Theorem serzf0i 8429

Description: If an infinite series converges, its underlying sequence converges to zero. Warning: The HTML proof page is 0.6 megabyte in size.

Hypotheses

Ref	Expression
serzf0.1	|- F e. _V
serzf0.2	|- M e. ZZ
serzf0.3	|- (k e. (ZZ>=` M) -> (F` k) e. CC)
serzf0.4	|- A e. _V
serzf0.5	|- (<.M, + >. seq F) ~~> A

Assertion

Ref	Expression
serzf0i	|- F ~~> 0

Distinct variable groups:   k,F   k,M

Proof of Theorem serzf0i

Step	Hyp	Ref	Expression
1		serzf0.2	. . . 4 |- M e. ZZ
2		peano2z 7375	. . . 4 |- (M e. ZZ -> (M + 1) e. ZZ)
3	1, 2	ax-mp 7	. . 3 |- (M + 1) e. ZZ
4		0cn 6481	. . 3 |- 0 e. CC
5		uzid 7596	. . . . . . . . 9 |- (M e. ZZ -> M e. (ZZ>=` M))
6	1, 5	ax-mp 7	. . . . . . . 8 |- M e. (ZZ>=` M)
7		peano2uz 7616	. . . . . . . 8 |- (M e. (ZZ>=` M) -> (M + 1) e. (ZZ>=` M))
8	6, 7	ax-mp 7	. . . . . . 7 |- (M + 1) e. (ZZ>=` M)
9		uzss 7600	. . . . . . 7 |- ((M + 1) e. (ZZ>=` M) -> (ZZ>=` (M + 1)) C_ (ZZ>=` M))
10	8, 9	ax-mp 7	. . . . . 6 |- (ZZ>=` (M + 1)) C_ (ZZ>=` M)
11	10	sseli 2617	. . . . 5 |- (h e. (ZZ>=` (M + 1)) -> h e. (ZZ>=` M))
12		fveq2 4681	. . . . . . 7 |- (k = h -> (F` k) = (F` h))
13	12	eleq1d 1963	. . . . . 6 |- (k = h -> ((F` k) e. CC <-> (F` h) e. CC))
14		serzf0.3	. . . . . 6 |- (k e. (ZZ>=` M) -> (F` k) e. CC)
15	13, 14	vtoclga 2352	. . . . 5 |- (h e. (ZZ>=` M) -> (F` h) e. CC)
16	11, 15	syl 12	. . . 4 |- (h e. (ZZ>=` (M + 1)) -> (F` h) e. CC)
17	16	rgen 2159	. . 3 |- A.h e. (ZZ>=` (M + 1))(F` h) e. CC
18		serzf0.1	. . . 4 |- F e. _V
19	18	clm4a 8350	. . 3 |- (((M + 1) e. ZZ /\ 0 e. CC /\ A.h e. (ZZ>=` (M + 1))(F` h) e. CC) -> (F ~~> 0 <-> A.x e. RR (0 < x -> E.j e. ZZ A.h e. (ZZ>=` (M + 1))(j <_ h -> (abs` ((F` h) - 0)) < x))))
20	3, 4, 17, 19	mp3an 1191	. 2 |- (F ~~> 0 <-> A.x e. RR (0 < x -> E.j e. ZZ A.h e. (ZZ>=` (M + 1))(j <_ h -> (abs` ((F` h) - 0)) < x)))
21		rehalfcl 7220	. . . . . 6 |- (x e. RR -> (x / 2) e. RR)
22	21	adantr 425	. . . . 5 |- ((x e. RR /\ 0 < x) -> (x / 2) e. RR)
23		halfpos2 7223	. . . . . 6 |- (x e. RR -> (0 < x <-> 0 < (x / 2)))
24	23	biimpa 460	. . . . 5 |- ((x e. RR /\ 0 < x) -> 0 < (x / 2))
25		serzf0.4	. . . . . 6 |- A e. _V
26		serzf0.5	. . . . . 6 |- (<.M, + >. seq F) ~~> A
27		0z 7355	. . . . . . 7 |- 0 e. ZZ
28		uzssz 7599	. . . . . . 7 |- (ZZ>=` 0) C_ ZZ
29		ssid 2634	. . . . . . 7 |- ZZ C_ ZZ
30	27, 28, 29	clmi2i 8347	. . . . . 6 |- (((A e. _V /\ (<.M, + >. seq F) ~~> A) /\ ((x / 2) e. RR /\ 0 < (x / 2))) -> E.m e. ZZ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))
31	25, 26, 30	mpanl12 773	. . . . 5 |- (((x / 2) e. RR /\ 0 < (x / 2)) -> E.m e. ZZ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))
32	22, 24, 31	syl11anc 524	. . . 4 |- ((x e. RR /\ 0 < x) -> E.m e. ZZ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))
33		peano2z 7375	. . . . . . . . 9 |- (m e. ZZ -> (m + 1) e. ZZ)
34	33	ad2antrl 442	. . . . . . . 8 |- ((x e. RR /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> (m + 1) e. ZZ)
