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Mirrors > Home > ILE Home > Th. List > nnrecl | Unicode version |
Description: There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.) |
Ref | Expression |
---|---|
nnrecl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . 4 | |
2 | gt0ap0 8515 | . . . 4 # | |
3 | 1, 2 | rerecclapd 8721 | . . 3 |
4 | arch 9102 | . . 3 | |
5 | 3, 4 | syl 14 | . 2 |
6 | recgt0 8736 | . . . . . 6 | |
7 | 3, 6 | jca 304 | . . . . 5 |
8 | nnre 8855 | . . . . . 6 | |
9 | nngt0 8873 | . . . . . 6 | |
10 | 8, 9 | jca 304 | . . . . 5 |
11 | ltrec 8769 | . . . . 5 | |
12 | 7, 10, 11 | syl2an 287 | . . . 4 |
13 | recn 7877 | . . . . . . . 8 | |
14 | 13 | adantr 274 | . . . . . . 7 |
15 | 14, 2 | recrecapd 8672 | . . . . . 6 |
16 | 15 | breq2d 3988 | . . . . 5 |
17 | 16 | adantr 274 | . . . 4 |
18 | 12, 17 | bitrd 187 | . . 3 |
19 | 18 | rexbidva 2461 | . 2 |
20 | 5, 19 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2135 wrex 2443 class class class wbr 3976 (class class class)co 5836 cc 7742 cr 7743 cc0 7744 c1 7745 clt 7924 cdiv 8559 cn 8848 |
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-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 |
This theorem depends on definitions: df-bi 116 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-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 |
This theorem is referenced by: qbtwnre 10182 trilpolemlt1 13761 |
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