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Mirrors > Home > ILE Home > Th. List > 2clim | Unicode version |
Description: If two sequences converge to each other, they converge to the same limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
2clim.1 | |
2clim.2 | |
2clim.3 | |
2clim.5 | |
2clim.6 | |
2clim.7 |
Ref | Expression |
---|---|
2clim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2clim.6 | . . . . . 6 | |
2 | rphalfcl 9652 | . . . . . 6 | |
3 | breq2 4002 | . . . . . . . 8 | |
4 | 3 | rexralbidv 2501 | . . . . . . 7 |
5 | 4 | rspccva 2838 | . . . . . 6 |
6 | 1, 2, 5 | syl2an 289 | . . . . 5 |
7 | 2clim.1 | . . . . . 6 | |
8 | 2clim.2 | . . . . . . 7 | |
9 | 8 | adantr 276 | . . . . . 6 |
10 | 2 | adantl 277 | . . . . . 6 |
11 | eqidd 2176 | . . . . . 6 | |
12 | 2clim.7 | . . . . . . 7 | |
13 | 12 | adantr 276 | . . . . . 6 |
14 | 7, 9, 10, 11, 13 | climi 11263 | . . . . 5 |
15 | 7 | rexanuz2 10968 | . . . . 5 |
16 | 6, 14, 15 | sylanbrc 417 | . . . 4 |
17 | 7 | uztrn2 9518 | . . . . . . . 8 |
18 | an12 561 | . . . . . . . . 9 | |
19 | simprr 531 | . . . . . . . . . . . . . . 15 | |
20 | 2clim.5 | . . . . . . . . . . . . . . . 16 | |
21 | 20 | ad2ant2r 509 | . . . . . . . . . . . . . . 15 |
22 | 19, 21 | abssubd 11170 | . . . . . . . . . . . . . 14 |
23 | 22 | breq1d 4008 | . . . . . . . . . . . . 13 |
24 | 23 | anbi1d 465 | . . . . . . . . . . . 12 |
25 | climcl 11258 | . . . . . . . . . . . . . . 15 | |
26 | 12, 25 | syl 14 | . . . . . . . . . . . . . 14 |
27 | 26 | ad2antrr 488 | . . . . . . . . . . . . 13 |
28 | rpre 9631 | . . . . . . . . . . . . . 14 | |
29 | 28 | ad2antlr 489 | . . . . . . . . . . . . 13 |
30 | abs3lem 11088 | . . . . . . . . . . . . 13 | |
31 | 21, 27, 19, 29, 30 | syl22anc 1239 | . . . . . . . . . . . 12 |
32 | 24, 31 | sylbid 150 | . . . . . . . . . . 11 |
33 | 32 | anassrs 400 | . . . . . . . . . 10 |
34 | 33 | expimpd 363 | . . . . . . . . 9 |
35 | 18, 34 | biimtrid 152 | . . . . . . . 8 |
36 | 17, 35 | sylan2 286 | . . . . . . 7 |
37 | 36 | anassrs 400 | . . . . . 6 |
38 | 37 | ralimdva 2542 | . . . . 5 |
39 | 38 | reximdva 2577 | . . . 4 |
40 | 16, 39 | mpd 13 | . . 3 |
41 | 40 | ralrimiva 2548 | . 2 |
42 | 2clim.3 | . . 3 | |
43 | eqidd 2176 | . . 3 | |
44 | 7, 8, 42, 43, 26, 20 | clim2c 11260 | . 2 |
45 | 41, 44 | mpbird 167 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wceq 1353 wcel 2146 wral 2453 wrex 2454 class class class wbr 3998 cfv 5208 (class class class)co 5865 cc 7784 cr 7785 clt 7966 cmin 8102 cdiv 8602 c2 8943 cz 9226 cuz 9501 crp 9624 cabs 10974 cli 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8603 df-inn 8893 df-2 8951 df-3 8952 df-4 8953 df-n0 9150 df-z 9227 df-uz 9502 df-rp 9625 df-seqfrec 10416 df-exp 10490 df-cj 10819 df-re 10820 df-im 10821 df-rsqrt 10975 df-abs 10976 df-clim 11255 |
This theorem is referenced by: mertensabs 11513 |
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