Theorem clim 13317

Description: Express the predicate: The limit of complex number sequence is , or converges to . This means that for any real , no matter how small, there always exists an integer such that the absolute difference of any later complex number in the sequence and the limit is less than . (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Hypotheses

Ref	Expression
clim.1
clim.3

Assertion

Ref	Expression
clim

Distinct variable groups:   ,,,   ,,,   ,,,

Proof of Theorem clim

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climrel 13315	. . . . 5
2	1	brrelex2i 5046	. . . 4
3	2	a1i 11	. . 3
4		elex 3118	. . . . 5
5	4	adantr 465	. . . 4
6	5	a1i 11	. . 3
7		clim.1	. . . 4
8		simpr 461	. . . . . . . 8
9	8	eleq1d 2526	. . . . . . 7
10		fveq1 5870	. . . . . . . . . . . . 13
11	10	adantr 465	. . . . . . . . . . . 12
12	11	eleq1d 2526	. . . . . . . . . . 11
13		oveq12 6305	. . . . . . . . . . . . . 14
14	10, 13	sylan 471	. . . . . . . . . . . . 13
15	14	fveq2d 5875	. . . . . . . . . . . 12
16	15	breq1d 4462	. . . . . . . . . . 11
17	12, 16	anbi12d 710	. . . . . . . . . 10
18	17	ralbidv 2896	. . . . . . . . 9
19	18	rexbidv 2968	. . . . . . . 8
20	19	ralbidv 2896	. . . . . . 7
21	9, 20	anbi12d 710	. . . . . 6
22		df-clim 13311	. . . . . 6
23	21, 22	brabga 4766	. . . . 5
24	23	ex 434	. . . 4
25	7, 24	syl 16	. . 3
26	3, 6, 25	pm5.21ndd 354	. 2
27		eluzelz 11119	. . . . . . 7
28		clim.3	. . . . . . . . 9
29	28	eleq1d 2526	. . . . . . . 8
30	28	oveq1d 6311	. . . . . . . . . 10
31	30	fveq2d 5875	. . . . . . . . 9
32	31	breq1d 4462	. . . . . . . 8
33	29, 32	anbi12d 710	. . . . . . 7
34	27, 33	sylan2 474	. . . . . 6
35	34	ralbidva 2893	. . . . 5
36	35	rexbidv 2968	. . . 4
37	36	ralbidv 2896	. . 3
38	37	anbi2d 703	. 2
39	26, 38	bitrd 253	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  cvv 3109   class class class wbr 4452  `cfv 5593  (class class class)co 6296  cc 9511  clt 9649  cmin 9828  cz 10889  cuz 11110  crp 11249  cabs 13067  cli 13307

This theorem is referenced by:  climcl  13322  clim2  13327  climshftlem  13397  climsuse  31614  ioodvbdlimc1lem2  31729  ioodvbdlimc2lem  31731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-cnex 9569  ax-resscn 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6299  df-neg 9831  df-z 10890  df-uz 11111  df-clim 13311