35		p1le 6995	. . . . . . . . . . . . . . . . . . 19 |- ((m e. RR /\ h e. RR /\ (m + 1) <_ h) -> m <_ h)
36	35	3expia 1069	. . . . . . . . . . . . . . . . . 18 |- ((m e. RR /\ h e. RR) -> ((m + 1) <_ h -> m <_ h))
37		zre 7348	. . . . . . . . . . . . . . . . . 18 |- (m e. ZZ -> m e. RR)
38		zre 7348	. . . . . . . . . . . . . . . . . 18 |- (h e. ZZ -> h e. RR)
39	36, 37, 38	syl2an 503	. . . . . . . . . . . . . . . . 17 |- ((m e. ZZ /\ h e. ZZ) -> ((m + 1) <_ h -> m <_ h))
40	39	ancoms 484	. . . . . . . . . . . . . . . 16 |- ((h e. ZZ /\ m e. ZZ) -> ((m + 1) <_ h -> m <_ h))
41	40	adantrr 431	. . . . . . . . . . . . . . 15 |- ((h e. ZZ /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> ((m + 1) <_ h -> m <_ h))
42		breq2 3342	. . . . . . . . . . . . . . . . . 18 |- (n = h -> (m <_ n <-> m <_ h))
43		fveq2 4681	. . . . . . . . . . . . . . . . . . . . 21 |- (n = h -> ((<.M, + >. seq F)` n) = ((<.M, + >. seq F)` h))
44	43	opreq1d 4897	. . . . . . . . . . . . . . . . . . . 20 |- (n = h -> (((<.M, + >. seq F)` n) - A) = (((<.M, + >. seq F)` h) - A))
45	44	fveq2d 4685	. . . . . . . . . . . . . . . . . . 19 |- (n = h -> (abs` (((<.M, + >. seq F)` n) - A)) = (abs` (((<.M, + >. seq F)` h) - A)))
46	45	breq1d 3348	. . . . . . . . . . . . . . . . . 18 |- (n = h -> ((abs` (((<.M, + >. seq F)` n) - A)) < (x / 2) <-> (abs` (((<.M, + >. seq F)` h) - A)) < (x / 2)))
47	42, 46	imbi12d 688	. . . . . . . . . . . . . . . . 17 |- (n = h -> ((m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)) <-> (m <_ h -> (abs` (((<.M, + >. seq F)` h) - A)) < (x / 2))))
48	47	rcla4va 2378	. . . . . . . . . . . . . . . 16 |- ((h e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2))) -> (m <_ h -> (abs` (((<.M, + >. seq F)` h) - A)) < (x / 2)))
49	48	adantrl 430	. . . . . . . . . . . . . . 15 |- ((h e. ZZ /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> (m <_ h -> (abs` (((<.M, + >. seq F)` h) - A)) < (x / 2)))
50	41, 49	syld 30	. . . . . . . . . . . . . 14 |- ((h e. ZZ /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> ((m + 1) <_ h -> (abs` (((<.M, + >. seq F)` h) - A)) < (x / 2)))
51		1re 6598	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
52		leaddsub 6816	. . . . . . . . . . . . . . . . . . 19 |- ((m e. RR /\ 1 e. RR /\ h e. RR) -> ((m + 1) <_ h <-> m <_ (h - 1)))
53	51, 52	mp3an2 1179	. . . . . . . . . . . . . . . . . 18 |- ((m e. RR /\ h e. RR) -> ((m + 1) <_ h <-> m <_ (h - 1)))
54	53, 37, 38	syl2an 503	. . . . . . . . . . . . . . . . 17 |- ((m e. ZZ /\ h e. ZZ) -> ((m + 1) <_ h <-> m <_ (h - 1)))
55	54	ancoms 484	. . . . . . . . . . . . . . . 16 |- ((h e. ZZ /\ m e. ZZ) -> ((m + 1) <_ h <-> m <_ (h - 1)))
56	55	adantrr 431	. . . . . . . . . . . . . . 15 |- ((h e. ZZ /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> ((m + 1) <_ h <-> m <_ (h - 1)))
57		peano2zm 7378	. . . . . . . . . . . . . . . . . 18 |- (h e. ZZ -> (h - 1) e. ZZ)
58		breq2 3342	. . . . . . . . . . . . . . . . . . . 20 |- (n = (h - 1) -> (m <_ n <-> m <_ (h - 1)))
