Theorem fsumabs 8303

Description: Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values.

Assertion

Ref	Expression
fsumabs	|- ((N e. (ZZ>=` M) /\ A.k e. (M...N)A e. CC) -> (abs` sum_k e. (M...N)A) <_ sum_k e. (M...N)(abs` A))

Distinct variable groups:   k,M   k,N

Proof of Theorem fsumabs

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. CC <-> A.k e. (M...M)A e. CC))
3	1	sumeq1d 8250	. . . . . 6 |- (j = M -> sum_k e. (M...j)A = sum_k e. (M...M)A)
4	3	fveq2d 4685	. . . . 5 |- (j = M -> (abs` sum_k e. (M...j)A) = (abs` sum_k e. (M...M)A))
5	1	sumeq1d 8250	. . . . 5 |- (j = M -> sum_k e. (M...j)(abs` A) = sum_k e. (M...M)(abs` A))
6	4, 5	breq12d 3351	. . . 4 |- (j = M -> ((abs` sum_k e. (M...j)A) <_ sum_k e. (M...j)(abs` A) <-> (abs` sum_k e. (M...M)A) <_ sum_k e. (M...M)(abs` A)))
7	2, 6	imbi12d 688	. . 3 |- (j = M -> ((A.k e. (M...j)A e. CC -> (abs` sum_k e. (M...j)A) <_ sum_k e. (M...j)(abs` A)) <-> (A.k e. (M...M)A e. CC -> (abs` sum_k e. (M...M)A) <_ sum_k e. (M...M)(abs` A))))
8		opreq2 4890	. . . . 5 |- (j = m -> (M...j) = (M...m))
9	8	raleqdv 2269	. . . 4 |- (j = m -> (A.k e. (M...j)A e. CC <-> A.k e. (M...m)A e. CC))
10	8	sumeq1d 8250	. . . . . 6 |- (j = m -> sum_k e. (M...j)A = sum_k e. (M...m)A)
11	10	fveq2d 4685	. . . . 5 |- (j = m -> (abs` sum_k e. (M...j)A) = (abs` sum_k e. (M...m)A))
12	8	sumeq1d 8250	. . . . 5 |- (j = m -> sum_k e. (M...j)(abs` A) = sum_k e. (M...m)(abs` A))
13	11, 12	breq12d 3351	. . . 4 |- (j = m -> ((abs` sum_k e. (M...j)A) <_ sum_k e. (M...j)(abs` A) <-> (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A)))
14	9, 13	imbi12d 688	. . 3 |- (j = m -> ((A.k e. (M...j)A e. CC -> (abs` sum_k e. (M...j)A) <_ sum_k e. (M...j)(abs` A)) <-> (A.k e. (M...m)A e. CC -> (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A))))
15		opreq2 4890	. . . . 5 |- (j = (m + 1) -> (M...j) = (M...(m + 1)))
16	15	raleqdv 2269	. . . 4 |- (j = (m + 1) -> (A.k e. (M...j)A e. CC <-> A.k e. (M...(m + 1))A e. CC))
17	15	sumeq1d 8250	. . . . . 6 |- (j = (m + 1) -> sum_k e. (M...j)A = sum_k e. (M...(m + 1))A)
18	17	fveq2d 4685	. . . . 5 |- (j = (m + 1) -> (abs` sum_k e. (M...j)A) = (abs` sum_k e. (M...(m + 1))A))
19	15	sumeq1d 8250	. . . . 5 |- (j = (m + 1) -> sum_k e. (M...j)(abs` A) = sum_k e. (M...(m + 1))(abs` A))
20	18, 19	breq12d 3351	. . . 4 |- (j = (m + 1) -> ((abs` sum_k e. (M...j)A) <_ sum_k e. (M...j)(abs` A) <-> (abs` sum_k e. (M...(m + 1))A) <_ sum_k e. (M...(m + 1))(abs` A)))
21	16, 20	imbi12d 688	. . 3 |- (j = (m + 1) -> ((A.k e. (M...j)A e. CC -> (abs` sum_k e. (M...j)A) <_ sum_k e. (M...j)(abs` A)) <-> (A.k e. (M...(m + 1))A e. CC -> (abs` sum_k e. (M...(m + 1))A) <_ sum_k e. (M...(m + 1))(abs` A))))
22		opreq2 4890	. . . . 5 |- (j = N -> (M...j) = (M...N))
23	22	raleqdv 2269	. . . 4 |- (j = N -> (A.k e. (M...j)A e. CC <-> A.k e. (M...N)A e. CC))
24	22	sumeq1d 8250	. . . . . 6 |- (j = N -> sum_k e. (M...j)A = sum_k e. (M...N)A)
25	24	fveq2d 4685	. . . . 5 |- (j = N -> (abs` sum_k e. (M...j)A) = (abs` sum_k e. (M...N)A))
