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Mirrors > Home > ILE Home > Th. List > climuni | Unicode version |
Description: An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
climuni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 9080 | . . 3 | |
2 | nnuz 9361 | . . . . . . 7 | |
3 | 1zzd 9081 | . . . . . . 7 # | |
4 | climcl 11051 | . . . . . . . . . . 11 | |
5 | 4 | 3ad2ant1 1002 | . . . . . . . . . 10 # |
6 | climcl 11051 | . . . . . . . . . . 11 | |
7 | 6 | 3ad2ant2 1003 | . . . . . . . . . 10 # |
8 | 5, 7 | subcld 8073 | . . . . . . . . 9 # |
9 | simp3 983 | . . . . . . . . . 10 # # | |
10 | 5, 7, 9 | subap0d 8406 | . . . . . . . . 9 # # |
11 | 8, 10 | absrpclapd 10960 | . . . . . . . 8 # |
12 | 11 | rphalfcld 9496 | . . . . . . 7 # |
13 | eqidd 2140 | . . . . . . 7 # | |
14 | simp1 981 | . . . . . . 7 # | |
15 | 2, 3, 12, 13, 14 | climi 11056 | . . . . . 6 # |
16 | simp2 982 | . . . . . . 7 # | |
17 | 2, 3, 12, 13, 16 | climi 11056 | . . . . . 6 # |
18 | 2 | rexanuz2 10763 | . . . . . 6 |
19 | 15, 17, 18 | sylanbrc 413 | . . . . 5 # |
20 | nnz 9073 | . . . . . . . . 9 | |
21 | uzid 9340 | . . . . . . . . 9 | |
22 | elex2 2702 | . . . . . . . . 9 | |
23 | r19.2m 3449 | . . . . . . . . . 10 | |
24 | 23 | ex 114 | . . . . . . . . 9 |
25 | 20, 21, 22, 24 | 4syl 18 | . . . . . . . 8 |
26 | simpr 109 | . . . . . . . . . . . . . . 15 | |
27 | simpll 518 | . . . . . . . . . . . . . . 15 | |
28 | 26, 27 | abssubd 10965 | . . . . . . . . . . . . . 14 |
29 | 28 | breq1d 3939 | . . . . . . . . . . . . 13 |
30 | simplr 519 | . . . . . . . . . . . . . . . 16 | |
31 | subcl 7961 | . . . . . . . . . . . . . . . . . 18 | |
32 | 31 | adantr 274 | . . . . . . . . . . . . . . . . 17 |
33 | 32 | abscld 10953 | . . . . . . . . . . . . . . . 16 |
34 | abs3lem 10883 | . . . . . . . . . . . . . . . 16 | |
35 | 27, 30, 26, 33, 34 | syl22anc 1217 | . . . . . . . . . . . . . . 15 |
36 | 33 | ltnrd 7875 | . . . . . . . . . . . . . . . 16 |
37 | 36 | pm2.21d 608 | . . . . . . . . . . . . . . 15 |
38 | 35, 37 | syld 45 | . . . . . . . . . . . . . 14 |
39 | 38 | expd 256 | . . . . . . . . . . . . 13 |
40 | 29, 39 | sylbid 149 | . . . . . . . . . . . 12 |
41 | 40 | impr 376 | . . . . . . . . . . 11 |
42 | 41 | adantld 276 | . . . . . . . . . 10 |
43 | 42 | expimpd 360 | . . . . . . . . 9 |
44 | 43 | rexlimdvw 2553 | . . . . . . . 8 |
45 | 25, 44 | sylan9r 407 | . . . . . . 7 |
46 | 45 | rexlimdva 2549 | . . . . . 6 |
47 | 5, 7, 46 | syl2anc 408 | . . . . 5 # |
48 | 19, 47 | mpd 13 | . . . 4 # |
49 | 48 | 3expia 1183 | . . 3 # |
50 | 1, 49 | mt2i 633 | . 2 # |
51 | apti 8384 | . . 3 # | |
52 | 4, 6, 51 | syl2an 287 | . 2 # |
53 | 50, 52 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 962 wceq 1331 wex 1468 wcel 1480 wral 2416 wrex 2417 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc 7618 cr 7619 c1 7621 clt 7800 cmin 7933 # cap 8343 cdiv 8432 cn 8720 c2 8771 cz 9054 cuz 9326 cabs 10769 cli 11047 |
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-in1 603 ax-in2 604 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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 |
This theorem is referenced by: fclim 11063 climeu 11065 climrecl 11093 summodclem2 11151 summodc 11152 prodmodclem2 11346 prodmodc 11347 ef0 11378 efcj 11379 efaddlem 11380 |
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