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Mirrors > Home > ILE Home > Th. List > clim | Unicode version |
Description: Express the predicate: The limit of complex number sequence is , or converges to . This means that for any real , no matter how small, there always exists an integer such that the absolute difference of any later complex number in the sequence and the limit is less than . (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
clim.1 | |
clim.3 |
Ref | Expression |
---|---|
clim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climrel 9801 | . . . . 5 | |
2 | 1 | brrelex2i 4385 | . . . 4 |
3 | 2 | a1i 9 | . . 3 |
4 | elex 2566 | . . . . 5 | |
5 | 4 | adantr 261 | . . . 4 |
6 | 5 | a1i 9 | . . 3 |
7 | clim.1 | . . . 4 | |
8 | simpr 103 | . . . . . . . 8 | |
9 | 8 | eleq1d 2106 | . . . . . . 7 |
10 | fveq1 5177 | . . . . . . . . . . . . 13 | |
11 | 10 | adantr 261 | . . . . . . . . . . . 12 |
12 | 11 | eleq1d 2106 | . . . . . . . . . . 11 |
13 | oveq12 5521 | . . . . . . . . . . . . . 14 | |
14 | 10, 13 | sylan 267 | . . . . . . . . . . . . 13 |
15 | 14 | fveq2d 5182 | . . . . . . . . . . . 12 |
16 | 15 | breq1d 3774 | . . . . . . . . . . 11 |
17 | 12, 16 | anbi12d 442 | . . . . . . . . . 10 |
18 | 17 | ralbidv 2326 | . . . . . . . . 9 |
19 | 18 | rexbidv 2327 | . . . . . . . 8 |
20 | 19 | ralbidv 2326 | . . . . . . 7 |
21 | 9, 20 | anbi12d 442 | . . . . . 6 |
22 | df-clim 9800 | . . . . . 6 | |
23 | 21, 22 | brabga 4001 | . . . . 5 |
24 | 23 | ex 108 | . . . 4 |
25 | 7, 24 | syl 14 | . . 3 |
26 | 3, 6, 25 | pm5.21ndd 621 | . 2 |
27 | eluzelz 8482 | . . . . . . 7 | |
28 | clim.3 | . . . . . . . . 9 | |
29 | 28 | eleq1d 2106 | . . . . . . . 8 |
30 | 28 | oveq1d 5527 | . . . . . . . . . 10 |
31 | 30 | fveq2d 5182 | . . . . . . . . 9 |
32 | 31 | breq1d 3774 | . . . . . . . 8 |
33 | 29, 32 | anbi12d 442 | . . . . . . 7 |
34 | 27, 33 | sylan2 270 | . . . . . 6 |
35 | 34 | ralbidva 2322 | . . . . 5 |
36 | 35 | rexbidv 2327 | . . . 4 |
37 | 36 | ralbidv 2326 | . . 3 |
38 | 37 | anbi2d 437 | . 2 |
39 | 26, 38 | bitrd 177 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 wrex 2307 cvv 2557 class class class wbr 3764 cfv 4902 (class class class)co 5512 cc 6887 clt 7060 cmin 7182 cz 8245 cuz 8473 crp 8583 cabs 9595 cli 9799 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-cnex 6975 ax-resscn 6976 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-ov 5515 df-neg 7185 df-z 8246 df-uz 8474 df-clim 9800 |
This theorem is referenced by: climcl 9803 clim2 9804 climshftlemg 9823 |
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