26	22	sumeq1d 8250	. . . . 5 |- (j = N -> sum_k e. (M...j)(abs` A) = sum_k e. (M...N)(abs` A))
27	25, 26	breq12d 3351	. . . 4 |- (j = N -> ((abs` sum_k e. (M...j)A) <_ sum_k e. (M...j)(abs` A) <-> (abs` sum_k e. (M...N)A) <_ sum_k e. (M...N)(abs` A)))
28	23, 27	imbi12d 688	. . 3 |- (j = N -> ((A.k e. (M...j)A e. CC -> (abs` sum_k e. (M...j)A) <_ sum_k e. (M...j)(abs` A)) <-> (A.k e. (M...N)A e. CC -> (abs` sum_k e. (M...N)A) <_ sum_k e. (M...N)(abs` A))))
29		csbfv2g 4700	. . . . . . 7 |- (M e. ZZ -> [_M / k]_(abs` A) = (abs` [_M / k]_A))
30	29	adantr 425	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)A e. CC) -> [_M / k]_(abs` A) = (abs` [_M / k]_A))
31		ra4csbela 2587	. . . . . . . 8 |- ((M e. (M...M) /\ A.k e. (M...M)(abs` A) e. RR) -> [_M / k]_(abs` A) e. RR)
32		elfz3 7661	. . . . . . . 8 |- (M e. ZZ -> M e. (M...M))
33		abscl 8084	. . . . . . . . 9 |- (A e. CC -> (abs` A) e. RR)
34	33	ralimi 2168	. . . . . . . 8 |- (A.k e. (M...M)A e. CC -> A.k e. (M...M)(abs` A) e. RR)
35	31, 32, 34	syl2an 503	. . . . . . 7 |- ((M e. ZZ /\ A.k e. (M...M)A e. CC) -> [_M / k]_(abs` A) e. RR)
36		leid 6701	. . . . . . 7 |- ([_M / k]_(abs` A) e. RR -> [_M / k]_(abs` A) <_ [_M / k]_(abs` A))
37	35, 36	syl 12	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)A e. CC) -> [_M / k]_(abs` A) <_ [_M / k]_(abs` A))
38	30, 37	eqbrtrrd 3359	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)A e. CC) -> (abs` [_M / k]_A) <_ [_M / k]_(abs` A))
39		fsum1s 8269	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)A e. CC) -> sum_k e. (M...M)A = [_M / k]_A)
40	39	fveq2d 4685	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)A e. CC) -> (abs` sum_k e. (M...M)A) = (abs` [_M / k]_A))
41		fsum1s 8269	. . . . . 6 |- ((M e. ZZ /\ A.k e. (M...M)(abs` A) e. CC) -> sum_k e. (M...M)(abs` A) = [_M / k]_(abs` A))
42	33	recnd 6468	. . . . . . 7 |- (A e. CC -> (abs` A) e. CC)
43	42	ralimi 2168	. . . . . 6 |- (A.k e. (M...M)A e. CC -> A.k e. (M...M)(abs` A) e. CC)
44	41, 43	sylan2 500	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)A e. CC) -> sum_k e. (M...M)(abs` A) = [_M / k]_(abs` A))
45	38, 40, 44	3brtr4d 3367	. . . 4 |- ((M e. ZZ /\ A.k e. (M...M)A e. CC) -> (abs` sum_k e. (M...M)A) <_ sum_k e. (M...M)(abs` A))
46	45	ex 402	. . 3 |- (M e. ZZ -> (A.k e. (M...M)A e. CC -> (abs` sum_k e. (M...M)A) <_ sum_k e. (M...M)(abs` A)))
47		eluzel2 7593	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> M e. ZZ)
48		eluzelz 7592	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> m e. ZZ)
49		fzssp1 7679	. . . . . . . . 9 |- ((M e. ZZ /\ m e. ZZ) -> (M...m) C_ (M...(m + 1)))
50	47, 48, 49	syl11anc 524	. . . . . . . 8 |- (m e. (ZZ>=` M) -> (M...m) C_ (M...(m + 1)))
51	50	sseld 2619	. . . . . . 7 |- (m e. (ZZ>=` M) -> (k e. (M...m) -> k e. (M...(m + 1))))
52	51	imim1d 33	. . . . . 6 |- (m e. (ZZ>=` M) -> ((k e. (M...(m + 1)) -> A e. CC) -> (k e. (M...m) -> A e. CC)))
53	52	ralimdv2 2173	. . . . 5 |- (m e. (ZZ>=` M) -> (A.k e. (M...(m + 1))A e. CC -> A.k e. (M...m)A e. CC))
