A))
67	66	adantll 428	. . . . . . . 8 |- ((((M e. ZZ /\ m e. ZZ /\ M < m) /\ (A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A))) /\ A.k e. (M...(m + 1))A e. CC) -> (sum_k e. (M...m)A + [_(m + 1) / k]_A) = (([_M / k]_A + sum_k e. ((M + 1)...m)A) + [_(m + 1) / k]_A))
68		fsump1s 8273	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
69		ltle 6690	. . . . . . . . . . . . 13 |- ((M e. RR /\ m e. RR) -> (M < m -> M <_ m))
70		zre 7348	. . . . . . . . . . . . 13 |- (M e. ZZ -> M e. RR)
71		zre 7348	. . . . . . . . . . . . 13 |- (m e. ZZ -> m e. RR)
72	69, 70, 71	syl2an 503	. . . . . . . . . . . 12 |- ((M e. ZZ /\ m e. ZZ) -> (M < m -> M <_ m))
73		eluz 7595	. . . . . . . . . . . 12 |- ((M e. ZZ /\ m e. ZZ) -> (m e. (ZZ>=` M) <-> M <_ m))
74	72, 73	sylibrd 221	. . . . . . . . . . 11 |- ((M e. ZZ /\ m e. ZZ) -> (M < m -> m e. (ZZ>=` M)))
75	74	3impia 1064	. . . . . . . . . 10 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> m e. (ZZ>=` M))
76	68, 75	sylan 497	. . . . . . . . 9 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
77	76	adantlr 429	. . . . . . . 8 |- ((((M e. ZZ /\ m e. ZZ /\ M < m) /\ (A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A))) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...(m + 1))A = (sum_k e. (M...m)A + [_(m + 1) / k]_A))
78		fzp1ss 7680	. . . . . . . . . . . . . . . . . 18 |- ((M e. ZZ /\ (m + 1) e. ZZ) -> ((M + 1)...(m + 1)) C_ (M...(m + 1)))
79		peano2z 7375	. . . . . . . . . . . . . . . . . 18 |- (m e. ZZ -> (m + 1) e. ZZ)
80	78, 79	sylan2 500	. . . . . . . . . . . . . . . . 17 |- ((M e. ZZ /\ m e. ZZ) -> ((M + 1)...(m + 1)) C_ (M...(m + 1)))
81	80	3adant3 896	. . . . . . . . . . . . . . . 16 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> ((M + 1)...(m + 1)) C_ (M...(m + 1)))
82	81	sseld 2619	. . . . . . . . . . . . . . 15 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> (k e. ((M + 1)...(m + 1)) -> k e. (M...(m + 1))))
83	82	imim1d 33	. . . . . . . . . . . . . 14 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> ((k e. (M...(m + 1)) -> A e. CC) -> (k e. ((M + 1)...(m + 1)) -> A e. CC)))
84	83	ralimdv2 2173	. . . . . . . . . . . . 13 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> (A.k e. (M...(m + 1))A e. CC -> A.k e. ((M + 1)...(m + 1))A e. CC))
85	84	imp 377	. . . . . . . . . . . 12 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. (M...(m + 1))A e. CC) -> A.k e. ((M + 1)...(m + 1))A e. CC)
86		fsump1s 8273	. . . . . . . . . . . . 13 |- ((m e. (ZZ>=` (M + 1)) /\ A.k e. ((M + 1)...(m + 1))A e. CC) -> sum_k e. ((M + 1)...(m + 1))A = (sum_k e. ((M + 1)...m)A + [_(m + 1) / k]_A))
87		zltp1le 7390	. . . . . . . . . . . . . . 15 |- ((M e. ZZ /\ m e. ZZ) -> (M < m <-> (M + 1) <_ m))
88		eluz 7595	. . . . . . . . . . . . . . . 16 |- (((M + 1) e. ZZ /\ m e. ZZ) -> (m e. (ZZ>=` (M + 1)) <-> (M + 1) <_ m))
89	88, 45	sylan 497	. . . . . . . . . . . . . . 15 |- ((M e. ZZ /\ m e. ZZ) -> (m e. (ZZ>=` (M + 1)) <-> (M + 1) <_ m))
90	87, 89	bitr4d 590	. . . . . . . . . . . . . 14 |- ((M e. ZZ /\ m e. ZZ) -> (M < m <-> m e. (ZZ>=` (M + 1))))
91	90	biimp3a 1194	. . . . . . . . . . . . 13 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> m e. (ZZ>=` (M + 1)))
