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Theorem rlim 13502
 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 13500 . . . . 5
21brrelex2i 4838 . . . 4
32a1i 11 . . 3
4 elex 3031 . . . . 5
54ad2antrl 732 . . . 4
65a1i 11 . . 3
7 rlim.1 . . . . 5
8 rlim.2 . . . . 5
9 cnex 9571 . . . . . 6
10 reex 9581 . . . . . 6
11 elpm2r 7444 . . . . . 6
129, 10, 11mpanl12 686 . . . . 5
137, 8, 12syl2anc 665 . . . 4
14 eleq1 2494 . . . . . . . . 9
15 eleq1 2494 . . . . . . . . 9
1614, 15bi2anan9 881 . . . . . . . 8
17 simpl 458 . . . . . . . . . . . 12
1817dmeqd 4999 . . . . . . . . . . 11
19 fveq1 5824 . . . . . . . . . . . . . . 15
20 oveq12 6258 . . . . . . . . . . . . . . 15
2119, 20sylan 473 . . . . . . . . . . . . . 14
2221fveq2d 5829 . . . . . . . . . . . . 13
2322breq1d 4376 . . . . . . . . . . . 12
2423imbi2d 317 . . . . . . . . . . 11
2518, 24raleqbidv 2978 . . . . . . . . . 10
2625rexbidv 2878 . . . . . . . . 9
2726ralbidv 2804 . . . . . . . 8
2816, 27anbi12d 715 . . . . . . 7
29 df-rlim 13496 . . . . . . 7
3028, 29brabga 4677 . . . . . 6
31 anass 653 . . . . . 6
3230, 31syl6bb 264 . . . . 5
3332ex 435 . . . 4
3413, 33syl 17 . . 3
353, 6, 34pm5.21ndd 355 . 2
3613biantrurd 510 . 2
37 fdm 5693 . . . . . . . 8
387, 37syl 17 . . . . . . 7
3938raleqdv 2970 . . . . . 6
40 rlim.4 . . . . . . . . . . 11
4140oveq1d 6264 . . . . . . . . . 10
4241fveq2d 5829 . . . . . . . . 9
4342breq1d 4376 . . . . . . . 8
4443imbi2d 317 . . . . . . 7
4544ralbidva 2801 . . . . . 6
4639, 45bitrd 256 . . . . 5
4746rexbidv 2878 . . . 4
4847ralbidv 2804 . . 3
4948anbi2d 708 . 2
5035, 36, 493bitr2d 284 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1872  wral 2714  wrex 2715  cvv 3022   wss 3379   class class class wbr 4366   cdm 4796  wf 5540  cfv 5544  (class class class)co 6249   cpm 7428  cc 9488  cr 9489   clt 9626   cle 9627   cmin 9811  crp 11253  cabs 13241   crli 13492 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-pm 7430  df-rlim 13496 This theorem is referenced by:  rlim2  13503  rlimcl  13510  rlimclim  13553  rlimres  13565  caurcvgr  13681  caurcvgrOLD  13682
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