59		fveq2 4681	. . . . . . . . . . . . . . . . . . . . . . 23 |- (n = (h - 1) -> ((<.M, + >. seq F)` n) = ((<.M, + >. seq F)` (h - 1)))
60	59	opreq1d 4897	. . . . . . . . . . . . . . . . . . . . . 22 |- (n = (h - 1) -> (((<.M, + >. seq F)` n) - A) = (((<.M, + >. seq F)` (h - 1)) - A))
61	60	fveq2d 4685	. . . . . . . . . . . . . . . . . . . . 21 |- (n = (h - 1) -> (abs` (((<.M, + >. seq F)` n) - A)) = (abs` (((<.M, + >. seq F)` (h - 1)) - A)))
62	61	breq1d 3348	. . . . . . . . . . . . . . . . . . . 20 |- (n = (h - 1) -> ((abs` (((<.M, + >. seq F)` n) - A)) < (x / 2) <-> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2)))
63	58, 62	imbi12d 688	. . . . . . . . . . . . . . . . . . 19 |- (n = (h - 1) -> ((m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)) <-> (m <_ (h - 1) -> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2))))
64	63	rcla4v 2376	. . . . . . . . . . . . . . . . . 18 |- ((h - 1) e. ZZ -> (A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)) -> (m <_ (h - 1) -> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2))))
65	57, 64	syl 12	. . . . . . . . . . . . . . . . 17 |- (h e. ZZ -> (A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)) -> (m <_ (h - 1) -> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2))))
66	65	imp 377	. . . . . . . . . . . . . . . 16 |- ((h e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2))) -> (m <_ (h - 1) -> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2)))
67	66	adantrl 430	. . . . . . . . . . . . . . 15 |- ((h e. ZZ /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> (m <_ (h - 1) -> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2)))
68	56, 67	sylbid 220	. . . . . . . . . . . . . 14 |- ((h e. ZZ /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> ((m + 1) <_ h -> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2)))
69	50, 68	jcad 661	. . . . . . . . . . . . 13 |- ((h e. ZZ /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> ((m + 1) <_ h -> ((abs` (((<.M, + >. seq F)` h) - A)) < (x / 2) /\ (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2))))
70		eluzelz 7592	. . . . . . . . . . . . 13 |- (h e. (ZZ>=` (M + 1)) -> h e. ZZ)
71	69, 70	sylan 497	. . . . . . . . . . . 12 |- ((h e. (ZZ>=` (M + 1)) /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> ((m + 1) <_ h -> ((abs` (((<.M, + >. seq F)` h) - A)) < (x / 2) /\ (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2))))
72	71	adantrl 430	. . . . . . . . . . 11 |- ((h e. (ZZ>=` (M + 1)) /\ (x e. RR /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2))))) -> ((m + 1) <_ h -> ((abs` (((<.M, + >. seq F)` h) - A)) < (x / 2) /\ (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2))))
73		subcl 6524	. . . . . . . . . . . . . . . . . 18 |- ((((<.M, + >. seq F)` h) e. CC /\ A e. CC) -> (((<.M, + >. seq F)` h) - A) e. CC)
74	18	serzcl2 8309	. . . . . . . . . . . . . . . . . . 19 |- ((h e. (ZZ>=` M) /\ A.k e. (ZZ>=` M)(F` k) e. CC) -> ((<.M, + >. seq F)` h) e. CC)
