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Mirrors > Home > ILE Home > Th. List > elnnz | Unicode version |
Description: Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.) |
Ref | Expression |
---|---|
elnnz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 8691 | . . . 4 | |
2 | orc 686 | . . . 4 | |
3 | nngt0 8709 | . . . 4 | |
4 | 1, 2, 3 | jca31 307 | . . 3 |
5 | idd 21 | . . . . . . 7 | |
6 | lt0neg2 8199 | . . . . . . . . . . . 12 | |
7 | renegcl 7991 | . . . . . . . . . . . . 13 | |
8 | 0re 7734 | . . . . . . . . . . . . 13 | |
9 | ltnsym 7818 | . . . . . . . . . . . . 13 | |
10 | 7, 8, 9 | sylancl 409 | . . . . . . . . . . . 12 |
11 | 6, 10 | sylbid 149 | . . . . . . . . . . 11 |
12 | 11 | imp 123 | . . . . . . . . . 10 |
13 | nngt0 8709 | . . . . . . . . . 10 | |
14 | 12, 13 | nsyl 602 | . . . . . . . . 9 |
15 | gt0ne0 8157 | . . . . . . . . . 10 | |
16 | 15 | neneqd 2306 | . . . . . . . . 9 |
17 | ioran 726 | . . . . . . . . 9 | |
18 | 14, 16, 17 | sylanbrc 413 | . . . . . . . 8 |
19 | 18 | pm2.21d 593 | . . . . . . 7 |
20 | 5, 19 | jaod 691 | . . . . . 6 |
21 | 20 | ex 114 | . . . . 5 |
22 | 21 | com23 78 | . . . 4 |
23 | 22 | imp31 254 | . . 3 |
24 | 4, 23 | impbii 125 | . 2 |
25 | elz 9014 | . . . 4 | |
26 | 3orrot 953 | . . . . . 6 | |
27 | 3orass 950 | . . . . . 6 | |
28 | 26, 27 | bitri 183 | . . . . 5 |
29 | 28 | anbi2i 452 | . . . 4 |
30 | 25, 29 | bitri 183 | . . 3 |
31 | 30 | anbi1i 453 | . 2 |
32 | 24, 31 | bitr4i 186 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 w3o 946 wceq 1316 wcel 1465 class class class wbr 3899 cr 7587 cc0 7588 clt 7768 cneg 7902 cn 8684 cz 9012 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8685 df-z 9013 |
This theorem is referenced by: nnssz 9029 elnnz1 9035 znnsub 9063 nn0ge0div 9096 msqznn 9109 elfz1b 9825 lbfzo0 9913 fzo1fzo0n0 9915 elfzo0z 9916 fzofzim 9920 elfzodifsumelfzo 9933 exp3val 10250 nnesq 10366 nnabscl 10827 cvgratnnlemabsle 11251 nndivdvds 11411 zdvdsdc 11426 oddge22np1 11490 evennn2n 11492 nno 11515 nnoddm1d2 11519 divalglemex 11531 divalglemeuneg 11532 divalg 11533 ndvdsadd 11540 sqgcd 11629 qredeu 11690 prmind2 11713 sqrt2irrlem 11751 sqrt2irrap 11769 qgt0numnn 11788 |
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