54	53	imim1d 33	. . . 4 |- (m e. (ZZ>=` M) -> ((A.k e. (M...m)A e. CC -> (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A)) -> (A.k e. (M...(m + 1))A e. CC -> (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A))))
55	53	imp 377	. . . . . . . . . . 11 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> A.k e. (M...m)A e. CC)
56		fsumcl 8275	. . . . . . . . . . 11 |- ((m e. (ZZ>=` M) /\ A.k e. (M...m)A e. CC) -> sum_k e. (M...m)A e. CC)
57	55, 56	syldan 516	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...m)A e. CC)
58		ra4csbela 2587	. . . . . . . . . . 11 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. CC) -> [_(m + 1) / k]_A e. CC)
59		peano2uz 7616	. . . . . . . . . . . 12 |- (m e. (ZZ>=` M) -> (m + 1) e. (ZZ>=` M))
60		eluzfz2 7659	. . . . . . . . . . . 12 |- ((m + 1) e. (ZZ>=` M) -> (m + 1) e. (M...(m + 1)))
61	59, 60	syl 12	. . . . . . . . . . 11 |- (m e. (ZZ>=` M) -> (m + 1) e. (M...(m + 1)))
62	58, 61	sylan 497	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> [_(m + 1) / k]_A e. CC)
63		abstri 8150	. . . . . . . . . 10 |- ((sum_k e. (M...m)A e. CC /\ [_(m + 1) / k]_A e. CC) -> (abs` (sum_k e. (M...m)A + [_(m + 1) / k]_A)) <_ ((abs` sum_k e. (M...m)A) + (abs` [_(m + 1) / k]_A)))
64	57, 62, 63	syl11anc 524	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> (abs` (sum_k e. (M...m)A + [_(m + 1) / k]_A)) <_ ((abs` sum_k e. (M...m)A) + (abs` [_(m + 1) / k]_A)))
65		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))
66	65	fveq2d 4685	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> (abs` sum_k e. (M...(m + 1))A) = (abs` (sum_k e. (M...m)A + [_(m + 1) / k]_A)))
67		oprex 4907	. . . . . . . . . . . 12 |- (m + 1) e. _V
68		csbfv2g 4700	. . . . . . . . . . . 12 |- ((m + 1) e. _V -> [_(m + 1) / k]_(abs` A) = (abs` [_(m + 1) / k]_A))
69	67, 68	ax-mp 7	. . . . . . . . . . 11 |- [_(m + 1) / k]_(abs` A) = (abs` [_(m + 1) / k]_A)
70	69	a1i 8	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> [_(m + 1) / k]_(abs` A) = (abs` [_(m + 1) / k]_A))
71	70	opreq2d 4898	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) = ((abs` sum_k e. (M...m)A) + (abs` [_(m + 1) / k]_A)))
72	64, 66, 71	3brtr4d 3367	. . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> (abs` sum_k e. (M...(m + 1))A) <_ ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)))
73	72	adantr 425	. . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) /\ (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A)) -> (abs` sum_k e. (M...(m + 1))A) <_ ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)))
74		abscl 8084	. . . . . . . . . . 11 |- (sum_k e. (M...m)A e. CC -> (abs` sum_k e. (M...m)A) e. RR)
75	57, 74	syl 12	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> (abs` sum_k e. (M...m)A) e. RR)
76	33	ralimi 2168	. . . . . . . . . . . 12 |- (A.k e. (M...m)A e. CC -> A.k e. (M...m)(abs` A) e. RR)
77	55, 76	syl 12	. . . . . . . . . . 11 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> A.k e. (M...m)(abs` A) e. RR)
78		fsumrecl 8277	. . . . . . . . . . 11 |- ((m e. (ZZ>=` M) /\ A.k e. (M...m)(abs` A) e. RR) -> sum_k e. (M...m)(abs` A) e. RR)