75	14	rgen 2159	. . . . . . . . . . . . . . . . . . 19 |- A.k e. (ZZ>=` M)(F` k) e. CC
76	74, 11, 75	sylancl 525	. . . . . . . . . . . . . . . . . 18 |- (h e. (ZZ>=` (M + 1)) -> ((<.M, + >. seq F)` h) e. CC)
77		climcl 8238	. . . . . . . . . . . . . . . . . . 19 |- ((A e. _V /\ (<.M, + >. seq F) ~~> A) -> A e. CC)
78	25, 26, 77	mp2an 761	. . . . . . . . . . . . . . . . . 18 |- A e. CC
79	73, 76, 78	sylancl 525	. . . . . . . . . . . . . . . . 17 |- (h e. (ZZ>=` (M + 1)) -> (((<.M, + >. seq F)` h) - A) e. CC)
80		abscl 8084	. . . . . . . . . . . . . . . . 17 |- ((((<.M, + >. seq F)` h) - A) e. CC -> (abs` (((<.M, + >. seq F)` h) - A)) e. RR)
81	79, 80	syl 12	. . . . . . . . . . . . . . . 16 |- (h e. (ZZ>=` (M + 1)) -> (abs` (((<.M, + >. seq F)` h) - A)) e. RR)
82	81	adantr 425	. . . . . . . . . . . . . . 15 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> (abs` (((<.M, + >. seq F)` h) - A)) e. RR)
83		subcl 6524	. . . . . . . . . . . . . . . . . 18 |- ((((<.M, + >. seq F)` (h - 1)) e. CC /\ A e. CC) -> (((<.M, + >. seq F)` (h - 1)) - A) e. CC)
84	18	serzcl2 8309	. . . . . . . . . . . . . . . . . . 19 |- (((h - 1) e. (ZZ>=` M) /\ A.k e. (ZZ>=` M)(F` k) e. CC) -> ((<.M, + >. seq F)` (h - 1)) e. CC)
85		eluzp1m1 7602	. . . . . . . . . . . . . . . . . . . 20 |- ((M e. ZZ /\ h e. (ZZ>=` (M + 1))) -> (h - 1) e. (ZZ>=` M))
86	1, 85	mpan 759	. . . . . . . . . . . . . . . . . . 19 |- (h e. (ZZ>=` (M + 1)) -> (h - 1) e. (ZZ>=` M))
87	84, 86, 75	sylancl 525	. . . . . . . . . . . . . . . . . 18 |- (h e. (ZZ>=` (M + 1)) -> ((<.M, + >. seq F)` (h - 1)) e. CC)
88	83, 87, 78	sylancl 525	. . . . . . . . . . . . . . . . 17 |- (h e. (ZZ>=` (M + 1)) -> (((<.M, + >. seq F)` (h - 1)) - A) e. CC)
89		abscl 8084	. . . . . . . . . . . . . . . . 17 |- ((((<.M, + >. seq F)` (h - 1)) - A) e. CC -> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) e. RR)
90	88, 89	syl 12	. . . . . . . . . . . . . . . 16 |- (h e. (ZZ>=` (M + 1)) -> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) e. RR)
91	90	adantr 425	. . . . . . . . . . . . . . 15 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) e. RR)
92		simpr 350	. . . . . . . . . . . . . . 15 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> x e. RR)
93		lt2halves 7228	. . . . . . . . . . . . . . 15 |- (((abs` (((<.M, + >. seq F)` h) - A)) e. RR /\ (abs` (((<.M, + >. seq F)` (h - 1)) - A)) e. RR /\ x e. RR) -> (((abs` (((<.M, + >. seq F)` h) - A)) < (x / 2) /\ (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2)) -> ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (((<.M, + >. seq F)` (h - 1)) - A))) < x))
94	82, 91, 92, 93	syl111anc 1100	. . . . . . . . . . . . . 14 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> (((abs` (((<.M, + >. seq F)` h) - A)) < (x / 2) /\ (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2)) -> ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (((<.M, + >. seq F)` (h - 1)) - A))) < x))
