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Mirrors > Home > ILE Home > Th. List > dvdsadd2b | Unicode version |
Description: Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Ref | Expression |
---|---|
dvdsadd2b |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 949 |
. . 3
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2 | simpl3l 1001 |
. . 3
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3 | simpl2 950 |
. . 3
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4 | simpl3r 1002 |
. . 3
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5 | simpr 109 |
. . 3
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6 | dvds2add 11257 |
. . . 4
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7 | 6 | imp 123 |
. . 3
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8 | 1, 2, 3, 4, 5, 7 | syl32anc 1189 |
. 2
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9 | simpl1 949 |
. . . 4
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10 | simp3l 974 |
. . . . . 6
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11 | simp2 947 |
. . . . . 6
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12 | zaddcl 8888 |
. . . . . 6
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13 | 10, 11, 12 | syl2anc 404 |
. . . . 5
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14 | 13 | adantr 271 |
. . . 4
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15 | 10 | znegcld 8969 |
. . . . 5
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16 | 15 | adantr 271 |
. . . 4
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17 | simpr 109 |
. . . 4
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18 | simpl3r 1002 |
. . . . 5
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19 | simpl3l 1001 |
. . . . . 6
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20 | dvdsnegb 11240 |
. . . . . 6
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21 | 9, 19, 20 | syl2anc 404 |
. . . . 5
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22 | 18, 21 | mpbid 146 |
. . . 4
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23 | dvds2add 11257 |
. . . . 5
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24 | 23 | imp 123 |
. . . 4
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25 | 9, 14, 16, 17, 22, 24 | syl32anc 1189 |
. . 3
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26 | simpl2 950 |
. . . 4
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27 | 12 | ancoms 265 |
. . . . . . 7
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28 | 27 | zcnd 8968 |
. . . . . 6
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29 | zcn 8853 |
. . . . . . 7
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30 | 29 | adantl 272 |
. . . . . 6
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31 | 28, 30 | negsubd 7896 |
. . . . 5
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32 | zcn 8853 |
. . . . . . 7
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33 | 32 | adantr 271 |
. . . . . 6
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34 | 30, 33 | pncan2d 7892 |
. . . . 5
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35 | 31, 34 | eqtrd 2127 |
. . . 4
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36 | 26, 19, 35 | syl2anc 404 |
. . 3
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37 | 25, 36 | breqtrd 3891 |
. 2
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38 | 8, 37 | impbida 564 |
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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-inn 8521 df-n0 8772 df-z 8849 df-dvds 11224 |
This theorem is referenced by: 3dvdsdec 11292 3dvds2dec 11293 |
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