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Mirrors > Home > MPE Home > Th. List > rlimcl | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
rlimcl | β’ (πΉ βπ π΄ β π΄ β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimf 15445 | . . . 4 β’ (πΉ βπ π΄ β πΉ:dom πΉβΆβ) | |
2 | rlimss 15446 | . . . 4 β’ (πΉ βπ π΄ β dom πΉ β β) | |
3 | eqidd 2734 | . . . 4 β’ ((πΉ βπ π΄ β§ π₯ β dom πΉ) β (πΉβπ₯) = (πΉβπ₯)) | |
4 | 1, 2, 3 | rlim 15439 | . . 3 β’ (πΉ βπ π΄ β (πΉ βπ π΄ β (π΄ β β β§ βπ¦ β β+ βπ§ β β βπ₯ β dom πΉ(π§ β€ π₯ β (absβ((πΉβπ₯) β π΄)) < π¦)))) |
5 | 4 | ibi 267 | . 2 β’ (πΉ βπ π΄ β (π΄ β β β§ βπ¦ β β+ βπ§ β β βπ₯ β dom πΉ(π§ β€ π₯ β (absβ((πΉβπ₯) β π΄)) < π¦))) |
6 | 5 | simpld 496 | 1 β’ (πΉ βπ π΄ β π΄ β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 βwral 3062 βwrex 3071 class class class wbr 5149 dom cdm 5677 βcfv 6544 (class class class)co 7409 βcc 11108 βcr 11109 < clt 11248 β€ cle 11249 β cmin 11444 β+crp 12974 abscabs 15181 βπ 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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