95		abssub 8146	. . . . . . . . . . . . . . . . . 18 |- ((((<.M, + >. seq F)` (h - 1)) e. CC /\ A e. CC) -> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) = (abs` (A - ((<.M, + >. seq F)` (h - 1)))))
96	95, 87, 78	sylancl 525	. . . . . . . . . . . . . . . . 17 |- (h e. (ZZ>=` (M + 1)) -> (abs` (((<.M, + >. seq F)` (h - 1)) - A)) = (abs` (A - ((<.M, + >. seq F)` (h - 1)))))
97	96	opreq2d 4898	. . . . . . . . . . . . . . . 16 |- (h e. (ZZ>=` (M + 1)) -> ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (((<.M, + >. seq F)` (h - 1)) - A))) = ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))))
98	97	breq1d 3348	. . . . . . . . . . . . . . 15 |- (h e. (ZZ>=` (M + 1)) -> (((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (((<.M, + >. seq F)` (h - 1)) - A))) < x <-> ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) < x))
99	98	adantr 425	. . . . . . . . . . . . . 14 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> (((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (((<.M, + >. seq F)` (h - 1)) - A))) < x <-> ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) < x))
100	94, 99	sylibd 219	. . . . . . . . . . . . 13 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> (((abs` (((<.M, + >. seq F)` h) - A)) < (x / 2) /\ (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2)) -> ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) < x))
101		subcl 6524	. . . . . . . . . . . . . . . . . 18 |- ((A e. CC /\ ((<.M, + >. seq F)` (h - 1)) e. CC) -> (A - ((<.M, + >. seq F)` (h - 1))) e. CC)
102	101, 78, 87	sylancr 526	. . . . . . . . . . . . . . . . 17 |- (h e. (ZZ>=` (M + 1)) -> (A - ((<.M, + >. seq F)` (h - 1))) e. CC)
103		abstri 8150	. . . . . . . . . . . . . . . . 17 |- (((((<.M, + >. seq F)` h) - A) e. CC /\ (A - ((<.M, + >. seq F)` (h - 1))) e. CC) -> (abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) <_ ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))))
104	79, 102, 103	syl11anc 524	. . . . . . . . . . . . . . . 16 |- (h e. (ZZ>=` (M + 1)) -> (abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) <_ ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))))
105	104	adantr 425	. . . . . . . . . . . . . . 15 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> (abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) <_ ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))))
106		addcl 6454	. . . . . . . . . . . . . . . . . . 19 |- (((((<.M, + >. seq F)` h) - A) e. CC /\ (A - ((<.M, + >. seq F)` (h - 1))) e. CC) -> ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1)))) e. CC)
107	79, 102, 106	syl11anc 524	. . . . . . . . . . . . . . . . . 18 |- (h e. (ZZ>=` (M + 1)) -> ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1)))) e. CC)
108		abscl 8084	. . . . . . . . . . . . . . . . . 18 |- (((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1)))) e. CC -> (abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) e. RR)