92	86, 91	sylan 497	. . . . . . . . . . . 12 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. ((M + 1)...(m + 1))A e. CC) -> sum_k e. ((M + 1)...(m + 1))A = (sum_k e. ((M + 1)...m)A + [_(m + 1) / k]_A))
93	85, 92	syldan 516	. . . . . . . . . . 11 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. ((M + 1)...(m + 1))A = (sum_k e. ((M + 1)...m)A + [_(m + 1) / k]_A))
94	93	opreq2d 4898	. . . . . . . . . 10 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. (M...(m + 1))A e. CC) -> ([_M / k]_A + sum_k e. ((M + 1)...(m + 1))A) = ([_M / k]_A + (sum_k e. ((M + 1)...m)A + [_(m + 1) / k]_A)))
95		ra4csbela 2587	. . . . . . . . . . . 12 |- ((M e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. CC) -> [_M / k]_A e. CC)
96		peano2uz 7616	. . . . . . . . . . . . . 14 |- (m e. (ZZ>=` M) -> (m + 1) e. (ZZ>=` M))
97	75, 96	syl 12	. . . . . . . . . . . . 13 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> (m + 1) e. (ZZ>=` M))
98		eluzfz1 7657	. . . . . . . . . . . . 13 |- ((m + 1) e. (ZZ>=` M) -> M e. (M...(m + 1)))
99	97, 98	syl 12	. . . . . . . . . . . 12 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> M e. (M...(m + 1)))
100	95, 99	sylan 497	. . . . . . . . . . 11 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. (M...(m + 1))A e. CC) -> [_M / k]_A e. CC)
101		fzp1ss 7680	. . . . . . . . . . . . . . . . . 18 |- ((M e. ZZ /\ m e. ZZ) -> ((M + 1)...m) C_ (M...m))
102	101, 58	sstrd 2627	. . . . . . . . . . . . . . . . 17 |- ((M e. ZZ /\ m e. ZZ) -> ((M + 1)...m) C_ (M...(m + 1)))
103	102	3adant3 896	. . . . . . . . . . . . . . . 16 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> ((M + 1)...m) C_ (M...(m + 1)))
104	103	sseld 2619	. . . . . . . . . . . . . . 15 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> (k e. ((M + 1)...m) -> k e. (M...(m + 1))))
105	104	imim1d 33	. . . . . . . . . . . . . 14 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> ((k e. (M...(m + 1)) -> A e. CC) -> (k e. ((M + 1)...m) -> A e. CC)))
106	105	ralimdv2 2173	. . . . . . . . . . . . 13 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> (A.k e. (M...(m + 1))A e. CC -> A.k e. ((M + 1)...m)A e. CC))
107	106	imp 377	. . . . . . . . . . . 12 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. (M...(m + 1))A e. CC) -> A.k e. ((M + 1)...m)A e. CC)
108		fsumcl 8275	. . . . . . . . . . . . 13 |- ((m e. (ZZ>=` (M + 1)) /\ A.k e. ((M + 1)...m)A e. CC) -> sum_k e. ((M + 1)...m)A e. CC)
109	108, 91	sylan 497	. . . . . . . . . . . 12 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. ((M + 1)...m)A e. CC) -> sum_k e. ((M + 1)...m)A e. CC)
110	107, 109	syldan 516	. . . . . . . . . . 11 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. ((M + 1)...m)A e. CC)
111		ra4csbela 2587	. . . . . . . . . . . 12 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. CC) -> [_(m + 1) / k]_A e. CC)
112		eluzfz2b 7660	. . . . . . . . . . . . 13 |- ((m + 1) e. (ZZ>=` M) <-> (m + 1) e. (M...(m + 1)))
113	97, 112	sylib 215	. . . . . . . . . . . 12 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> (m + 1) e. (M...(m + 1)))