79	77, 78	syldan 516	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...m)(abs` A) e. RR)
80		ra4csbela 2587	. . . . . . . . . . 11 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))(abs` A) e. RR) -> [_(m + 1) / k]_(abs` A) e. RR)
81	33	ralimi 2168	. . . . . . . . . . 11 |- (A.k e. (M...(m + 1))A e. CC -> A.k e. (M...(m + 1))(abs` A) e. RR)
82	80, 61, 81	syl2an 503	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> [_(m + 1) / k]_(abs` A) e. RR)
83		leadd1 6808	. . . . . . . . . 10 |- (((abs` sum_k e. (M...m)A) e. RR /\ sum_k e. (M...m)(abs` A) e. RR /\ [_(m + 1) / k]_(abs` A) e. RR) -> ((abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A) <-> ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) <_ (sum_k e. (M...m)(abs` A) + [_(m + 1) / k]_(abs` A))))
84	75, 79, 82, 83	syl111anc 1100	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> ((abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A) <-> ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) <_ (sum_k e. (M...m)(abs` A) + [_(m + 1) / k]_(abs` A))))
85	84	biimpa 460	. . . . . . . 8 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) /\ (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A)) -> ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) <_ (sum_k e. (M...m)(abs` A) + [_(m + 1) / k]_(abs` A)))
86		fsump1s 8273	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(abs` A) e. CC) -> sum_k e. (M...(m + 1))(abs` A) = (sum_k e. (M...m)(abs` A) + [_(m + 1) / k]_(abs` A)))
87	42	ralimi 2168	. . . . . . . . . 10 |- (A.k e. (M...(m + 1))A e. CC -> A.k e. (M...(m + 1))(abs` A) e. CC)
88	86, 87	sylan2 500	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...(m + 1))(abs` A) = (sum_k e. (M...m)(abs` A) + [_(m + 1) / k]_(abs` A)))
89	88	adantr 425	. . . . . . . 8 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) /\ (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A)) -> sum_k e. (M...(m + 1))(abs` A) = (sum_k e. (M...m)(abs` A) + [_(m + 1) / k]_(abs` A)))
90	85, 89	breqtrrd 3363	. . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) /\ (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A)) -> ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) <_ sum_k e. (M...(m + 1))(abs` A))
91		fsumcl 8275	. . . . . . . . . . 11 |- (((m + 1) e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...(m + 1))A e. CC)
92	91, 59	sylan 497	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...(m + 1))A e. CC)
93		abscl 8084	. . . . . . . . . 10 |- (sum_k e. (M...(m + 1))A e. CC -> (abs` sum_k e. (M...(m + 1))A) e. RR)
94	92, 93	syl 12	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> (abs` sum_k e. (M...(m + 1))A) e. RR)
95		readdcl 6455	. . . . . . . . . 10 |- (((abs` sum_k e. (M...m)A) e. RR /\ [_(m + 1) / k]_(abs` A) e. RR) -> ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) e. RR)
96	75, 82, 95	syl11anc 524	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) e. RR)
97		fsumrecl 8277	. . . . . . . . . 10 |- (((m + 1) e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(abs` A) e. RR) -> sum_k e. (M...(m + 1))(abs` A) e. RR)
