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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.
StepHypRef Expression
1 climrel 13315 . . . . 5
21brrelex2i 5046 . . . 4
32a1i 11 . . 3
4 elex 3118 . . . . 5
54adantr 465 . . . 4
65a1i 11 . . 3
7 clim.1 . . . 4
8 simpr 461 . . . . . . . 8
98eleq1d 2526 . . . . . . 7
10 fveq1 5870 . . . . . . . . . . . . 13
1110adantr 465 . . . . . . . . . . . 12
1211eleq1d 2526 . . . . . . . . . . 11
13 oveq12 6305 . . . . . . . . . . . . . 14
1410, 13sylan 471 . . . . . . . . . . . . 13
1514fveq2d 5875 . . . . . . . . . . . 12
1615breq1d 4462 . . . . . . . . . . 11
1712, 16anbi12d 710 . . . . . . . . . 10
1817ralbidv 2896 . . . . . . . . 9
1918rexbidv 2968 . . . . . . . 8
2019ralbidv 2896 . . . . . . 7
219, 20anbi12d 710 . . . . . 6
22 df-clim 13311 . . . . . 6
2321, 22brabga 4766 . . . . 5
2423ex 434 . . . 4
257, 24syl 16 . . 3
263, 6, 25pm5.21ndd 354 . 2
27 eluzelz 11119 . . . . . . 7
28 clim.3 . . . . . . . . 9
2928eleq1d 2526 . . . . . . . 8
3028oveq1d 6311 . . . . . . . . . 10
3130fveq2d 5875 . . . . . . . . 9
3231breq1d 4462 . . . . . . . 8
3329, 32anbi12d 710 . . . . . . 7
3427, 33sylan2 474 . . . . . 6
3534ralbidva 2893 . . . . 5
3635rexbidv 2968 . . . 4
3736ralbidv 2896 . . 3
3837anbi2d 703 . 2
3926, 38bitrd 253 1
 Colors of variables: wff setvar class 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
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