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Mirrors > Home > ILE Home > Th. List > dvdsmul1 | Unicode version |
Description: An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdsmul1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9027 | . . 3 | |
2 | zcn 9027 | . . 3 | |
3 | mulcom 7717 | . . 3 | |
4 | 1, 2, 3 | syl2anr 288 | . 2 |
5 | zmulcl 9075 | . . 3 | |
6 | dvds0lem 11430 | . . . . 5 | |
7 | 6 | ex 114 | . . . 4 |
8 | 7 | 3com12 1170 | . . 3 |
9 | 5, 8 | mpd3an3 1301 | . 2 |
10 | 4, 9 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 class class class wbr 3899 (class class class)co 5742 cc 7586 cmul 7593 cz 9022 cdvds 11420 |
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-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-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-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
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-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-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-dvds 11421 |
This theorem is referenced by: dvdsmultr1 11458 3dvdsdec 11489 3dvds2dec 11490 2teven 11511 opoe 11519 omoe 11520 z4even 11540 ndvdsi 11557 mulgcd 11631 dvdsmulgcd 11640 lcmval 11671 lcmcllem 11675 lcmgcdlem 11685 qredeq 11704 cncongr2 11712 nprm 11731 exprmfct 11745 evenennn 11833 |
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