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Theorem rlimcl 15447
Description: Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
rlimcl (𝐹 β‡π‘Ÿ 𝐴 β†’ 𝐴 ∈ β„‚)

Proof of Theorem rlimcl
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimf 15445 . . . 4 (𝐹 β‡π‘Ÿ 𝐴 β†’ 𝐹:dom πΉβŸΆβ„‚)
2 rlimss 15446 . . . 4 (𝐹 β‡π‘Ÿ 𝐴 β†’ dom 𝐹 βŠ† ℝ)
3 eqidd 2734 . . . 4 ((𝐹 β‡π‘Ÿ 𝐴 ∧ π‘₯ ∈ dom 𝐹) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘₯))
41, 2, 3rlim 15439 . . 3 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐹 β‡π‘Ÿ 𝐴 ↔ (𝐴 ∈ β„‚ ∧ βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ βˆ€π‘₯ ∈ dom 𝐹(𝑧 ≀ π‘₯ β†’ (absβ€˜((πΉβ€˜π‘₯) βˆ’ 𝐴)) < 𝑦))))
54ibi 267 . 2 (𝐹 β‡π‘Ÿ 𝐴 β†’ (𝐴 ∈ β„‚ ∧ βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ βˆ€π‘₯ ∈ dom 𝐹(𝑧 ≀ π‘₯ β†’ (absβ€˜((πΉβ€˜π‘₯) βˆ’ 𝐴)) < 𝑦)))
65simpld 496 1 (𝐹 β‡π‘Ÿ 𝐴 β†’ 𝐴 ∈ β„‚)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   class class class wbr 5149  dom cdm 5677  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  β„cr 11109   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444  β„+crp 12974  abscabs 15181   β‡π‘Ÿ crli 15429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-pm 8823  df-rlim 15433
This theorem is referenced by:  rlimi  15457  rlimclim1  15489  rlimuni  15494  rlimresb  15509  rlimcld2  15522  rlimabs  15553  rlimcj  15554  rlimre  15555  rlimim  15556  rlimo1  15561  rlimadd  15587  rlimaddOLD  15588  rlimsub  15589  rlimmul  15590  rlimmulOLD  15591  rlimdiv  15592  rlimsqzlem  15595  fsumrlim  15757  dchrisum0lem2a  27020  mulog2sumlem2  27038  mulog2sumlem3  27039
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