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Mirrors > Home > ILE Home > Th. List > climcl | Unicode version |
Description: Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
climcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climrel 11017 | . . . . 5 | |
2 | 1 | brrelex1i 4552 | . . . 4 |
3 | eqidd 2118 | . . . 4 | |
4 | 2, 3 | clim 11018 | . . 3 |
5 | 4 | ibi 175 | . 2 |
6 | 5 | simpld 111 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 wral 2393 wrex 2394 cvv 2660 class class class wbr 3899 cfv 5093 (class class class)co 5742 cc 7586 clt 7768 cmin 7901 cz 9022 cuz 9294 crp 9409 cabs 10737 cli 11015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-cnex 7679 ax-resscn 7680 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-neg 7904 df-z 9023 df-uz 9295 df-clim 11016 |
This theorem is referenced by: climuni 11030 fclim 11031 climeu 11033 climreu 11034 2clim 11038 climcn1lem 11056 climrecl 11061 climadd 11063 climmul 11064 climsub 11065 climaddc2 11067 climcau 11084 geoisum1c 11257 |
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