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Mirrors > Home > ILE Home > Th. List > dvdsmulcr | Unicode version |
Description: Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdsmulcr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 964 |
. . . . 5
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2 | simp3l 992 |
. . . . 5
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3 | 1, 2 | zmulcld 9083 |
. . . 4
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4 | simp2 965 |
. . . . 5
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5 | 4, 2 | zmulcld 9083 |
. . . 4
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6 | 3, 5 | jca 302 |
. . 3
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7 | 3simpa 961 |
. . 3
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8 | simpr 109 |
. . 3
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9 | 8 | zcnd 9078 |
. . . . . . 7
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10 | 1 | zcnd 9078 |
. . . . . . . 8
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11 | 10 | adantr 272 |
. . . . . . 7
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12 | 2 | adantr 272 |
. . . . . . . 8
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13 | 12 | zcnd 9078 |
. . . . . . 7
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14 | 9, 11, 13 | mulassd 7713 |
. . . . . 6
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15 | 14 | eqeq1d 2123 |
. . . . 5
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16 | 9, 11 | mulcld 7710 |
. . . . . 6
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17 | 4 | adantr 272 |
. . . . . . 7
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18 | 17 | zcnd 9078 |
. . . . . 6
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19 | simpl3r 1020 |
. . . . . . 7
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20 | 0z 8969 |
. . . . . . . . . . 11
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21 | zapne 9029 |
. . . . . . . . . . 11
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22 | 20, 21 | mpan2 419 |
. . . . . . . . . 10
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23 | 22 | adantr 272 |
. . . . . . . . 9
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24 | 23 | 3ad2ant3 987 |
. . . . . . . 8
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25 | 24 | adantr 272 |
. . . . . . 7
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26 | 19, 25 | mpbird 166 |
. . . . . 6
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27 | 16, 18, 13, 26 | mulcanap2d 8336 |
. . . . 5
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28 | 15, 27 | bitr3d 189 |
. . . 4
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29 | 28 | biimpd 143 |
. . 3
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30 | 6, 7, 8, 29 | dvds1lem 11352 |
. 2
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31 | dvdsmulc 11369 |
. . 3
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32 | 31 | 3adant3r 1196 |
. 2
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33 | 30, 32 | impbid 128 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-mulrcl 7644 ax-addcom 7645 ax-mulcom 7646 ax-addass 7647 ax-mulass 7648 ax-distr 7649 ax-i2m1 7650 ax-0lt1 7651 ax-1rid 7652 ax-0id 7653 ax-rnegex 7654 ax-precex 7655 ax-cnre 7656 ax-pre-ltirr 7657 ax-pre-ltwlin 7658 ax-pre-lttrn 7659 ax-pre-apti 7660 ax-pre-ltadd 7661 ax-pre-mulgt0 7662 ax-pre-mulext 7663 |
This theorem depends on definitions: df-bi 116 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-id 4175 df-po 4178 df-iso 4179 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 df-le 7730 df-sub 7858 df-neg 7859 df-reap 8255 df-ap 8262 df-inn 8631 df-n0 8882 df-z 8959 df-dvds 11342 |
This theorem is referenced by: mulgcddvds 11621 |
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