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Mirrors > Home > ILE Home > Th. List > divides | Unicode version |
Description: Define the divides
relation. ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
divides |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4019 |
. . 3
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2 | df-dvds 11830 |
. . . 4
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3 | 2 | eleq2i 2256 |
. . 3
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4 | 1, 3 | bitri 184 |
. 2
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5 | oveq2 5905 |
. . . . 5
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6 | 5 | eqeq1d 2198 |
. . . 4
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7 | 6 | rexbidv 2491 |
. . 3
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8 | eqeq2 2199 |
. . . 4
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9 | 8 | rexbidv 2491 |
. . 3
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10 | 7, 9 | opelopab2 4288 |
. 2
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11 | 4, 10 | bitrid 192 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-iota 5196 df-fv 5243 df-ov 5900 df-dvds 11830 |
This theorem is referenced by: dvdsval2 11832 dvds0lem 11843 dvds1lem 11844 dvds2lem 11845 0dvds 11853 dvdsle 11885 divconjdvds 11890 odd2np1 11913 even2n 11914 oddm1even 11915 opeo 11937 omeo 11938 m1exp1 11941 divalgb 11965 modremain 11969 zeqzmulgcd 12006 gcddiv 12055 dvdssqim 12060 coprmdvds2 12128 congr 12135 divgcdcoprm0 12136 cncongr2 12139 dvdsnprmd 12160 prmpwdvds 12390 |
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