Theorem clim 8237

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 4680	. . . . . . . . 9 |- (f = F -> (f` k) = (F` k))
2	1	eleq1d 1963	. . . . . . . 8 |- (f = F -> ((f` k) e. CC <-> (F` k) e. CC))
3	1	opreq1d 4897	. . . . . . . . . 10 |- (f = F -> ((f` k) - y) = ((F` k) - y))
4	3	fveq2d 4685	. . . . . . . . 9 |- (f = F -> (abs` ((f` k) - y)) = (abs` ((F` k) - y)))
5	4	breq1d 3348	. . . . . . . 8 |- (f = F -> ((abs` ((f` k) - y)) < x <-> (abs` ((F` k) - y)) < x))
6	2, 5	anbi12d 690	. . . . . . 7 |- (f = F -> (((f` k) e. CC /\ (abs` ((f` k) - y)) < x) <-> ((F` k) e. CC /\ (abs` ((F` k) - y)) < x)))
7	6	imbi2d 674	. . . . . 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 2142	. . . . 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 674	. . . 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 2123	. . 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 678	. 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 1957	. . 3 |- (y = A -> (y e. CC <-> A e. CC))
13		opreq2 4890	. . . . . . . . . 10 |- (y = A -> ((F` k) - y) = ((F` k) - A))
14	13	fveq2d 4685	. . . . . . . . 9 |- (y = A -> (abs` ((F` k) - y)) = (abs` ((F` k) - A)))
15	14	breq1d 3348	. . . . . . . 8 |- (y = A -> ((abs` ((F` k) - y)) < x <-> (abs` ((F` k) - A)) < x))
16	15	anbi2d 678	. . . . . . 7 |- (y = A -> (((F` k) e. CC /\ (abs` ((F` k) - y)) < x) <-> ((F` k) e. CC /\ (abs` ((F` k) - A)) < x)))
17	16	imbi2d 674	. . . . . 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 2142	. . . . 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 674	. . . 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 2123	. . 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 690	. 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 8235	. 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 3568	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 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   - cmin 6445   <_ cle 6448  ZZcz 6451   < clt 6653  abscabs 8000   ~~> cli 8234

This theorem is referenced by:  climcl 8238  clm1i 8337  lmclim 9241

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-clim 8235

Copyright terms: Public domain