109	107, 108	syl 12	. . . . . . . . . . . . . . . . 17 |- (h e. (ZZ>=` (M + 1)) -> (abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) e. RR)
110	109	adantr 425	. . . . . . . . . . . . . . . 16 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> (abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) e. RR)
111		abscl 8084	. . . . . . . . . . . . . . . . . . 19 |- ((A - ((<.M, + >. seq F)` (h - 1))) e. CC -> (abs` (A - ((<.M, + >. seq F)` (h - 1)))) e. RR)
112	102, 111	syl 12	. . . . . . . . . . . . . . . . . 18 |- (h e. (ZZ>=` (M + 1)) -> (abs` (A - ((<.M, + >. seq F)` (h - 1)))) e. RR)
113		readdcl 6455	. . . . . . . . . . . . . . . . . 18 |- (((abs` (((<.M, + >. seq F)` h) - A)) e. RR /\ (abs` (A - ((<.M, + >. seq F)` (h - 1)))) e. RR) -> ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) e. RR)
114	81, 112, 113	syl11anc 524	. . . . . . . . . . . . . . . . 17 |- (h e. (ZZ>=` (M + 1)) -> ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) e. RR)
115	114	adantr 425	. . . . . . . . . . . . . . . 16 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) e. RR)
116		lelttr 6693	. . . . . . . . . . . . . . . 16 |- (((abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) e. RR /\ ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) e. RR /\ x e. RR) -> (((abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) <_ ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) /\ ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) < x) -> (abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) < x))
117	110, 115, 92, 116	syl111anc 1100	. . . . . . . . . . . . . . 15 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> (((abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) <_ ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) /\ ((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) < x) -> (abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) < x))
118	105, 117	mpand 765	. . . . . . . . . . . . . 14 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> (((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) < x -> (abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) < x))
119		eluzp1l 7603	. . . . . . . . . . . . . . . . . . . . . 22 |- ((M e. ZZ /\ h e. (ZZ>=` (M + 1))) -> M < h)
120	1, 119	mpan 759	. . . . . . . . . . . . . . . . . . . . 21 |- (h e. (ZZ>=` (M + 1)) -> M < h)
121		addex 6470	. . . . . . . . . . . . . . . . . . . . . . 23 |- + e. _V
122	121, 18	seqzm1 7792	. . . . . . . . . . . . . . . . . . . . . 22 |- ((M e. ZZ /\ h e. ZZ /\ M < h) -> ((<.M, + >. seq F)` h) = (((<.M, + >. seq F)` (h - 1)) + (F` h)))
123	1, 122	mp3an1 1178	. . . . . . . . . . . . . . . . . . . . 21 |- ((h e. ZZ /\ M < h) -> ((<.M, + >. seq F)` h) = (((<.M, + >. seq F)` (h - 1)) + (F` h)))
124	70, 120, 123	syl11anc 524	. . . . . . . . . . . . . . . . . . . 20 |- (h e. (ZZ>=` (M + 1)) -> ((<.M, + >. seq F)` h) = (((<.M, + >. seq F)` (h - 1)) + (F` h)))