98	97, 59, 81	syl2an 503	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> sum_k e. (M...(m + 1))(abs` A) e. RR)
99		letr 6695	. . . . . . . . 9 |- (((abs` sum_k e. (M...(m + 1))A) e. RR /\ ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) e. RR /\ sum_k e. (M...(m + 1))(abs` A) e. RR) -> (((abs` sum_k e. (M...(m + 1))A) <_ ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) /\ ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) <_ sum_k e. (M...(m + 1))(abs` A)) -> (abs` sum_k e. (M...(m + 1))A) <_ sum_k e. (M...(m + 1))(abs` A)))
100	94, 96, 98, 99	syl111anc 1100	. . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) -> (((abs` sum_k e. (M...(m + 1))A) <_ ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) /\ ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) <_ sum_k e. (M...(m + 1))(abs` A)) -> (abs` sum_k e. (M...(m + 1))A) <_ sum_k e. (M...(m + 1))(abs` A)))
101	100	adantr 425	. . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) /\ (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A)) -> (((abs` sum_k e. (M...(m + 1))A) <_ ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) /\ ((abs` sum_k e. (M...m)A) + [_(m + 1) / k]_(abs` A)) <_ sum_k e. (M...(m + 1))(abs` A)) -> (abs` sum_k e. (M...(m + 1))A) <_ sum_k e. (M...(m + 1))(abs` A)))
102	73, 90, 101	mp2and 767	. . . . . 6 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. CC) /\ (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A)) -> (abs` sum_k e. (M...(m + 1))A) <_ sum_k e. (M...(m + 1))(abs` A))
103	102	exp31 407	. . . . 5 |- (m e. (ZZ>=` M) -> (A.k e. (M...(m + 1))A e. CC -> ((abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A) -> (abs` sum_k e. (M...(m + 1))A) <_ sum_k e. (M...(m + 1))(abs` A))))
104	103	a2d 16	. . . 4 |- (m e. (ZZ>=` M) -> ((A.k e. (M...(m + 1))A e. CC -> (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A)) -> (A.k e. (M...(m + 1))A e. CC -> (abs` sum_k e. (M...(m + 1))A) <_ sum_k e. (M...(m + 1))(abs` A))))
105	54, 104	syld 30	. . 3 |- (m e. (ZZ>=` M) -> ((A.k e. (M...m)A e. CC -> (abs` sum_k e. (M...m)A) <_ sum_k e. (M...m)(abs` A)) -> (A.k e. (M...(m + 1))A e. CC -> (abs` sum_k e. (M...(m + 1))A) <_ sum_k e. (M...(m + 1))(abs` A))))
106	7, 14, 21, 28, 46, 105	uzind4 7619	. 2 |- (N e. (ZZ>=` M) -> (A.k e. (M...N)A e. CC -> (abs` sum_k e. (M...N)A) <_ sum_k e. (M...N)(abs` A)))
107	106	imp 377	1 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)A e. CC) -> (abs` sum_k e. (M...N)A) <_ sum_k e. (M...N)(abs` A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  [_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  ZZ>=cuz 7586  ...cfz 7637  abscabs 8000  sum_csu 8239

This theorem is referenced by:  fsumabs2mul 8304  iserzabslem 8438  efaddlem19 8618

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-sup 5664  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-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-sum 8240

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