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Theorem clim 9802
 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 9801 . . . . 5
21brrelex2i 4385 . . . 4
32a1i 9 . . 3
4 elex 2566 . . . . 5
54adantr 261 . . . 4
65a1i 9 . . 3
7 clim.1 . . . 4
8 simpr 103 . . . . . . . 8
98eleq1d 2106 . . . . . . 7
10 fveq1 5177 . . . . . . . . . . . . 13
1110adantr 261 . . . . . . . . . . . 12
1211eleq1d 2106 . . . . . . . . . . 11
13 oveq12 5521 . . . . . . . . . . . . . 14
1410, 13sylan 267 . . . . . . . . . . . . 13
1514fveq2d 5182 . . . . . . . . . . . 12
1615breq1d 3774 . . . . . . . . . . 11
1712, 16anbi12d 442 . . . . . . . . . 10
1817ralbidv 2326 . . . . . . . . 9
1918rexbidv 2327 . . . . . . . 8
2019ralbidv 2326 . . . . . . 7
219, 20anbi12d 442 . . . . . 6
22 df-clim 9800 . . . . . 6
2321, 22brabga 4001 . . . . 5
2423ex 108 . . . 4
257, 24syl 14 . . 3
263, 6, 25pm5.21ndd 621 . 2
27 eluzelz 8482 . . . . . . 7
28 clim.3 . . . . . . . . 9
2928eleq1d 2106 . . . . . . . 8
3028oveq1d 5527 . . . . . . . . . 10
3130fveq2d 5182 . . . . . . . . 9
3231breq1d 3774 . . . . . . . 8
3329, 32anbi12d 442 . . . . . . 7
3427, 33sylan2 270 . . . . . 6
3534ralbidva 2322 . . . . 5
3635rexbidv 2327 . . . 4
3736ralbidv 2326 . . 3
3837anbi2d 437 . 2
3926, 38bitrd 177 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243   wcel 1393  wral 2306  wrex 2307  cvv 2557   class class class wbr 3764  cfv 4902  (class class class)co 5512  cc 6887   clt 7060   cmin 7182  cz 8245  cuz 8473  crp 8583  cabs 9595   cli 9799 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-cnex 6975  ax-resscn 6976 This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-ov 5515  df-neg 7185  df-z 8246  df-uz 8474  df-clim 9800 This theorem is referenced by:  climcl  9803  clim2  9804  climshftlemg  9823
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