Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dvdsmultr2 | Unicode version |
Description: If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
Ref | Expression |
---|---|
dvdsmultr2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsmul2 11710 | . . . 4 | |
2 | 1 | biantrud 302 | . . 3 |
3 | 2 | 3adant1 1000 | . 2 |
4 | simp1 982 | . . 3 | |
5 | simp3 984 | . . 3 | |
6 | zmulcl 9220 | . . . 4 | |
7 | 6 | 3adant1 1000 | . . 3 |
8 | dvdstr 11724 | . . 3 | |
9 | 4, 5, 7, 8 | syl3anc 1220 | . 2 |
10 | 3, 9 | sylbid 149 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wcel 2128 class class class wbr 3965 (class class class)co 5824 cmul 7737 cz 9167 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-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 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-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 |
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-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 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-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-sub 8048 df-neg 8049 df-inn 8834 df-n0 9091 df-z 9168 df-dvds 11684 |
This theorem is referenced by: ordvdsmul 11728 bezoutlemstep 11881 mulgcddvds 11971 |
Copyright terms: Public domain | W3C validator |