114	111, 113	sylan 497	. . . . . . . . . . 11 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. (M...(m + 1))A e. CC) -> [_(m + 1) / k]_A e. CC)
115		addass 6460	. . . . . . . . . . 11 |- (([_M / k]_A e. CC /\ sum_k e. ((M + 1)...m)A e. CC /\ [_(m + 1) / k]_A e. CC) -> (([_M / k]_A + sum_k e. ((M + 1)...m)A) + [_(m + 1) / k]_A) = ([_M / k]_A + (sum_k e. ((M + 1)...m)A + [_(m + 1) / k]_A)))
116	100, 110, 114, 115	syl111anc 1100	. . . . . . . . . 10 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. (M...(m + 1))A e. CC) -> (([_M / k]_A + sum_k e. ((M + 1)...m)A) + [_(m + 1) / k]_A) = ([_M / k]_A + (sum_k e. ((M + 1)...m)A + [_(m + 1) / k]_A)))
117	94, 116	eqtr4d 1928	. . . . . . . . 9 |- (((M e. ZZ /\ m e. ZZ /\ M < m) /\ A.k e. (M...(m + 1))A e. CC) -> ([_M / k]_A + sum_k e. ((M + 1)...(m + 1))A) = (([_M / k]_A + sum_k e. ((M + 1)...m)A) + [_(m + 1) / k]_A))
118	117	adantlr 429	. . . . . . . 8 |- ((((M e. ZZ /\ m e. ZZ /\ M < m) /\ (A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A))) /\ A.k e. (M...(m + 1))A e. CC) -> ([_M / k]_A + sum_k e. ((M + 1)...(m + 1))A) = (([_M / k]_A + sum_k e. ((M + 1)...m)A) + [_(m + 1) / k]_A))
119	67, 77, 118	3eqtr4d 1937	. . . . . . 7 |- ((((M e. ZZ /\ m e. ZZ /\ M < m) /\ (A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A))) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...(m + 1))A = ([_M / k]_A + sum_k e. ((M + 1)...(m + 1))A))
120	119	exp31 407	. . . . . 6 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> ((A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A)) -> (A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...(m + 1))A = ([_M / k]_A + sum_k e. ((M + 1)...(m + 1))A))))
121	63, 120	syld 30	. . . . 5 |- ((M e. ZZ /\ m e. ZZ /\ M < m) -> ((A.k e. (M...m)A e. CC -> sum_k e. (M...m)A = ([_M / k]_A + sum_k e. ((M + 1)...m)A)) -> (A.k e. (M...(m + 1))A e. CC -> sum_k e. (M...(m + 1))A = ([_M / k]_A + sum_k e. ((M + 1)...(m + 1))A))))
122	8, 16, 24, 32, 57, 121	uzind2 7418	. . . 4 |- ((M e. ZZ /\ N e. ZZ /\ M < N) -> (A.k e. (M...N)A e. CC -> sum_k e. (M...N)A = ([_M / k]_A + sum_k e. ((M + 1)...N)A)))
123	122	3expa 1067	. . 3 |- (((M e. ZZ /\ N e. ZZ) /\ M < N) -> (A.k e. (M...N)A e. CC -> sum_k e. (M...N)A = ([_M / k]_A + sum_k e. ((M + 1)...N)A)))
124		eluzel2 7593	. . . 4 |- (N e. (ZZ>=` M) -> M e. ZZ)
125		eluzelz 7592	. . . 4 |- (N e. (ZZ>=` M) -> N e. ZZ)
126	124, 125	jca 310	. . 3 |- (N e. (ZZ>=` M) -> (M e. ZZ /\ N e. ZZ))
127	123, 126	sylan 497	. 2 |- ((N e. (ZZ>=` M) /\ M < N) -> (A.k e. (M...N)A e. CC -> sum_k e. (M...N)A = ([_M / k]_A + sum_k e. ((M + 1)...N)A)))
128	127	3impia 1064	1 |- ((N e. (ZZ>=` M) /\ M < N /\ A.k e. (M...N)A e. CC) -> sum_k e. (M...N)A = ([_M / k]_A + sum_k e. ((M + 1)...N)A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  [_csb 2540   C_ wss 2593   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  1c1 6387   + caddc 6389   <_ cle 6448  ZZcz 6451   < clt 6653  ZZ>=cuz 7586  ...cfz 7637  sum_csu 8239

This theorem is referenced by:  fsum1p 8279  fsumsplit 8280  fsumrev 8289  binomlem1 8326  binomlem4 8329

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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