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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climrel 10655 |
. . . . 5
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2 | 1 | brrelexi 4476 |
. . . 4
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3 | eqidd 2089 |
. . . 4
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4 | 2, 3 | clim 10656 |
. . 3
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5 | 4 | ibi 174 |
. 2
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6 | 5 | simpld 110 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-pow 4007 ax-pr 4034 ax-cnex 7426 ax-resscn 7427 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-sbc 2841 df-un 3003 df-in 3005 df-ss 3012 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-br 3844 df-opab 3898 df-mpt 3899 df-id 4118 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-rn 4447 df-res 4448 df-ima 4449 df-iota 4975 df-fun 5012 df-fn 5013 df-f 5014 df-fv 5018 df-ov 5647 df-neg 7646 df-z 8741 df-uz 9010 df-clim 10654 |
This theorem is referenced by: climuni 10668 fclim 10669 climeu 10671 climreu 10672 2clim 10676 climcn1lem 10694 climrecl 10699 climadd 10701 climmul 10702 climsub 10703 climaddc2 10705 climcau 10723 geoisum1c 10901 |
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