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Mirrors > Home > ILE Home > Th. List > recgt0 | Unicode version |
Description: The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
recgt0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt1 7889 | . . . . 5 | |
2 | 0re 7766 | . . . . . 6 | |
3 | 1re 7765 | . . . . . 6 | |
4 | 2, 3 | ltnsymi 7863 | . . . . 5 |
5 | 1, 4 | ax-mp 5 | . . . 4 |
6 | simpll 518 | . . . . . . . . . 10 | |
7 | gt0ap0 8388 | . . . . . . . . . . 11 # | |
8 | 7 | adantr 274 | . . . . . . . . . 10 # |
9 | 6, 8 | rerecclapd 8593 | . . . . . . . . 9 |
10 | 9 | renegcld 8142 | . . . . . . . 8 |
11 | simpr 109 | . . . . . . . . 9 | |
12 | simpl 108 | . . . . . . . . . . . 12 | |
13 | 12, 7 | rerecclapd 8593 | . . . . . . . . . . 11 |
14 | 13 | adantr 274 | . . . . . . . . . 10 |
15 | 14 | lt0neg1d 8277 | . . . . . . . . 9 |
16 | 11, 15 | mpbid 146 | . . . . . . . 8 |
17 | simplr 519 | . . . . . . . 8 | |
18 | 10, 6, 16, 17 | mulgt0d 7885 | . . . . . . 7 |
19 | 12 | recnd 7794 | . . . . . . . . . . 11 |
20 | 19 | adantr 274 | . . . . . . . . . 10 |
21 | recclap 8439 | . . . . . . . . . 10 # | |
22 | 20, 8, 21 | syl2anc 408 | . . . . . . . . 9 |
23 | 22, 20 | mulneg1d 8173 | . . . . . . . 8 |
24 | recidap2 8447 | . . . . . . . . . 10 # | |
25 | 20, 8, 24 | syl2anc 408 | . . . . . . . . 9 |
26 | 25 | negeqd 7957 | . . . . . . . 8 |
27 | 23, 26 | eqtrd 2172 | . . . . . . 7 |
28 | 18, 27 | breqtrd 3954 | . . . . . 6 |
29 | 1red 7781 | . . . . . . 7 | |
30 | 29 | lt0neg1d 8277 | . . . . . 6 |
31 | 28, 30 | mpbird 166 | . . . . 5 |
32 | 31 | ex 114 | . . . 4 |
33 | 5, 32 | mtoi 653 | . . 3 |
34 | lenlt 7840 | . . . 4 | |
35 | 2, 13, 34 | sylancr 410 | . . 3 |
36 | 33, 35 | mpbird 166 | . 2 |
37 | recap0 8445 | . . . 4 # # | |
38 | 19, 7, 37 | syl2anc 408 | . . 3 # |
39 | 19, 7, 21 | syl2anc 408 | . . . 4 |
40 | 0cn 7758 | . . . 4 | |
41 | apsym 8368 | . . . 4 # # | |
42 | 39, 40, 41 | sylancl 409 | . . 3 # # |
43 | 38, 42 | mpbid 146 | . 2 # |
44 | ltleap 8394 | . . 3 # | |
45 | 2, 13, 44 | sylancr 410 | . 2 # |
46 | 36, 43, 45 | mpbir2and 928 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 class class class wbr 3929 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 cmul 7625 clt 7800 cle 7801 cneg 7934 # cap 8343 cdiv 8432 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 |
This theorem is referenced by: prodgt0gt0 8609 ltdiv1 8626 ltrec1 8646 lerec2 8647 lediv12a 8652 recgt1i 8656 recreclt 8658 recgt0i 8664 recgt0ii 8665 recgt0d 8692 nnrecgt0 8758 nnrecl 8975 |
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