125	124	eqcomd 1889	. . . . . . . . . . . . . . . . . . 19 |- (h e. (ZZ>=` (M + 1)) -> (((<.M, + >. seq F)` (h - 1)) + (F` h)) = ((<.M, + >. seq F)` h))
126		subadd 6532	. . . . . . . . . . . . . . . . . . . 20 |- ((((<.M, + >. seq F)` h) e. CC /\ ((<.M, + >. seq F)` (h - 1)) e. CC /\ (F` h) e. CC) -> ((((<.M, + >. seq F)` h) - ((<.M, + >. seq F)` (h - 1))) = (F` h) <-> (((<.M, + >. seq F)` (h - 1)) + (F` h)) = ((<.M, + >. seq F)` h)))
127	76, 87, 16, 126	syl111anc 1100	. . . . . . . . . . . . . . . . . . 19 |- (h e. (ZZ>=` (M + 1)) -> ((((<.M, + >. seq F)` h) - ((<.M, + >. seq F)` (h - 1))) = (F` h) <-> (((<.M, + >. seq F)` (h - 1)) + (F` h)) = ((<.M, + >. seq F)` h)))
128	125, 127	mpbird 213	. . . . . . . . . . . . . . . . . 18 |- (h e. (ZZ>=` (M + 1)) -> (((<.M, + >. seq F)` h) - ((<.M, + >. seq F)` (h - 1))) = (F` h))
129		npncan 6560	. . . . . . . . . . . . . . . . . . . 20 |- ((((<.M, + >. seq F)` h) e. CC /\ A e. CC /\ ((<.M, + >. seq F)` (h - 1)) e. CC) -> ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1)))) = (((<.M, + >. seq F)` h) - ((<.M, + >. seq F)` (h - 1))))
130	78, 129	mp3an2 1179	. . . . . . . . . . . . . . . . . . 19 |- ((((<.M, + >. seq F)` h) e. CC /\ ((<.M, + >. seq F)` (h - 1)) e. CC) -> ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1)))) = (((<.M, + >. seq F)` h) - ((<.M, + >. seq F)` (h - 1))))
131	76, 87, 130	syl11anc 524	. . . . . . . . . . . . . . . . . 18 |- (h e. (ZZ>=` (M + 1)) -> ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1)))) = (((<.M, + >. seq F)` h) - ((<.M, + >. seq F)` (h - 1))))
132		subid1 6556	. . . . . . . . . . . . . . . . . . 19 |- ((F` h) e. CC -> ((F` h) - 0) = (F` h))
133	16, 132	syl 12	. . . . . . . . . . . . . . . . . 18 |- (h e. (ZZ>=` (M + 1)) -> ((F` h) - 0) = (F` h))
134	128, 131, 133	3eqtr4d 1937	. . . . . . . . . . . . . . . . 17 |- (h e. (ZZ>=` (M + 1)) -> ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1)))) = ((F` h) - 0))
135	134	fveq2d 4685	. . . . . . . . . . . . . . . 16 |- (h e. (ZZ>=` (M + 1)) -> (abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) = (abs` ((F` h) - 0)))
136	135	breq1d 3348	. . . . . . . . . . . . . . 15 |- (h e. (ZZ>=` (M + 1)) -> ((abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) < x <-> (abs` ((F` h) - 0)) < x))
137	136	adantr 425	. . . . . . . . . . . . . 14 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> ((abs` ((((<.M, + >. seq F)` h) - A) + (A - ((<.M, + >. seq F)` (h - 1))))) < x <-> (abs` ((F` h) - 0)) < x))
138	118, 137	sylibd 219	. . . . . . . . . . . . 13 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> (((abs` (((<.M, + >. seq F)` h) - A)) + (abs` (A - ((<.M, + >. seq F)` (h - 1))))) < x -> (abs` ((F` h) - 0)) < x))
139	100, 138	syld 30	. . . . . . . . . . . 12 |- ((h e. (ZZ>=` (M + 1)) /\ x e. RR) -> (((abs` (((<.M, + >. seq F)` h) - A)) < (x / 2) /\ (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2)) -> (abs` ((F` h) - 0)) < x))
