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Mirrors > Home > ILE Home > Th. List > lmres | Unicode version |
Description: A function converges iff its restriction to an upper integers set converges. (Contributed by Mario Carneiro, 31-Dec-2013.) |
Ref | Expression |
---|---|
lmres.2 | TopOn |
lmres.4 | |
lmres.5 |
Ref | Expression |
---|---|
lmres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmres.2 | . . . . . . 7 TopOn | |
2 | toponmax 12570 | . . . . . . 7 TopOn | |
3 | 1, 2 | syl 14 | . . . . . 6 |
4 | cnex 7868 | . . . . . 6 | |
5 | ssid 3157 | . . . . . . 7 | |
6 | uzssz 9476 | . . . . . . . 8 | |
7 | zsscn 9190 | . . . . . . . 8 | |
8 | 6, 7 | sstri 3146 | . . . . . . 7 |
9 | pmss12g 6632 | . . . . . . 7 | |
10 | 5, 8, 9 | mpanl12 433 | . . . . . 6 |
11 | 3, 4, 10 | sylancl 410 | . . . . 5 |
12 | zex 9191 | . . . . . . 7 | |
13 | 12, 6 | ssexi 4114 | . . . . . 6 |
14 | lmres.4 | . . . . . 6 | |
15 | pmresg 6633 | . . . . . 6 | |
16 | 13, 14, 15 | sylancr 411 | . . . . 5 |
17 | 11, 16 | sseldd 3138 | . . . 4 |
18 | 17, 14 | 2thd 174 | . . 3 |
19 | eqid 2164 | . . . . . . . . . 10 | |
20 | 19 | uztrn2 9474 | . . . . . . . . 9 |
21 | dmres 4899 | . . . . . . . . . . . 12 | |
22 | 21 | elin2 3305 | . . . . . . . . . . 11 |
23 | 22 | baib 909 | . . . . . . . . . 10 |
24 | fvres 5504 | . . . . . . . . . . 11 | |
25 | 24 | eleq1d 2233 | . . . . . . . . . 10 |
26 | 23, 25 | anbi12d 465 | . . . . . . . . 9 |
27 | 20, 26 | syl 14 | . . . . . . . 8 |
28 | 27 | ralbidva 2460 | . . . . . . 7 |
29 | 28 | rexbiia 2479 | . . . . . 6 |
30 | 29 | imbi2i 225 | . . . . 5 |
31 | 30 | ralbii 2470 | . . . 4 |
32 | 31 | a1i 9 | . . 3 |
33 | 18, 32 | 3anbi13d 1303 | . 2 |
34 | lmres.5 | . . 3 | |
35 | 1, 19, 34 | lmbr2 12761 | . 2 |
36 | 1, 19, 34 | lmbr2 12761 | . 2 |
37 | 33, 35, 36 | 3bitr4rd 220 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wcel 2135 wral 2442 wrex 2443 cvv 2721 wss 3111 class class class wbr 3976 cdm 4598 cres 4600 cfv 5182 (class class class)co 5836 cpm 6606 cc 7742 cz 9182 cuz 9457 TopOnctopon 12555 clm 12734 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pm 6608 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-top 12543 df-topon 12556 df-lm 12737 |
This theorem is referenced by: (None) |
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