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Mirrors > Home > ILE Home > Th. List > dvdsval3 | Unicode version |
Description: One nonzero integer divides another integer if and only if the remainder upon division is zero, see remark in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.) |
Ref | Expression |
---|---|
dvdsval3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 9186 | . . . 4 | |
2 | nnne0 8861 | . . . 4 | |
3 | 1, 2 | jca 304 | . . 3 |
4 | dvdsval2 11686 | . . . 4 | |
5 | 4 | 3expa 1185 | . . 3 |
6 | 3, 5 | sylan 281 | . 2 |
7 | zq 9535 | . . . 4 | |
8 | 7 | adantl 275 | . . 3 |
9 | nnq 9542 | . . . 4 | |
10 | 9 | adantr 274 | . . 3 |
11 | nngt0 8858 | . . . 4 | |
12 | 11 | adantr 274 | . . 3 |
13 | modq0 10228 | . . 3 | |
14 | 8, 10, 12, 13 | syl3anc 1220 | . 2 |
15 | 6, 14 | bitr4d 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 wne 2327 class class class wbr 3965 (class class class)co 5824 cc0 7732 clt 7912 cdiv 8545 cn 8833 cz 9167 cq 9528 cmo 10221 cdvds 11683 |
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 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 |
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-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-po 4256 df-iso 4257 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-n0 9091 df-z 9168 df-q 9529 df-rp 9561 df-fl 10169 df-mod 10222 df-dvds 11684 |
This theorem is referenced by: dvdsmod0 11689 dvdsdc 11694 moddvds 11695 summodnegmod 11718 mulmoddvds 11755 mod2eq0even 11769 odzdvds 12120 m1dvdsndvds 12123 |
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