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Mirrors > Home > ILE Home > Th. List > modqlt | Unicode version |
Description: The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Ref | Expression |
---|---|
modqlt |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qcn 9322 |
. . . . . 6
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2 | 1 | 3ad2ant1 983 |
. . . . 5
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3 | qcn 9322 |
. . . . . 6
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4 | 3 | 3ad2ant2 984 |
. . . . 5
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5 | qre 9313 |
. . . . . . 7
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6 | 5 | 3ad2ant2 984 |
. . . . . 6
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7 | simp3 964 |
. . . . . 6
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8 | 6, 7 | gt0ap0d 8303 |
. . . . 5
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9 | 2, 4, 8 | divcanap2d 8459 |
. . . 4
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10 | 9 | oveq1d 5741 |
. . 3
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11 | 7 | gt0ne0d 8187 |
. . . . . 6
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12 | qdivcl 9331 |
. . . . . 6
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13 | 11, 12 | syld3an3 1242 |
. . . . 5
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14 | qcn 9322 |
. . . . 5
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15 | 13, 14 | syl 14 |
. . . 4
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16 | 13 | flqcld 9937 |
. . . . 5
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17 | 16 | zcnd 9072 |
. . . 4
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18 | 4, 15, 17 | subdid 8089 |
. . 3
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19 | modqval 9984 |
. . 3
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20 | 10, 18, 19 | 3eqtr4rd 2156 |
. 2
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21 | qfraclt1 9940 |
. . . . 5
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22 | 13, 21 | syl 14 |
. . . 4
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23 | 4, 8 | dividapd 8453 |
. . . 4
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24 | 22, 23 | breqtrrd 3919 |
. . 3
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25 | qre 9313 |
. . . . . 6
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26 | 13, 25 | syl 14 |
. . . . 5
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27 | 16 | zred 9071 |
. . . . 5
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28 | 26, 27 | resubcld 8056 |
. . . 4
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29 | ltmuldiv2 8537 |
. . . 4
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30 | 28, 6, 6, 7, 29 | syl112anc 1201 |
. . 3
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31 | 24, 30 | mpbird 166 |
. 2
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32 | 20, 31 | eqbrtrd 3913 |
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 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-mulrcl 7638 ax-addcom 7639 ax-mulcom 7640 ax-addass 7641 ax-mulass 7642 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-1rid 7646 ax-0id 7647 ax-rnegex 7648 ax-precex 7649 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-apti 7654 ax-pre-ltadd 7655 ax-pre-mulgt0 7656 ax-pre-mulext 7657 ax-arch 7658 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rmo 2396 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-po 4176 df-iso 4177 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-reap 8249 df-ap 8256 df-div 8340 df-inn 8625 df-n0 8876 df-z 8953 df-q 9308 df-rp 9338 df-fl 9930 df-mod 9983 |
This theorem is referenced by: modqelico 9994 zmodfz 10006 modqid2 10011 modqabs 10017 modqmuladdim 10027 modaddmodup 10047 modqsubdir 10053 divalglemnn 11457 divalgmod 11466 bezoutlemnewy 11524 bezoutlemstep 11525 eucalglt 11578 |
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