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Mirrors > Home > ILE Home > Th. List > elnn0z | Unicode version |
Description: Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
elnn0z |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 8944 | . . . 4 | |
2 | elnn0 8937 | . . . . . . 7 | |
3 | 2 | biimpi 119 | . . . . . 6 |
4 | 3 | orcomd 703 | . . . . 5 |
5 | 3mix1 1135 | . . . . . 6 | |
6 | 3mix2 1136 | . . . . . 6 | |
7 | 5, 6 | jaoi 690 | . . . . 5 |
8 | 4, 7 | syl 14 | . . . 4 |
9 | elz 9014 | . . . 4 | |
10 | 1, 8, 9 | sylanbrc 413 | . . 3 |
11 | nn0ge0 8960 | . . 3 | |
12 | 10, 11 | jca 304 | . 2 |
13 | 9 | simprbi 273 | . . . 4 |
14 | 13 | adantr 274 | . . 3 |
15 | 0nn0 8950 | . . . . . 6 | |
16 | eleq1 2180 | . . . . . 6 | |
17 | 15, 16 | mpbiri 167 | . . . . 5 |
18 | 17 | a1i 9 | . . . 4 |
19 | nnnn0 8942 | . . . . 5 | |
20 | 19 | a1i 9 | . . . 4 |
21 | simpr 109 | . . . . . . 7 | |
22 | 0red 7735 | . . . . . . . 8 | |
23 | zre 9016 | . . . . . . . . 9 | |
24 | 23 | adantr 274 | . . . . . . . 8 |
25 | 22, 24 | lenltd 7848 | . . . . . . 7 |
26 | 21, 25 | mpbid 146 | . . . . . 6 |
27 | nngt0 8709 | . . . . . . 7 | |
28 | 24 | lt0neg1d 8245 | . . . . . . 7 |
29 | 27, 28 | syl5ibr 155 | . . . . . 6 |
30 | 26, 29 | mtod 637 | . . . . 5 |
31 | 30 | pm2.21d 593 | . . . 4 |
32 | 18, 20, 31 | 3jaod 1267 | . . 3 |
33 | 14, 32 | mpd 13 | . 2 |
34 | 12, 33 | impbii 125 | 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 cle 7769 cneg 7902 cn 8684 cn0 8935 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-n0 8936 df-z 9013 |
This theorem is referenced by: nn0zrab 9037 znn0sub 9077 nn0ind 9123 fnn0ind 9125 fznn0 9848 elfz0ubfz0 9857 elfz0fzfz0 9858 fz0fzelfz0 9859 elfzmlbp 9864 difelfzle 9866 difelfznle 9867 elfzo0z 9916 fzofzim 9920 ubmelm1fzo 9958 flqge0nn0 10021 zmodcl 10072 modqmuladdnn0 10096 modsumfzodifsn 10124 uzennn 10164 zsqcl2 10325 nn0abscl 10812 geolim2 11236 cvgratnnlemabsle 11251 oexpneg 11486 oddnn02np1 11489 evennn02n 11491 nn0ehalf 11512 nn0oddm1d2 11518 divalgb 11534 dfgcd2 11614 algcvga 11644 hashgcdlem 11814 ennnfoneleminc 11835 |
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