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Mirrors > Home > ILE Home > Th. List > divcnv | Unicode version |
Description: The sequence of reciprocals of positive integers, multiplied by the factor , converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Jim Kingdon, 22-Oct-2022.) |
Ref | Expression |
---|---|
divcnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . . . 7 | |
2 | 1 | abscld 11076 | . . . . . 6 |
3 | simpr 109 | . . . . . 6 | |
4 | 2, 3 | rerpdivcld 9630 | . . . . 5 |
5 | arch 9082 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 1 | ad3antrrr 484 | . . . . . . . . . 10 |
8 | eluzelz 9443 | . . . . . . . . . . . 12 | |
9 | 8 | adantl 275 | . . . . . . . . . . 11 |
10 | 9 | zcnd 9282 | . . . . . . . . . 10 |
11 | 9 | zred 9281 | . . . . . . . . . . 11 |
12 | 0red 7874 | . . . . . . . . . . . 12 | |
13 | simpllr 524 | . . . . . . . . . . . . 13 | |
14 | 13 | nnred 8841 | . . . . . . . . . . . 12 |
15 | 13 | nngt0d 8872 | . . . . . . . . . . . 12 |
16 | eluzle 9446 | . . . . . . . . . . . . 13 | |
17 | 16 | adantl 275 | . . . . . . . . . . . 12 |
18 | 12, 14, 11, 15, 17 | ltletrd 8293 | . . . . . . . . . . 11 |
19 | 11, 18 | gt0ap0d 8499 | . . . . . . . . . 10 # |
20 | 7, 10, 19 | absdivapd 11090 | . . . . . . . . 9 |
21 | 12, 11, 18 | ltled 7989 | . . . . . . . . . . 11 |
22 | 11, 21 | absidd 11062 | . . . . . . . . . 10 |
23 | 22 | oveq2d 5837 | . . . . . . . . 9 |
24 | 20, 23 | eqtrd 2190 | . . . . . . . 8 |
25 | 2 | ad3antrrr 484 | . . . . . . . . 9 |
26 | 3 | ad3antrrr 484 | . . . . . . . . 9 |
27 | 11, 18 | elrpd 9595 | . . . . . . . . 9 |
28 | 4 | ad3antrrr 484 | . . . . . . . . . 10 |
29 | simplr 520 | . . . . . . . . . 10 | |
30 | 28, 14, 11, 29, 17 | ltletrd 8293 | . . . . . . . . 9 |
31 | 25, 26, 27, 30 | ltdiv23d 9659 | . . . . . . . 8 |
32 | 24, 31 | eqbrtrd 3986 | . . . . . . 7 |
33 | 32 | ralrimiva 2530 | . . . . . 6 |
34 | 33 | ex 114 | . . . . 5 |
35 | 34 | reximdva 2559 | . . . 4 |
36 | 6, 35 | mpd 13 | . . 3 |
37 | 36 | ralrimiva 2530 | . 2 |
38 | nnuz 9469 | . . 3 | |
39 | 1zzd 9189 | . . 3 | |
40 | nnex 8834 | . . . . 5 | |
41 | 40 | mptex 5692 | . . . 4 |
42 | 41 | a1i 9 | . . 3 |
43 | simpr 109 | . . . 4 | |
44 | simpl 108 | . . . . 5 | |
45 | 43 | nncnd 8842 | . . . . 5 |
46 | 43 | nnap0d 8874 | . . . . 5 # |
47 | 44, 45, 46 | divclapd 8658 | . . . 4 |
48 | oveq2 5829 | . . . . 5 | |
49 | eqid 2157 | . . . . 5 | |
50 | 48, 49 | fvmptg 5543 | . . . 4 |
51 | 43, 47, 50 | syl2anc 409 | . . 3 |
52 | 38, 39, 42, 51, 47 | clim0c 11178 | . 2 |
53 | 37, 52 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 wral 2435 wrex 2436 cvv 2712 class class class wbr 3965 cmpt 4025 cfv 5169 (class class class)co 5821 cc 7725 cr 7726 cc0 7727 c1 7728 clt 7907 cle 7908 cdiv 8540 cn 8828 cz 9162 cuz 9434 crp 9555 cabs 10892 cli 11170 |
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 4495 ax-iinf 4546 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-mulrcl 7826 ax-addcom 7827 ax-mulcom 7828 ax-addass 7829 ax-mulass 7830 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-1rid 7834 ax-0id 7835 ax-rnegex 7836 ax-precex 7837 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-apti 7842 ax-pre-ltadd 7843 ax-pre-mulgt0 7844 ax-pre-mulext 7845 ax-arch 7846 ax-caucvg 7847 |
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 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-frec 6335 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-reap 8445 df-ap 8452 df-div 8541 df-inn 8829 df-2 8887 df-3 8888 df-4 8889 df-n0 9086 df-z 9163 df-uz 9435 df-rp 9556 df-seqfrec 10340 df-exp 10414 df-cj 10737 df-re 10738 df-im 10739 df-rsqrt 10893 df-abs 10894 df-clim 11171 |
This theorem is referenced by: trireciplem 11392 expcnvap0 11394 |
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