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Mirrors > Home > ILE Home > Th. List > caubnd2 | Unicode version |
Description: A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.) |
Ref | Expression |
---|---|
cau3.1 |
Ref | Expression |
---|---|
caubnd2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1rp 8587 | . . 3 | |
2 | breq2 3768 | . . . . . 6 | |
3 | 2 | anbi2d 437 | . . . . 5 |
4 | 3 | rexralbidv 2350 | . . . 4 |
5 | 4 | rspcv 2652 | . . 3 |
6 | 1, 5 | ax-mp 7 | . 2 |
7 | eluzelz 8482 | . . . . . . . . . . 11 | |
8 | cau3.1 | . . . . . . . . . . 11 | |
9 | 7, 8 | eleq2s 2132 | . . . . . . . . . 10 |
10 | uzid 8487 | . . . . . . . . . 10 | |
11 | 9, 10 | syl 14 | . . . . . . . . 9 |
12 | simpl 102 | . . . . . . . . . 10 | |
13 | 12 | ralimi 2384 | . . . . . . . . 9 |
14 | fveq2 5178 | . . . . . . . . . . 11 | |
15 | 14 | eleq1d 2106 | . . . . . . . . . 10 |
16 | 15 | rspcva 2654 | . . . . . . . . 9 |
17 | 11, 13, 16 | syl2an 273 | . . . . . . . 8 |
18 | abscl 9649 | . . . . . . . 8 | |
19 | 17, 18 | syl 14 | . . . . . . 7 |
20 | 1re 7026 | . . . . . . 7 | |
21 | readdcl 7007 | . . . . . . 7 | |
22 | 19, 20, 21 | sylancl 392 | . . . . . 6 |
23 | simpr 103 | . . . . . . . . . . . . 13 | |
24 | simplr 482 | . . . . . . . . . . . . 13 | |
25 | abs2dif 9702 | . . . . . . . . . . . . 13 | |
26 | 23, 24, 25 | syl2anc 391 | . . . . . . . . . . . 12 |
27 | abscl 9649 | . . . . . . . . . . . . . . 15 | |
28 | 23, 27 | syl 14 | . . . . . . . . . . . . . 14 |
29 | 24, 18 | syl 14 | . . . . . . . . . . . . . 14 |
30 | 28, 29 | resubcld 7379 | . . . . . . . . . . . . 13 |
31 | 23, 24 | subcld 7322 | . . . . . . . . . . . . . 14 |
32 | abscl 9649 | . . . . . . . . . . . . . 14 | |
33 | 31, 32 | syl 14 | . . . . . . . . . . . . 13 |
34 | lelttr 7106 | . . . . . . . . . . . . . 14 | |
35 | 20, 34 | mp3an3 1221 | . . . . . . . . . . . . 13 |
36 | 30, 33, 35 | syl2anc 391 | . . . . . . . . . . . 12 |
37 | 26, 36 | mpand 405 | . . . . . . . . . . 11 |
38 | ltsubadd2 7428 | . . . . . . . . . . . . 13 | |
39 | 20, 38 | mp3an3 1221 | . . . . . . . . . . . 12 |
40 | 28, 29, 39 | syl2anc 391 | . . . . . . . . . . 11 |
41 | 37, 40 | sylibd 138 | . . . . . . . . . 10 |
42 | 41 | expimpd 345 | . . . . . . . . 9 |
43 | 42 | ralimdv 2388 | . . . . . . . 8 |
44 | 43 | impancom 247 | . . . . . . 7 |
45 | 17, 44 | mpd 13 | . . . . . 6 |
46 | breq2 3768 | . . . . . . . 8 | |
47 | 46 | ralbidv 2326 | . . . . . . 7 |
48 | 47 | rspcev 2656 | . . . . . 6 |
49 | 22, 45, 48 | syl2anc 391 | . . . . 5 |
50 | 49 | ex 108 | . . . 4 |
51 | 50 | reximia 2414 | . . 3 |
52 | rexcom 2474 | . . 3 | |
53 | 51, 52 | sylib 127 | . 2 |
54 | 6, 53 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 wrex 2307 class class class wbr 3764 cfv 4902 (class class class)co 5512 cc 6887 cr 6888 c1 6890 caddc 6892 clt 7060 cle 7061 cmin 7182 cz 8245 cuz 8473 crp 8583 cabs 9595 |
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-in1 544 ax-in2 545 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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 ax-arch 7003 ax-caucvg 7004 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 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-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 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-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-2 7973 df-3 7974 df-4 7975 df-n0 8182 df-z 8246 df-uz 8474 df-rp 8584 df-iseq 9212 df-iexp 9255 df-cj 9442 df-re 9443 df-im 9444 df-rsqrt 9596 df-abs 9597 |
This theorem is referenced by: (None) |
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