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Theorem rlim 12289
 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:   ,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,)   (,,)

Proof of Theorem rlim
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimrel 12287 . . . . 5
21brrelex2i 4919 . . . 4
32a1i 11 . . 3
4 elex 2964 . . . . 5
54ad2antrl 709 . . . 4
65a1i 11 . . 3
7 rlim.1 . . . . 5
8 rlim.2 . . . . 5
9 cnex 9071 . . . . . 6
10 reex 9081 . . . . . 6
11 elpm2r 7034 . . . . . 6
129, 10, 11mpanl12 664 . . . . 5
137, 8, 12syl2anc 643 . . . 4
14 eleq1 2496 . . . . . . . . 9
15 eleq1 2496 . . . . . . . . 9
1614, 15bi2anan9 844 . . . . . . . 8
17 simpl 444 . . . . . . . . . . . 12
1817dmeqd 5072 . . . . . . . . . . 11
19 fveq1 5727 . . . . . . . . . . . . . . 15
20 oveq12 6090 . . . . . . . . . . . . . . 15
2119, 20sylan 458 . . . . . . . . . . . . . 14
2221fveq2d 5732 . . . . . . . . . . . . 13
2322breq1d 4222 . . . . . . . . . . . 12
2423imbi2d 308 . . . . . . . . . . 11
2518, 24raleqbidv 2916 . . . . . . . . . 10
2625rexbidv 2726 . . . . . . . . 9
2726ralbidv 2725 . . . . . . . 8
2816, 27anbi12d 692 . . . . . . 7
29 df-rlim 12283 . . . . . . 7
3028, 29brabga 4469 . . . . . 6
31 anass 631 . . . . . 6
3230, 31syl6bb 253 . . . . 5
3332ex 424 . . . 4
3413, 33syl 16 . . 3
353, 6, 34pm5.21ndd 344 . 2
3613biantrurd 495 . 2
37 fdm 5595 . . . . . . . 8
387, 37syl 16 . . . . . . 7
3938raleqdv 2910 . . . . . 6
40 rlim.4 . . . . . . . . . . 11
4140oveq1d 6096 . . . . . . . . . 10
4241fveq2d 5732 . . . . . . . . 9
4342breq1d 4222 . . . . . . . 8
4443imbi2d 308 . . . . . . 7
4544ralbidva 2721 . . . . . 6
4639, 45bitrd 245 . . . . 5
4746rexbidv 2726 . . . 4
4847ralbidv 2725 . . 3
4948anbi2d 685 . 2
5035, 36, 493bitr2d 273 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2705  wrex 2706  cvv 2956   wss 3320   class class class wbr 4212   cdm 4878  wf 5450  cfv 5454  (class class class)co 6081   cpm 7019  cc 8988  cr 8989   clt 9120   cle 9121   cmin 9291  crp 10612  cabs 12039   crli 12279 This theorem is referenced by:  rlim2  12290  rlimcl  12297  rlimclim  12340  rlimres  12352  caurcvgr  12467 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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-pm 7021  df-rlim 12283
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