140	139	adantrr 431	. . . . . . . . . . 11 |- ((h e. (ZZ>=` (M + 1)) /\ (x e. RR /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2))))) -> (((abs` (((<.M, + >. seq F)` h) - A)) < (x / 2) /\ (abs` (((<.M, + >. seq F)` (h - 1)) - A)) < (x / 2)) -> (abs` ((F` h) - 0)) < x))
141	72, 140	syld 30	. . . . . . . . . 10 |- ((h e. (ZZ>=` (M + 1)) /\ (x e. RR /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2))))) -> ((m + 1) <_ h -> (abs` ((F` h) - 0)) < x))
142	141	expcom 403	. . . . . . . . 9 |- ((x e. RR /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> (h e. (ZZ>=` (M + 1)) -> ((m + 1) <_ h -> (abs` ((F` h) - 0)) < x)))
143	142	r19.21aiv 2175	. . . . . . . 8 |- ((x e. RR /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> A.h e. (ZZ>=` (M + 1))((m + 1) <_ h -> (abs` ((F` h) - 0)) < x))
144		breq1 3341	. . . . . . . . . . 11 |- (j = (m + 1) -> (j <_ h <-> (m + 1) <_ h))
145	144	imbi1d 675	. . . . . . . . . 10 |- (j = (m + 1) -> ((j <_ h -> (abs` ((F` h) - 0)) < x) <-> ((m + 1) <_ h -> (abs` ((F` h) - 0)) < x)))
146	145	ralbidv 2123	. . . . . . . . 9 |- (j = (m + 1) -> (A.h e. (ZZ>=` (M + 1))(j <_ h -> (abs` ((F` h) - 0)) < x) <-> A.h e. (ZZ>=` (M + 1))((m + 1) <_ h -> (abs` ((F` h) - 0)) < x)))
147	146	rcla4ev 2381	. . . . . . . 8 |- (((m + 1) e. ZZ /\ A.h e. (ZZ>=` (M + 1))((m + 1) <_ h -> (abs` ((F` h) - 0)) < x)) -> E.j e. ZZ A.h e. (ZZ>=` (M + 1))(j <_ h -> (abs` ((F` h) - 0)) < x))
148	34, 143, 147	syl11anc 524	. . . . . . 7 |- ((x e. RR /\ (m e. ZZ /\ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)))) -> E.j e. ZZ A.h e. (ZZ>=` (M + 1))(j <_ h -> (abs` ((F` h) - 0)) < x))
149	148	exp32 408	. . . . . 6 |- (x e. RR -> (m e. ZZ -> (A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)) -> E.j e. ZZ A.h e. (ZZ>=` (M + 1))(j <_ h -> (abs` ((F` h) - 0)) < x))))
150	149	r19.23adv 2215	. . . . 5 |- (x e. RR -> (E.m e. ZZ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)) -> E.j e. ZZ A.h e. (ZZ>=` (M + 1))(j <_ h -> (abs` ((F` h) - 0)) < x)))
151	150	adantr 425	. . . 4 |- ((x e. RR /\ 0 < x) -> (E.m e. ZZ A.n e. ZZ (m <_ n -> (abs` (((<.M, + >. seq F)` n) - A)) < (x / 2)) -> E.j e. ZZ A.h e. (ZZ>=` (M + 1))(j <_ h -> (abs` ((F` h) - 0)) < x)))
152	32, 151	mpd 29	. . 3 |- ((x e. RR /\ 0 < x) -> E.j e. ZZ A.h e. (ZZ>=` (M + 1))(j <_ h -> (abs` ((F` h) - 0)) < x))
153	152	ex 402	. 2 |- (x e. RR -> (0 < x -> E.j e. ZZ A.h e. (ZZ>=` (M + 1))(j <_ h -> (abs` ((F` h) - 0)) < x)))
154	20, 153	mprgbir 2163	1 |- F ~~> 0

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  <.cop 3046   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   - cmin 6445   / cdiv 6447   <_ cle 6448  ZZcz 6451   < clt 6653  2c2 7145  ZZ>=cuz 7586   seq cseqz 7774  abscabs 8000   ~~> cli 8234

This theorem is referenced by:  ser1f0i 8430

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-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-clim 8235  df-sum 8240

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