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Mirrors > Home > ILE Home > Th. List > dvdscmulr | Unicode version |
Description: Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdscmulr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3l 1025 |
. . . . 5
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2 | simp1 997 |
. . . . 5
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3 | 1, 2 | zmulcld 9367 |
. . . 4
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4 | simp2 998 |
. . . . 5
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5 | 1, 4 | zmulcld 9367 |
. . . 4
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6 | 3, 5 | jca 306 |
. . 3
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7 | 2, 4 | jca 306 |
. . 3
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8 | simpr 110 |
. . 3
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9 | 1 | adantr 276 |
. . . . . . . 8
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10 | 9 | zcnd 9362 |
. . . . . . 7
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11 | 8 | zcnd 9362 |
. . . . . . 7
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12 | 2 | adantr 276 |
. . . . . . . 8
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13 | 12 | zcnd 9362 |
. . . . . . 7
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14 | 10, 11, 13 | mul12d 8096 |
. . . . . 6
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15 | 14 | eqeq1d 2186 |
. . . . 5
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16 | 11, 13 | mulcld 7965 |
. . . . . 6
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17 | 4 | adantr 276 |
. . . . . . 7
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18 | 17 | zcnd 9362 |
. . . . . 6
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19 | simpl3r 1053 |
. . . . . . 7
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20 | 0z 9250 |
. . . . . . . 8
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21 | zapne 9313 |
. . . . . . . 8
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22 | 9, 20, 21 | sylancl 413 |
. . . . . . 7
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23 | 19, 22 | mpbird 167 |
. . . . . 6
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24 | 16, 18, 10, 23 | mulcanapd 8604 |
. . . . 5
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25 | 15, 24 | bitr3d 190 |
. . . 4
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26 | 25 | biimpd 144 |
. . 3
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27 | 6, 7, 8, 26 | dvds1lem 11790 |
. 2
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28 | dvdscmul 11806 |
. . 3
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29 | 28 | 3adant3r 1235 |
. 2
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30 | 27, 29 | impbid 129 |
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-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-mulrcl 7898 ax-addcom 7899 ax-mulcom 7900 ax-addass 7901 ax-mulass 7902 ax-distr 7903 ax-i2m1 7904 ax-0lt1 7905 ax-1rid 7906 ax-0id 7907 ax-rnegex 7908 ax-precex 7909 ax-cnre 7910 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 ax-pre-apti 7914 ax-pre-ltadd 7915 ax-pre-mulgt0 7916 ax-pre-mulext 7917 |
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-nel 2443 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 4290 df-po 4293 df-iso 4294 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-sub 8117 df-neg 8118 df-reap 8519 df-ap 8526 df-inn 8906 df-n0 9163 df-z 9240 df-dvds 11776 |
This theorem is referenced by: modmulconst 11811 mulgcd 11997 oddpwdclemxy 12149 oddpwdclemodd 12152 pcpremul 12273 |
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