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Theorem clim 12276
 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:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem clim
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climrel 12274 . . . . 5
21brrelex2i 4910 . . . 4
32a1i 11 . . 3
4 elex 2956 . . . . 5
54adantr 452 . . . 4
65a1i 11 . . 3
7 clim.1 . . . 4
8 simpr 448 . . . . . . . 8
98eleq1d 2501 . . . . . . 7
10 fveq1 5718 . . . . . . . . . . . . 13
1110adantr 452 . . . . . . . . . . . 12
1211eleq1d 2501 . . . . . . . . . . 11
13 oveq12 6081 . . . . . . . . . . . . . 14
1410, 13sylan 458 . . . . . . . . . . . . 13
1514fveq2d 5723 . . . . . . . . . . . 12
1615breq1d 4214 . . . . . . . . . . 11
1712, 16anbi12d 692 . . . . . . . . . 10
1817ralbidv 2717 . . . . . . . . 9
1918rexbidv 2718 . . . . . . . 8
2019ralbidv 2717 . . . . . . 7
219, 20anbi12d 692 . . . . . 6
22 df-clim 12270 . . . . . 6
2321, 22brabga 4461 . . . . 5
2423ex 424 . . . 4
257, 24syl 16 . . 3
263, 6, 25pm5.21ndd 344 . 2
27 eluzelz 10485 . . . . . . 7
28 clim.3 . . . . . . . . 9
2928eleq1d 2501 . . . . . . . 8
3028oveq1d 6087 . . . . . . . . . 10
3130fveq2d 5723 . . . . . . . . 9
3231breq1d 4214 . . . . . . . 8
3329, 32anbi12d 692 . . . . . . 7
3427, 33sylan2 461 . . . . . 6
3534ralbidva 2713 . . . . 5
3635rexbidv 2718 . . . 4
3736ralbidv 2717 . . 3
3837anbi2d 685 . 2
3926, 38bitrd 245 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698  cvv 2948   class class class wbr 4204  cfv 5445  (class class class)co 6072  cc 8977   clt 9109   cmin 9280  cz 10271  cuz 10477  crp 10601  cabs 12027   cli 12266 This theorem is referenced by:  climcl  12281  clim2  12286  climshftlem  12356  climsuse  27648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-cnex 9035  ax-resscn 9036 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-fv 5453  df-ov 6075  df-neg 9283  df-z 10272  df-uz 10478  df-clim 12270
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