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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9234 |
. . 3
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2 | zcn 9234 |
. . 3
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3 | mulcom 7918 |
. . 3
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4 | 1, 2, 3 | syl2anr 290 |
. 2
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5 | zmulcl 9282 |
. . 3
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6 | dvds0lem 11779 |
. . . . 5
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7 | 6 | ex 115 |
. . . 4
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8 | 7 | 3com12 1207 |
. . 3
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9 | 5, 8 | mpd3an3 1338 |
. 2
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10 | 4, 9 | mpd 13 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-sub 8107 df-neg 8108 df-inn 8896 df-n0 9153 df-z 9230 df-dvds 11766 |
This theorem is referenced by: dvdsmultr1 11809 3dvdsdec 11840 3dvds2dec 11841 2teven 11862 opoe 11870 omoe 11871 z4even 11891 ndvdsi 11908 mulgcd 11987 dvdsmulgcd 11996 lcmval 12033 lcmcllem 12037 lcmgcdlem 12047 qredeq 12066 cncongr2 12074 nprm 12093 exprmfct 12108 prmdiv 12205 difsqpwdvds 12307 expnprm 12321 pockthlem 12324 evenennn 12364 lgsdir 14069 2sqlem4 14087 |
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