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Theorem rlim 13318
Description: Express the predicate: The limit of complex number function is , or converges to , in the real sense. This means that for any real , no matter how small, there always exists a number such that the absolute difference of any number in the function beyond and the limit is less than . (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypotheses
Ref Expression
rlim.1
rlim.2
rlim.4
Assertion
Ref Expression
rlim
Distinct variable groups:   ,   , , ,   , , ,   , , ,

Proof of Theorem rlim
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimrel 13316 . . . . 5
21brrelex2i 5046 . . . 4
32a1i 11 . . 3
4 elex 3118 . . . . 5
54ad2antrl 727 . . . 4
65a1i 11 . . 3
7 rlim.1 . . . . 5
8 rlim.2 . . . . 5
9 cnex 9594 . . . . . 6
10 reex 9604 . . . . . 6
11 elpm2r 7456 . . . . . 6
129, 10, 11mpanl12 682 . . . . 5
137, 8, 12syl2anc 661 . . . 4
14 eleq1 2529 . . . . . . . . 9
15 eleq1 2529 . . . . . . . . 9
1614, 15bi2anan9 873 . . . . . . . 8
17 simpl 457 . . . . . . . . . . . 12
1817dmeqd 5210 . . . . . . . . . . 11
19 fveq1 5870 . . . . . . . . . . . . . . 15
20 oveq12 6305 . . . . . . . . . . . . . . 15
2119, 20sylan 471 . . . . . . . . . . . . . 14
2221fveq2d 5875 . . . . . . . . . . . . 13
2322breq1d 4462 . . . . . . . . . . . 12
2423imbi2d 316 . . . . . . . . . . 11
2518, 24raleqbidv 3068 . . . . . . . . . 10
2625rexbidv 2968 . . . . . . . . 9
2726ralbidv 2896 . . . . . . . 8
2816, 27anbi12d 710 . . . . . . 7
29 df-rlim 13312 . . . . . . 7
3028, 29brabga 4766 . . . . . 6
31 anass 649 . . . . . 6
3230, 31syl6bb 261 . . . . 5
3332ex 434 . . . 4
3413, 33syl 16 . . 3
353, 6, 34pm5.21ndd 354 . 2
3613biantrurd 508 . 2
37 fdm 5740 . . . . . . . 8
387, 37syl 16 . . . . . . 7
3938raleqdv 3060 . . . . . 6
40 rlim.4 . . . . . . . . . . 11
4140oveq1d 6311 . . . . . . . . . 10
4241fveq2d 5875 . . . . . . . . 9
4342breq1d 4462 . . . . . . . 8
4443imbi2d 316 . . . . . . 7
4544ralbidva 2893 . . . . . 6
4639, 45bitrd 253 . . . . 5
4746rexbidv 2968 . . . 4
4847ralbidv 2896 . . 3
4948anbi2d 703 . 2
5035, 36, 493bitr2d 281 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  C_wss 3475   class class class wbr 4452  domcdm 5004  -->wf 5589  `cfv 5593  (class class class)co 6296   cpm 7440   cc 9511   cr 9512   clt 9649   cle 9650   cmin 9828   crp 11249   cabs 13067   crli 13308
This theorem is referenced by:  rlim2  13319  rlimcl  13326  rlimclim  13369  rlimres  13381  caurcvgr  13496
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-un 6592  ax-cnex 9569  ax-resscn 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-pm 7442  df-rlim 13312
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