Theorem clim 6977

Description: Express the predicate: The limit of complex number sequence F is A, or F converges to A. This means that for any real x, no matter how small, there always exists an integer j such that the absolute difference of any later complex number in the sequence and the limit is less than x.

Assertion

Ref	Expression
clim	|- ((F e. C /\ A e. D) -> (F ~~> A <-> (A e. CC /\ A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - A)) < x))))))

Distinct variable groups:   j,k,x,F   A,j,k,x

Proof of Theorem clim

Step	Hyp	Ref	Expression
1		fveq1 3723	. . . . . . . . 9 |- (f = F -> (f` k) = (F` k))
2	1	eleq1d 1540	. . . . . . . 8 |- (f = F -> ((f` k) e. CC <-> (F` k) e. CC))
3	1	opreq1d 3975	. . . . . . . . . 10 |- (f = F -> ((f` k) - y) = ((F` k) - y))
4	3	fveq2d 3728	. . . . . . . . 9 |- (f = F -> (abs` ((f` k) - y)) = (abs` ((F` k) - y)))
5	4	breq1d 2629	. . . . . . . 8 |- (f = F -> ((abs` ((f` k) - y)) < x <-> (abs` ((F` k) - y)) < x))
6	2, 5	anbi12d 628	. . . . . . 7 |- (f = F -> (((f` k) e. CC /\ (abs` ((f` k) - y)) < x) <-> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x)))
7	6	imbi2d 612	. . . . . 6 |- (f = F -> ((j <_ k -> ((f` k) e. CC /\ (abs` ((f` k) - y)) < x)) <-> (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x))))
8	7	rexralbidv 1682	. . . . 5 |- (f = F -> (E.j e. ZZ A.k e. ZZ (j <_ k -> ((f` k) e. CC /\ (abs` ((f` k) - y)) < x)) <-> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x))))
9	8	imbi2d 612	. . . 4 |- (f = F -> ((0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((f` k) e. CC /\ (abs` ((f` k) - y)) < x))) <-> (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x)))))
10	9	ralbidv 1663	. . 3 |- (f = F -> (A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((f` k) e. CC /\ (abs` ((f` k) - y)) < x))) <-> A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x)))))
11	10	anbi2d 616	. 2 |- (f = F -> ((y e. CC /\ A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((f` k) e. CC /\ (abs` ((f` k) - y)) < x)))) <-> (y e. CC /\ A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x))))))
12		eleq1 1534	. . 3 |- (y = A -> (y e. CC <-> A e. CC))
13		opreq2 3969	. . . . . . . . . 10 |- (y = A -> ((F` k) - y) = ((F` k) - A))
14	13	fveq2d 3728	. . . . . . . . 9 |- (y = A -> (abs` ((F` k) - y)) = (abs` ((F` k) - A)))
15	14	breq1d 2629	. . . . . . . 8 |- (y = A -> ((abs` ((F` k) - y)) < x <-> (abs` ((F` k) - A)) < x))
16	15	anbi2d 616	. . . . . . 7 |- (y = A -> (((F` k) e. CC /\ (abs` ((F` k) - y)) < x) <-> ((F` k) e. CC /\ (abs` ((F` k) - A)) < x)))
17	16	imbi2d 612	. . . . . 6 |- (y = A -> ((j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x)) <-> (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - A)) < x))))
18	17	rexralbidv 1682	. . . . 5 |- (y = A -> (E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x)) <-> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - A)) < x))))
19	18	imbi2d 612	. . . 4 |- (y = A -> ((0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x))) <-> (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - A)) < x)))))
20	19	ralbidv 1663	. . 3 |- (y = A -> (A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x))) <-> A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - A)) < x)))))
21	12, 20	anbi12d 628	. 2 |- (y = A -> ((y e. CC /\ A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x)))) <-> (A e. CC /\ A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - A)) < x))))))
22		df-clim 6975	. 2 |- ~~> = {<.f, y>. | (y e. CC /\ A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((f` k) e. CC /\ (abs` ((f` k) - y)) < x))))}
23	11, 21, 22	brabg 2818	1 |- ((F e. C /\ A e. D) -> (F ~~> A <-> (A e. CC /\ A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - A)) < x))))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646   class class class wbr 2619  ` cfv 3182  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234   - cmin 5292   <_ cle 5295  ZZcz 5298   < clt 5486  abscabs 6750   ~~> cli 6974

This theorem is referenced by:  climcl 6978  clm1 7077  lmclim 7963

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965  df-clim 6975

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