Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  clim Structured version   Unicode version

Theorem clim 13464
 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 13462 . . . . 5
21brrelex2i 4864 . . . 4
32a1i 11 . . 3
4 elex 3067 . . . . 5
54adantr 463 . . . 4
65a1i 11 . . 3
7 clim.1 . . . 4
8 simpr 459 . . . . . . . 8
98eleq1d 2471 . . . . . . 7
10 fveq1 5847 . . . . . . . . . . . . 13
1110adantr 463 . . . . . . . . . . . 12
1211eleq1d 2471 . . . . . . . . . . 11
13 oveq12 6286 . . . . . . . . . . . . . 14
1410, 13sylan 469 . . . . . . . . . . . . 13
1514fveq2d 5852 . . . . . . . . . . . 12
1615breq1d 4404 . . . . . . . . . . 11
1712, 16anbi12d 709 . . . . . . . . . 10
1817ralbidv 2842 . . . . . . . . 9
1918rexbidv 2917 . . . . . . . 8
2019ralbidv 2842 . . . . . . 7
219, 20anbi12d 709 . . . . . 6
22 df-clim 13458 . . . . . 6
2321, 22brabga 4703 . . . . 5
2423ex 432 . . . 4
257, 24syl 17 . . 3
263, 6, 25pm5.21ndd 352 . 2
27 eluzelz 11135 . . . . . . 7
28 clim.3 . . . . . . . . 9
2928eleq1d 2471 . . . . . . . 8
3028oveq1d 6292 . . . . . . . . . 10
3130fveq2d 5852 . . . . . . . . 9
3231breq1d 4404 . . . . . . . 8
3329, 32anbi12d 709 . . . . . . 7
3427, 33sylan2 472 . . . . . 6
3534ralbidva 2839 . . . . 5
3635rexbidv 2917 . . . 4
3736ralbidv 2842 . . 3
3837anbi2d 702 . 2
3926, 38bitrd 253 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   wceq 1405   wcel 1842  wral 2753  wrex 2754  cvv 3058   class class class wbr 4394  cfv 5568  (class class class)co 6277  cc 9519   clt 9657   cmin 9840  cz 10904  cuz 11126  crp 11264  cabs 13214   cli 13454 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-cnex 9577  ax-resscn 9578 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-ov 6280  df-neg 9843  df-z 10905  df-uz 11127  df-clim 13458 This theorem is referenced by:  climcl  13469  clim2  13474  climshftlem  13544  climsuse  36963  ioodvbdlimc1lem2  37078  ioodvbdlimc2lem  37080
 Copyright terms: Public domain W3C validator