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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 11042 | . . . . 5 | |
2 | 1 | brrelex2i 4578 | . . . 4 |
3 | 2 | a1i 9 | . . 3 |
4 | elex 2692 | . . . . 5 | |
5 | 4 | adantr 274 | . . . 4 |
6 | 5 | a1i 9 | . . 3 |
7 | clim.1 | . . . 4 | |
8 | simpr 109 | . . . . . . . 8 | |
9 | 8 | eleq1d 2206 | . . . . . . 7 |
10 | fveq1 5413 | . . . . . . . . . . . . 13 | |
11 | 10 | adantr 274 | . . . . . . . . . . . 12 |
12 | 11 | eleq1d 2206 | . . . . . . . . . . 11 |
13 | oveq12 5776 | . . . . . . . . . . . . . 14 | |
14 | 10, 13 | sylan 281 | . . . . . . . . . . . . 13 |
15 | 14 | fveq2d 5418 | . . . . . . . . . . . 12 |
16 | 15 | breq1d 3934 | . . . . . . . . . . 11 |
17 | 12, 16 | anbi12d 464 | . . . . . . . . . 10 |
18 | 17 | ralbidv 2435 | . . . . . . . . 9 |
19 | 18 | rexbidv 2436 | . . . . . . . 8 |
20 | 19 | ralbidv 2435 | . . . . . . 7 |
21 | 9, 20 | anbi12d 464 | . . . . . 6 |
22 | df-clim 11041 | . . . . . 6 | |
23 | 21, 22 | brabga 4181 | . . . . 5 |
24 | 23 | ex 114 | . . . 4 |
25 | 7, 24 | syl 14 | . . 3 |
26 | 3, 6, 25 | pm5.21ndd 694 | . 2 |
27 | eluzelz 9328 | . . . . . . 7 | |
28 | clim.3 | . . . . . . . . 9 | |
29 | 28 | eleq1d 2206 | . . . . . . . 8 |
30 | 28 | oveq1d 5782 | . . . . . . . . . 10 |
31 | 30 | fveq2d 5418 | . . . . . . . . 9 |
32 | 31 | breq1d 3934 | . . . . . . . 8 |
33 | 29, 32 | anbi12d 464 | . . . . . . 7 |
34 | 27, 33 | sylan2 284 | . . . . . 6 |
35 | 34 | ralbidva 2431 | . . . . 5 |
36 | 35 | rexbidv 2436 | . . . 4 |
37 | 36 | ralbidv 2435 | . . 3 |
38 | 37 | anbi2d 459 | . 2 |
39 | 26, 38 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 wrex 2415 cvv 2681 class class class wbr 3924 cfv 5118 (class class class)co 5767 cc 7611 clt 7793 cmin 7926 cz 9047 cuz 9319 crp 9434 cabs 10762 cli 11040 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-cnex 7704 ax-resscn 7705 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-neg 7929 df-z 9048 df-uz 9320 df-clim 11041 |
This theorem is referenced by: climcl 11044 clim2 11045 climshftlemg 11064 |
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