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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 9193 | . . 3 | |
2 | nnuz 9474 | . . . . . . 7 | |
3 | 1zzd 9194 | . . . . . . 7 # | |
4 | climcl 11179 | . . . . . . . . . . 11 | |
5 | 4 | 3ad2ant1 1003 | . . . . . . . . . 10 # |
6 | climcl 11179 | . . . . . . . . . . 11 | |
7 | 6 | 3ad2ant2 1004 | . . . . . . . . . 10 # |
8 | 5, 7 | subcld 8186 | . . . . . . . . 9 # |
9 | simp3 984 | . . . . . . . . . 10 # # | |
10 | 5, 7, 9 | subap0d 8519 | . . . . . . . . 9 # # |
11 | 8, 10 | absrpclapd 11088 | . . . . . . . 8 # |
12 | 11 | rphalfcld 9616 | . . . . . . 7 # |
13 | eqidd 2158 | . . . . . . 7 # | |
14 | simp1 982 | . . . . . . 7 # | |
15 | 2, 3, 12, 13, 14 | climi 11184 | . . . . . 6 # |
16 | simp2 983 | . . . . . . 7 # | |
17 | 2, 3, 12, 13, 16 | climi 11184 | . . . . . 6 # |
18 | 2 | rexanuz2 10891 | . . . . . 6 |
19 | 15, 17, 18 | sylanbrc 414 | . . . . 5 # |
20 | nnz 9186 | . . . . . . . . 9 | |
21 | uzid 9453 | . . . . . . . . 9 | |
22 | elex2 2728 | . . . . . . . . 9 | |
23 | r19.2m 3480 | . . . . . . . . . 10 | |
24 | 23 | ex 114 | . . . . . . . . 9 |
25 | 20, 21, 22, 24 | 4syl 18 | . . . . . . . 8 |
26 | simpr 109 | . . . . . . . . . . . . . . 15 | |
27 | simpll 519 | . . . . . . . . . . . . . . 15 | |
28 | 26, 27 | abssubd 11093 | . . . . . . . . . . . . . 14 |
29 | 28 | breq1d 3975 | . . . . . . . . . . . . 13 |
30 | simplr 520 | . . . . . . . . . . . . . . . 16 | |
31 | subcl 8074 | . . . . . . . . . . . . . . . . . 18 | |
32 | 31 | adantr 274 | . . . . . . . . . . . . . . . . 17 |
33 | 32 | abscld 11081 | . . . . . . . . . . . . . . . 16 |
34 | abs3lem 11011 | . . . . . . . . . . . . . . . 16 | |
35 | 27, 30, 26, 33, 34 | syl22anc 1221 | . . . . . . . . . . . . . . 15 |
36 | 33 | ltnrd 7988 | . . . . . . . . . . . . . . . 16 |
37 | 36 | pm2.21d 609 | . . . . . . . . . . . . . . 15 |
38 | 35, 37 | syld 45 | . . . . . . . . . . . . . 14 |
39 | 38 | expd 256 | . . . . . . . . . . . . 13 |
40 | 29, 39 | sylbid 149 | . . . . . . . . . . . 12 |
41 | 40 | impr 377 | . . . . . . . . . . 11 |
42 | 41 | adantld 276 | . . . . . . . . . 10 |
43 | 42 | expimpd 361 | . . . . . . . . 9 |
44 | 43 | rexlimdvw 2578 | . . . . . . . 8 |
45 | 25, 44 | sylan9r 408 | . . . . . . 7 |
46 | 45 | rexlimdva 2574 | . . . . . 6 |
47 | 5, 7, 46 | syl2anc 409 | . . . . 5 # |
48 | 19, 47 | mpd 13 | . . . 4 # |
49 | 48 | 3expia 1187 | . . 3 # |
50 | 1, 49 | mt2i 634 | . 2 # |
51 | apti 8497 | . . 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 963 wceq 1335 wex 1472 wcel 2128 wral 2435 wrex 2436 class class class wbr 3965 cfv 5170 (class class class)co 5824 cc 7730 cr 7731 c1 7733 clt 7912 cmin 8046 # cap 8456 cdiv 8545 cn 8833 c2 8884 cz 9167 cuz 9439 cabs 10897 cli 11175 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-frec 6338 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-n0 9091 df-z 9168 df-uz 9440 df-rp 9561 df-seqfrec 10345 df-exp 10419 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899 df-clim 11176 |
This theorem is referenced by: fclim 11191 climeu 11193 climrecl 11221 summodclem2 11279 summodc 11280 prodmodclem2 11474 prodmodc 11475 ef0 11569 efcj 11570 efaddlem 11571 |
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