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Mirrors > Home > ILE Home > Th. List > nndivdvds | Unicode version |
Description: Strong form of dvdsval2 11697 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Ref | Expression |
---|---|
nndivdvds |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 9191 | . . . . 5 | |
2 | 1 | adantl 275 | . . . 4 |
3 | nnne0 8866 | . . . . 5 | |
4 | 3 | adantl 275 | . . . 4 |
5 | nnz 9191 | . . . . 5 | |
6 | 5 | adantr 274 | . . . 4 |
7 | dvdsval2 11697 | . . . 4 | |
8 | 2, 4, 6, 7 | syl3anc 1220 | . . 3 |
9 | 8 | anbi1d 461 | . 2 |
10 | nnre 8845 | . . . . 5 | |
11 | 10 | adantr 274 | . . . 4 |
12 | nnre 8845 | . . . . 5 | |
13 | 12 | adantl 275 | . . . 4 |
14 | nngt0 8863 | . . . . 5 | |
15 | 14 | adantr 274 | . . . 4 |
16 | nngt0 8863 | . . . . 5 | |
17 | 16 | adantl 275 | . . . 4 |
18 | 11, 13, 15, 17 | divgt0d 8811 | . . 3 |
19 | 18 | biantrud 302 | . 2 |
20 | elnnz 9182 | . . 3 | |
21 | 20 | a1i 9 | . 2 |
22 | 9, 19, 21 | 3bitr4d 219 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2128 wne 2327 class class class wbr 3967 (class class class)co 5826 cr 7733 cc0 7734 clt 7914 cdiv 8549 cn 8838 cz 9172 cdvds 11694 |
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-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 ax-pre-mulext 7852 |
This theorem depends on definitions: df-bi 116 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-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-id 4255 df-po 4258 df-iso 4259 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-div 8550 df-inn 8839 df-n0 9096 df-z 9173 df-dvds 11695 |
This theorem is referenced by: nndivides 11704 dvdsdivcl 11754 divgcdnn 11874 lcmgcdlem 11969 isprm6 12037 oddpwdclemodd 12062 oddpwdclemdc 12063 divnumden 12086 hashgcdlem 12128 hashgcdeq 12129 |
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