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Mirrors > Home > ILE Home > Th. List > dvdsval2 | Unicode version |
Description: One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.) |
Ref | Expression |
---|---|
dvdsval2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divides 11531 |
. . 3
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2 | 1 | 3adant2 1001 |
. 2
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3 | zcn 9083 |
. . . . . . . . . . 11
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4 | 3 | 3ad2ant3 1005 |
. . . . . . . . . 10
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5 | 4 | adantr 274 |
. . . . . . . . 9
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6 | zcn 9083 |
. . . . . . . . . 10
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7 | 6 | adantl 275 |
. . . . . . . . 9
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8 | zcn 9083 |
. . . . . . . . . . 11
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9 | 8 | 3ad2ant1 1003 |
. . . . . . . . . 10
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10 | 9 | adantr 274 |
. . . . . . . . 9
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11 | simpl2 986 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 0z 9089 |
. . . . . . . . . . . . 13
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13 | zapne 9149 |
. . . . . . . . . . . . 13
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14 | 12, 13 | mpan2 422 |
. . . . . . . . . . . 12
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15 | 14 | 3ad2ant1 1003 |
. . . . . . . . . . 11
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16 | 15 | adantr 274 |
. . . . . . . . . 10
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17 | 11, 16 | mpbird 166 |
. . . . . . . . 9
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18 | 5, 7, 10, 17 | divmulap3d 8609 |
. . . . . . . 8
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19 | eqcom 2142 |
. . . . . . . 8
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20 | 18, 19 | syl6bb 195 |
. . . . . . 7
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21 | 20 | biimprd 157 |
. . . . . 6
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22 | 21 | impr 377 |
. . . . 5
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23 | simprl 521 |
. . . . 5
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24 | 22, 23 | eqeltrd 2217 |
. . . 4
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25 | 24 | rexlimdvaa 2553 |
. . 3
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26 | simpr 109 |
. . . . 5
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27 | simp2 983 |
. . . . . . . 8
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28 | 27, 15 | mpbird 166 |
. . . . . . 7
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29 | 4, 9, 28 | divcanap1d 8575 |
. . . . . 6
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30 | 29 | adantr 274 |
. . . . 5
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31 | oveq1 5789 |
. . . . . . 7
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32 | 31 | eqeq1d 2149 |
. . . . . 6
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33 | 32 | rspcev 2793 |
. . . . 5
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34 | 26, 30, 33 | syl2anc 409 |
. . . 4
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35 | 34 | ex 114 |
. . 3
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36 | 25, 35 | impbid 128 |
. 2
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37 | 2, 36 | bitrd 187 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-n0 9002 df-z 9079 df-dvds 11530 |
This theorem is referenced by: dvdsval3 11533 nndivdvds 11535 divconjdvds 11583 zeo3 11601 evend2 11622 oddp1d2 11623 fldivndvdslt 11668 divgcdz 11696 dvdsgcdidd 11718 mulgcd 11740 sqgcd 11753 lcmgcdlem 11794 mulgcddvds 11811 qredeu 11814 prmind2 11837 divgcdodd 11857 divnumden 11910 hashdvds 11933 hashgcdlem 11939 |
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